closer look at the derivations of transport equations from the first principles. ...... Zienkiewicz O. C. and Codina R., 'A general algorithm for compressible and.
Extended Navier-Stokes Equations: Derivations and Applications to Fluid Flow Problems Erweiterte Navier-Stokes-Gleichungen: Ableitungen und Anwendungen auf Strömungsprobleme Der Technischen Fakultät der Universität Erlangen-Nürnberg zur Erlangung des Grades
DOKTOR-INGENIEUR
vorgelegt von
Rajamani Sambasivam
Erlangen, 2013
i
Als Dissertation genehmigt von der Technischen Fakultät der Universität Erlangen – Nürnberg
Tag der Einreichung
: 14.09.2011
Tag der Promotion
: 09.03.2012
Dekan
: Prof. Dr.-Ing. habil. Marion Merklein
Berichterstatter
: Prof. i. R. Dr. Dr. hc. Franz Durst Prof. Dr. rer. nat. habil.Ulrich Rüde PD. Dr.–Ing. habil. Stefan Becker
ii
Extended Navier-Stokes Equations: Derivations and Applications to Fluid Flow Problems Erweiterte Navier-Stokes-Gleichungen: Ableitungen und Anwendungen auf Strömungsprobleme
PhD Thesis
Rajamani Sambasivam
ii
I dedicate this work To the memory of my father, Mr. Rajamani To my mother, Mrs. Gandhi To my wife, Debjani To my daughter, Aishani To my son, Aadharsh To my brother, Krishnan
iii
iv
ACKNOWLEDGEMENTS This thesis is based on the work I carried out partly during my stay at the Institute of Fluid Mechanics (LSTM), University of Erlangen, Germany during a collaborative project and mostly during the brief stay at FMP Technology GmbH, Erlangen and at Jamshedpur, India. First of all, I would like to express my gratitude to my mentor and guide, Prof. Dr. Dr. h. c. Franz Durst, for the guidance and support regarding this work and beyond. I am also grateful to him for choosing me as a student and I will always cherish, throughout my life, the interactions I had with him both in Germany and India. During the last stages of my stay at LSTM, in the summer of 2005, he introduced me to this fascinating world of extended Navier-Stokes equations in the sidelines of our meeting regarding supersonic jet simulations expanding into hot environments. From that day, he continued to support me in the pursuit of this exciting research domain and showed child-like enthusiasm whenever I came up with some interesting results. Without his constant encouragement and guidance, this work would not have reached this stage. Further, I would also like to express my gratitude to him for providing insights into fluid mechanics and transport processes which helped me to understand the underlying physical phenomena ‘differently’. The next person, who showed equal enthusiasm in my research, is Dr. Sanjay Chandra, Chief of Technology (Global Wires and Longs), and my superior at TATA Steel. He has been a constant source of encouragement and motivating force for me to pursue my research interests in fundamental fluid mechanics along with applied research for TATA Steel. I always value the support and guidance he has been providing all through my tenure at TATA Steel. Further, I express my gratitude to all my teachers during my schooling and also in graduation and post-graduation engineering studies. Especially, I would like to name (i) Prof. K Subramanian, who taught me engineering thermodynamics and thermal engineering at the undergraduate level at Government College of Engineering, Tirunelveli, India and (ii) Prof. V. V. Satyamurty, Indian Institute of Technology, Khargpur, who was my guide during my post-graduation thesis on heat transfer. I thank them for the great insight they provided in two of my favourite subjects. I also would like to thank Dr. T Venugopalan, Chief Technology Officer, TATA Steel for the constant encouragement and his frequent inquiries about my PhD. Further, I would like to thank the management of TATA Steel for giving me an opportunity to work in Germany for a collaborative project in the area of designing supersonic oxygen lance for LD steelmaking and I might not have met Prof. Durst without this opportunity. For this, I would like to thank Dr. T Mukherjee, erstwhile Deputy Managing Director, TATA Steel, Dr. D. Bhattacharjee, then Chief of R&D and Scientific Services, Dr. S K Ajmani, Head – Steelmaking and Casting research group and most importantly, once v
again to Dr. Sanjay Chandra, then Chief of R&D. I also thank Mr. Sudhir Malavade of TATA Steel who helped me with the corrections during submission of the thesis. I would like to express my gratitude to Mrs. Heidi Durst, wife of Prof. Durst for providing a homely environment for my family in Germany and also for accepting my intrusions on Prof. Durst’s time on holidays and weekends. I also like to thank Ms. Susanne Braun and Mrs. Johanna Grasser, FMP Technology for making all the long written communications between me and Prof. Durst possible and also for sending corrections and suggestions on the thesis from Prof. Durst available to me in time. I would also like to thank Mr. Nishanth Dongari and Mr. C Karthik with whom I had long discussions about the extended Navier-Stokes equations at FMP Technology GmbH in Erlangen. I also express my thanks to Mr. Navaneetha Krishnan, Mr. D. Filimonov and Dr. T. Adachi for making the analytical solutions to microchannel flows possible and also for various technical discussions which helped me to understand the microchannel flows better. I express my gratitude to Prof. Suman Chakraborty, Indian Institute of Technology, Kharagpur, India for providing the DSMC simulations data for the microchannel with a backward facing step. I would like to thank another important person, who was one of my great motivators during the progress of this work and that is Prof. Manoranjan Maiti, my father in-law. I thank him for his constant encouragement, frequent inquiries about the status of the work and suggesting corrections and changes during the writing of the thesis. Further, I also thank Mr. M. B. Das who also provided a great degree of support and encouragement during the course of this work. Last but not the least, I express my gratitude and love to my whole family for being the driving force in my life. I would like to thank my wife, Debjani for managing the house and children during the frequent absence from home during various stages of this thesis and also during business trips. I am grateful to her for keeping me focused in my work during tough times with her support, suggestions and love. I thank my children, Aishani and Aadharsh for providing the much needed diversion from the work with their love and affection; my brother, Krishnan for his constant support and encouragement and to my mother, Mrs. Gandhi Rajamani for her unwavering and impeccable support throughout my life. Finally, I would like to dedicate this work and all my successes in life to my late father, Mr. S. K. S. Rajamani who had been and will remain the greatest father one could ever have. I thank him for his faith and confidence in me and for urging me to do well in my studies and in whatever I venture into. He was the person who I looked up to right from my childhood in my hours of need and now, his memories have taken up that place. 14 September 2011
Rajamani Sambasivam
th
vi
ABSTRACT The present thesis summarises the author’s research work, carried out under the supervision of Prof. Franz Durst, Professor Emeritus, University of Erlangen-Nürnberg, in the field of extended Navier-Stokes equations. Through some of the earlier collaborative work, the author realised that under certain flow situations, the description provided by the classical Navier-Stokes equations did not corroborate with the corresponding experimental measurements. This had puzzled the author and led to a closer look at the derivations of transport equations from the first principles. It was found out that the derivations of the classical equations had certain flaws when employed to solve gas flows with strong density/pressure and temperature gradients which resulted in additional diffusion mass fluxes. It was argued that this additional mass flux needed to be superimposed onto the convective mass fluxes treated in the classical Navier-Stokes equations. Further, it was also necessary to incorporate modified constitutive relationships of molecular transport of momentum and heat in the governing equations. Hence it may be argued that the classical Navier-Stokes equations are only valid when there are no density and temperature gradients in ideal gas flows. The present thesis commences with a summary of development of the classical NavierStokes equations based on the existing historical knowledge of fluid mechanics. A brief history of the development of the classical Navier-Stokes equations, consisting of the continuity, momentum and energy equations, are provided, however, taking most recent research efforts into account, providing a summary of the knowledge that existed when the author commenced his own derivations of the basic fluid flow equations in the presence of density and temperature gradients in ideal gas flows. These derivations resulted in a new set of equations, referred to as the extended Navier-Stokes equations. The equations, thus derived in this thesis, were compared with other extended forms of fluid flow equations based on completely different considerations for the causes of differences between the classical and extended equations. Subsequently, investigations were carried out for ideal gas flows through microchannels and capillaries employing the extended Navier-Stokes equations, derived in this thesis. The author’s initial work was on the numerical predictions of gas flows through microchannels and excellent agreement was obtained with the experimental measurements without invoking the Maxwell slip boundary condition at the solid walls. Based on the insight obtained from the numerical simulations, he finally also helped a group of researchers to derive a complete analytical solution to gas flows through microchannels and capillaries. Based on this analytical solution procedure, it is not only possible to obtain accurate description of the velocity distributions but also the pressure distribution along the micro-conduit. More importantly, a characteristic pressure was introduced based on the geometry and physical properties of the gas which was found to describe the characteristics of the flow accurately. Further, accurate numerical descriptions of microchannel flows with separation could also be provided based on the vii
extended Navier-Stokes equations. A backward facing step flow was chosen as an example to demonstrate the usefulness of the extended equations in predicting gas flows with separation and excellent agreements were obtained with DSMC simulations. Subsequently, numerical predictions of gas flows subjected to strong temperature gradients were carried out and the prediction of one dimensional supersonic and hypersonic shock waves in a monoatomic gas was chosen to demonstrate the usefulness of the extended equations. Excellent agreement of computed shock structures with experimental measurements was attained. Comparisons of various shock parameters such as inverse density thickness, density asymmetry quotient and temperature – density separation, were also provided and the extended Navier-Stokes equations were found to perform satisfactorily. Further, the inadequacies of the classical and extended NavierStokes equations in predicting hypersonic shock waves were stated. Interestingly, as explained in this thesis, it was possible to provide physically meaningful explanations, employing the extended equations, to some of the unsolved problems such as thermophoresis and thermal transpiration. Further, it was evidently demonstrated that one needed to consider the extended equations even in certain large scale gas flow problems in order to obtain accurate and detailed description of flow and heat transfer characteristics. Finally, the possibilities of further research work, employing these powerful equations, were also identified.
viii
Table of Contents (v)
ACKNOWLEDGEMENTS
ABSTRACT
(vii)
TABLE OF CONTENTS
(ix)
1
1 INTRODUCTION AND AIM OF WORK 1.1 Background and overview 1.2 Outline of the thesis
1 7
2 THE NAVIER-STOKES EQUATIONS FOR FLUIDS WITH CONSTANT PROPERTIES 11 18
3 THE EXTENDED NAVIER-STOKES EQUATIONS 3.1 Diffusion transport of mass 3.2 Diffusion transport of heat 3.3 Diffusion transport of momentum 3.4 The extended Navier-Stokes equations 3.4.1 Total velocity form of the extended equations 3.4.2 Extended total energy equation 3.5 Brenner’s extended Navier-Stokes equations 3.6 Comparison of the extended Navier-Stokes equations 4
NUMERICAL PREDICTIONS MICROCHANNELS
OF
GAS
20 22 19 24 25 26 32 34 FLOWS
THROUGH 39
4.1 Introduction and literature review 4.2 Analytical solutions based on the CNSE with no-slip boundary condition 4.2.1 Derivations for microchannel flows 4.2.2 Derivations for capillary flows 4.2.3 Importance of diffusion mass transport 4.3 Governing equations, geometry and boundary conditions 4.4 Results and discussions 4.4.1 Velocity profiles 4.4.2 Pressure profiles 4.4.3 The ‘Knudsen-Paradox’ ix
39 41 43 46 48 50 54 54 57 58
4.4.4 Comparison of mass flow rates 4.5 Insights into the physics of microchannel flows 5
60 61
ANALYTICAL TREATMENTS OF GAS FLOWS THROUGH MICROCONDUITS 67 5.1 Introduction 67 5.2 Analytical solution procedure 68 5.2.1 Order of magnitude analysis 68 5.2.2 Analytical solutions of microchannel flows 72 5.2.3 Analytical solutions of capillary flows 74 5.2.4 Semi-analytical solutions of pressure in microchannels and capillaries 77 5.3 Results and discussions of gas flows through microchannels 78 5.3.1 Regimes of flows in microchannels 78 5.3.2 Discussions on pressure profiles 80 5.3.3 Discussions on pressure gradient profiles 84 5.3.4 Comparison of total mass flow rates 86 5.3.5 Discussions on the total velocity profiles 90 5.3.6 Conductance of the microchannel and the ‘Knudsen-paradox’ 93 5.4 Results and discussions of flows through capillaries 96
6.
GAS FLOWS THROUGH COMPLEX MICROCHANNEL GEOMETRIES 101 6.1 Introduction 101 6.2 Comparisons of the ENSE solutions with Beskok [2001] and Celik and Edis [2006] 104 6.2.1 Geometry and boundary conditions 104 6.2.2 Comparisons of results with Beskok [2001] 108 6.2.3 Comparisons of results with Celik and Edis [2006] 122 6.3 Comparisons of the ENSE solutions with Chakraborty [2010] 124 6.3.1 Geometry and boundary conditions 125 6.3.2 Results and Discussions for the case of Kno=0.01 126 6.3.3 Results and Discussions for the case of Kno=1 134
7.
SOLUTIONS TO PROBLEMS WITH STRONG TEMPERATURE GRADIENTS: SHOCK WAVES 143 7.1 Introduction 7.2 Computational inconsistencies faced in other methods 7.3 Geometry and boundary conditions 7.3.1 Variation of dynamic viscosity with temperature 7.4 Results and discussions 7.4.1 Density and temperature profiles x
143 144 147 150 151 152
7.4.2 Velocity and Mach number profiles 7.4.3 Inverse density thickness, L1 7.4.4 Density asymmetry quotient, Q 7.4.5 Temperature-density separation, T 7.4.6 Discussions on the definition of Knudsen number 7.4.7 Discussions on violation of laws of thermodynamics 8
IMPORTANT RESULTS AND SUGGESTIONS FOR FURTHER RESEARCH 8.1 Introductory remarks 8.2 Outlook towards future research 8.2.1 Gas flows through microchannels 8.2.2 Computations of hypersonic shock structures 8.2.3 Thermophoresis 8.2.4 Thermal transpiration 8.2.5 Strongly heated pipe flows 8.3 Concluding Remarks
158 163 165 167 168 170
177 177 180 180 181 184 187 191
REFERENCES
195
APPENDIX
203
ZUSAMMENFASSUNG
215
xi
xii
Chapter 1
INTRODUCTION AND AIM OF WORK 1.1 Background and Overview Fluid mechanics, one of the oldest and well-researched sciences, has wide spread applications in many industries. The history of development of fluid mechanics and the salient contributions from various eminent scientists is depicted in Figure 1.1. The basic dynamic laws such as Newton’s laws of motion, Newton’s law of viscosity, Pascal’s law etc. were developed in the 17th century. The development of these fundamental laws had culminated in the formulation of the governing equations of fluid mechanics in the nineteenth century, famously known as the Navier-Stokes equations, named after the two distinguished scientists. These equations are referred to as the classical NavierStokes equations (CNSE) in this thesis. The classical equations are believed to be complete and capable of solving all kinds of fluid flow problems. This view point has also been strengthened by decades of intense theoretical research contributions and their validations with accurate measurements in sophisticated experiments. Hence, the interests and focus of the modern fluid mechanical research community have primarily been in the direction of developing more accurate measurement techniques and implementing the various observed physical phenomena into computational fluid dynamics software.
Figure 1.1 History of fluid mechanical research of various eminent scientists [Durst (2008)]
Modern measurement techniques such as Laser Doppler Anemometry (LDA) are capable of measuring even minute fluid velocities and turbulent fluctuations to a great degree of accuracy. The ongoing development of high performance computing machines and the availability of sophisticated computational fluid dynamics software, armed with capabilities to handle complex geometries and novel grid generation techniques, have revolutionized research and development activities in many fields of engineering, science and medicine. Further, the development of turbulent models based on the Reynolds averaged Navier-Stokes equations (RANS) has simplified the treatments of complex turbulent flows of engineering significance without any appreciable loss of accuracy. The development of modern transport systems such as aircrafts and automobiles employs high-end sophisticated computational simulation tools based on CNSE, as shown in Figure 1.2 and the agreement between the computational results and experimental measurements has been exemplary. Further, the computational and experimental measurement tools are widely employed as the preferred choice in designing complex manufacturing systems with a great degree of reliability and acceptability. Hence, one may say that this efficacy of the computational fluid dynamics and measurement techniques in providing accurate answers to complex fluid flow problems is one of the most significant achievements of the modern fluid mechanics research.
Figure 1.2 Application of classical Navier-Stokes equations to engineering flow problems [Durst (2008)] On the contrary, there are certain problems in fluid mechanics that remain unsolved in spite of the wealth of knowledge generated in these topics. The ‘slip-flow’ theory 2
Masss flow rate, Kg/s
employed in gas flows through micro-conduits and thermophoresis are just two of the examples. The classical equations failed to predict either the characteristics of ideal gas flows through microchannels at high Knudsen numbers as shown in Figure 1.3, see Sambasivam and Durst [2011(b)], Arkilic et al [1997, 2001] and Gad-el-Hak [1997, 2002] or the physical reasons behind a phenomenon known as thermophoresis, a process occurring very much within the continuum range, see Brenner [2005(a)-(d)]. The present state-of-the-art theories explaining the flow of gases through microchannels and capillaries are replete with empiricism and lack sound theoretical basis. The theoretical and numerical treatments of microchannels require a tuning parameter, known as Tangential Momentum Accommodation Coefficient (TMAC), to match the experimentally observed higher mass flow rates. The introduction of TMAC leads to slip at the wall, i.e. the velocity of gas at the solid wall is not zero, and this is the only possible way, as of today, one can match the experimentally observed mass flow rate of an ideal gas through a microchannel with the theoretical or numerical analysis. The bulk of the research in this field is devoted to tuning or fitting the results of theoretical models based on the slip-velocity with the help of experimental measurements.
Experimental data of Maurer et al [2003]
CNSE
0.5Pin2 Po2
Figure 1.3 Discrepancies of classical solutions and experimental measurements in gas flows through microchannels However, it can be shown categorically that the typical roughness values of microchannel walls will result in the no-slip boundary condition at the wall; see Mo and Rosenberger [1990]. Furthermore, prediction of flow characteristics based on the ‘slipflow’ theory becomes difficult, rather impossible, as soon as a simple complexity in geometry, say a backward facing step, is introduced. The prevailing conclusion of the research community is that CNSE do not give accurate predictions for flows through microchannels in the slip-flow regime and beyond, i.e. for Knudsen number values more than 0.01. Further, at Knudsen numbers beyond 0.1, it is generally considered that 3
CNSE even with the slip boundary condition cease to be valid; see Zhang [2005]. Therefore, the Direct Simulation Monte Carlo (DSMC) or Lattice Boltzman Modelling (LBM) techniques are employed to numerically predict the flow of ideal gases through microchannels and capillaries at larger Knudsen numbers. In short, it may be said that the sound physical understanding of flow of gases through complex network of microchannels, a basic requirement for the ongoing development of Micro-ElectroMechanical Systems (MEMS), still remains evasive.
