Effects of two transversal finite dimensions in long microchannel: Analytical approach in slip regime J. G. Méolans, M. Hadj Nacer, M. Rojas, P. Perrier, and I. Graur Citation: Phys. Fluids 24, 112005 (2012); doi: 10.1063/1.4767514 View online: http://dx.doi.org/10.1063/1.4767514 View Table of Contents: http://pof.aip.org/resource/1/PHFLE6/v24/i11 Published by the American Institute of Physics.
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PHYSICS OF FLUIDS 24, 112005 (2012)
Effects of two transversal finite dimensions in long microchannel: Analytical approach in slip regime J. G. Meolans, M. Hadj Nacer, M. Rojas, P. Perrier, and I. Graura) ´ Aix-Marseille Universit´e, Ecole Polytechnique Universitaire de Marseille, D´epartement de M´ecanique Energ´etique, UMR CNRS 7343, 5 rue Enrico Fermi, 13453 Marseille Cedex 13, France (Received 19 April 2012; accepted 4 October 2012; published online 30 November 2012)
An analytical approach has been developed to take account of the influence of the lateral walls on a stationary isothermal gas flow through a rectangular microchannel. The study concerns pressure-gradient-driven flows in channels where the length is large compared to the critical smallest dimension, namely, the channel height. The calculation of the bulk velocity is based on the Stokes equation treatment and uses the property of the Laplace operator. This novel method remains very easy to use when the second order term with respect to the Knudsen number is taken into account in the wall boundary conditions. The method is notably of high practical interest when applied to rectangular-cross-section microchannels that connect upstream and downstream high capacity reservoirs. The mass flow rates measured along such systems are fitted to first or second order polynomial forms following the mean Knudsen number of the flow. The present calculation also leads to a completely explicit second order expression for the mass flow rate. Thus, the first and second order experimental and theoretical coefficients can be identified immediately, allowing direct evaluations of physical gas-wall interaction features, in particular, the first, second order slip and the accommodation coefficients. An example of the implementation of the proposed technique is given. The slip and accommodation coefficients are extracted from the measurements of the mass flow rate through microchannels of rectangular crossC 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4767514] section.
I. INTRODUCTION
With the arrival of novel micro-technologies that permit applications in diverse areas such as engineering and medicine, gas flows through long microchannels of rectangular cross-section are now of great practical interest. Because of the small dimension of the microchannel cross-section, the gas flow inside is very often under slip regime conditions when the devices are used at atmospheric pressure. Theoretical and numerical approaches for the channel’s cross-section have long been mainly one-dimensional.1–4 However, several authors have studied the two-dimensional flow behaviour in a channel cross-section and the influence of the lateral walls on the main characteristic parameter of the flow, e.g., the mass flow rate. If we disregard some specific cases, like the flow through an equilateral triangular duct,5 the main two-dimensional approaches can be found in the previous works.6–12 The solving method proposed by Ebert and Sparrow11 for rectangular channels in slip regime shows some rough similarities with the present study. These authors used a velocity expansion, following basis functions of functional space, but using a separation variable technique to solve the Stokes equation. Thus the authors could not prove, a priori, the validity of their expansion. Moreover, only the case of a constant pressure gradient in the flow direction seems to have been considered. Finally, the local velocity and the mean velocity on the cross-section were given through very
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24, 112005-1
C 2012 American Institute of Physics
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general expressions: the integration constants were not explicitly related to the boundary conditions, where no second order term was present. Furthermore, no expansion of the velocity was considered following Knudsen number orders: thus the slip coefficients were not evaluated. Later, using an integral transform technique Yu and Amel13 studied the energy and momentum equations for a rectangular channel and obtained a very complicated formal expression for the velocity. The aim of these authors was to calculate the mean Nusselt numbers. Again, no second order boundary conditions and no expansion according to Knudsen number orders was given. The same limits can be pointed out in the work of Morini and Spiga14 who also used an integral transform method to obtain the velocity profile in rectangular channels. The authors’ purpose was to obtain “two-dimensional correcting factors” to be applied to the shear stress, the momentum flux, and the kinetic energy. Closer to our objectives was a study by Aubert and Colin.9 These authors undertook a slip regime study of the pressure-gradient-driven flows in rectangular microchannels using a second order boundary condition. They also obtained a velocity expansion on basis functions of the L2 functional vector space. Two calculation methods were tested. The first one was related to a separable variable method and did not completely satisfy the authors. The second