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Copyright © 2016 by American Scientific Publishers

Journal of Nanofluids Vol. 5, pp. 1–9, 2016 (www.aspbs.com/jon)

All rights reserved. Printed in the United States of America

Numerical Investigation on Boundary Layer Flow of a Nanofluid Towards an Inclined Plate with Convective Boundary: Boungiorno Nanofluid Model B. Mahanthesh1, 2 , B. J. Gireesha1, 3, ∗ , Thammanna1 , R. S. R. Gorla3 , B. C. Prasannakumara1 , and P. Venkatesh4 1

Department of Studies and Research in Mathematics, Kuvempu University, Shankaraghatta 577451, Shimoga, Karnataka, India 2 Department of Mathematics and Statistics, Christ University, Bangalore 29, India 3 Department of Mechanical Engineering, Cleveland State University, Cleveland-44114, OHIO, USA 4 Sahyadri Science College, Shimoga 577203, Karnataka, India

KEYWORDS: Convective Boundary Condition, Inclined Plate, Nanofluid, Thermal Radiation, Brownian Motion, Viscous Dissipation.

1. INTRODUCTION The theory of embedding small solid particles into a conventional base fluid to increase the thermal conductivity of the suspension has been practiced for a long time. But, most of the studies have been carried out using suspensions of millimeter- or micrometer-sized particles, which turns to problems such as stability, sedimentation and channel clogging. To overcome this challenge, Choi1 proposed an innovative method of suspension of solid particles smaller than 100 nm into a base fluid. He termed it as a nanofluid and it is defined as a dilute suspension of nanoparticles in a base fluid. Some important advantages like high stability, reduced particle clogging and importantly, high heat transfer capabilities over the conventional colloidal suspensions were presented by nanofluids. In view of this, the study of nanofluid flow problems has wide ∗

Author to whom correspondence should be addressed. Emails: [email protected], [email protected] Received: 30 May 2016 Accepted: 22 June 2016

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range of applications ranging from transport to industrial applications such as engine cooling/vehicle thermal management, radiators, heat exchangers, micro channel heat sinks, nuclear reactors, electronic cooling systems, lubricants, fuel cells, chemical industry, domestic refrigerator and etc. Its relevance can also be seen in biomedical and pharmaceutical fields.2 A theoretical study on thermal conductivity of nanofluid with copper nanoparticles was presented by Choi and Eastman.3 They found that, to improve the heat transfer by a factor of 2, in conventional fluid the pumping power should be enhanced by a factor of 10. However, if a nanoparticle-based fluid with a thermal conductivity of three times that of a conventional fluid were used in the same heat transfer equipment, the nanoparticle-based fluid would double the rate of heat transfer without raising pumping power. The experimental studies by Refs. [4–6] showed that, even with a small volumetric fraction of nanoparticles ( 0 corresponds to heating of fluid/cooling the plate, T < 0 corresponds to cooling of the fluid/heating of the plate and T = 0 corresponds to absence of mixed convection. The velocity

0.3

= /2

Cf

Bi 0.05 0.2 0.6 1 2 5 10 50 100 500 1000 10000 50000 100000

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Numerical Investigation on Boundary Layer Flow of a Nanofluid Towards an Inclined Plate with Convective Boundary

Mahanthesh et al.

Fig. 1. Influence of Nt on f , and profiles (Blue-f ; Red-; Black-).

Fig. 3. Influence of on f , and profiles (Blue-f ; Red-; Black-).

boundary layer thickness reduces significantly. Further, it is observed that, the velocity and temperature profile response qualitatively opposite behavior with increasing values of Le. Figure 7 is sketched to show the variations of velocity, temperature and nanoparticle volume fraction against heat source/sink parameter. It is noticed that, Q > 0 corresponds to an internal heat generation, Q < 0 corresponds to heat absorption and Q = 0 corresponds to the absence of heat source/sink. As expected the temperature profile intensified due to increase in heat/source sink parameter. This intensification of the thermal boundary layer is responsible for increasing momentum boundary layer. Additionally, the nanoparticle volume fraction profile decreases near the plate and then it is found to be increase away from the plate as can be seen in Figure 7. Figure 8 represents the response of flow, thermal and solutal boundary layer as Eckert number increases. Interestingly, the same behavior

is observed for increasing Ec as compared to the influence of Q. Figure 9 is plotted to depict the variation of f , and distributions for different R. It is observed that, the velocity and temperature fields decreases near the plate, and increases significantly away from the plate. Also, close to the plate solutal boundary layer is thicker whereas the thermal boundary is thinner away from the plate. The variations in both velocity and nanoparticle volume fraction due to the Biot number Bi is shown in graph 10. By increasing the Biot number, an increase in both velocity and nanoparticle volume fraction fields was observed. Consequently, the momentum and solutal boundary layer is thicker for larger values of Biot number. In this study we intend to study the effect of Biot number on temperature profile in three trends. The first is Bi < 1, which represents uniform thermal field inside the surface. It is clarified through the plot 12 that, for smaller values of Bi (Bi < 1, the variation of temperature within the body is very small

Fig. 2. Influence of Nb on f , and profiles (Blue-f ; Red-; Black-).

Fig. 4. Influence of T on f , and profiles (Blue-f ; Red-; Black-).