CNSE
Figure 1.4 Discrepancy in computed shock structure employing classical equations and experimental measurements by Alsmeyer [1976]; Inlet Mach number 3.81 The other inadequately treated phenomenon, known as thermophoresis, is defined as the movement of small non-Brownian particles suspended in a gas from hotter to colder regions without any perceptible motion of the fluid. Interestingly, this phenomenon occurs well within the continuum regime, i.e. the fluid can be accurately defined and treated as continuum instead of molecules or atoms, and yet CNSE cannot provide any valid explanation to this phenomenon; see Brenner [2005(a)-(d)]. Once again, the explanations provided by other prevailing theories are based on empirical parameters to match experimental observations and questionable thermal slip assumptions at the surface of even a microscopic particle. To further substantiate the incompleteness of the basic equations of fluid mechanics, the observed discrepancy between experimental measurements and numerical predictions of the structure of shock waves is depicted in Figure 1.4; see Alsmeyer [1976] and Greenshields and Reese [2007(a) and (b)]. According to Greenshields and Reese [2007(a)], the numerically computed spread of the shock wave employing CNSE is less than the experimental measurements. In order to get good agreement with the experimental measurements, it is required to employ extensions to the classical equations with higher order terms, for example Burnett equations. It is essential to note 4
that the extended hydrodynamic models, available as of today, are based on mathematical perturbations rather than derivations based on physics based arguments and hence suffer from convergence difficulties and oscillations. More importantly, it is well known that the shock structures obtained with such extended hydrodynamic models violate the second law of thermodynamics. 40 CNSE
Local Nusselt number
35
Experiment
30 25 20 15 10 5 0
5
10
15
20
25
Nondimensional axial distance Figure 1.5 Discrepancy in computed local Nusselt number employing classical equations and experimental measurements by Shehata and McEligot [1998] The classical equations fail not only in the fringe regions of the near-continuum or moderately-high Knudsen number flows but also in certain large scale engineering systems of very small Knudsen numbers. For example, the local Nusselt number values of a strongly heated turbulent pipe flow obtained by CNSE are compared with the experimental measurements of Shehata and McEligot [1998] in Figure 1.5. It can be observed that the classical RANS equations with k turbulence model failed to predict the heat transfer characteristics with a large discrepancy with the experimental measurements. In general, the inadequate modelling of buoyancy driven turbulence production and dissipation terms will be suspected to be the reason for the failure of RANS based turbulence model in predicting the flow characteristics properly. Hence, based on the experimental results, the turbulence model constants will be adjusted so that good agreements between the numerical predictions and experimental measurements are achieved. However, it can also be argued that such large discrepancies may not happen because of poor modelling of buoyancy driven production of turbulent kinetic energy alone and it can also be because of some other physical phenomenon which has not been incorporated in the classical governing equations, as shown later in this thesis. 5
From the above mentioned examples, discussed based on Figures 1.3–1.5, one may conclude that CNSE are not capable of handling every ideal gas flow problem accurately or in other words, the classical equations are, in some respect, incomplete. Obviously, one is expected to get disturbed by the aforesaid two contrasting view points of the present-day status of fluid mechanics. On one hand, we have a thorough welldeveloped system of equations, aptly supported by the developments of experimental and computational tools that help to solve complex flow situations with a great degree of accuracy. On the other hand, the classical equations struggle to provide appropriate answers to some of the intriguing problems of practical interest that remain unsolved for decades. The aim of this thesis is to develop physically sound answers to some of the above mentioned problems by revisiting the derivations of CNSE. In CNSE, it is customary to introduce the constitutive relationships proposed for the molecular momentum and heat transfer given by the following equations.
Molecular momentum transfer ij U i x j
Molecular transport of heat q i
T xi
U j 2 ij U k x i 3 x k
(1.1) (1.2)
Equations (1.1) and (1.2) were derived from the Stoke’s hypothesis of molecular momentum transport based on the Newton’s law of viscosity and Fourier’s law of heat diffusion, respectively. These relationships are considered to be applicable to all Newtonian fluids and the coefficients in the constitutive relationships, dynamic viscosity and thermal conductivity are modelled as functions of temperature. However, Durst et al [2006] proposed an extension to CNSE during the period when the research work for this thesis was commenced. The derivations of the extended NavierStokes equations (ENSE) were revisited as part of the research work summarized in this thesis and finally, the extended equations were obtained in the present form discussed in this thesis. Moreover, the total velocity, defined as the sum of convective and diffusion velocities, based formulation of the Navier-Stokes equations was also derived. This form of the equations was found to be easy to employ in obtaining physically meaningful solutions to gas flows through closed conduits. Further, Brenner [2005(a)(c)] also proposed another set of extensions to the Navier-Stokes equations to explain the thermophoretic motion by altering the classical constitutive relationships for the molecular momentum and heat transfer. On comparing with the extended equations suggested by Brenner, it is stressed here that the extended equations derived in this thesis was found to provide physically-correct explanations for the flow of ideal gases in the presence of temperature and density gradients. In this thesis, the solutions obtained with ENSE for a number of problems are described in detail and the results were found to have excellent agreement with the experimental measurements.
6
1.2 Outline of the thesis The thesis is structured in the following way. In Chapter 2, the well-known derivations of CNSE are presented. Though the derivations of the classical constitutive relationships, given by equations (1.1) and (1.2), are available in many standard text books on the subject, it was felt that the derivations of the classical equations could be used as a starting point to obtain the extended form of the equations discussed later in the thesis. In Chapter 3, it is further argued that the discrepancies observed between the experimental and theoretical/numerical predictions of certain ideal gas flow problems directly point towards the incompleteness of CNSE. It is shown that CNSE are valid only for flow fields free of mass diffusion, i.e. gas flows without density and temperature gradients. Thereafter, it is evidently established that when density and temperature gradients are present in ideal gas flows, CNSE need to be extended by incorporating self-diffusion transport of mass and the allied additional diffusion transport of heat and momentum. Furthermore, the concept of ‘total velocity’, defined as the vector sum of convective and diffusion velocities, is, thereby, introduced and the derivations of the extended equations based on the total velocity formulation are also described in chapter 3. Furthermore, the extended Navier-Stokes equations proposed by Brenner [2005(a)] are also presented and the observed discrepancies between the two sets of equations are elaborated. It is argued that ENSE derived in this thesis are complete and have a strong physical basis and hence, it can be employed in solving gas flow problems with density and/or temperature gradients. As the first application of ENSE, numerical solutions of ideal gas flows through microchannels are presented in chapter 4. The available literature on gas flows through microchannels clearly concluded that CNSE failed to determine the characteristics of gas flows through micro-conduits in the alleged ‘slip-flow’ regime since the experimentally measured mass flow rates through the conduits were found to be more than those predicted by the theoretical/numerical analysis. As it could be observed from the published literature, the ‘slip-flow’ theory was introduced in the theoretical analysis of microchannel flows based on only the above mentioned observation and the determination of the ‘slip coefficient’ employing experimental measurements based on the ‘Maxwell-slip’ theory was mandatory for any plausible theoretical analysis. However, it can evidently be shown that the roughness of solid surfaces employed in manufacturing micro-conduits will necessitate the no-slip boundary condition to be satisfied and hence the introduction of the Maxwell slip velocity, applicable only in the case of molecularly smooth walls, in the theoretical/numerical analysis is empirical and lacks sound theoretical foundation. Further, the introduction of minor complexity in the geometry, such as a backward facing step, will cease the usability of the slip flow theory. Before employing the ENSE to solve gas flows through micro-conduits, it was required to estimate the importance of the acceleration terms in the momentum equations. In the published literature, there are analytical solutions for fully-developed gas flows through microchannels with both the no-slip and the Maxwell-slip boundary conditions. In this 7
chapter, analytical solutions based on integral analysis of CNSE including the nonlinear acceleration terms with the no-slip boundary condition for microchannels and capillaries are obtained. It has been found that the effect of the non-linear inertial terms can be neglected and gas flows through micro-conduits can be treated almost like fullydeveloped flows. Further, the numerically predicted results obtained with the ENSE have been found to agree excellently with the experimental measurements of gas flows through microchannels. It is claimed that this conclusively proves the physical mechanism causing the additional mass flow rate through microchannels to be the selfdiffusion of mass driven by density gradients. It can also be observed in this chapter that the numerical predictions employing the ENSE are able to determine each and every characteristic of gas flows through microchannels without making any empirical or modelling assumptions, even at high Knudsen numbers of O 1 . Further, phenomena such as the ‘Knudson paradox’, i.e. the presence of a minimum in the conductance curve, are also predicted and explained with ease. It was felt that the insights obtained from numerical simulations of gas flows through microchannels could lead to developing an analytical solution, as shown in chapter 5. The extended equations in total velocity form were employed to develop the analytical solutions for microchannels and capillary flows and the results agree very well with the experimental measurements. Further, the introduction of characteristic pressure has been observed to be an invaluable tool in the analysis of gas flows through microconduits. Furthermore, a semi-analytical solution procedure was also developed to obtain the pressure distribution along microchannels and capillaries and the results also agree well with the experimental measurements and numerical predictions alike. Further, the presence of complex geometries in micro-conduits limits the usefulness of the slip-flow theory. Since the slip-velocity at the solid wall needs to be determined from the experimental measurements, the complex geometry, say a sudden change in the cross-section, limits the application of the slip-flow theory. On the contrary, treatment of complex geometry did not pose any difficulty in the analysis with the ENSE, as shown in chapter 6 for the case of prediction of flow characteristics of microchannels with a backward facing step. Based on the results presented here, it is possible to conclude that the problem of gas flows through micro-conduits of any arbitrary shape can be solved by employing the ENSE. As stated in the previous subsection, the classical equations fail to predict the hypersonic shock structures accurately and the presence of non-continuum effects at the supposedly high Knudsen number is stated to be the reason for the failure of the CNSE in predicting such strongly compressible flows. In chapter 7, the structures of one dimensional supersonic and hypersonic shock waves predicted employing the ENSE have been compared with experimental measurements and the profiles agree exceedingly well. The modified extended equations proposed by Brenner [2005(a)] were found to fail at higher inlet Mach number values. Further, shock parameters, such as inverse density thickness and density asymmetry quotient, predicted by the ENSE also agree very well with experimental measurements and the temperature-density separation distance has been found to be under-predicted by the ENSE in comparison 8
with the DSMC simulations. Moreover, in this chapter, some important discussions on the choice of characteristic length in defining Knudsen number in shock waves and observed thermodynamic violations in shock wave simulations are also presented. In chapter 8, the summary and conclusions of results obtained for all the problems considered in this thesis are provided. Further, the possibilities of employing the ENSE in solving a number of flow cases of physical significance are also outlined along with sample results. These include the descriptions of thermophoretic motion and thermal transpiration and heat transfer from a strongly heated pipe to a gas. Further, the agreement of numerical predictions of shock waves employing the ENSE with experimental measurements has been found to improve when the self-diffusion coefficient is modified based on the Chapman-Enskog theory. It is finally stated that the ENSE has been found to be a valuable tool in providing physically sound answers to a number of unsolved gas flow problems.
9
10
Chapter 2
THE NAVIER-STOKES EQUATIONS FOR FLUIDS OF CONSTANT PROPERTIES As shown in a number of standard text books on fluid mechanics, see Durst [2008], Bird Stewart and Lightfoot [1960] and White [1974], the classical governing equations of fluid mechanics for an isothermal ideal gas flow can be written in the following way.
U iC 0 Continuity equation: t xi C U Cj U iCU Cj P ij Momentum equation: g j t xi x j xi
Equation of state: P T
(2.1) (2.2) (2.3)
In the above mentioned equations, U iC is the convection velocity, the local density, P the local static pressure, T the static temperature, the gas constant, ijC the classical molecular momentum transport terms and g j the acceleration due to gravity in the j direction. Assuming isothermal flow conditions and taking into consideration the symmetry of the term ijC , i.e. ijC Cji , it can be observed that there are eleven unknowns in the C C above questions, namelyU 1C , U 2C , U 3C , P , , 11C , 12C , 13C , 22C , 23 . For these and 33 unknowns, there are only five partial differential equations, in equations (2.1) to (2.3), available. Hence, this is an incomplete system of equations and it is therefore necessary to state additional equations, i.e. to express the unknown terms ijC in a physically well-founded manner, as functions of the velocity components. For the whole class of fluids whose molecular momentum transport properties can be classified as ‘Newtonian’, the constitutive relationships can be obtained from the kinetic theory of gases, see Durst [2008]. In the standard text books, the velocity components and molecular momentum transfer tensor do not have any superscripts and are denoted by U i ,U j and ij , respectively. However, the superscripts are used in this thesis to differentiate the convective velocity from the total and diffusion velocities introduced in the following chapter. Considering a control volume in the space as shown in Figure 2.1, with all boundaries parallel to the Cartesian coordinate system, the j-momentum transported in the idirection by an instantaneous velocity field Uˆ iC can be stated as follows: (2.4) Iij Uˆ iC Uˆ Cj Fi
The instantaneous velocity components Uˆ iC and Uˆ Cj can be decomposed into their respective velocity components of the flow U iC and U Cj and the molecular velocities u iM and u Mj , respectively and hence, equation (2.4) can be rewritten as:
Figure 2.1 j-Momentum input due to flow through the plane
[Durst (2008)]
Uˆ iCUˆ Cj U iC u iM U Cj u Mj U iCU Cj u iM U Cj u Mj U iC u iM u Mj
(2.5)
By time averaging, the time-averaged total momentum change of the volume can be obtained as: Uˆ iCUˆ Cj U iCU Cj uiM U Cj u Mj U iC uiM u Mj Term - I Term - II Term - III Term - IV
(2.6)
The total momentum input, given by equation (2.6), consists of four terms as given below: Term I: -Momentum input in the direction due to the velocity field U iC of the fluid. Term II: j-Momentum input in the direction due to the time-averaged molecular velocity in the direction u iM . Term III: -Momentum input in the direction due to the time-averaged molecular velocity in the direction u jM . Term IV: For i j , u iM u Mj 0 as the molecular motion in the three coordinate directions are not correlated. Further, when i=j, the term u iM u Mj results in the time-averaged molecular motion caused pressure. 12
In equation (2.6), Term I represents the convective transport of j-momentum in the idirection due to the convective velocityU iC . In gases, the molecular mean free path l is of finite dimensions, i.e. ≠ 0, and for this reason the time averages u iM u Mj and u Mj u iM are not equal to zero when i=j. The diffusive transport of momentum in the i and j directions are given by Terms II and III in equation (2.6). In order to calculate these two terms, the following considerations have been carried out. When x1 x2 x3 a the number of molecules moving in the direction and passing the plane in Figure 2.2 in the time duration t , can be expressed as: 1 zi na 2uiM t (2.7) 6 where n is equal to the number of molecules per unit volume, a 2 the magnitude of the area , and u iM the mean velocity of the molecules in the direction. Further, one can also write the following expression for the mass transport through as follows:
l l
Figure 2.2 j-Momentum input in the xi direction caused by the mean molecular velocity u iM [Durst (2008)] mM zi
1 mM n a 2 u iM t 6
(2.8)
where m M represents the mass of a molecule and thus the density can be expressed as m M n . Here, it is necessary to note that the mean molecular velocity u M is constant in all the directions, i.e. u M u iM u jM . However, the subscripts i and j are used for the mean molecular velocity to denote only the motion in that particular direction. As shown in Figure 2.2, let us consider two planes located at a distance ± above and below the plane Fi . Further, the velocity components in the j direction in these two planes are given by U Cj ( xi l ) and U Cj ( xi l ) . The -momentum inputs in the positive and negative directions arriving at Fi can be stated as: Iij z i m M U Cj ( xi l )
(2.9) 13
Iij z i m M U Cj ( xi l )
(2.10)
Therefore, the net momentum input can be expressed as the vector sum of the values given in equations (2.9) and (2.10) as given below: Iij z i mM U Cj xi l U Cj xi l (2.11) By incorporating the expression given for z i in equation (2.7), equation (2.11) can be rewritten as: Iij
1 mM n a 2u M t U Cj xi l U Cj xi l 6
(2.12)
The net momentum input per unit area and unit time can be obtained by Taylor series expansion of the velocity terms around x i . Neglecting higher order terms, equation (2.12) can be expressed as: Term II ijI
U Cj U Cj 1 Iij 1 M C C u U x l U x j i j i xi xi a 2 t 6
l
(2.13)
Further, when one substitutes the expression of dynamic viscosity given by the kinetic theory of gases, equation (2.13) can be expressed as follows: C
C
U j U j 1 ijI u M l 3 xi xi
(2.14)
Analogous to this, it is possible to carry out considerations to derive the expression for Term III in equation (2.6). The number of molecules moving in the j-direction can be written as: 1 z j na 2u jM t (2.15) 6 Subsequently, the j-momentum inputs from the planes located at distances l from the plane under consideration B shown in Figure 2.3, can be expressed as given below: I ji z j m M U iC ( x j l )
(2.16) (2.17)
I ji z j m M U iC ( x j l )
Therefore, the net momentum input can be expressed as:
Iij z j m M U iC x j l U iC x j l
(2.18)
Analogous to the derivations shown in equations (2.12) to (2.14), the following equation can be obtained by simplifying equation (2.18). Term III
II ij
U iC 1 M U iC u l 3 x j x j
(2.19)
14
B Fj
B Fj
l l
Figure 2.3 i-Momentum input in the x j direction caused by with molecular velocity u jM [Durst (2008)] Hence, the total molecular transport of momentum in the control volume can be written as given below: C ij
I ij
II ij
U Cj U iC x j xi
(2.20)
Equation (2.20) provides the total momentum input ijC when =constant, i.e. when d V 0 . However, when constant, an additional term needs to be added to dt C shown in equation (2.20) which is caused by the volume increase of the control ij volume. For the volume increase of a control volume at point xi and time , see Durst [2008], it can be written as: d V U iC V (2.21) dt
xi
Further, the corresponding surface increase can be given as: d F 2 U iC F dt
3
xi
(2.22)
With this increased surface area, an increased momentum input results which can be expressed as given below: 2 U kC ijIII ij 3 x k
(2.23)
This term has to be added to equation (2.20) in order to obtain the general ijC relationship for the total momentum input per unit time and unit area for ideal gases as given below: 15
U Cj U iC xi x j C ij
C 2 ij U k 3 xk
(2.24)
When equation (2.24) is considered for ijC , the following closed system of basic equations of fluid mechanics can be obtained for isothermal flow conditions and Newtonian fluids. U iC 0 Continuity equation: (2.1) t xi C C U j U iCU Cj P ij Momentum equation: g j t x i x j x i
Equation of state: P T Molecular momentum transport:
(2.2) (2.3)
C ij
U Cj U iC 2 U kC ij x j 3 xk xi
(2.24)
When non-isothermal conditions are encountered, it is necessary to incorporate the thermal energy equation (2.25), into the system of equations shown in equations (2.1) - (2.3) and (2.24) and the thermal energy equation is given below: U Cj U Cj q i e U iC e C (2.25) P ij t
x i
xi
x j
x i
In equation (2.25), the internal energy e is given by e CV T where CV is the specific heat at constant volume. The unknown term q i for the molecular transport of heat is closed by the Fourier’s law of conduction as: q i q iF
T xi
(2.26)
where is the local thermal conductivity of the fluid. Equations (2.1)-(2.3) and (2.24)-(2.26) have successfully been employed in solving a number of ideal gas flow problems in the past. However, it is stressed that when strong density and temperature gradients are encountered in the flow, an important transport mechanism, i.e. self-diffusion of mass, is expected to assume significance and this is completely neglected in the classical governing equations mentioned above. When the extent of self-diffusion of mass is negligible, the classical equations do not pose any serious errors and have been used successfully to predict non-isothermal gas flow problems in the past. However, there are gas flow problems with strong density and temperature gradients of practical interest where the self-diffusion of mass becomes important. Therefore, the self-diffusion transport of mass was incorporated into the classical Navier-Stokes equations and the extended equations were obtained to solve gas flow problems with strong temperature and density gradients, as discussed in the subsequent chapters. 16
Chapter 3
THE EXTENDED NAVIER-STOKES EQUATIONS When the classical Navier-Stokes equations (2.25) to (2.30), also referred to as the CNSE, were employed to solve ideal gas flow problems such as flow through microconduits and prediction of shock waves, one could observe discrepancies between the experimental measurements and theoretical and/or numerical predictions, as shown in Figure 1.1. In order to address such discrepancies, extensions to the CNSE have been suggested, for example Durst et al [2006] and Brenner [2005(a)]. Extensions to the Navier-Stokes equations are needed when flows of ideal gases are solved in the presence of strong density and temperature gradients. Brenner [2005(a)] proposed a set of extended equations in order to describe the thermophoretic motion in gases, and these are provided at the end of this chapter for comparisons with the extended equations derived in this thesis. Durst et al [2006] argued that when strong density and/or temperature gradients are present in the flow, the influence of self-diffusion of mass cannot be neglected, as explained in detail in subsequent sections. The presence of density and temperature gradients in ideal gas flows results in an additional molecular diffusive transport of mass caused by the self-diffusion in the direction opposite to the gradients of density and temperature. Further, it was also claimed that the said mass diffusion also gave raise to additional heat and momentum diffusion. Hence Durst et al [2006] suggested that the classical constitutive relationships for the diffusive transport of heat and momentum, given in equations (2.24) and (2.26), also had to be enhanced with the additional terms caused by the self-diffusion of mass. When the effects of self-diffusion of mass are incorporated in the CNSE, one can obtain the extended Navier-Stokes equations as described in this chapter, also referred to as the ENSE. The equations derived in this thesis are slightly different from those presented by Durst et al [2006], but the basic derivation procedure remains the same. In this chapter, expressions for the extended diffusion transport of mass, momentum and energy are derived. Further, the extended constitutive relationships obtained for the diffusion transport are incorporated in the CNSE to obtain the ENSE. As a special case, the extended equations are also obtained in the total velocity form and this form was found to be useful in certain flow simulations with solid boundaries. The extended equations suggested by Brenner [2005(a)], referred to as the BNSE, are also provided for comparison.