one, related to a variational approach, required a numerical calculation to determine the basis functions. But, in any case Aubert and Colin9 used a second order slip condition given by Deissler15 and based on a phenomenological approach. This condition differs from the second order condition used in the present paper which was established theoretically on the basis of the kinetic theory.6, 8, 16, 22 Sharipov10 solved the two-dimensional Stokes equation in the hydrodynamic flow regime with non-slip boundary condition and obtained the explicit expression for the longitudinal component of the bulk velocity and the mass flow rate through a rectangular-cross-section channel. However, in the slip flow regime, when the Stokes equation was subjected to the first order slip boundary condition with respect to the Knudsen number, the bulk velocity and the mass flow rate was obtained only numerically. The following three works6, 7, 17 have made a fundamental contribution to theory involved in the theme of this paper: Loyalka et al.7 presented a general theoretical kinetic study of Poiseuille and thermal creep flows in long rectangular channels, especially in molecular and transitional flow regimes. Thus, their results do not concern the macroscopic formulation to be used in slip flow regime. In contrast, in Ref. 6, the purpose of the calculation was to obtain a high order slip boundary condition usable at macroscopic level to describe isothermal microflows. Furthermore, Cercignani6 focused his interest on the case where two components of the velocity gradient exist in the plane normal to the flow axis. On this basis, the author of Ref. 6 shows that, in the second order term of the velocity slip expansion the second derivative of the velocity according to the normal coordinate is changed into a Laplacian operator reduced to the coordinates describing the points of the channel cross-section. The assumptions used in Ref. 6 and the criticisms of the results, notably formulated in Ref. 17, will be found in Sec. II. Some of the previous approaches basically allow an evaluation of the influence of a twodimensional geometry on the procedure for extracting the gas-surface accommodation coefficient or on other features, such as the second order effect in slip regime. Nevertheless, up to now for flows from slip to free molecular regime, the existing two-dimensional approaches required a numerical treatment of the two-dimensional contribution. The aim of present study is to develop a new method giving an explicit analytical second order expression for the gas flow parameters for hydrodynamic and slip regimes in long microchannels characterized by a two-dimensional cross-section. The approach developed here has been used to solve equations involving the Laplace operator in fields such as neutron diffusion and, more recently, heat diffusion. But up to now, it has been poorly employed for gas dynamic problems and, in this sense, it may be considered as a novel approach in this domain. Here, the analytical expression of the mass flow rate will be compared to Sharipov’s10 numerical results and the importance of the influence of the lateral walls will be shown. Finally, the proposed approach will be implemented to extract the velocity slip and accommodation coefficients from the experimental data on the mass flow rate through two microchannels of rectangular cross-section.
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z y h/2 pout
pin L
-w/2
w/2 x -h/2
(a)
(b) FIG. 1. Physical domain.
II. PROBLEM FORMULATION A. Conservation equation
A long channel, containing a gas, connects two tanks which are maintained at pressures pin and pout , see Fig. 1. The channel axis coincides with the z axis. The transversal cross-section of the channel, normal to the z axis, is a rectangle: the height h of this rectangle is the smallest critical length of the system. The rectangle width w verifies the relation h ≤ w. The channel height to length ratio h/L verifies: h/L = εhL , where εhL 1. The flow along this channel is engendered by a pressure difference obtained by keeping the inlet and outlet reservoirs at pin and pout , respectively. As is well known, in long microchannels under an isothermal flow condition, the Navier-Stokes equation for the tangential component uz of the bulk velocity u = (u x , u y , u z ) may be approximated in the form2, 18 ∂ 2u z ∂ 2u z 1 dp , + = ∂x2 ∂ y2 μ dz
(1)
where p and μ are the pressure and the viscosity coefficient, respectively. From a strict formal point of view equation (1) is obtained using the following assumptions: r r r r
the channel height-to-length ratio h/L = εhL is small enough, εhL 1; the two components of bulk velocity ux and uy are of the order of εhL ; following on the two previous assumptions the flow Mach number is small, of the order of εhL ; (pin − pout )/pin ∼ 1, which means that the pressure change between the tanks is of the same order as the pressure itself. In addition, the pressure remains of the same order all along the tube.
In the framework of our approach, it is not necessary to assume that the flow is incompressible or that the pressure gradient is constant. B. Boundary conditions
As mentioned previously, we consider here the slip flow regime through a channel characterized by a Knudsen number varying in the range 0.01 ≤ Kn ≤ 0.3. The Knudsen number is defined as the ratio between the molecular mean free path λ, μ√ λ = kλ 2RT (2) p and the characteristic length of the system, namely, h, which is the smaller dimension of the channel’s cross-section K n = λ/ h.