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Fig. 7. Influence of Q on f , and profiles (Blue-f ; Red-; Black-).

and can be approximated reasonably. The second one is, Bi > 1, which indicates that the thermal field inside the surface is non-uniform. This is also clarified via graph 11 that, for larger values of Bi (Bi > 1, the behavior of temperature within the body is non-uniform. Finally, the third trend is Bi → , which represents constant surface temperature. Further it is observed that, the thermal boundary layer thickness significantly increases with Bi. The Table II illustrate the effect of the Biot number on skin-friction co-efficient, Nusselt number and Sherwood number for three different values of inclination angle. It is observed that, by increasing Bi from 0.05–50, the profiles of Cf , Nu and Sh get increases notably, whereas a very slight/no variation (enhancement) is found after 50–100000. It is also seen that, the drag force is higher in vertical plate case ( = 0 than other cases. Further, the Nusselt number is higher and Sherwood number is lower in the case of horizontal plate ( = /2 than in other cases.

Table III presents the responses of skin-friction coefficient and Nusselt number for the variation of Nb, Nt and Le in heat sink (Q < 0 and heat source (Q > 0 cases. It is observed that, the skin-friction co-efficient and Nusselt number have qualitatively opposite behavior for an increase in Nb, Nt and Le. It also illustrates that, the drag force as well as the rate of heat transfer is lower in heat source case than in sink case. Finally the Table IV is evident to analyze the variation of T , C , R and Q on skin-friction co-efficient, Nusselt number and Sherwood number for two cases namely Ec = 0 and Ec = 1. It is observed that, both Cf and Sh are increasing functions of T , C and Q and they are decreasing functions of R. While Nu is increasing function of T , C and R and is a decreasing function of Q. Further, it shows that, both skin-friction co-efficient and rate of heat transfer is higher in the absence of viscous dissipation as compared with

Fig. 6. Influence of Le on f , and profiles (Blue-f ; Red-; Black-).

Fig. 8. Influence of Ec on f , and profiles (Blue-f ; Red-; Black-).


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Fig. 5. Influence of C on f , and profiles (Blue-f ; Red-; Black-).


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its presence. This trend is quite opposite for Sherwood number.

4. CONCLUDING REMARKS

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Fig. 9. Influence of R on f , and profiles (Blue-f ; Red-; Black-).

Fig. 10. Influence of Bi on f and profiles (Blue-f ; Red-.

Fig. 11. Influence of Bi on profiles.

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A steady boundary layer flow using Boungiorno nanofluid model over a convectively heated inclined plate with viscous dissipation, heat source/sink and thermal radiation effects is investigated numerically. In the flow and heat transport equation system a new nanoparticle volume transport equation is introduced. Numerical solutions are obtained using Shooting method along with fourth-fifth order Runge-Kutta Fehlberg method. Moreover, the physical interpretations and the behavior of interesting parameters on different flow fields are presented, analyzed and discussed. It is found that, thermal boundary layer thickness increased by increasing Brownian motion and thermophoresis parameter. An opposite behavior is predicted for nanoparticle volume fraction with increase in Nb and Nt. The Biot number provides the increase in flow, thermal and solutal boundary layer thickness. The effects of Ec and Q are qualitatively same for velocity, temperature and nanoparticle volume fraction fields. The friction force is higher in the vertical plate as compared with both inclined and horizontal plates. It is also concluded that, the rate of heat transfer is higher in horizontal plate, but the rate of nanoparticle mass transfer is lower on the same plate as compared with vertical and an inclined plate. Nomenclature C Nanoparticle volume fraction (kg/m3 Cf Skin friction co-efficient Cw Concentration at the wall (kg/m3 C Ambient nanofluid volume fraction (kg/m3 c Constant cp f Specific heat coefficient of fluid (J/kg K) cp p Specific heat coefficient of nanoparticles (J/kg K) DB Brownian diffusion coefficient DT Thermophoretic diffusion coefficient Ec Eckert number f Dimensionless velocity component g Acceleration due to gravity (m/s2 hf Heat transfer coefficient jw Nanoparticles mass flux k Thermal conductivity (W/m K) k1 Mean absorption coefficient (m−1 Le Lewis number Nb Brownian motion parameter Nt Thermophoresis parameter Nux Local Nusselt number Pr Prandtl number Q Heat source/sink parameter Q0 Heat source/sink co-efficient Q0∗ Constant qw Heat flux qr Radiative heat flux (Wm−2 J. Nanofluids, 5, 1–9, 2016

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R Rex Sh T t Tf T V u v x y


Thermal radiation parameter Local Reynolds number Sherwood number Fluid temperature (K) Time (s) Surface temperature (K) Surface temperature (K) Velocity of the fluid (m/s) Velocity components along x and y directions (ms−1 Coordinate along the plate (m) Coordinate normal to the plate (m).

Superscript Derivative with respect to . Acknowledgment: The author B. J. Gireesha gratefully acknowledges the financial support of UGC, New Delhi, India for pursuing this work under Raman Fellowship for Post Doctoral Research for Indian Scholars in USA 2014–2015.

References and Notes 1. S. U. S. Choi, Enhancing thermal conductivity of fluids with nanoparticle, Developments and Applications of Non-Newtonian Flows, edited by D. A. Siginer and H. P. Wang, ASME. MD. 231 and FED 66 (1995), pp. 99–105. 2. R. Saidur, K. Y. Leong, and H. A. Mahammad, Renewable and Sustainable Energy Reviews 15, 1646 (2011).


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Greek Symbols Kinematic viscosity (m2 s−1 ) Volumetric volume expansion coefficient Dynamic viscosity (kg m−1 s−1 Dimensionless nanoparticle volume fraction Stefan–Boltzmann constant (W m−2 K−4 Dimensionless temperature Stream function Similarity variable w Surface shear stress Inclination angle f Density of the base fluid (kg/m3 p Density of the particles (kg/m3 f Ambient density of the fluid T Thermal Grashof number C Solutal Grashof number.

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