3.1 Diffusion transport of mass Usually, the diffusion transport of mass in gases is considered in the classical governing equations, shown in equations (2.1)-(2.3), when multiple species are present in the problem under consideration. This well-known problem is usually referred to as multi-component diffusion. However, diffusion transport of mass also occurs in gases when only one gas is present, and this is known as self-diffusion. Self-diffusion is always neglected in gases with the presumption that its influence is insignificant in comparison with the convective transport of mass. However, in the presence of strong density and temperature gradients, one can easily demonstrate that the self-diffusion of mass cannot be neglected and at times can attain magnitudes of the order of convective mass transport. Therefore, it is argued that incorporation of the selfdiffusion transport of mass in the classical continuity equation (2.25) is mandatory for consistent treatment of ideal gas flows.
T xi l x l i M u xi l U C x l j i
T xi l xi l M u xi l U C x l j i
l l
Figure 3.1: Self-diffusion mass transport due to density and temperature gradients [Durst (2008)] To analyze the molecular transport processes, one can consider Figure 3.1. The selfdiffusion transport of mass can be estimated from two planes separated by a distance l from the plane under consideration. Applying the kinetic theory of gases, the 1 molecular transport of mass can be written as m xDi l xi l u M xi l and 6 1 m xDi l xi l u M xi l from the planes located at xi l and xi l , respectively. 6 Here is the local density of the gas and u M is the molecular mean velocity, defined as: 18
uM
8T mM
(3.1)
where T is the absolute temperature, the Boltzmann constant, mM the molecular mass and l the local molecular mean free path of the considered ideal gas. The net diffusive mass flux can be expressed as: 1 m iD xi l u M xi l xi l u M xi l (3.2) 6 One can clearly observe from equations (3.1) and (3.2) that when there are no spatial density and temperature gradients in the flow field, the net diffusive mass flux m iD 0 . However, when density and temperature gradients are present, the net diffusive mass flux m iD is present and it can be evaluated based on the derivations given below.
When equation (3.2) is expanded using Taylor series, one can obtain the following expression after truncating the higher order terms: m iD
M M 1 l u M xi u l xi l u M xi u l (3.3) xi 6 x i xi x i x i
Taking into consideration the isotropy of the molecular motion and by including only the product terms that contain first-order derivatives, the diffusive mass transport in the xi direction can be written from equation (3.3) as: 1 u M m iD l u M 3 xi xi
(3.4)
The diffusion coefficient D can be written as:
D 1 lu M 3
(3.5)
By substituting equation (3.5), equation (3.4) can be rewritten as: u M D m i D M xi xi u
(3.6)
Further, substituting D for ideal gases and employing equation (3.1), the following form can be obtained: 1 T 1 T m iD D D x i 2T x i x i 2T x i
(3.7)
Equation (3.7) shows that the diffusive transport of mass is driven by both the local density and temperature gradients. Further, assuming local thermodynamic equilibrium, from the ideal gas law given by equation (2.3), one can write: 1 1 P 1 T P T (3.8) xi P xi T xi 19
Substituting equation (3.8) in equation (3.7), one can also obtain the following: 1 1 P 1 T 1 T m iD D x i 2T x i P x i 2T x i
(3.9)
The results of the above derivations are well known and they yield the following terms: Fick’s mass diffusion term: m iF D (3.10) xi
D T Soret’s mass diffusion term: m iS 2T xi where D S 2T is the Soret diffusion coefficient.
(3.11)
Hence it can be seen that the derivations proposed in this section have only yielded well-known relationships for the diffusive mass fluxes in gas flows and not something unknown to the fluid mechanical research community. However, the terms in equation (3.9) are not taken into account in the conventional theoretical or numerical analysis of ideal gas flows, since they are considered negligible with respect to the convective mass transport terms in the continuity equation (2.1).
3.2 Diffusion transport of heat One can clearly observe from Figure 3.1 that for every component of the diffusive transport of mass, a corresponding diffusive transport of heat is associated. Hence one can write the following equations using quantities for the mean molecular motion: 1 1 q xDi l xi l u M xi l exi l and q xDi l xi l u M xi l e xi l (3.12) 6 6 where e is the local internal energy of the gas and q xDi l and q xDi l are the molecular heat transport from the xi l and xi l planes, respectively. Further, for an ideal gas, the internal energy e can be written as e CV T , where CV is the specific heat at constant volume. Now, equation (3.12) can be rewritten as: C C q xDi l V xi l u M xi l T xi l and q xDi l V xi l u M xi l T xi l (3.13) 6 6 Similar to equation (3.2), one can write for the net diffusive heat flux: C qiD V xi l u M xi l T xi l xi l u M xi l T xi l 6
(3.14)
Further, by a Taylor series expansion of the terms in square brackets in equation (3.14) and truncating the series after the first-order derivatives, the following expression can be obtained: 20
M u M T x l u x l T x l i i i xi xi xi CV (3.15) D q i M 6 M xi l u xi u l T xi T l xi xi xi
Equation (3.15) can be rewritten, neglecting all terms that contain products of two first-order derivatives or more, as: T C T u M T q iD V lu M M (3.16) 3 x x x u i i i M From the definition of u M given by equation (3.1), one can derive u x as: i
u M 8 1 T xi mM 2 T xi
(3.17)
and subsequently the following relationship can be obtained: 1 u M 1 T M 2T xi u xi
(3.18)
Substituting equation (3.18) in equation (3.16), the following expression for qiD is obtained: 1 C T 1 T 1 T q iD V lu M 3 xi 2T xi T xi 1 M 1 T 1 1 T CV T l u 3 T xi xi 2T xi D
(3.19)
Equation (3.19) can be rewritten as: T 1 1 T q iD CV D CV T D xi xi 2T xi
(3.20)
m iD
Hence one can obtain the final equation for the total diffusive heat transport in an ideal gas as: T q iT m iD CV T xi qiSD
(3.21)
qiF
which consists of the following. T xi
Fourier’s heat diffusion term: q iF
21
[3.22, same as equation (2.26)]
An additional heat diffusion term as a result of the self-diffusion of mass: (3.23)
q iSD m iD C V T
Interestingly, one can also interpret equation (3.20) differently as follows: 3 T T q iD CV T D eff Df 2 xi
xi
Temperatur e-driven heat transport term
Density -driven heat transport term Dufour - term
xi
xi
(3.24)
The Dufour term, in equation (3.24) is often considered in treatments of binary diffusion problems, see Coelho and Silva Telles [2002], but not for “self-diffusion” problems. Furthermore, one can observe in equation (3.24) the change in the effective 3 thermal conductivity eff , whereas the Dufour diffusion coefficient D f depicts 2 its dependence on the temperature, T.
3.3 Diffusion transport of momentum From Figure 3.1, one can write the following expressions for the diffusive transport of momentum from the two planes: 1 1 I xDi l xi l u M xi l U Cj xi l and I xDi l xi l u M xi l U Cj xi l (3.25) 6 6 where U Cj is the local convective velocity in the j direction and I xDi l and I xDi l are the molecular momentum transport from the xi l and xi l planes, respectively. Further, one can write the net diffusive momentum flux of momentum j in the i direction as: 1 ij xi l u M xi l U Cj xi l xi l u M xi l U Cj xi l (3.26) 6
Further, by expanding the terms in square brackets in equation (3.26) employing Taylor series and truncating the series after the first-order derivatives, the following expression can be obtained: U Cj C M u M l u xi l U j xi l xi x x x i i i 1 ij 6 U Cj M u M C xi l u xi l U j xi l x x x i i i
(3.27)
Neglecting all terms that contain products of two or more first-order derivatives, equation (3.27) can be rewritten as: 22
C C C 1 M U j U j u M U j ij lu M 3 xi xi xi u
(3.28)
By substituting equation (3.18), equation (3.28) can be rewritten as: C 1 1 1 T 1 U j C M ij U j lu C 3 x i 2T x i U j x i
(3.29)
Further, equation (3.29) can be rearranged as: U Cj 1 1 T ij U Cj x i x i 2T x i
(3.30)
Employing equation (3.9), the final form of equation (3.30) can be expressed as: ij
U Cj xi
m iDU Cj
(3.31)
Similarly, one can also write the expression for the net diffusive flux of momentum i in the j direction as: 1 ji x j l u M x j l U iC x j l x j l u M x j l U iC x j l (3.32) 6
Following the steps mentioned earlier, one can obtain the following expression: ji
U iC m DjU iC x j
(3.33)
Since the flow is considered to be compressible, the expansion in time of the fluid element yields an increase in momentum input to the fluid element. Therefore, similar to the derivations in equations (2.20) and (2.21), one can write the following expression: 2 1 1 ijVD ij x k l u M xk l U kC x k l x k l u M x k l U kC xk l (3.34) 3 6 6
Expanding the terms in parentheses into a Taylor series and manipulating equation (3.34) as in previous steps, one can obtain the following momentum transport tensor: 1 U kC 2 2 1 T ijVD ij ij U kC (3.35) 3 xk 3 xk 2T xk Employing equation (3.9), equation (3.35) can be rewritten as: U kC 2 2 ijVD ij ij m kDU kC 3 x k 3
(3.36) 23
Combining equations (3.31), (3.33) and (3.36), one can obtain the complete constitutive diffusive momentum transport tensor as:
T ij
U Cj U iC xi x j
C 2 ij U k m iDU Cj m DjU iC 2 ij m kDU kC 3 xk 3
(3.37)
This expression for the total momentum diffusion due to molecular action ijT contains the classical momentum diffusion terms given in equation (2.24) and also three additional terms arising due to the self-diffusion transport of mass. It is interesting that the well-known derivation procedure presented in this section yields additional terms when one takes property variations into account.
3.4 The extended Navier-Stokes equations When one considers the extended diffusion transport terms shown in equations (3.9), (3.21) and (3.37) as the constitutive relationships in the CNSE presented in equations (2.1), (2.2) and (2.25), the ENSE are obtained as given below. When the self-diffusive mass transport, equation (3.9), is included in the classical continuity equation, shown in equation (2.1), one can obtain the extended continuity equation: m D U iC i (3.38) t
xi
xi
Equation (3.38) shows that only in those flows where the molecular mass diffusion m iD is negligible in comparison with the convective mass transport can the classical continuity equation be obtained. Similarly, when the extended constitutive relationship for the molecular momentum transport, equation (3.37), is used in the classical momentum equations, equation (2.2), one can obtain the extended momentum equation as:
U Cj t
U
C i
U Cj
xi
U Cj U C 2 U kC i ij P xi x j 3 x k g j x j xi 2 D C D C D C m i U j m j U i ij m k U k 3
(3.39)
Equation (3.39) can also be rewritten as:
U Cj t
U
C i
x i
U Cj
P x j
T ij
x i
(3.40)
g j
Similarly, substituting the extended molecular energy transport term, equation (3.21), and the extended molecular momentum transport terms, equation (3.37), in the classical energy equation (2.25), one can obtain the extended energy equation as: U Cj q iT U iC e U iC e T P ij (3.41) t
xi
xi
xi
xi
24
It is stressed that the set of extended Navier-Stokes equations, given by equations (3.38), (3.40) and (3.41), must be solved to obtain accurate and physically meaningful solutions when strong temperature and density gradients are present in gas flows. Further, in some special flow geometries with solid walls, such as straight microchannels, it may be helpful to express the extended Navier-Stokes equations in the total velocity form, as shown in the following section. 3.4.1 Total velocity form of the extended equations One can express the self-diffusive mass transport m iD as a function of diffusion velocity which is defined as: (3.42) m iD U iD where U iD is the diffusion velocity in the i-direction. Employing equation (3.9), one can express the diffusion velocity U iD as: U iD
1 1 T 1 P 1 T x i 2T x i P xi 2T x i
(3.43)
By substituting equation (3.42), the extended continuity equation (3.38) can be written as: U iD U iC (3.44) t
xi
xi
Now, let us introduce a ‘total velocity’, defined as the vector sum of convective and diffusion velocities, as shown below: (3.45) U iT U iC U iD By employing this definition of the total velocity, the extended continuity equation (3.38) can be rewritten as: U iT 0 (3.46) t xi It is interesting that the extended continuity equation (3.46) is similar to the classical continuity equation (2.1). Similarly, the extended momentum equations (3.40) can also be rewritten, by rearranging some of the terms, as:
U Cj t
P U iC U Cj m iDU Cj m DjU iC x i x j x i
C 2 D C ij 3 ij m k U k g j
(3.47)
Employing the definition of total velocity given in equation (3.45), one can write the following expression: 25
U iT U Tj U iC U Cj U iDU Cj U DjU iC U iDU Dj U iCU Cj m iDU Cj m DjU iC m iD U Dj
(3.48) Using equation (3.48), equation (3.47) can be rewritten as:
U Tj t
U Dj P C 2 T T D D D C U i U j g j ij mi U j ij m k U k x i x j xi 3 t
(3.49)
The newly added term m iDU Dj within the total momentum diffusion term has the product of pressure and temperature derivatives; see equations (3.9) and (3.43), which were neglected earlier as higher order terms in the Taylor series expansion. Therefore, one can also drop this term from the right hand side of equation (3.49). Similarly, one can also rewrite the extended energy equation given in equation (3.41) in terms of the total velocity as:
e U iC e t x i x i
U Cj U iC T D T m i e P ij x i x i x i
(3.50)
By readjusting the total heat diffusion terms, one can obtain the extended energy equation in terms of the total velocity as given below:
T e U i e t x i x i
T x i
U Cj U Cj T P ij x j x i
(3.51)
The extended governing equations given in the total velocity form shown in equations (3.46), (3.49) and (3.51) may be extremely useful in certain flow configurations as shown later in this thesis. 3.4.2 Extended total energy equation In many compressible flow problems, it is preferable to solve the energy equation with the total energy E as a variable which is defined as: 2 1 E e U Cj e Ek (3.52) 2
As the first step in obtaining the energy equation in terms of the total energy E, one needs to derive the mechanical energy equation, which is obtained by multiplying the momentum equation by the velocity. It is interesting that one can derive two completely different forms of the extended mechanical energy equation. When the extended momentum equation (3.40) is multiplied byU Cj , one can obtain: U
C j
U Cj t
U U C j
C i
x i
U Cj
U
C j
ijT P C U j g jU Cj x j x i
where ijT is given by equation (3.37). 26
(3.53)
By employing the chain rule of differentiation, equation (3.53) can be rewritten as:
U Cj U Cj t
U
C j
U Cj t
U iC U Cj U Cj x i
U
C i
U
C j
U Cj x i
U
C j
ijT P C U j g j U Cj x j x i
(3.54) Equation (3.54) can be further simplified as given below: U Cj U Cj U Cj U iC U Cj U Cj U Cj ijT C P C U Cj U iC U U g jU Cj j j t
xi
t
xi
x j
xi
(3.55) Equation (3.55) can be rewritten once again by expressing the terms inside the square brackets on the left-hand side using the chain rule of differentiation as: C U Cj U Cj U Cj U iC U Cj U Cj U iC U Cj C C C U i
t
x i
U j
t
U j
t
x i
U Cj
Equation (3.56) is simplified further as shown below: U Cj U iC U Cj U CjU Cj U iC U CjU Cj C
t
U j
xi
t
xi U Cj
U j
x i
ijT
(3.56)
P U Cj g jU Cj x j x i
U iC U xi t C j
ijT P U Cj g jU Cj x j xi
(3.57)
U iC It is interesting to note that in equation (3.57) is not zero based on the t xi
extended continuity equation shown in equation (3.38). It is important to observe from equation (3.38) and (3.46) that the material derivative is zero only when written in terms of the total velocity. Therefore, equation (3.57) can be expressed as: U Cj U iC U Cj m iD C 2 C C 2 C U j U i U j U j U Cj t xi xi xi t ijT (3.58) C P C U j U j g jU Cj x j xi
C Equation (3.58) can be rewritten in terms of local kinetic energy Ek
C
convective velocityU j , as:
27
1 C 2 U j , due to 2
E kC U iC E kC 2 1 U Cj t xi t 2 xi
1 C C 2 2 U i U j
U Cj U iC U Cj ijT iD C m C P C U U j U j g jU Cj U j t x x x x i i j i C j
(3.59)
Further, equation (3.59) can be expanded using the chain rule of differentiation as: C U Cj 1 C U iC U Cj 1 U Cj E kC U iC E kC 1 C U j 1 C C C Uj U j Uj U i U j t
xi
U
t
2
C j
U Cj t
t
2
U U C j
C i
U Cj
xi
xi
2
U
C 2 j
2
xi
ijT m iD C P C U j U j g jU Cj xi x j xi
(3.60) The following equation can be obtained by simplifying equation (3.60): C C C E kC U iC E kC 1 C U j 1 C U i U j Uj Uj t xi 2 t 2 xi U Cj U Cj 1 C C C U j Ui Uj 2 xi t
2
ijT m iD C P C U j U j g jU Cj xi x j xi
(3.61)
By expanding the terms in square brackets in equation (3.61) using the chain rule of differentiation, equation (3.61) can be simplified to:
U U t x
E kC U iC E kC 1 C - Uj t x i 2
C i
2
i
C 2 j
ijT m iD P U Cj U Cj g jU Cj x i x j x i
(3.62) Further, employing the extended continuity equation (3.38), equation (3.62) can be further simplified to obtain the final form of the mechanical energy equation as:
C U Cj ijT U Cj U Cj E kC U iC E kC m iD PU j E kC P ijT g j U Cj (3.63) t x i x i x j x j x i x i
The extended mechanical energy equation (3.63) can be added to the extended thermal energy equation (3.41) to obtain the extended total energy equation as: T C q iT m iD PU iC ij U j E C U iC E C Ek g jU Cj t xi xi xi xi xi (3.64) C where the total energy E is given by: 28
EC e
1 C Uj 2
2
e EkC
(3.65)
Further, one can also derive the total energy equation based on the total velocity-based momentum equation (3.49). One can obtain the total mechanical energy equation by multiplying equation (3.49) by the total velocity U Tj as: U
T j
U Tj
U
T j
~ijT U Dj T T T P T T U i U j U j U j U j g jU Tj x i x j x i t
t where ~T is given by:
(3.66)
ij
2 ~ijT ijC m iDU Dj ij m kDU kC 3
(3.67)
By employing the chain rule of differentiation, one can rewrite equation (3.66) as: U Tj U Tj U Tj U iT U Tj U Tj U Tj T T T U j U i U j t
t
xi
xi
(3.68)
~ijT U Dj P T T U U j U j g jU Tj x j xi t T j
Equation (3.68) can be simplified further as given below: U Tj U Tj U Tj U iT U Tj U Tj U Tj T T U j Ui t
x i
t
x i ~ijT T
U Dj P T U U j U j g jU Tj x j x i t T j
(3.69)
Equation (3.69) can be rewritten once again by expressing the terms inside the square brackets on the left-hand side using the chain rule of differentiation as: T U Tj U Tj U Tj U iT U Tj U Tj U iT U Tj T T T U i t
x i
U j
t U Tj
U j
t
t
xi
U j
t
x i
~ijT U Dj P U Tj U Tj g jU Tj x j x i t
Equation (3.70) is simplified further as shown below: U Tj U iT U Tj U Tj U Tj U iT U Tj U Tj T
U j
x i
xi
U iT U Tj xi t
~ijT U Dj P T T U U j U j g jU Tj x j xi t T j
29
(3.70)
(3.71)
U iT Based on the total continuity equation (3.46), one can write the terms t xi
in
equation (3.71) as zero and hence equation (3.71) can be simplified to: U Tj U iT U Tj U Tj U Tj U iT U Tj U Tj T t
U j
xi
t
xi
~ijT U Dj P T T U U j U j g jU Tj x j xi t T j
T By introducing the local total kinetic energy E k
(3.72)
1 T 2 U j due to the total velocity U Tj , 2
equation (3.72) can be rewritten as: E kT U iT E kT 1 2 1 T T 2 U Tj U i U j t
xi
t 2
xi 2
U Tj U iT U Tj ~ijT U Dj T P T T U Tj U U U g jU Tj j j j xi x j xi t t
(3.73)
One can simplify equation (3.73) to obtain the following equation: T T T U Tj U Tj E kT U iT E kT 1 T U j 1 T U i U j 1 T T Uj Uj U j Ui t
x i
t
2
x i
2
U Tj
2
~ijT
t U Dj T
P U Tj U j x j x i
t
xi
g U j
(3.74)
T j
By expanding the terms in square brackets in equation (3.74) using the chain rule of differentiation, equation (3.74) can be simplified to:
PU Tj U Tj ~ijT U Tj U Tj U Dj E kT U iT E kT T T ~ P ij U j g j U Tj t x i x j x j x i x i t
(3.75) The extended mechanical energy equation (3.75) can be added to the extended thermal energy equation (3.41) to obtain another form of the extended total energy equation:
30
U
E T t
T i
ET
x i
q
T T U Tj PU iT U iD ~ij U j T ~ P ij x i x i x i x i x i i
ijT
U Cj x i
U Tj
U Dj
g U j
t
(3.76) T j