(3)
In the previous relations, R is the specific gas constant, T is the gas temperature, and kλ is a coefficient that depends on the intermolecular interaction model. From expressions (2) and (3), it is clear that the Knudsen number is a local quantity which depends on the z coordinate. Moreover, it is important to remember that, since we have an axis (or a plane) of symmetry, the second order terms with respect
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to the Knudsen number are negligible in the conservation equation system and are important only when they result from the slip boundary condition.6, 19 Therefore, in the geometry considered here, according to the results given in Ref. 6, a pertinent slip velocity boundary condition may be written as ∂u z (4) u z = ±A1 λ − A2 λ2 u z , s s ∂ n s where the operator nabla is equal to ∂ 2 /∂x2 + ∂ 2 /∂y2 , A1 and A2 are the coefficients introduced in Ref. 6. The subscript s characterizes values of the tangential component of the velocity at the channel surface, while n identifies the direction normal to the wall, i.e., successively either x or y. Finally, as usual, at any point of the solid surface, n is oriented by the entering normal. Consequently in front of A1 , the sign + is chosen when relation (4) is for a solid surface located at a positive value of the x (or y) coordinate. Some additional comments are needed regarding Eq. (4). In Ref. 6, the results were obtained using some assumptions notably a Bhatnagar-Gross-Krook (BGK) equation model and a gas of Maxwell molecules model. Moreover, the velocity gradient in the flow direction was neglected. Finally, for simplification, the author considered flow over a plane wall. Nevertheless, he applied his result to calculate the mass flow rate in a cylindrical tube. Cercignani’s6 calculation generated critical comments: the authors of Refs. 16 and 17 claimed that the curvature of the cylindrical tube was not correctly taken into account. The same criticism is to be found in Ref. 20. However, Hadjiconstantinou20 considered that the assumption of constant velocity in the flow direction led to rather correct results. In any case, no criticisms were voiced against the Laplacian functional form of the second order term. Furthermore, in the rectangular channel case considered here, the wall surface materials, the surface roughness, and the surface curvatures are the same on the vertical and horizontal walls and we can thus consider the same coefficient A2 for all these walls. Consequently, Cercignani’s6 analysis leads directly to Eq. (4). Finally, a question remains to be clarified concerning the accuracy of the theoretical coefficient A2 when defined in Ref. 6. In fact the author himself underlined that the calculation developed in Ref. 6 does not take into account the Knudsen layer effect; moreover, this effect appears of the same order as the second order terms present in Eq. (4). More detailed comments on the influence of the Knudsen layer will be given in Sec. VIII. C. New formulation
Now we will resolve Eq. (1) subject to boundary condition (4). It results from Eq. (1) that uz does not depend on the x and y coordinates whatever the point considered in the channel cross-section. Thus u z in Eq. (4) will be constant at any point of a section perimeter. The same s comment may be applied to the molecular mean free path λ, owing to the pressure property. Then it is convenient to use the function change u ∗ = u z + A2 λ2 u z . (5) s
Consequently, Eq. (4) is changed into ∂u ∗ u ∗ = ±A1 λ . s ∂n s
(6)
Naturally, the form of Eq. (1) does not change and it may be rewritten using the new function u*, u ∗ =
1 dp . μ dz
(7)
In Sec. III, we will present the general approach that will be used to solve Eq. (7) subject to the boundary condition (6) to determine the unknown function u* and, further uz , by using relation (5).