where E T e 12 U Tj is the local total energy based on the total kinetic energy of the fluid. This form of the total energy equation was found to be helpful in solving wallbounded flows. 2
The extended Navier-Stokes equations derived above are reproduced again below as a summary and these equations are solved to obtain the solutions presented in the subsequent chapters for gas flow problems in the presence of strong density and temperature gradients. Continuity equation: m D U iC i (3.38) t
xi
xi
Momentum equation:
U Cj t
U
C i
U Cj
x i
P x j
T ij
x i
(3.40)
g j
Momentum equation in the total velocity form:
U Tj t
U Dj P C 2 T T D D D C U i U j g j ij m i U j ij m k U k x i x j xi 3 t
Thermal energy equation: q iT e U iC e t
xi
U Cj U iC T P ij xi xi xi
Mechanical energy equation:
(3.49)
(3.41)
C U Cj ijT U Cj U Cj E kC U iC E kC m iD PU j Ek P ijT g jU Cj (3.63) t x i x i x j x j x i x i
Mechanical energy equation based on the total kinetic energy:
PU Tj U Tj ~ijT U Tj U Tj U Dj E kT U iT E kT P ~ijT U Tj g j U Tj t x i x j x j x i x i t
(3.75) 31
Total energy equation: q iT E C U iC E C t
xi
T
C
ij U j m iD PU iC Ek g jU Cj xi xi xi xi
(3.64)
Total energy equation based on the total kinetic energy: T T T E T U iT E T U Cj q i PU iT U iD ~ij U j ~ T U j T P ij ij t
xi
xi
xi
xi
xi
U Tj
xi
U t
D j
xi
g U j
(3.76)
T j
Total molecular momentum transport: U Cj U iC 2 U kC 2 ij m iDU Cj m DjU iC ij m kDU kC xi x j 3 x k 3 2 ~ijT ijC m iDU Dj ij m kDU kC 3 T ij
Total molecular transport of heat: T q iT m iD e xi
(3.37) (3.67)
(3.21)
3.5 Brenner’s extended Navier-Stokes equations Brenner [2005(a), (b)] also proposed a set of extensions to the classical Navier-Stokes equations based on his observations of thermophoretic motion. Thermophoresis, defined as the observed motion of particles against the temperature gradient in an otherwise quiescent stratified gas, is not completely explained by the CNSE. Brenner therefore introduced two type of velocities, namely the mass velocity and volume velocity in his formulation. The velocity appearing in the CNSE, shown in equations (2.1)–(2.3) and (2.24)–(2.26), was referred to as the mass velocity U im (same as the convective velocityU iC ). Further, a fictional volume velocity U iV was introduced in the governing equations. Brenner’s suggestion was to include the volume velocityU iV in the constitutive relationship of the classical diffusive momentum transport instead of the mass velocityU im . Hence the constitutive relationship for the molecular momentum transfer proposed by Brenner is given below:
Br ij
U Vj U iV xi x j
V 2 ij U k 3 x k
(3.77)
The volume and mass velocities are interrelated by the following expression: (3.78)
U iV U im j iV
32
where j iV is defined as: 1 ln (3.79) x i x i where and are the local thermal diffusivity and density of the gas under jiV
consideration. Further, by incorporating the definition of jiV
C p
, one can rewrite equation (3.79) as: C p
1 x i
(3.80)
In equation (3.80), is the local thermal conductivity of the fluid. By substituting equation (3.78) in equation (3.77), the following equation for the molecular transport of momentum can be obtained:
Br ij
U mj U im 2 j Vj j iV U km ij xi xi x j x 3 x j k
V 2 ij j k 3 x k
(3.81)
Equation (3.81) can be rewritten as:
Br ij
C ij
j Vj j iV xi x j
V 2 ij j k 3 x k
(3.82)
where ijC is the classical molecular transport of momentum terms based on the mass velocity, as given in equation (2.24). By incorporating the extended molecular momentum transport terms, given by equation (3.82), in the classical momentum equation (2.2), the extended momentum equation due to Brenner was obtained as given below:
U mj t
U
m i
x i
U mj
P
x j
ijBr x i
(3.83)
g j
Further, Brenner also modified the classical heat diffusion term as: j iq q i Pj iV
(3.84) where q i is the classical heat diffusion term given in equation (2.26) and P the local pressure in the flow field. The modified heat diffusion term in equation (3.84) and the extended momentum diffusion term in equation (3.82) can be incorporated in the classical energy equation (2.25) in order to obtain the extended energy equation due to Brenner as:
U mj U mj j q e U im e i P ijBr t x i x i x j x i
33
(3.85)
Furthermore, a similar U im to U iV modification for the velocity appearing in the no-slip tangential velocity boundary condition at solid surfaces was also suggested by Brenner. However, the normal velocity at the solid wall was retained as the mass velocity U im . The extensions suggested by Brenner retained the classical continuity equation given in equation (2.1) without any further modifications. The extended equations proposed by Brenner are summarized here for ease of comparison with the set of extended equations derived in this thesis. Continuity equation: U im t
xi
(2.1)
0
Momentum equations:
U mj t
U
m i
U mj
x i
P x j
Br ij
x i
(3.83)
g j
Thermal energy equation:
U mj U mj j q e U im e i P ijBr t x i x i x j x i
(3.85)
Molecular momentum transport:
Br ij
U mj U im xi x j
m V 2 j V ij U k j j i 3 x i x j x k
V 2 ij j k 3 x k
(3.81)
Molecular transport of heat: (3.84)
j iq q i Pj iV
3.6 Comparison of the extended Navier-Stokes equations Another set of extended Navier-Stokes equations was proposed by Oettinger [2005] in his phenomenological generic theory. He demonstrated that in the generic formulation, one would obtain the classical Navier-Stokes equations when the terms associated with mass density in the friction matrix were identically zero. Then, by including non-zero terms associated with mass density in the friction matrix, a revised set of governing equations was derived that included two velocities, similar to the volume and mass velocities proposed by Brenner [2005(a), (b)]. He further argued that the classical governing equations historically ignored mass diffusivity on the basis that the diffusive mass flux was assumed to be zero. However, he observed that the associated momentum and energy fluxes might not be zero. His formulation included the momentum and energy fluxes, both of which are entropy producing, making the process 34
of mass diffusion irreversible. He felt that the ability of mass diffusion to produce entropy was missing from the classical Navier-Stokes equations. Table 3.1 Comparison of the extended equations proposed in the present work and by Brenner [2005(a),(b)] Type of Equation
Present work
U Cj t
U
C i
U Cj
xi
P
ijT xi
U
m i
U mj
xi
g j
x j
P
ijBr xi
g j
m
U j jiq e U im e P t xi xi x j
U Cj
xi
1 1 T m iD x i 2T x i
U Cj U iC 2 U kC T Molecular ij ij xi x j 3 xk transport of 2 momentum m iDU Cj m DjU iC ij m kDU kC 3
Molecular transport of heat
t
Molecular transport of mass
U mj
q T U iC e U iC e i P t xi xi xi T ij
U im 0 t xi x j
Energy
m iD U iC t xi xi
Continuity
Momentum
Brenner
j iV
Br ij
C p
Br ij
U mj xi
1 x i
U mj U im xi x j j Vj jiV xi x j
m 2 ij U k 3 x k V 2 ij j k 3 xk
j iq q i Pj iV
T q m iD CV T xi T i
Based on the work of Brenner [2005(a),(b)] and Oettinger [2005], Greenshields and Reese [2007] predicted shock structures using the extended form of the Navier-Stokes equations given by Brenner [2005(a)] and obtained better agreements with experimental measurements of Alsmeyer [1976]. However, they had to include the mass diffusion in the continuity equation (2.1) proposed by Brenner [2005(a),(b)]. It is interesting that it was not possible for them to obtain convergence of the solution without the inclusion of the mass diffusion. It can be easily observed that the inclusion of mass diffusion in 35
the continuity equation to obtain converged solutions was a major deviation from the work of Brenner [2005(a)]. Further, they also reported that the results obtained with modified form of the equations suggested by Brenner [2005(a)] (with inclusion of mass diffusion in the continuity equation) displayed unphysical behaviour when the coefficient of volume diffusion, in equation (3.79) exceeded the kinematic viscosity . They speculatively claimed that this could be attributed both to instabilities in temporal disturbances and to a spurious phase velocity-frequency relationship. Later, Chakraborty and Durst [2007] compared the two independent and apparently dissimilar considerations proposed by Durst et al [2006] and Brenner [2005(a), (b)]. It was concluded from this study that the constitutive relationships of heat and momentum transport in ideal gas flows with strong density and temperature gradients, obtained by these two different proposals, possessed physical and mathematical equivalences at small gradients. Here, the similarity was achieved by casting the momentum flux in an equivalent linear form based on a gradient-diffusion hypothesis under the assumption of constant pressure condition. Although there have been a number of attempts to compare the two sets of extended equations, there are conspicuous discrepancies between them: 1. The extended equations proposed by Brenner [2005(a),(b)] and Oettinger [2005] were obtained based on phenomenological observations, i.e. they were obtained to explain certain physical or experimental observations. For example, Brenner [2005(a),(b)] intended to explain the thermophoretic motion and, hence, adjusted the constitutive relationships to that phenomenon only. However, the extended equations presented in this thesis were obtained based on the first principles and are generic in nature, i.e. the formulation and derivations of these equations were not carried out to explain a particular phenomenon. Therefore, one can attempt to solve any gas flow problem with density and temperature gradients with the set of equations presented in this thesis. 2. It is interesting to observe from equation (3.80) and Table 3.1 that the diffusion velocity is in the direction of the density gradient in the case of Brenner’s extension whereas it is in the opposite direction to the density and temperature gradients, as shown in equation (3.7), in the present derivations. As Brenner attempted to explain the thermophoretic motion, he observed the motion of the particles to be in the direction of the density gradient and, hence, he expressed the diffusion velocity to be in the same direction as the density gradient. However, in general, it is customary to express diffusion transport as functions of negative gradients of field variables, see equations (2.20), (2.26) and (3.7). It is argued, as shown in chapter 8, that the incomplete understanding of thermophoretic motion led to the representation of diffusion velocity as shown in equation (3.80).
36
3. The diffusion velocity in Brenner’s derivations is a function only of the density gradient, see equation (3.80), whereas the diffusion velocity is a function of both density/pressure and temperature gradients in the present derivations, see equation (3.7). Further, the coefficient of diffusion is also different in both cases. 4. One more striking discrepancy is the absence of mass diffusion in the continuity equation in Brenner’s extensions, as shown in equation (2.1). As there is no apparant diffusion mass transport observed in the thermophoretic motion, Brenner’s continuity equation did not include any diffusion transport. Further, it is interesting that since Brenner‘s extended equations were derived to explain thermophoretic motion, the absence of mass diffusion from the continuity equation also made the volume velocity fictious. However, the diffusion transport of mass, shown in equation (3.7), is a physical and real quantity and also measureable. It is also further argued that the presence of mass diffusion leads to the inclusion of extended constitutive relationships for momentum and heat, as shown in equations (3.37) and (3.21), respectively, in the present derivations. 5. Furthermore, the constitutive relationships of diffusion transport of momentum and heat in both cases are different, as shown in equations (3.37) and (3.81) and equations (3.21) and (3.84), respectively. The above differences suggest that the extended form of the Navier-Stokes equations derived in this thesis is completely different from those suggested by Brenner [2005(a),(b)] and Oettinger [2005]. It is argued that the extended equations derived in this chapter have a strong physical basis and, hence, can be employed in solving gas flows with strong temperature and density gradients in order to obtain accurate predictions. A number of such flow problems are solved in subsequent chapters in order to substantiate this argument.
37
38
Chapter 4
NUMERICAL PREDICTIONS OF GAS FLOWS THROUGH MICROCHANNELS 4.1 Introduction and Literature Review The entire field of flows through microchannels has become a very popular research area due to an increasing interest in Micro-Electro-Mechanical Systems (MEMS), as well as bio-mechanical, Lab-on-the-Chip Systems (LCS); see Karniadakis et al (2005). These flows remain a major interest to fluid mechanics research since the very first experimental investigations; see Arkilic et al [2001] and Maurer et al [2003]. The MEMS systems typically employ small channels with characteristic dimensions in the range of 1-100µm; see Gad-el-Hak [2002]. Analysis of flow of gases through straight channels of such dimensions is an integral part of designing MEMS systems. A lot of literature is available on the gas flow experiments conducted in microchannels; see Pfahler et al (1991), Pong et al (1994), Harley et al (1995), Liu et al (1995) and Shih et al (1995, 1996). It can be observed from these literatures that under some inlet and outlet pressure conditions, the experimentally measured mass flow rates of gases were higher than those computed/calculated employing the classical Navier-Stokes equations (CNSE). This vital observation had puzzled and excited researchers all over the world over the last few decades and a large number of publications on this topic have emerged. In the absence of any other sound physics based answers to the lower mass flow rates obtained with CNSE, employing the no-slip boundary condition at the solid wall, in comparison with the corresponding experimentally measured values, the theoretical treatments in the published literature suggested employing a slip velocity at the wall; see Arkilic et al [2001] and Maurer et al [2003]. Further, this velocity was assumed to be the slip velocity proposed by Maxwell (1879) who suggested that the velocity at a molecularly smooth solid wall can be calculated by: C 2 dU 1 US (4.1) l dx 2 In equation (4.1), l is the molecular mean free path, U1C the tangential velocity component and x2 the normal direction to the wall. Further, is the tangential momentum accommodation coefficient. The expression for the slip velocity shown in equation (4.1) is based on the assumption that one fraction of the molecules interacting with the wall is reflected in a diffusive way and the rest is reflected specularly. Thus, by assuming that a certain fraction of molecules retain their tangential momentum after bombarding the solid wall, it was possible to obtain a nonzero wall velocity with the help of equation (4.1). However, it is important to note
that the fraction had to be experimentally determined, i.e. the theoretical predictions had to be empirically fitted to the experimental measurements in gas flow treatments based on slip-velocity in micro-conduits. So far, the successful theoretical and numerical treatments of microchannel flows were based on the CNSE, shown in equations (2.1)–(2.3) and (2.24), subjected to the Maxwell slip velocity boundary condition at the solid walls, shown in equation (4.1); see Arkilic et al (1994) and Arkilic and Schmidt [1997]. As shown by Arkilic and Schmidt [1997], it is possible to obtain an analytical solution for the mass flow rate of an isothermal ideal gas flow through two-dimensional microchannels with the slip boundary condition at the walls as: 2 Pin H 3 wPo2 Pin m 1 12 Kn (4.2) 1 24LT Po Po where H, L and w are the height, length and width of the considered microchannel, respectively. Further, Pin and Po are the inlet and outlet static pressures, respectively and T the absolute static temperature of the fluid in the channel. is the dynamic viscosity of the fluid and the specific gas constant. Kn is the Knudsen number and 2 is given by where is the tangential momentum accommodation coefficient; see Arkilic and Schmidt [1997]. It is well-known that if becomes zero in equation (4.2), one obtains the conventional no-slip boundary condition for the flow velocity and hence, one can obtain the corresponding no-slip mass flow rate through the microchannel as: H 3 wPo2 m 24 LT
P in Po
2 1
(4.3)
Though the research community has widely accepted the ‘slip-velocity’ based models for predicting gas flows through microchannels, it can be proven easily that employing the Maxwell slip-velocity, valid only for the case of molecularly smooth walls, in microchannel flows is highly objectionable. Typical wall roughness values in microchannels are of the order of 10 8 - 10 7 m whereas the diameter of molecules is of the order of 10 10 m. As shown by Mo and Rosenberger [1990], the apparently smaller wall roughness values of microchannels are at least 2-3 orders greater than the molecular diameter and hence, this wall roughness values are good enough to enforce the no-slip boundary condition to the flow. Therefore, the slip-velocity assumption in analyzing gas flows through microchannels is questionable and needs to be scrutinized thoroughly. When the research work leading to this thesis was underway, it was proposed that the presence of self-diffusion of mass, driven by strong density/pressure gradients, in gas flows through microchannels could be the reason for the increased mass flow rates observed in the experiments. It is interesting to note that since the height of 40
microchannels is very small in comparison to the length, i.e. h L , the streamwise pressure gradients are in general, very high and the inlet and outlet pressures can also be less than the atmospheric pressure under typical operating conditions of the channel. Hence, as observed in equation (3.9), the self-diffusion caused mass transport in gas flows through microchannels can be significant and needs to be incorporated in the theoretical or numerical analysis. With this intuitive supposition, it was decided to employ the total velocity form of the ENSE, presented in equations (3.46), (3.49) and (3.67) to numerically analyse flow of gases through microchannels. Interestingly, as shown later in this chapter, the numerical results obtained with these equations without any empirical tuning parameters agree excellently with the experimental measurements. Further, it is argued that the approach presented here accurately handles the physics behind gas flows through microchannels. It is also interesting that all the salient characteristics of microchannel flows are also explained with great insight based on this analysis, thus proving convincingly that the observed additional mass flow rate in experiments is nothing but the self-diffusion mass transport driven by the pressure/density gradients. Further, because of the small dimensions of micro-conduits, it is known that the Knudsen number Kn, defined as the ratio of the molecular mean free path l to the characteristic length Lc becomes a key parameter to determine the characteristics of these flows. Conventionally, the height of the channel is taken as the characteristic length of flows through microchannels and the Knudsen number is defined as: l Kn (4.4) H However, it is also possible to define the characteristic length based on the gradients of macroscopic quantities such as density or pressure, and the Knudsen number can also be expressed as: 1 1 P l Kn l (4.5) x P x 1 1 Equation (4.5) is considered as a more precise choice for the characteristic length; see Gad-el-Hak [1999]. Interestingly, the length scale for the diffusive mass transport of isothermal gas flows, shown in equation (3.9), is similar to the one shown in equation (4.5). Hence, it is important to deduce which definition of Knudsen number provides a clear description of the flow characteristics in micro-conduits.
4.2 Analytical solutions based on the CNSE with the no-slip boundary condition While obtaining the analytical expressions shown in equations (4.2) and (4.3), the gas flow through the channel was assumed to be fully-developed and the nonlinear acceleration terms were neglected by Arkilic and Schmidt [1997]. It is generally assumed that the low Reynolds number creeping flows in microchannels behave like 41
incompressible flows; see Zhang et al [2005]. However, it is easy to observe that the acceleration terms in the left-hand side of the momentum equations, presented in equation (2.2), are non-zero because of the variation of density and the velocity profile does not become fully-developed. Therefore, under certain operating conditions, the nonlinear terms may assume significance and cannot be neglected. Hence it is required to estimate the relative influence of the acceleration terms in gas flows through micro-conduits and neglecting of inertial terms needs to be justified. Therefore, an analytical solution procedure based on the integral form of the CNSE with the no-slip boundary condition was developed and the results were compared with equation (4.3) and experimental measurements of straight microchannels and capillaries. x2
H
x1 x2
(a)
H
x1 x2
(b)
H
x1 (c)
Figure 4.1 Schematic representation of the coupling of density and velocity profiles in gas flows through micro-conduits; Profiles of (a) density (b) streamwise velocity and (c) mass flow rate 42
The schematic representation of the density and streamwise velocity profiles of gas flows through microchannels are shown in Figure 4.1. The pressure-gradient driven gas flow through the channel has higher density at the inlet than that at the outlet which results in a gradual increase of the streamwise velocity towards the outlet. Because of large aspect ratios, the development length is negligibly small in comparison to length of microchannels and hence the cross-stream velocity is zero everywhere. Therefore, one can easily deduce that the mass flow rate profiles do not vary along the streamwise direction, i.e. the flow is fully-developed in terms of the mass flow rate and not in terms of the streamwise velocity. The geometry and boundary conditions employed are shown in Figure 4.2. In this figure, h is half-height of the channel and R the radius of the capillary. U m is the maximum velocity at any cross-section and Pi and Po are the static pressures at the inlet and outlet boundaries, respectively. 4.2.1 Derivations for microchannel flows The governing equations for isothermal gas flows through a microchannel are given below. Continuity equation:
U1C 0 x1
(4.6)
U 1C U 1C dP U 1C Momentum equation: x1 dx1 x 2 x 2
(4.7)
In equation (4.7), the streamwise momentum diffusion is neglected in order to simplify the analytical solution procedure. Since the cross-stream velocity is zero throughout the domain, the mass flow rate at any given cross-stream location x2 is constant, i.e. x1 U 1C x1 , x 2 is constant. It is also important to note that the temperature can be x2
assumed to be constant for low Mach number flows through micro-conduits. Therefore, the pressure and hence, the density are functions of only the streamwise coordinate x1 . To obtain integral solutions, it is customary to employ the profile method, see Schlichting [1979], Mills [1992] and von Karman [1921], by assuming a polynomial function for the mass flow rate through the channel as: U 1C A B x2 C x2 2 (4.8) U mC h h subjected to the boundary conditions given below: At x2 0 ; U 1C U mC (4.9a) At x2 0 ;
U 1C 0 x 2
(4.9b) 43
At x2 h ; U 1C 0
(4.9c)
x2
Wall,
Inlet =const
h Symmetry,
,
Outlet =const x1
L (a)
r Wall, Inlet =const
R Axis,
Outlet =const
, x1 L (b)
Figure 4.2 Geometry and boundary conditions employed in the analytical solutions based on the integral form of the equations: (a) microchannel and (b) capillary
In equation (4.9a), U mC is the maximum value of the velocity at any given streamwise location x1 . By applying the boundary conditions given in equation (4.9), it is possible to evaluate the coefficients in equation (4.8) and the mass flow rate profile can be expressed as: U1C 1 x2 2 (4.10) U mC h Further, the momentum equation, given in equation (4.7), can be expressed in the integral form as: h h h U 1C U 1C dP U 1C dx2 dx 2 (4.11) dx2 x1 0
dx 01
Term - I
Term - II
x 0 x 2 2 Term - III
The three-terms given in equation (4.11) can be integrated individually as shown below:
U 1C U 1C d h U 1C Term – I = dx 2 = x1 dx1 0 0 h
2
dx2
44
(4.12)
By introducing the mass flow rate profile given by equation (4.10) in equation (4.12), Term-I can be written as: Term – I = U
C 2 m
2 2 d h 1 x2 1 dx 2 dx1 0 h
(4.13)
Since the density is only a function of the streamwise coordinate x1 , it is possible to rewrite equation (4.13) as: 2
Term – I = U
C 2 m
2 d 1 h x 2 1 dx 2 dx1 0 h
(4.14)
After performing the integration, Term-I becomes: Term – I =
2 8 1 dP U mC hT 2 15 P dx1
(4.15)
where is the specific gas constant. Similarly, the other terms in equation (4.11) can be integrated as: dP dx1 2 T U mC Term – III = hP
Term – II = h
(4.16)
(4.17)
By substituting equations (4.15)-(4.17), equation (4.11) can be expressed as: 2 8 1 dP dP 2T U mC hT 2 h U mC 0 15
P dx1
dx1
hP
(4.18)
After separating the variables, it is possible to integrate equation (4.18) in x1 to obtain the following form: 2 P 16h 2 T ln in U mC 60TL U mC 15h 2 Pin2 Po2 0 (4.19) Po
The quadratic equation in U mC can easily be solved to obtain the peak value of the mass flow rate profile as given below: 60 TL C m
U
60 TL 2 960h 4 T ln Pin P
P 32 h 2 T ln in Po
P2 P2 in o o
(4.20)
It is well-known that in fully-developed channel flows, the ratio of average to maximum mass flow rates is given by: 2 U 1C U mC (4.21) 3 45
Further, the mass flow rate through the channel is given by: m 2hw U1C Hw U1C (4.22) where H and w are the height and width of the channel, respectively. To compare the mass flow rate with the one provided by Arkilic and Schmidt [1997] as given in equation (4.3), equation (4.11) can be solved neglecting Term – I, to obtain the following expression for the peak mass flow rate: h 2 Pin2 Po2 C U m (4.23) 4LT
It can be observed that the identical expression of the average mass flow rate is obtained to the one shown in equation (4.3), given by Arkilic and Schmidt [1997], by employing the peak mass flow rate, given by equation (4.23). Further, when comparing the results obtained with equations (4.22) and (4.3), it can be concluded that there are no significant differences in terms of mass flow rates for the operating pressure values provided by Arkilic et al [2001] and the inclusion of convective terms did not change the results in any appreciable manner. Hence, it can be concluded that the nonlinear convective acceleration terms can be neglected from the theoretical or numerical analysis of isothermal gas flows through straight microchannels considered in this study. However, for larger channels with higher Reynolds number gas flows, the acceleration terms may have to be incorporated and the result presented in equation (4.20) can be used in such cases. It is necessary to stress here that this important conclusion helps in simplifying the numerical solutions described in this chapter and ensuring the analytical solutions feasible with ENSE, as described in the next chapter. 4.2.2 Derivations for capillary flows The geometry and boundary conditions employed to obtain the integral solutions for gas flows through capillaries are shown in Figure 4.2(b). The CNSE for the case of isothermal axisymmetric compressible ideal gas flows are given below. U1C 0 Continuity equation: (4.24) x1
U 1CU 1C dP 1 U 1C Momentum equation: r x1 dx1 r r r
(4.25)
The profile method explained in section 4.2.1 resulted in a mass flow rate profile similar to the one shown in equation (4.10) and hence, the following mass flow rate profile was employed in the analysis: U 1C 1 r 2 (4.26) U mC R subjected to the boundary conditions given below: At r 0 ; U 1C U mC (4.27a) 46
U1C 0 r At x2 R ; U1C 0
At r 0 ;
(4.27b)
(4.27c)
The integral form of the momentum equation shown in equation (4.25) can be written as given below: R R R U 1C U 1C dP 1 U 1C (4.28) dr dr r dr x1 0
0 dx1
r r r 0
Term I
Term _ II
Term _ III
On integrating, equation (4.28) results in the following form: 2 8 1 dP dP 4T U mC RT 2 R U mC 0 P dx1
15
dx1
(4.29)
RP
After separating the variables and integrating, the peak value of the mass flow rate profile is given by: 120 TL C m
U
120 TL 2 960 R 4 T ln Pin P
P2 P2 in o o
P 32 R T ln in Po
(4.30)
2
It is well-known in fully-developed pipe flows that the ratio of the average to the maximum mass flow rate is given by: 1 U 1C U mC (4.31) 2 Further, the mass flow rate through the capillary can be obtained by: m R 2 U1C
(4.32)
Furthermore, when equation (4.28) was solved by employing terms II and III only, the expression for the peak value of the mass flow rate profile could be obtained as: R 2 Pin2 Po2 U mC (4.33) 8LT Therefore, the average mass flow rate through the capillary can be expressed as:
U 1C
R 2 Pin2 Po2 D 2 Pin2 Po2 16LT 64LT
(4.34)
where D is the diameter of the capillary considered.
47
4.2.3 Importance of diffusion mass transport As shown above, one can express the mass flow rates of ideal gas flows through straight microchannels and capillaries employing equations (4.22) and (4.32), respectively. It is well known that under certain operating conditions, the mass flow rates obtained with the no-slip boundary condition at the solid wall deviate from the experimentally measured values, see Maurer et al [2003], Arkilic et al [2001] and Gad-el-Hak [1999, 2002], which led to the introduction of the ‘slip-flow’ theory as described in section 4.1. As shown in the subsequent section, the ENSE were employed to predict the characteristics of gas flows through microchannels in this thesis. Before embarking on the numerical simulations with the ENSE, it was required to establish the flow conditions in which the additional mass flow rate due to selfdiffusion needed to be considered in the analysis. x2
x1 H Diffusive flow
Convective flow
Figure 4.3 Convective and diffusive parts of the velocity profile in microchannel flows
The additional diffusion mass flow rate in a straight microchannel can be computed employing equation (3.9) as given below: 1 dP H m 1D U 1D H D P dx1
(4.35)
For the case of microchannel flows, the ratio of mass flow rates due to diffusion and convection can be obtained employing equations (4.35) and (4.3) as: m D 12 2 C (4.36) m PH 2 Employing the following well-known relationship for the Knudsen number Kn: Ma l Kn (4.37) 2 Re H
48
and employing the definitions of the Mach and Reynolds numbers given by HU 1C UC Ma 1 Re , respectively, one can derive the expression for the mean C molecular free path l , as: l (4.38) 2 C where C is the local velocity of sound and the kinematic viscosity of the fluid. Using the expression for the mean molecular free path l given in equation (4.38), it is possible to deduce the ratio of diffusion and convective mass flow rates for microchannel flows from equation (4.36) as: m D 24 C Kn2 (4.39) m Similarly, the ratio of diffusion and convective mass flow rates for gas flows through capillaries can be expressed, employing equations (3.9), (4.32), (4.34) and (4.36), as: m D 64 C Kn 2 (4.40) m where the diameter of capillary D is used as a characteristic dimension in defining the Knudsen number Kn. Employing equations (4.39) and (4.40), one can evaluate the Knudsen number at which the self-diffusion mass transport assumes significance over the convective transport. Table 4.1 provides values of the ratio of mass flow rates at various Knudsen numbers for ideal gas flows through microchannels and capillaries. One can clearly observe that the error of around 5% in mass flow rate will be present if the selfdiffusion mass transport is not considered in the analysis, even at a relatively smaller Knudsen number of 0.05. Further, the influence of self-diffusion increases rapidly beyond a Knudsen number value of 0.25. It is also interesting to note in Table 4.1 that the influence of self-diffusion assumes significance at even smaller Knudsen numbers in capillary flows than flows through microchannels. Table 4.1 Ratio of mass flow rates as a function of Knudsen number
0.01
0.05
0.1
0.25
0.5
0.75
1
2
5
10
Kn Channel 0.036 0.081 0.114 0.181 0.256 0.313 0.362 0.512 0.809 1.144 flows Kn Capillary 0.022 flows
0.05
0.07
0.111 0.157 0.192 0.222 0.313 0.495 0.701
49
In order to validate the important result shown in Table 4.1, the convective mass flow rates obtained with equations (4.3) and (4.22) are compared with the experimental measurements of Maurer et al [2003] in Figure 4.4. As mentioned earlier, it was found that the mass flow rates predicted by equations (4.3) and (4.22) were near identical and hence only one set of values is shown. The Knudsen number was evaluated at the average of the inlet and outlet pressures. It is clearly evident from Figure 4.4 that the theoretical estimate of the influence of diffusion mass flow rate shown in equation (4.39) and Table 4.1 is accurate and one can clearly observe the deviation of the experimental measurements from the values predicted by the CNSE employing the no-slip boundary condition beyond a Knudsen number of 0.1. Hence, one is encouraged to take the self-diffusion driven mass transport into account in predicting the characteristics of gas flows through straight microchannels and capillaries employing the ENSE.
CNSE Expt._Maurer et al [2003]
Figure 4.4 Deviation of mass flow rate predicted by the CNSE employing the no-slip boundary condition at the wall from the experimental measurements of Maurer et al [2003]
4.3 Governing equations, geometry and boundary conditions The total velocity based ENSE presented below, simplified from equations (3.46), (3.49) and (3.9), were employed to solve steady isothermal gas flows through straight microchannels. U iT 0 Continuity equation: (4.41) xi 50
~T P ij T T Momentum equations: U i U j x i x j x i
(4.42)
Molecular momentum transport: C
U j U iC ~ijT xi x j
C 2 ij U k m iDU Dj 2 ij m kDU kC 3 x k 3
[4.43, same as equation (3.67)] 1 1 P xi P xi
Self-diffusive transport of mass: m iD
(4.44)
It is well-known that the available computational fluid dynamics (CFD) codes have been developed for the CNSE employed with the no-slip boundary conditions, presented in equations (2.1)-(2.3). The computations mentioned here were carried out in the commercially available CFD software, Fluent 6.3. The additional terms in the ENSE were incorporated into the solver as mass and momentum source terms employing User Defined Functions (UDF). The UDF code is provided in Appendix. x2
,
Wall,
Outlet
Inlet
h Symmetry, x1 L (a)
x2 Wall,
, Outlet
Inlet
h Symmetry, x1 L (b)
Figure 4.5 Geometry and boundary conditions employed in the numerical simulations of straight two-dimensional microchannels: (a) CNSE and (b) ENSE The two-dimensional geometry and boundary conditions employed in the simulations of straight microchannels employing both the CNSE and the ENSE are presented in Figure 4.5. The published experimental measurements of Maurer et al [2003] were 51
used to compare the numerical results obtained for the straight microchannels and the salient experimental conditions are summarized in Table 4.2. The grid counts in the streamwise and cross stream directions were 30 and 5000, respectively. At the inlet and outlet boundaries, constant static pressure values estimated based on the experimental conditions shown in Table 4.2 were specified. Symmetry boundary condition was employed at the middle plane of the channel since only half of the channel was simulated. Second order discretization schemes were employed for all variables whereas the PISO (Pressure Implicit Splitting of Operators) scheme was used for pressure-velocity coupling. Table 4.2 Summary of experimental conditions employed by Maurer et al [2003] Experimental parameters
Maurer et al [2003]
Reported channel height, H μm
1.14
Width, w μm
200
Length, L mm
10
Gas used
Helium
Gas constant [J/Kg.K]
2077
Temperature, [K]
293
Viscosity of the gas [10-5 Pa s]
1.99
Inlet Pressure range [kPa]
26-507
Outlet Pressure [kPa]
12-101
Outlet Knudsen number Kno
1.509-0.179
It is necessary to explain the wall boundary condition employed in the simulations with the ENSE. It is a well-known practice to employ the no-slip boundary condition at the solid wall in the classical fluid mechanics as shown in Figure 4.5(a). Interestingly, based on experimentally observed higher mass flow rates in comparison to those obtained with the classical no-slip boundary condition, the slip-flow theory was introduced in numerical simulations of gas flows through micro-conduits where the slip-coefficient needed to be experimentally determined. It is shown clearly by Mo and Rosenberger [1990] that unless the wall is atomically smooth, the no-slip boundary condition for the velocity is valid at all the engineering surfaces. To understand the nature of molecules-wall interactions, one can refer to the schematics shown in Figure 4.6. As shown in Figure 4.6(a), unlike the molecules moving in far-away distances from the solid wall, the molecules caught in between the 52
roughness elements are expected to lose their momentum due to multiple bombardments onto the wall roughness elements and hence the molecules tend to attain the velocity of the wall. The roughness elements are a minimum of two orders of magnitude higher than the diameter of molecules in a typical microchannel; see Arkilic et al [2001]. Therefore, as shown schematically in Figure 4.6(b)-(d), the convection, diffusion and total velocity profiles are expected to satisfy the no-slip boundary condition at the wall.
Molecule Mean roughness height Wall
Wall
(a)
(b)
Mean roughness height Wall
Wall
(c)
(d)
Figure 4.6 Schematic representation of flow characteristics near the solid wall in gas flows through microchannels (a) Effect of wall roughness; (b) Convective velocity profile; (c) Diffusion velocity profile; (d) Total velocity profile However, unlike the convection velocity profile, depicted in Figure 4.6(b), which gradually increases from a value of zero at the mean height of roughness elements at the wall to the maximum at the mid-section of the channel, the diffusion velocity depicts a sharp jump immediately beyond the roughness elements of the wall. Since gas flows through straight microchannels behave like fully-developed flows, the pressure remains constant along the cross-stream direction and hence, the pressure gradient values at the wall and mid-section of the channel are identical. Therefore, from equation (4.44), one can observe that the magnitude of diffusion velocity is the 53
same in the cross-stream direction at any given streamwise location. It is easy to conclude that when the boundary conditions are employed at the mid-height of roughness elements of the solid wall, the total velocity profile, defined as the sum of convective and diffusive velocity profiles, displays a slip-like behaviour at the solid wall as shown in Figure 4.6(d). It is important to note that the physical reason for this ‘slip-like’ non-zero velocity at the wall, i.e. at the mean height of roughness elements, is the self-diffusion mass transport driven by pressure/density gradients and not the tangential momentum accommodation based slip concept adopted by the ‘slip-flow’ models proposed in the microchannel literature like Arkilic et al [2001] and Maurer et al [2003]. Based on the above arguments, the boundary conditions at the wall were introduced in the simulations as shown in Figure 4.5(b). Unlike the convection velocity, the normal gradient of the diffusion velocity at the solid wall is zero in straight micro-conduits, i.e. the self-diffusion caused mass flow results in insignificant momentum transfer to the solid wall or the diffusion flow experiences negligible resistance from the solid wall. This important observation helps in providing physically sound explanations to some of the apparently puzzling characteristics of micro-conduit flows such as the ‘Knudsen paradox’, as described later in this chapter.
4.4 Results and discussions 4.4.1 Velocity profiles Since the Knudsen number increases along the length of the channel, the relative significance of the self-diffusion mass transport with respect to the convective mass transport can also vary significantly as shown in equation (4.39). Therefore, the flow characteristics can be governed purely by convection in some regions of the channel whereas the self-diffusion can assume huge significance at some other regions of the channel. As examples, two velocity profiles are discussed here to explain the changing dynamics at different sections of the channel. In Figures 4.7 and 4.8, the total velocity profiles are presented at two different axial locations and pressure ratios. The convection and diffusion velocity components of the total velocity are also shown in these figures. Further, the velocity profiles obtained numerically by employing the CNSE with the no-slip boundary condition are also provided for comparison. In Figure 4.7(a), at half of the length of the channel, i.e. x1 0.005m, one can observe that there is almost no difference between the solutions of the CNSE and the ENSE and the diffusion velocity is negligibly small. The local value of pressure at this location was found to be 4.9 bar which resulted in the local Knudsen number of 0.035. One can observe from equation (4.39) that the diffusion mass flow rate is negligibly small at this Knudsen number and hence, the velocity profiles obtained with the CNSE and the ENSE are near-identical. On the contrary, in Figure 4.7(b), it can be observed that there exists a significant deviation in the velocity 54
profiles at the outlet plane of the channel. The convective velocity component obtained with the ENSE is much less than that calculated with the CNSE. 6.E-07 Convection_component Diffusion_component Total_velocity Classical_conv_velocity
Cross-stream coordinate, m
5.E-07
4.E-07
3.E-07
2.E-07
1.E-07
0.E+00 0
0.1
0.2
0.3
0.4
0.5
0.6
Axial velocity, m/s
(a) 6.E-07
Cross-stream coordinate, m
5.E-07
4.E-07
3.E-07
2.E-07 Convection_component Diffusion_component Total_velocity Classical_conv_velocity
1.E-07
0.E+00 0
2
4
6
8
10
12
14
16
18
20
Axial velocity, m/s
(b)
Figure 4.7 Comparison of streamwise velocity profiles obtained with the CNSE and the ENSE for Helium gas flow through microchannel, Pin 7 bar; Po 0.12 bar; Streamwise locations (a) 0.005m and (b) 0.01m 55
6.E-07 Convection_component Diffusion_component Total_velocity Classical_conv._velocity
Cross-stream coordinate, m
5.E-07
4.E-07
3.E-07
2.E-07
1.E-07
0.E+00 0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.14
0.16
0.18
Axial velocity, m/s
(a) 6.E-07 Convection_component Diffusion_component Total_velocity Classical_conv._velocity
Cross-stream coordinate, m
5.E-07
4.E-07
3.E-07
2.E-07
1.E-07
0.E+00 0
0.02
0.04
0.06
0.08
0.1
0.12
Axial velocity, m/s
(b)
Figure 4.8 Comparison of streamwise velocity profiles obtained with the CNSE and the ENSE of Helium flow through microchannel, Pin 0.4 bar; Po 0.12 bar; Streamwise locations (a) 0.005m and (b) 0.01m
56
Further, it can also be observed that the maximum value of the velocity profile at the centre of the channel obtained with the CNSE is slightly more than the total velocity obtained with the ENSE. However, it is evident that the average velocity obtained with the ENSE is higher than that of the CNSE, similar to the observations in the experiments. It is also interesting to notice in Figure 4.7(b) that the diffusion velocity is much higher than its convection counterpart since the local Knudsen number at the outlet is 1.42. It is evident from the above discussions that the self-diffusion mass transport need not influence the characteristics of the channel at all sections for the total mass flow rate through the channel to increase. When the local Knudsen number remains low, the convective transport dominates the diffusion and determines the flow rate through the channel. However, when the Knudsen number assumes a relatively higher value, the diffusion begins to influence the characteristics of the flow. This reveals a very intimate coupling between the convection and diffusion mechanisms governing the dynamics of gas flows through microchannels. Unlike the case shown in Figure 4.7, it is also possible that the self-diffusion transport plays a dominant role throughout the length of the channel as shown in Figure 4.8. For the case considered here, the local Knudsen number was found to be 0.43, 0.71 and 1.42 at the entrance, middle and outlet sections of the channel, respectively. One can immediately deduce from equation (4.39) that the mass transport due to selfdiffusion is expected to be much higher than that of the convection at these Knudsen numbers. As evidently seen in Figure 4.8, the CNSE with the no-slip boundary condition predict very less mass flow rate through the channel in comparison to the solutions obtained with the ENSE. Further, one can also observe that the total velocity, dominated by the self-diffusion mass transport, is more than three times than the convective velocity predicted by the CNSE. 4.4.2 Pressure profiles From the discussions presented above, one can readily infer that the combined influence of self-diffusion and convective mass flow rates will change the pressure profiles along the length of the channel. In Figure 4.9, the dimensionless pressure profiles, defined as P P Po Pin Po , are plotted as a function of the streamwise coordinate for different pressure ratios. It is important to note that the outlet pressure was kept at a constant value of 0.12 bar, resulting in an exit Knudsen number of 1.42 at all the simulations. The dimensionless pressure profiles obtained employing the CNSE with the no-slip boundary condition are also plotted in this figure for comparison.
57
(a)
(b)
(c)
(d) CNSE
ENSE
Figure 4.9 Comparison of dimensionless pressure profiles along the length of the channel; Po = 0.12 bar; PR Pin Po (a) 58.3; (b) 10; (c) 3.33; (d) 1.67 As the first observation, in Figure 4.9, one can clearly see that the pressure profiles obtained with the CNSE are convex in nature for all the pressure ratios. On the contrary, the pressure profiles obtained with the ENSE show a gradual variation from convex to concave as the pressure ratio decreases. At high pressure ratios, the mass flow rate is governed by the convective transport in most parts of the channel and hence, the difference between the pressure profiles obtained by the CNSE and the ENSE is negligible, as observed in Figure 4.9(a). As the pressure ratio is gradually decreased, the self-diffusion mass transport starts assuming significance over convection and influences the flow characteristics. Since there is no cross-stream gradient of the diffusion velocity, the diffusion transport of mass does not cause any momentum loss to the wall and hence, the pressure drop through the channel becomes negligible. Therefore, the prescribed pressure difference across the channel is used to enhance only the convective transport through the channel. One may say that the convex shape of the pressure profile is prevalent in the convection dominated region of the channel and a concave profile is obtained when 58
the diffusion transport is dominant. This combined influence of self-diffusion and convective mass transports alters the curvature of the pressure profile from convex to concave, as seen in Figure 4.9(b). As shown in the encircled region of Figure 4.9(b), the convex profile near the convection dominated inlet region slowly gets transformed into the diffusion dominated concave profile near the outlet. Detailed analysis on the nature of pressure and pressure gradient profiles are given in the next chapter. When the pressure ratio is reduced further, the mass transport due to self-diffusion dominates the convective transport throughout the channel and the resistance for the flow in the channel goes down significantly. Here, one obtains the complete concave shape of the pressure profiles, as shown in Figure 4.9 (c) and (d). These interesting characteristics in microchannel flows are not predicted by the CNSE. 4.4.3 The ‘Knudsen Paradox’ In his experimental study of rarefied gas flows, Knudsen [1909] discovered that as the pressure was reduced to very low values, the conductance of the gas did not get reduced continually. On the contrary, the conductance through the channel reached a minimum and then started increasing as the pressure was further reduced. Since it was not possible to explain this phenomenon by the CNSE, this is referred to as the ‘Knudsen paradox’. In other words, the ‘Knudsen paradox’ is defined as the occurrence of a minimum in the conductance of the channel before rising again when plotted against the Knudsen number. Both the CNSE and the first order slip models based on the ‘Maxwellian-slip’ theory are unable to predict the ‘Knudsen paradox’ at all whereas the predictions based on the second order slip models are not accurate enough; see Zhang [2005]. Hence, it is generally believed that one needs to employ the Lattice Boltzmann Equation (LBE) based methods or the Direct Simulation Monte Carlo (DSMC) techniques to predict the ‘Knudsen paradox’ phenomena in microchannels. Needless to mention, these techniques also involve a number of assumptions, for example the parameters for molecule-wall interactions etc, which need to be experimentally verified and determined and hence, there is certain empiricism involved in these microscopic models, as well. The microchannel results obtained with the ENSE were employed to predict the ‘Knudsen paradox’. As it has already been shown in this chapter, there are no underlying assumptions either in the derivations or in the simulations in the case of the ENSE. In Figure 4.10, the conductance of the microchannel predicted by the ENSE, defined as Cˆ M T Pin Po , is plotted against the average Knudsen number, evaluated at the average of the inlet and outlet pressures. Here, M T is the total mass flow rate through the channel. The conductance values obtained in the experiments by Maurer et al [2003] and by the CNSE are also shown in the figure for comparison. As shown in Figure 4.10, the classical equations do not predict the paradox at all and the conductance decreases continually with increasing Knudsen number. However, the extended equations predict the ‘Knudsen paradox’ and the results also excellently 59
match with the experimental measurements. The physical reasons for the presence of the ‘Knudsen paradox’ are given below.
CNSE ENSE Experiment_Maurer et al [2003]
Figure 4.10 Conductance of the microchannel, Cˆ M T Pin Po as a function of average Knudsen number depicting the ‘Knudsen paradox’; Comparisons with experimental measurements of Maurer et al [2003]
As the Knudsen number increases, the significance of diffusion mass transport in comparison to convection increases dramatically as shown by equation (4.39). As mentioned in sections 4.4.1 and 4.4.2, unlike the convective transport, the diffusion mass transport does not cause any significant transport of momentum to the solid walls because of the absence of normal diffusion velocity gradient at the wall and hence the pressure drop due to the diffusive mass transport in the channel is negligible. However, the convective velocity gradient in the normal direction to the wall is nonzero and hence the convective flow results in momentum transport to the wall and leads to pressure drop. It is interesting to note that at the critical Knudsen number where the minimum occurs in the conductance profile, a balance is reached between the convective and self-diffusive mass transports. Beyond this critical value, any further increase in the Knudsen number results in an imbalance favouring enhanced diffusion transport that requires no pressure drop in the channel. For a given pressure difference across the channel, the presence of higher diffusion transport of mass, with almost negligible loss of momentum to the wall, results in an enhanced mass flow 60
through the channel. Or in other words, one can obtain the same mass flow rate obtained at a given Knudsen number with a lower pressure gradient at higher Knudsen numbers because of enhanced diffusion transport. Therefore, the conductance of the channel increases with increasing Knudsen number beyond the minimum value as observed by Knudsen [1909]. This physically sound and rather simple explanation of the ‘Knudsen paradox’, which was not readily available all these decades, cannot be given on the basis of the CNSE. Further, the ENSE can indeed predict this experimental observation without any empirical tuning parameters. 4.4.4 Comparison of mass flow rates One of the important aims of any experiment involving micro-conduits is the accurate prediction of the mass flow rate for a given pressure ratio and channel dimensions since prior knowledge of the mass flow characteristics through the channel is mandatory to determine the performance of any MEMS equipment. In Figure 4.11, the experimentally measured mass flow rates through a straight microchannel by Maurer et al [2003] are compared with the numerically obtained flow rates with the ENSE. The mass flow rates predicted by the CNSE with the no-slip boundary condition are also shown for comparison. Since the outlet pressure is constant in all the simulations, a higher value of difference of square of the pressures represents lower Knudsen number and vice versa.
Experiments_Maurer et al [2003] CNSE ENSE
1 2 Pin2
2
Po2 , bar
Figure 4.11 Mass flow rate M T comparisons obtained with the CNSE and the ENSE with the experimental measurements of Maurer et al [2003]; Po=0.12bar It can be observed in Figure 4.11 that the CNSE predict the mass flow rates accurately at higher values of squares of pressure, i.e. at lower Knudsen numbers. As discussed 61
earlier, the diffusion mass flow rate is negligible when the Knudsen number is small and hence the classical equations are valid in this region. However, as the Knudsen number increases or the difference of square of pressures decreases, the deviation in the mass flow rates predicted by the CNSE and measured in the experiments continually increases and even the error in the theoretical prediction reaches more than one order of magnitude. On the contrary, the ENSE accurately predict the mass flow rate through the microchannel at all the Knudsen numbers. This conclusively proves that the additional mass flow rate observed in gas flow experiments in microchannels is caused by the self-diffusion of mass.
4.5 Insights into the physics of microchannel flows It is widely believed that the Navier-Stokes equations, even with the ‘slip-velocity’ at the wall, cannot be used beyond a Knudsen number of 0.1 in predicting the characteristics of gas flows through microchannels since the continuum assumption employed in deriving these equations cannot hold at these high Knudsen number regimes. It is also stated in the published literature that at conditions where Kn>0.1, it is required to employ microscopic non-continuum computational tools such as the Lattice Boltzmann Modelling (LBM) or the Direct Simulation Monte Carlo (DSMC) techniques to obtain accurate description of the characteristics of gas flows through microchannels; see Zhang et al [2005] and Gad-el-Hak [2002]. Interestingly, in contrast, the results obtained with the continuum based ENSE agree excellently well with the experimental measurements even for Kn>2, a moderately rarefied freemolecular regime. In order to understand these apparent conflicting observations, it is required to understand the definition of the Knudsen number. As per the classical definition, see Gad-el-Hak [2002], the Knudsen number, defined in equation (4.45), defines whether the fluid can be dealt as a continuum or not: l (4.45) Lc where l is the molecular mean free path and Lc the characteristic length of the flow. Kn
Needless to mention, it is imperative that one needs to choose the right characteristic length in equation (4.45) in order to obtain meaningful insights into the dynamics of flows. Conventionally, in all internal flow cases, the hydraulic diameter for circular or near-circular geometries and the cross-stream height for two dimensional channels are chosen as the characteristic length for estimating dimensionless parameters such as Reynolds number. Based on convention, the cross-stream height of the channel is also employed as the characteristic length to define the Knudsen number in microchannel flows, as shown in equation (4.4). Another definition, which is rather not used because of difficulties in its determination, employs the gradient of pressure or density as the characteristic length, as defined in equation (4.5); see Gad-el-Hak [1999]. As one can observe, it may be difficult to estimate the local characteristic length scale employing equation (4.5) in experimental measurements. Now it is necessary to choose from the 62
two definitions, the correct characteristic length of the gas flows through microchannel that describes the flow characteristics properly.
Molecular mean free path
1.6E-06
Molecular mean free path, m
1.6
Pressure ratio
1.4E-06
3.333 3.333
1.667 1.667
Local Knudsen number
10 10
1.4 1.2
1.2E-06
1
1.0E-06 0.8 8.0E-07 0.6
6.0E-07
0.4
4.0E-07
Local Knudsen number
1.8E-06
0.2
2.0E-07 0.0E+00
0
0
0.002
0.004 0.006 0.008 Streamwise distance from the inlet, m
0.01
(a) 0.025
1.2E-03 Pressure ratio 1.667 1.667
Diffusion length scale, m
0.02
3.333 3.333
10 1.0E-03
10
8.0E-04 0.015 6.0E-04 0.01 4.0E-04 0.005
Local Knudsen number
Diffusion length scale Local Knudsen number
2.0E-04
0
0.0E+00 0
0.002
0.004
0.006
0.008
0.01
Streamwise distance from the inlet, m
(b)
Figure 4.12 Comparison of Knudsen number at various pressure ratios; Po=0.12 bar (a) Variation of mean free path and local Knudsen number Kn l H ; (b) Variation of 1 P inverse diffusion length scale Lc P x1
1
63
and local Knudsen number Kn
l P P x1
In Figure 4.12(a), the molecular mean free path, l and the local Knudsen number Kn l H are plotted as a function of the streamwise coordinate for various pressure ratios. In all the cases, the outlet absolute pressure was kept at a constant value of 0.12 bar. As observed in this figure, the molecular mean free path and local Knudsen number profiles are similar and the maximum local Knudsen number value at the outlet of the channel for all the cases was found to be 1.42, since the outlet pressure was kept the same in all the simulations. It is interesting to note that as per the classification of gas flows into different regimes based on Knudsen number, see Gadel-Hak [1999], the Knudsen number at the outlet in Figure 4.12(a) falls under the category of transition or moderately rarefied regime. As stated earlier, the continuum assumption must have seized to provide any meaningful results beyond a Knudsen number of 0.1. Since one can obtain excellent results with the ENSE even beyond a Knudsen number of unity, one can conclude that the Knudsen number definition employed in Figure 4.12(a) does not describe the characteristics of microchannel flows properly. The local Knudsen number defined with the characteristic length given by equation (4.4) is shown as a function of streamwise coordinate is shown in Figure 4.12(b). 1 P Further, the diffusion length scale defined as Lc P x1
1
is also plotted in the same
figure. It can be observed in this figure that the diffusion length scale is much higher than the molecular mean free path l shown in Figure 4.12(a). Therefore, one can understand that the self-diffusion present in microchannels influences flow characteristics over longer distances in comparison to the length scales of mean molecular motion. Evidently, one can observe in Figure 4.12(b) that the Knudsen number defined by equation (4.5) is less by three or four orders in comparison to the conventional definition, shown in Figure 4.12(a). Hence, one may conclude that the height of the channel may be used as a characteristic length in microchannel flows when the flow is dominated by only convection. However, when the flow characteristics in micro-conduits are also determined by diffusion, the height of the channel cannot be used to characterize the flow. Further, it is interesting to observe that the diffusion does not vary in the cross-stream direction and is only the function of the streamwise direction. This also points out that the characteristic length has to be chosen along the streamwise direction. Typically, the Knudsen number is a function of only the local pressure when the characteristic length is taken to be the channel height. As seen in Figure 4.13(a), the Knudsen number at the end of the channel is constant for all values of pressure ratio since the outlet pressure has been fixed to be the same value in all the simulations. However, when the Knudsen number is defined based on the characteristic length presented in equation (4.5), it is a function of both the local value of pressure and its gradient in the channel. For example, it can be observed in Figure 4.13(b) that the local Knudsen number for the case with a pressure ratio of 10 is less near the inlet of the channel in comparison to that of the cases with pressure ratios of 1.67 and 3.33. 64
However, towards the outlet of the channel, the local Knudsen number with the pressure ratio of 10 is one order of magnitude more than that of the other two cases.
(a)
(b)
Figure 4.13 Comparison of conductance, Cˆ m T Pin Po of the microchannel as a function of local Knudsen number; (a) Kn
l P l and (b) Kn P x1 H
Hence, one can observe that the local value of the absolute pressure alone does not determine the Knudsen number of the flow and the local pressure gradient is also equally important. From the above discussions, it can be concluded that the conventional definition of Knudsen number does not provide precise insight into the 65
characteristics of gas flows through microchannels. Further, the definition based on the characteristic length presented in equation (4.5) suggests that the continuum assumption can still be valid in most of the gas flow situations experienced in microchannels. It is proposed that the gas flow characteristics through micro-conduits need to be carefully revisited once again in the light of above discussions. One can summarize the discussions presented in this chapter as given below. The convective acceleration terms in the Navier-Stokes equations can be neglected in the analysis of low Reynolds number creeping flow through the microchannels. The ENSE predict all the characteristics of gas flows through microchannels accurately without any empirical tuning parameters and provide physically sound explanations. Hence, it can be taken as the proof that the additional mass flow rate observed in the experiments is the self-diffusive mass transport. The classical definition of Knudsen number does not characterize the gas flows through microchannels properly and the diffusion length is the correct length scale to be used in microchannel flows.
66
Chapter 5
ANALYTICAL TREATMENTS OF GAS FLOWS THROUGH MICRO-CONDUITS 5.1 Introduction As shown in the previous chapter, the extended Navier-Stokes equations (ENSE) were successfully employed to numerically solve gas flows through microchannels and excellent agreements were obtained with the experimental measurements. These results conclusively proved that the excess mass flow rate, observed in experiments, was indeed caused by the self-diffusion of mass driven by the pressure gradient along the length of the channel. Further, the discussions on the right definition of the Knudsen number also clarified that the continuum-based ENSE are very much valid even at ‘apparently high’ Knudsen numbers defined by the mean molecular mean free path and the height of channel. Buoyed up by these interesting and revealing results, an analytical solution for the flow of gases through micro-conduits was attempted, as explained in this chapter. Needless to mention, the availability of an analytical solution to determine the flow characteristics of microchannels and capillaries will completely eliminate the high computational efforts required in the currently employed simulation techniques, such as the Lattice Boltzmann Method (LBM) or the Direct Simulation Monte Carlo (DSMC) computations. In this chapter, the analytical solution procedure employing the ENSE is explained in detail and a similar treatment is also available in the articles co-authored by the author; see Filimonov et al [2010] and Navaneetha Krishnan et al [2010]. The comparisons of mass flow rates of gas flows through microchannels obtained based on the analytical solutions are carried out with the numerical solutions described in chapter 4 and the experimental measurements by Maurer et al [2003], Arkilic et al [1994] and Colin et al [2004] are shown. Furthermore, a semi-analytical solution of the pressure profile along the length of the channel is also obtained and the same is compared with corresponding results of the numerical solutions and the experimental measurements. Furthermore, the analytical results obtained for micro-capillary flows are also compared with the experimental measurements by Yang and Garimella [2009]. In addition, a characteristic pressure is defined, based on the dimensions and fluid properties for both microchannels and micro-capillaries and it is shown to be a very important parameter in determining the characteristics of gas flows through micro-conduits.
5.2 Analytical solution procedure 5.2.1 Order of magnitude analysis In the analytical solution procedure, the gas flow is assumed to be two-dimensional for microchannels and axisymmetric for capillaries in rectangular and cylindrical coordinate systems, respectively. Furthermore, the flows are also assumed to be steady and isothermal in nature. Since the fluid is assumed to be isothermal, the viscosity is also considered to be constant. The schematic representation of the convective and total velocity profiles obtained from the numerical solutions of the CNSE and the ENSE mentioned in the previous chapter are shown in Figure 5.1. The ENSE in the total velocity form, obtained from equations (3.46) and (3.49), can be written for steady, isothermal gas flows, as given below: Continuity equation: U iT 0 xi Momentum equations:
(5.1)
P U iT U Tj xi x j xi
2 C D D D C ij m i U j 3 ij m k U k
(5.2)
The density is calculated based on the equation of state given by equation (2.3). Further, equations (5.1) and (5.2) can be expanded for a two-dimensional flow situation, employing equation (2.24), as Continuity equation: U1T U 2T 0 x1 x2
(5.3)
x1 - momentum equation:
U 1T U 1T U 1T U 2T x1 x 2
4 U 1C 2 U 2C m 1DU 1D P 3 x1 3 x 2 x1 x1 2 D C m 1 U 1 m 2DU 2C 3 U 1C U 2C D D m U 2 2 x 2 x 2 x1
68
(5.4)
x 2 - momentum equation:
U 1T U 2T U 2T U 2T U 1C U 2C P D D m U 1 2 x1 x 2 x 2 x1 x 2 x1
x 2
4 U 2C 2 U 1C 2 m 2DU 2D m 1DU 1C m 2DU 2C 3 x1 3 3 x 2
(5.5)
x2 or r
x1
(a)
x2 or r
x1
(b)
Figure 5.1 Schematic representation of velocity profiles obtained from the numerical solutions presented in chapter 4; (a) Convective velocity from the CNSE (b) Total velocity from the ENSE For the case of gas flows through straight microchannels, there is no diffusion transport of mass in the cross-stream direction since the pressure is constant in this direction. Similarly, the convective velocity in the cross-stream direction is also zero since the flow is assumed to be fully developed. Further, as shown in chapter 4, the convective acceleration terms can be neglected in the momentum equations for the case of gas flows through micro-conduits. Therefore, equations (5.3) to (5.5) can be simplified as follows: 69
U1T 0 x1
(5.6)
4 U 1C U 1C 2 D C D D m U m U 1 1 1 1 3 x 2 3 x1 x2 C U 1C P 2 U 1 2 0 m 1DU 1C x 2 x1 x 2 x 2 3 x1 3 0
P x1 x1
(5.7)
(5.8)
To further simplify equations (5.6) – (5.8), an order of magnitude analysis was employed. The characteristic velocity and length scales for this analysis had to be carefully chosen. The convective velocity can be scaled with the average velocity at the exit of the channel U and the diffusion velocity is scaled as ~ L based on the expression for the self-diffusion velocity given by equation (3.43). Further, the characteristic length for the streamwise and cross-stream directions are the length L and height H of the channel, respectively where H 0 and this region has also been masked by the normalization. 7.4.3 Inverse density thickness, L1 Apart from the comparisons of shock structures, one can compare the results obtained from different methods based on other shock parameters. As the first parameter, the dimensionless shock inverse density thickness was chosen which is defined as given below. L1
lM 1 2 1
(7.14)
max
where l M 1 is the molecular mean free path in the upstream of the shock region and 1 and 2 are the densities at the upstream and downstream regions, respectively. Further, max represents the maximum value of the density gradient within the shock. In the case of shock wave problem, there is no easily determinable characteristic length scale and hence, it is not easy to define the Knudsen number. It is customary to use the actual thickness of the shock layer itself as the characteristic length scale. Therefore, the dimensional inverse density thickness can be used to define the Knudsen number of the shock wave, as l (7.15) max 2 1 where l is the local molecular mean free path of the shock wave. Greenshields and Kn
Reese [2007(a)] observed that based on the above mentioned definition of the Knudsen number Kn , one would obtain very high Knudsen numbers and hence any hydrodynamic model should fail to predict the characteristics of the shock structures. 163
They, however, reasoned that one could still assess any extended hydrodynamic equations for their usefulness as engineering models.
Inverse density thickness, 1/m
0.5 0.45 0.4 0.35 0.3 0.25 0.2 CNSE ENSE Experiment MBNSE_S=0.72
0.15 0.1 0.05 1
2
3
4
5
6
Mach number
7
8
9
10
11
Figure 7.15 Comparisons of inverse density thickness, L1 of the shock structure obtained with different methods In Figure 7.15, the dimensionless inverse density thickness L1 values obtained with different models are compared. The values obtained with the CNSE and the ENSE are shown along with the values predicted by the MBNSE. Greenshields and Reese [2007(a)] employed the MBNSE and predicted the dimensionless inverse density thickness L1 values with three values of the power law exponent, S of dynamic viscosity of Argon. In Figure 7.15, the values obtained by the power law exponent S of 0.72 are only shown. Further, Alsmeyer [1976] provided the most comprehensive collection of experimental shock data, comprising previously published work as well as his own results. The dimensionless inverse density thickness L1 values collated from the experimental measurements from Steinhilper [1972], Alsmeyer [1976] and Torecki and Walenta [1993] are shown in Figure 7.15, as shown in Figure 10 of Greenshields and Reese [2007(a)]. As it can be observed in this figure, the CNSE fail to predict this important parameter of the shock wave characteristics and the values predicted are much more than the experimental predictions. The values obtained with the MBNSE with the power law exponent of 0.72 closely match the experimental measurements. It can be observed in this figure that the ENSE predict 8-10% higher values of inverse density thickness as 164
compared to the experimental measurements. Though it may be considered that the agreement of the numerical results obtained with the ENSE and experimental measurements is reasonable, it is intuitively felt that one can improve the predictions of the ENSE further because of the following reasons. As observed by Greenshields and Reese [2007(a)], the numerical predictions of the inverse density thickness was found to be influenced by the chosen power law exponent since with increasing value of the power law exponent, the prediction was found to improve and resemble the experimental measurements. They also found out that the power law exponent S of 0.74 seemed to represent the best-fit line of the experimental measurements of the inverse density thickness, shown in Figure 7.15. It can also be observed in this figure that the predicted values of inverse density thickness by the ENSE are closer to the experimental measurements at smaller inlet Mach numbers than the larger values. Since the chosen power law exponent S of the viscosity in this study was 0.72 and this value was found to agree well with the experimental measurements of viscosity at lower temperatures. In the simulations by Greenshields and Reese [2007(a)], they mentioned that the power law exponent S of 0.76 was found to be suitable for inlet Mach number values more than 8. In short, if one chooses the right viscosity model, the predictions of the ENSE can be improved further. Since the objective of the studies mentioned in this chapter is to show that the ENSE can be employed to predict the characteristics of shock waves accurately and the self-diffusion of mass is the sole reason for the enhanced spread of shock profiles observed in the experiments. Therefore, the additional studies involving choosing the correct power law model of viscosity was not carried out, as part of this thesis. Since the temperature in the upstream region of the shock is very low, it is not correct to choose a single power law exponent S for the whole shock wave. It is required to use different values of the power law exponent at different sections of the shock wave depending on the temperature range prevailing in the region. Further, there are other viscosity models, such as Sutherland’s model and it may be possible that the predictions can be improved further by employing these models, as well. Moreover, there are some other pertinent issues with the ENSE as discussed later in this chapter, which demand some more detailed attention. The author is interested to work on each of the above mentioned aspects in order to improve the predictions of shock waves by employing the ENSE. 7.4.4 Density asymmetry quotient, Q Another important quality parameter of shock wave predictions is known as the density asymmetry quotient Q which is defined as, 0 dx
Q
(7.16)
1 dx
0
165
where is the local value of dimensionless density. Since the inverse density thickness comparison, shown in Figure 7.15, depends only on the density gradient at the midpoint of the shock structure, it does not express anything about the overall shape of the profile. Therefore, the density asymmetry quotient Q can be used to describe the complete shock profile and some experimental data is also available for this parameter for effective comparisons.
Density Asymmetry Quotient
1.5
1.4 CNSE ENSE
1.3
Experiment MBNSE_1 1.2
MBNSE_2
1.1
1
0.9 1
2
3
4
5
6
7
8
9
10
11
Mach number
Figure 7.16 Comparisons of density asymmetry quotient Q of the shock structure obtained with different methods As shown in equation (7.16), the density asymmetry quotient Q is a measure of the skewness of dimensionless density profile of shocks about the midpoint. Therefore, a symmetric shock would consequently have the density asymmetry quotient Q of unity but real shock waves are found to be not completely symmetrical about their midpoint. It is observed from the experimental measurements that the density profile is skewed a little towards the downstream direction at smaller inlet Mach numbers, i.e. Q 1 . However, as the Mach number increases, the spread of the shock increases, as discussed earlier. Hence, the value of the density asymmetry coefficient Q becomes more than unity and it increases with increasing inlet Mach number within the range studied in the experiments. In Figure 7.16, the density asymmetry quotient Q values obtained with the CNSE and the ENSE are compared with the experimental measurements of Alsmeyer [1976]. Further, the values obtained by the MBNSE, the data extracted from Figures 11 and 4 166
from Greenshields and Reese [2007(a)] and [2007(b)], respectively, are also shown in the same figure. Since the plots of density asymmetry quotient employing the MBNSE differed significantly in Greenshields and Reese [2007(a)] and [2007(b)], both profiles are presented in Figure 7.16 as the MBNSE_1 and the MBNSE_2, respectively. It is evidently seen in Figure 7.16 that both the CNSE and the MBNSE do not at all predict the asymmetry in the density profile properly. The CNSE overpredict the density asymmetry quotient Q by ~30% whereas MBNSE_1 underpredict it by ~20%. The profile shown as the MBNSE_2 over-predicts the density asymmetry quotient at lower Mach numbers and it approaches the experimental measurements at higher inlet Mach numbers. However, the predictions by the ENSE agree excellently with the experimental measurements at all Mach numbers. This result indicates that the ENSE developed in this thesis can be one of the stronger competitors for predicting the structure of the shock waves accurately.
Dimensionless temperature - density seperation distance
7.4.5 Temperature-density separation, T 3.5 CNSE 3
ENSE DSMC Brenner
2.5
2
1.5
1
0.5 1
2
3
4
5
6
7
8
9
10
11
Mach number
Figure 7.17 Comparisons of dimensionless temperature - density separation distance T of the shock structure obtained with different methods Another shock structure parameter is known as the temperature–density separation T and this distance is measured between the midpoints of the normalized density and temperature profiles. It is understood from the available literature that the shock structure of density happens to be in the downstream of that of temperature and this is believed to be due to the finite relaxation times for momentum and energy transport. However, there is no experimental data for this parameter since it is very difficult to 167
measure temperature profiles in a shock. Typically, one has to employ the DSMC simulations to generate this data for comparison with hydrodynamic models. In Figure 7.17, the calculated temperature–density separation distances obtained by the CNSE, the ENSE and the MBNSE are compared against the values obtained by the DSMC simulations of Lumpkin and Chapman [1991]. It is interesting to note that none of the models are able to predict the temperature–density separation distance calculated by the DSMC simulations. The separation thickness T calculated by the DSMC simulations increases monotonically with Mach number. The values calculated from all the hydrodynamic models are lower than those predicted by the DSMC simulations for almost all the Mach numbers. The predictions of the CNSE and the MBNSE show a decreasing trend up to the Mach number of ~2.3 and then the values gradually increase with increasing Mach number. The separation thickness values calculated from the ENSE also increase monotonically with Mach number similar to the DSMC results, i.e. both the profiles are almost parallel, but the absolute values are much less in comparison to the DSMC simulations. Though the differences between the results obtained by the DSMC simulations and the ENSE are strikingly evident, one needs to consider the following before concluding about the suitability of the ENSE in predicting shock structures. In the comparisons with experimental measurements of inverse density thickness L1 and density asymmetry quotient Q , the ENSE performed excellently, as shown in Figures 7.15 and 7.16. The temperature–density separation thickness is not an experimentally measurable parameter and it might be possible that the underlying assumptions in the DSMC simulations could be the reason for this large discrepancy observed in the comparisons shown in Figure 7.17. At this juncture, it is difficult to favour one method over the other based on the set of comparisons provided in this chapter. However, the availability of the ENSE can revive the studies on shock structures and characteristics and it is believed that promising new understanding of the underlying physics of shock waves can be obtained from such studies. 7.4.6 Discussions on the definition of Knudsen number In the absence of solid walls, it is always difficult to choose the correct characteristic length to define dimensionless parameters and the definition of Knudsen number for the case of shock waves is no exception. It is well-known for a long time that the CNSE do not predict shock structures accurately and historically, the high Knudsen number of the flow was blamed as the reason for the failure of the CNSE. It was argued that the local equilibrium assumption would no longer be valid in high Knudsen number flows and hence the continuum based CNSE was expected to fail in the predictions. Based on this argument, many complex extensions were proposed to the CNSE to incorporate the experimentally observed spread of the shock structures in the computational simulations. In order to understand the characteristics of the shock structures and 168
Knudsen number, defined by equation (7.15)
choose the right model to simulate the shock waves, it was mandatory to select a suitable characteristic length to define the Knudsen number. 0.35 Ma 2.31 Ma 2.84 Ma 3.8 Ma 8 Ma 9 Ma 11
0.3 0.25 0.2 0.15 0.1 0.05 0
Knudsen number, defined by equation (7.17)
-10
-8
-6
-4
-2
0
2
4
6
Nondimensional distance (a) 0.25 Ma 2.31 Ma 2.84 Ma 3.8 Ma 8 Ma 9 Ma 11
0.2
0.15
0.1
0.05
0 -10
-8
-6
-4
-2
0
2
4
6
Nondimensional distance (b)
Figure 7.18 Comparison of Knudsen number profiles at different inlet Mach numbers As mentioned by Greenshields and Reese [2007(a)], it is customary to choose the Knudsen number definition given by equation (7.15) in shock wave simulations. The Knudsen number profiles, calculated using equation (7.15), are shown in Figure 7.18(a) for various inlet Mach numbers. It can be observed in this figure that the Knudsen number profiles resemble the profiles of dimensional variables shown in Figures (7.4)– (7.10). For a given Mach number, the maximum value of the Knudsen number is at the 169
upstream section of the shock where the pressure is the minimum. Though the lower temperature prevailing at the upstream section of the shock attempts to reduce the Knudsen number, the influence of pressure is stronger. The minimum Knudsen number occurs at the downstream region of the shock, as expected. However, it is possible to observe the following ambiguity in the profiles shown in Figure 7.18(a). The inlet pressure and temperature were the same for all the Mach numbers simulated in Figure 7.18(a) and therefore, the molecular mean free path computed using equation (7.14) is also identical for all the cases. Since the selected characteristic length, based on the maximum value of normalized density gradient, is a function of the inlet Mach number, one obtains different Knudsen number values at the upstream of the shock. It was felt that this does not characterize the shock wave correctly. Therefore, the Knudsen number was redefined using the local diffusion length scale and molecular mean free path l, as given by, 1 1 T Kn l (7.17) x1 2T x1 The calculated Knudsen number profiles employing equation (7.17) are plotted at different inlet Mach number values in Figure 7.18(b). It is evident from this figure that the Knudsen number values identically tend to zero in both upstream and downstream regions for all the Mach numbers. As the inlet Mach number increases, one might expect that the tendency to deviate from local equilibrium condition and the peak value of Knudsen number profile will also increase. However, as observed in Figure 7.18(b), such a monotonic increase in the peak Knudsen number is not observed and the maximum value of Knudsen number is limited to ~0.22 only at all inlet Mach numbers. It is argued that this particular characteristic of the shock wave indicates the reason why the excellent prediction of shock structures was possible by employing the continuumbased the ENSE proposed in this thesis. 7.4.7 Discussions on violation of laws of thermodynamics It is, in general, believed that as the flow becomes rarefied, the predictions of continuum formulations, such as the CNSE, become inaccurate since these formulations employ linear approximations to the molecular momentum and heat transport, as shown in equations (2.24) and (2.30), respectively. Therefore, higher order terms in the constitutive relationships of molecular momentum and heat transports were incorporated in the governing equations to improve predictive capabilities under rarefied conditions and so the second-order systems of hydrodynamic equations, such as the Burnett and the Woods equations, were obtained. While it was observed that these equations provided a better description of the shock structure on coarse grids, they were prone to instabilities, believed to be small wavelength instabilities, when the grids were refined. It was conjectured that an inherent entropy inconsistency might be the cause of computational instability and subsequently it was proven that the one-dimensional Burnett equations, when applied 170
to the hypersonic shock structure problem, can violate the second law of thermodynamics as the local Knudsen number increases above a critical limit. For a detailed description of the above mentioned problem, one can refer to Balakrishnan [2004].
Dimensionless specific entropy,
S
Dimensionless length, x1 l M 1
Dimensionless length, x1 l M 1
Figure 7.19 Dimensionless specific entropy, S profiles of a shock with inlet Mach number of 20, as obtained by Balakrishnan [2004]; Grid fineness ratios, x1 l M 1 : (a) 8 and (b) 4 In Figure 7.19, S indicates the change of entropy. As shown in Figure 7.19, the dimensionless entropy S profiles portrayed a dip in the upstream region of the shock and the extent of this dip increased with increasing grid fineness ratio x l M 1 . For an inlet Mach number of 20, it was found that the critical grid fineness ratio x l M 1 is 4 and the temperature became negative for values less than 4. In Figure 7.20, the dimensionless entropy profiles of a shock wave, calculated with the CNSE and the ENSE, with an inlet Mach number of 11 are shown. Interestingly, the grid fineness ratio x l M 1 employed in the simulations was 0.033 and no negative entropy dip was observed in the upstream region of the shock. Subsequently, the grid fineness ratio x l M 1 was decreased to a value of 0.01 and even then, no abnormalities in the specific entropy profiles were observed. Further, the entropy profiles obtained with a number of inlet Mach numbers are also presented in Figure 7.21 and the solutions obtained were entropy-consistent in each and every case. As explained by Balakrishnan [2004], by employing the modified Burnett equations, one could choose the right set of equations and coefficients to obtain an entropy consistent shock structure. Even after that elaborate careful selection of terms, one could go up to a grid fineness ratio x l M 1 of only 0.1 and the solution showed inconsistency below the value of 0.1. On the 171
contrary, the ENSE based solutions faced no such constraints and the solution was entropy-consistent at all grid fineness levels. 5
Dimensionless specific entropy
4.5 4 3.5 3 CNSE
2.5
ENSE
2 1.5 1 0.5 0 -16
-14
-12
-10
-8
-6
-4
-2
0
2
4
Nondimensional distance
Figure 7.20 Dimensionless specific entropy, S profiles obtained with the CNSE and the ENSE of a shock with inlet Mach number of 11; Grid fineness ratio x l M 1 0.033 4.5 Ma 2.31
Dimensionless specific entropy
4
Ma 2.84 Ma 3.8
3.5
Ma 8 3
Ma 9 Ma 11
2.5 2 1.5 1 0.5 0 -16
-14
-12
-10
-8
-6
-4
-2
0
2
4
6
8
Nondimensional distance
Figure 7.21 Dimensionless specific entropy, S profiles of a shock obtained with the ENSE with different inlet Mach numbers; Grid fineness ratio x l M 1 0.033 172
1140
Total temperature, K
1120 1100 CNSE ENSE
1080 1060 1040 1020 1000 -10
-8
-6
-4
-2
0
2
4
6
8
10
Nondimensional distance (a) 6900 6800
Total temperature, K
6700 6600
CNSE
6500
ENSE
6400 6300 6200 6100 6000 5900 -10
-8
-6
-4
-2
0
2
4
6
8
10
Nondimensional distance (b)
Figure 7.22 Comparison of total temperature profiles obtained with the CNSE and the ENSE; Inlet Mach number (a) 2.34 and (b) 8 One more interesting observation about the shock wave results can be made related to the first law of thermodynamics. Shock waves can, in general, be considered as adiabatic systems and one dimensional shock waves are certainly adiabatic. It is wellknown that the total temperature at all sections must remain constant in adiabatic flows. In the case of the CNSE, the total temperature of the fluid T0 is defined by the following equation: 173
T0
U T
C 2 1
(7.18)
2C P
In equation (7.18), T and U 1C are the local static temperature and convective velocity of the fluid, respectively. Similarly, the total temperature of the fluid T0 can be defined for the case of the ENSE as, T0
U T
T 2 1
(7.19)
2C P
where U 1T is the local total velocity of the fluid. The total velocity is the vector sum of the convective and diffusion velocities as shown in equation (3.51). In Figure (7.22), the total temperature profiles obtained with the CNSE and the ENSE are presented for two inlet Mach numbers. It is interesting to note that the total temperature does not remain constant and reaches a maximum value at the shock in the case of the CNSE. One can note that it is an undisputable violation of the first law of thermodynamics and the author does not know any published literature discussing this discrepancy. Surprisingly, the total temperature does also not remain constant in the case of the ENSE. In this case, the total temperature drops at the shock to a minimum value and this is also a violation of the first law of thermodynamics. It is known that the CNSE do not predict shock structures properly and hence it is no wonder that the results obtained with the equations show certain thermodynamic inconsistency. However, the predictions of the ENSE were pretty close to the experimental measurements and therefore it is surprising to find similar thermodynamic inconsistency in this case as well. There may be a number of reasons for this inconsistency and few of the possibilities are enlisted here. (i) The selection of the correct power law exponent of the viscosity model can influence the spread of the shock wave and therefore, can reduce the deviation from the thermodynamically consistent condition of constant total temperature across the shock. Further, one need not assume constant power law exponent throughout the shock for a given Mach number and can choose the exponent based on the local temperature values, as discussed before. (ii) In the simulations presented in this chapter, Argon was assumed to be adhering to the ideal gas law. Since very high temperatures are encountered in the downstream region of strong shock waves, one can expect some deviation from the ideal gas law. One can incorporate real gas behaviour in the computations in order to study its influence on this observed violation of the first law of thermodynamics. (iii) More importantly, as discussed in the beginning of section 7.2.3, the second m D term, Ek i in the right hand side of equation (7.3) had to be dropped from the xi total energy equation and only then converged solutoin could be obtained with the ENSE. The reason for this behaviour of the equations is not known yet. 174
(iv) Further, the thermal energy equation, shown in equation (3.56), is given once again here with a slight modification.
e U iC e t x i x i
U j U i T D T m i e P ij x i x i x i
(7.20)
In the last two terms in the right hand side of equation (7.20), U i has been introduced instead of the convective velocity U iC . Since the diffusion velocity is a macroscopic quantity like the convective velocity, it can also take part in the expansion work and viscous dissipation. Therefore, in equation (7.20), U i can be replaced with U iT in one or both of the terms. The author tried to change the two terms in terms of the total velocity to see whether the consistency with respect to the first law of thermodynamics could be achieved but faced with poor quality of predictions of the shock structures. Based on the above mentioned discussions, it can be concluded that the problem of numerical computation of shock structures is not yet understood completely and there remains a number of open questions. However, based on the results discussed in this chapter, it needs to be mentioned here that the ENSE can be employed as a new and effective tool to understand this complex flow structure.
175
176
Chapter 8
IMPORTANT RESULTS AND SUGGESTIONS FOR FURTHER RESEARCH 8.1 Introductory remarks The governing equations of fluid mechanics, known as the CNSE were, in general, considered to be well-developed and capable of handling all sorts of complex problems accurately. The excellent agreement between the experimental measurements and theoretical and computational predictions were deemed to be the proof of exactness of the governing equations and the employed constitutive relationships of diffusion transports of momentum and heat, provided by the Newton’s law of viscosity and Fourier’s law of heat diffusion, respectively. Interestingly, certain problems, such as gas flows through microchannels and capillaries, accurate predictions of shock structures, physically convincing explanations to thermophoresis and thermal transpiration etc., remained unexplained by the CNSE. These problems were thought to be in the fringe region where the continuum assumption, upon which the CNSE were based, itself was not valid and hence special treatments, such as invoking ‘slip-flow’ theory in the case of microchannel flows and introduction of extended hydrodynamic models such as Burnett equations in the case of shock wave simulations etc., were provided to solve these so called ‘high Knudsen number flows’. Since one could obtain excellent agreements with experimental measurements in most of the flow cases, the completeness of the CNSE was never questioned in the past. In the recent past, there is an increasing discomfort amongst the scientific community about the comprehensiveness of the CNSE and there were number recent attempts to derive what is known as ‘the extended Navier-Stokes equations’ in order to broaden the validity of the continuum based the CNSE, the most recent one was proposed by Brenner [2005(a)]. Though there was a general consensus on ‘something is amiss’ with the CNSE, no missing physical process was identified as the reason for the observed deviations of theoretical predictions from corresponding experimental measurements and hence, the proposals were extemporized and remained fundamentally questionable, for example the ‘volume diffusion’ theory proposed by Brenner [2005(a)]. As claimed by Brenner [2005(a)], it was possible for the volume to diffuse without any corresponding mass diffusion and this theory had been accepted because of the explanation provided to problems such as thermophoresis, even though the basic assumption was flawed. At the same time Brenner [2005(a)] proposed his version of the extended equations (BNSE), Durst et al [2006] proposed the ENSE based on the argument that the self-diffusion transport of mass needed to be incorporated in the continuity equation. Though the presence of self-diffusion in liquids and gases was known for centuries, it was not included in the CNSE because of its apparent smallness.
Durst et al [2006] argued that though the self-diffusion transport of mass remained negligible in comparison to the convective transport in most flow situations, it could assume greater significance in gas flows where strong density and temperature gradients were present such as gas flows through microchannels and shock waves. Further, the additional diffusive transports of momentum and energy caused by the self-diffusion mass transport were also incorporated in the momentum and energy equations, respectively and subsequently, the ENSE were obtained. In the present thesis, the ENSE were derived based on the procedure followed by Durst et al [2006], as shown in chapter 3. Further, comparisons between the ENSE proposed in this thesis and the BNSE were also provided and the absence of diffusive transport in the continuity equation was stated to be the major flow in the derivations by Brenner [2005(a)]. Any theoretical model needs to be validated with experimental measurements in a number of flow situations to prove its validity. Therefore, the ENSE proposed in this thesis was also employed to solve problems such as gas flows through straight microchannels and capillaries, gas flows through microchannels with a backward facing step and predictions of shock structures in a monoatomic gas. In chapter 4, the characteristics of gas flows through straight microchannels were numerically computed and compared with the experimental measurements of Arkilic et al [2001] and Maurer et al [2003]. The analytical solutions obtained with the CNSE and the no-slip boundary condition at the solid wall proved that the convective acceleration terms could be neglected in gas flows through straight microchannels. The numerical predictions obtained with the ENSE agreed excellently with the experimental measurements of Maurer et al [2003]. This conclusively proved that the reason for the observed additional mass transport in the experiments was indeed the self-diffusion mass transport driven by the density gradient in the channel and not because of the widely believed and accepted Maxwell-slip based ‘slip-flow’ theory. It could be observed in this chapter that both the convective and diffusion velocity profiles obtained with the ENSE satisfied the no-slip boundary condition at the wall. However, since the diffusion velocity reached its maximum value immediately at the outer region of the wall roughness elements, the total velocity profile, the vector sum of the convective and diffusion velocities, resembled the slip-flow profile. Further, it was stressed that it was essential to understand that the origin to this slip-like behaviour observed in the simulations with the ENSE was different from the ‘slip-flow’ theory. In the case of the ‘slip-flow’ theory, it was necessary to conduct experiments to measure the value of slipcoefficient whereas the diffusion velocity was completely deterministic from the first principles. Based on the ENSE, it was possible to predict the ‘Knudsen paradox’, defined as the occurrence of the minimum in the conductance profile, and physical reasons behind the apparent ‘paradoxical’ behaviour of the conductance profile were also explained in detail. Since there was an excellent agreement between numerical predictions based on the ENSE and experimental measurements, it could be argued that gas flows through microchannels, in the range studied in this thesis, were essentially within the continuum region. Further, to elaborate this aspect, the Knudsen number was defined based on the characteristic length obtained from the ‘diffusion length scale’ of the flow instead of the often employed height of the channel. Interestingly, it was 178
observed that the Knudsen number defined based on the diffusion length scale was two to three orders of magnitude smaller than that obtained based on the classical definition as shown in Figure 4.13. Subsequently, based on the insights obtained from the numerical simulations of microchannel flows employing the ENSE, analytical solutions to gas flows through microchannels and capillaries were attempted, as described in chapter 5. The convective acceleration terms were dropped from the momentum equations based on the detailed order of magnitude analysis. The resulting equation could be integrated to obtain the analytical solutions for the mass flow rate and velocity through microchannels and capillaries. More significantly, a characteristic pressure PC was introduced which was found to describe the nature of flow characteristics through microchannels and capillaries accurately. When the local dimensionless pressure obtained based on the characteristic pressure, P PC was equal to unity, the convective and diffusive mass transports were found to be equal. Similarly, when the local pressure was more than the characteristic pressure PC , then the convective mass transport was dominant over the diffusion transport and vice versa. The mass flow rates obtained by the analytical solutions for both microchannels and capillaries were found to agree excellently with the corresponding experimental measurements. Furthermore, one semi-analytical solution procedure was developed to obtain the pressure profiles along the length of the channel and capillaries and the theoretically obtained profiles agreed remarkably with the experimental values, as well. It can be convincingly stated here that one can use the analytical solutions of the ENSE to obtain all the characteristics of isothermal gas flows through microchannels and capillaries accurately, up to an outlet Knudsen number of unity. One of the major bottlenecks of the ‘slip-flow’ theory is its inability to predict gas flows through complex microchannel geometries, say for example, a backward facing step. Since the tangential momentum accommodation coefficient needs to be determined experimentally in order to employ the ‘slip-flow’ theory, one would find it difficult to determine the same in complex geometries. This inability of the ‘slip-flow’ theory led to the application of the Lattice Boltzmann Simulations (LBS) and the Direct Simulation Monte Carlo (DSMC) simulations in complex microchannel geometries. Interestingly, the ENSE based simulations could predict gas flow characteristics through microchannels with a backward facing step exceedingly well and the excellent comparisons were obtained with the DSMC simulations over a wide Knudsen number range. It was found that the velocity values at the wall reported in the DSMC simulations were not evaluated exactly ‘at the wall’ but a small normal distance away from it. This led to some discrepancy in comparing the wall velocity magnitudes obtained with the ENSE. Other than this minor discrepancy, all the flow characteristics of microchannel flows with a backward facing step were accurately determined employing the ENSE.
179
As shown in the derivations of the ENSE in this thesis, the self-diffusion mass flow rate does also depend on the temperature gradients present in gas flows and hence, it was necessary to assess the equations in predicting gas flows with strong temperature gradients. Prediction of one dimensional shock structures of supersonic and hypersonic gas flows of a monoatomic gas was taken to be the test problem for which extensive experimental measurements and results from the DSMC simulations are available for comparison. Further, the numerical predictions, based on the modified extended Navier-Stokes equations proposed by Brenner [2005(a)] (MBNSE), provided by Greenshields and Reese [2007(a)], were also employed for comparisons. As explained in chapter 7, the shock structures were predicted accurately by the ENSE for all the Mach numbers studied in both upstream and downstream regions of the shock whereas the MBNSE failed to predict the upstream characteristics of the shock accurately at higher Mach numbers. Shock parameters obtained by the ENSE such as inverse density thickness L1 and density asymmetry quotient Q were found to be in excellent agreement with the experimental measurements. However, the ENSE were found to under-predict the temperature–density separation thickness T in comparison to the predictions of the DSMC simulations. Since this is not an experimentally determined parameter, it is necessary to employ elaborate validations to verify the predictions of temperature–density separation thickness T based on both the ENSE and the DSMC simulations. Furthermore, as detailed by Balakrishnan [2004], the shock structures obtained employing the Burnett equations were found to violate the second law of thermodynamics at the upstream location of the shock for lower grid fineness values. The predictions by the ENSE were free of this thermodynamic violation at any grid fineness values. More importantly, the simulations carried out employing both the CNSE and the ENSE were found to violate the first law of thermodynamics as the total temperature was found to vary along the length of the shock. One can notice that the characteristics of shock structures have not yet been understood completely and it is stressed here that the ENSE can be employed as a useful tool in exploring the underlying physics of hypersonic shock waves.
8.2 Outlook towards future research It is felt that the problems solved in this thesis provide a strong foundation for analyzing many diffusion dominated gas flows in detail. In this section, some of the potential research areas that can be explored employing the ENSE are outlined. 8.2.1 Gas flows through microchannels One can observe from the various results discussed in chapters 4 and 5 that the problem of isothermal gas flows through straight microchannels and capillaries has been completely resolved employing the ENSE. It has also been completely proven that the additional mass flow rates observed in experiments is indeed the self-diffusion transport of mass and not because of the Maxwellian slip at the solid boundary. It is felt that the following problems can be dealt further in the case of straight microchannels. 180
(i)
Gas flows through straight channels with heat transfer: A number of experimental measurements are available in the published literautre in the area of heat transfer in microchannel gas flows. The ENSE can be employed to predict the heat transfer rates in microchannels and compare with the experimental measurements. (ii) It is generally felt that the noncontinuum effects will be very significant even at the Knudsen number value of around 0.01-0.05. On the contrary, the ENSE, which was derived based on the continuum hypothesis, were found to work satisfactorily at Knudsen numbers of O(1) , as shown in chapters 4 and 5. One can compare experimental measurements at higher Knudsen numbers with the simulated results to determine the upper limit of the validity of the ENSE. (iii) The ENSE were found to satisfactorily predict the characteristics of gas flows in microchannels with a single backward facing step. However, if the experimental measurements are made available, one can evaluate gas flows in very complex flow situations. As of today, there is no other continuum based models available which can handle such complex flows efficiently and further employing the DSMC simulations or any other molecule-based simulations will be computationally prohibitive. 8.2.2 Computations of hypersonic shock structures Needless to mention, the shock structures obtained with the ENSE were superior to the MBNSE at all values of inlet Mach number. The shock structures, obtained with the ENSE, in the upstream direction of the shock wave was found to be smooth and closer to the experimental measurements whereas the profiles simulated with the MBNSE could not calculate the upstream region properly. However, it was felt that the predictions of density profiles could be improved further if the correct self-diffusion coefficient in the simulations with the ENSE. As mentioned by Greenshields and Reese [2007(a)], the self-diffusion coefficient was equal to Dm 1.32 1.32 when calculated using the Chapman–Enskog theory employing the Lennard – Jones potential. However, they faced convergence difficulties while using this value of the diffusion coefficient and finally, settled for Dm . When one employs the classical kinetic theory of gases, the self-diffusion coefficient is predicted to be equal to the kinematic viscosity of the gas, i.e., Dm as employed in the simulations presented in chapters 4–7. However, the real atoms and molecules do not behave exactly identical to the assumptions of the kinetic theory of gases and the self-diffusion coefficient is modelled, under these conditions, as Dm C where C varies between 1.2 – 1.5; see Kennard [1938]. As stated by Kennard [1938], the value of the constant C was found to be 1.2 and 1.543 when the hard spheres assumption and the inter-molecular repulsion based on the 5th power of the distance were considered, respectively. 181
Nondimensional Density/Temperature
1 0.9 0.8 0.7 Rho_CNSE
0.6
Rho_ENSE 0.5
Rho_MBNSE
0.4
Rho_Expt
0.3
Temp_CNSE
0.2
Temp_ENSE
0.1
Temp_MBNSE
0 -6
-5
-4
-3
-2
-1
0
1
2
3
4
5
Nondimensional distance (a)
Nondimensional Density/Temperature
1 0.9 0.8 0.7 0.6
Rho_CNSE Rho_ENSE Rho_MBNSE Rho_DSMC Temp_CNSE Temp_ENSE Temp_DSMC Temp_MBNSE
0.5 0.4 0.3 0.2 0.1 0 -10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
Nondimensional distance (b)
Figure 8.1 Comparison of shock structures obtained with modified self-diffusion coefficient; Inlet Mach number (a) 2.84 and (b) 11 In order to verify the influence of the modified self-diffusion coefficient, predictions of shock structures were done for two Mach numbers with the self-diffusion coefficient of Dm 1.32 , mentioned by Greenshields and Reese[2007(a)] and the modified selfdiffusion mass transport was calculated as, 182
1 1 T m iD 1.32 xi 2T xi
(8.1)
The shock structures calculated using the modified self-diffusion coefficient are presented in Figures 8.1. While comparing with the shock structures presented in Figures 7.5 and 7.10, it can be noticed that the agreement between the density profiles obtained with the ENSE and experimental measurements is excellent in Figure 8.1(a). Similarly, the profiles obtained with the ENSE also compare favourably with the values obtained by the DSMC simulations in Figure 8.1(b). This interesting result clearly indicates that the right selection of the self-diffusion coefficient can indeed assist in predicting shock structures accurately employing the ENSE. This result again confirms that the self-diffusion mass transport is the solitary reason for the observed spread of shock structures in experimental measurements and one need to consider the ENSE in order to obtain physically consistent shock structures in hypersonic flows. Furthermore, it can be noted in chapter 4 that the height of microchannel was adjusted slightly in the numerical simulations employing the ENSE to obtain exact agreements with the experimental measurements. If the modified self-diffusion coefficient had been employed, such a correction in the channel dimension would not have been required and similar to shock wave simulations; one would obtain excellent agreement with the experiments in simulations of microchannels as well. Apart from the above mentioned domain of research in hypersonic shock waves employing the ENSE, the following areas can also be explored further. (i)
As indicated in Figure 7.22, the total temperature profiles obtained by both the ENSE and the CNSE do not remain constant across the shock wave and violate the first law of thermodynamics. It is intuitively argued that by choosing the correct form of the energy equation, as indicated in equation (7.24), one can address this violation positively. m D (ii) Further, the term Ek i was neglected in the total energy equation presented xi in equation (7.8). When this particular term was included, it was not possible to obtain converged shock structures employing the ENSE and the exact reasons for this behaviour has not yet been understood. It is necessary to revisit the derivations of the total energy equation and also the solution procedure to find out the underlying reasons for this problem. (iii) Interestingly, the temperature – density sepearation distance T predicted by the ENSE did not agree with the values obtained by the DSMC simulations whereas there was excellent agreement of the inverse density thickness L1 and density asymmetry quotient Q obtained by the ENSE and experimental measurements. It is widely accepted in the compressible flow literature that the density lacks behind temperature in a hypersonic shock, though there are no 183
experimental verification of this phenomenon. This particular problem is considered to be an excellent puzzle related to fundamentals of compressible flow physics which needs to be solved. (iv) Further, the viloation of the second law of thermodynamics in the case of the Burnett’s equations was discussed in the upstream region of shock waves, as shown in Figure 7.19. As stated in section 7.4.7, it was demonstrated that one could easily mitigate this thermodynamic violation by employing the ENSE. Interestingly, one can also observe another entropy inconsistency in the downstream region of the shock in Figure 7.19. It can be observed in this figure that the entropy reaches a maximum before reaching a slightly lesser value in the downstream of the shock. Further, it can be noticed in Figure 7.20 that this hump is less in the case of the ENSE than that of the CNSE. It is obvious that the entropy can only increase or stay constant across the shock and this observed hump is generally overseen in the literature. When the upstream entropy inconsistency was discussed elaborately by Balakrishnan [2004], there was no mention of this downstream violation. However, it is clearly evident that this is another violation of second law of thermodynamics and it is intuitively stated here that this violation can also be removed by choosing the correct form of the energy equation. 8.2.3 Thermophoresis Thermophoresis is an observed physical phenomenon and needs to be considered while studying deposition of soot particles in industrial gas cleaning applications, determination of exhaust gas particle trajectories from combustion devices, study of particulate material deposition on turbine blades, studies on the efficiency of air filters and aerosol scrubbers and also in a number of other areas involving deposition of particles such as chemical vapour deposition, see Davis [2006], Garg and Jayaraj [1988], Homsy et al [1981]. Thermophoresis is defined as the motion of non-Brownian particles, suspended in an otherwise quiescent medium, from hotter to colder regions, i.e. the particles move against the externally imposed temperature gradient on the static fluid. Brenner [2005(a)] proposed his extended Navier-Stokes equations only to explain this phenomenon and he attributed thermophoretic particle motion, in the so-called ‘‘near-continuum’’ range of Knudsen numbers Kn 1 , to small non-continuum effects. He further elaborated that these non-continuum effects are caused by the action of thermal stresses existing in the gas proximate to the particle surface, resulting in the Maxwell slip (‘thermal creep’) at the surface. It is well-known that the Maxwell slip condition is usually applied to flow cases where the Knudsen number is large, i.e. Kn>0.01. On the contrary, the phenomenon of thermophoresis can be observed in quiescent gases at all pressures subjected to temperature gradients where the continuum assumption is very much valid and the Knudsen number is very small, i.e. Kn = 0.0) { F_PROFILE(f,t,position) = 624014.0+(0.5*rho*vel*vel); } else F_PROFILE(f,t,position) = 624014.0; } } end_f_loop(f,t) } DEFINE_PROFILE (inlet_temperature, t, position) { real cp, vel,Ma,T; face_t f; cp=1040.67; begin_f_loop(f,t) { vel=F_U(f,t); T=F_T(f,t); Ma=vel/sqrt(1.4*8314.5*T/28.0134); { if (vel >= 0.0) F_PROFILE(f,t,position) = 330.0+(0.5*F_U(f,t)*F_U(f,t)/1040.67); else F_PROFILE(f,t,position) = 330.0; } } end_f_loop(f,t) } 204
DEFINE_PROFILE (outlet_pressure, t, position) { real rho, vel,u,v; face_t f; begin_f_loop(f,t) { rho = F_R(f,t); u = F_U(f,t); v = F_V(f,t); vel = sqrt((u*u)+(v*v)); { if (u