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III. THE THEORETICAL BASES OF OUR APPROACH
The present approach is mainly based on the spectral properties of the Laplace operator included in the Stokes equation (7). When applied to the vector space of L2 functions defined on a finite spatial domain D, the Laplacian is characterized by an eigenfunction discrete spectrum. Now on the border of this spatial domain we define a “condition of cancellation” for the L2 functions, i.e., either the nullity of the function, or the nullity of its normal derivative, or a constant ratio between the function and its derivative. If we require the Laplacian eigenfunctions to verify this “condition of cancellation” on the border of the spatial domain, then their countable set provides a basis of the vector space of L2 functions defined on the spatial domain D and subjected to the same “cancellation condition.” Moreover, each function of the set is orthogonal to the other set functions in the sense of the Hermitian product defined on the functional space. Finally, the eigenvalues associated with these eigenfunctions form a discrete countable series of real negative numbers. Then, in the finite spatial domain of a microchannel cross-section, it appears convenient to search for the solution of the Stokes equation (7), i.e., the velocity u* satisfying cancellation conditions (6), in the form of an expansion following the Laplacian eigenfunctions which verify Eq. (6) since: r The basis property of these eigenfunctions guarantees the uniqueness of the solution thus found. r Generally, the boundary condition verified by the eigenfunctions insures the velocity boundary conditions, because of their respective similarity. Here, for example, u* will verify the cancellation condition (6). r The use of the Laplacian eigenfunction basis allows significant simplifications in the Stokes equation: the Laplace operator vanishes and the partial derivative equation reduces to a simple differential equation, generally easy to solve. But the general process described above concerns an expansion following the eigenfunctions of the complete Laplace operator. Thus, this process would rather be suitable for treating non-stationary flows governed by a conservation equation corresponding to a modified Eq. (7) also including time derivatives. In the present case of a flow through a channel rectangular cross-section, under steady and isothermal conditions, the momentum conservation equation reduces to the Poisson equation (7), where the (x, y) Laplace operator is applied to the u* function. Thus, the knowledge of the spectrum of the complete operator is not of practical interest here. To simplify the resolution of Eq. (7), it is convenient to split the operator into different parts, respectively, regarding the different spatial variables. Then it is easily shown that all the properties quoted above, concerning the spectral elements of the complete Laplace operator acting on the functional space L2 (x, y), may be transposed on each partial operator ∂ 2 /∂x2 or ∂ 2 /∂y2 . Of course, each partial operator acts on the restricted functional space that depends on the spatial variable considered in each partial derivation. Therefore, the most convenient partial operator will be chosen for investigation in regard to its spectral elements. Thus, the solution will again be sought in the form of a series of functions. But for the present stationary study, the velocity will be expanded according to the eigenfunctions of the chosen partial Laplace operator.
IV. ANALYSIS OF THE BOUNDARY CONDITIONS: CHOICE OF AN ORTHOGONAL FUNCTION SET
We now have to choose the partial space function depending on the most convenient space parameter leading to the easiest resolution of the problem. This choice requires us to analyze the boundary conditions verified by the velocity u*. It is obvious that the slip boundary condition (6), which relates u* and its normal derivative at the wall, is a “cancellation condition” as discussed previously in Sec. III. This boundary condition may be rewritten on the wall, at the y limits (±h/2) and at the x limits (±w/2), see Fig. 1. But it is more convenient, using the symmetry planes (x = 0 and y = 0) of the system to consider only the reduced spatial domain defined by: 0 ≤ x ≤ w/2 and
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0 ≤ y ≤ h/2. Then the boundary conditions in a quarter of the channel cross-section read ∂u ∗ (0, y) = 0, ∂x u ∗ (w/2, y) = −A1 λ
∂u ∗ (w/2, y), ∂x
(8)
∂u ∗ (x, h/2). ∂y
(9)
and ∂u ∗ (x, 0) = 0, ∂y u ∗ (x, h/2) = −A1 λ
Thus the functional space L2 (x) (associated with the boundary condition at x = w/2 and with the partial Laplacian ∂ 2 /∂x2 ) or the functional space L2 (y) (associated with the boundary condition at y = h/2 and with the partial Laplacian ∂ 2 /∂y2 ) are both convenient to construct either an orthogonal normal basis of x functions or an orthogonal normal basis of y functions. Thus we may choose to seek the u* function in the series form u∗ =
∞
gn (x) f n (y),
(10)
n=0
where the summation index n is a natural integer, and the set of functions fn (y) is provided by the eigenfunctions of the ∂ 2 /∂y2 operator verifying, at the y limits, conditions similar to (8), and where gn (x) are functions of L2 (x) space, which will be found by solving equation (7) subject to conditions (9). We will see in Sec. V that gn (x) and fn (y) functions are parametrically dependent on z due notably to the boundary “cancellation conditions.” This dependence will appear through the “constants” introduced in the integration process. V. SEARCH FOR THE REDUCED VELOCITY
Now we will obtain the explicit expressions for the fn (y) and gn (x) functions involved in expansion (10). The fn (y) basis will be obtained by solving the system d 2 f n (y) = −νn2 f n (y), dy 2 f n (h/2) = −A1 λ
d fn (h/2), dy
d fn (0) = 0, dy
(11)
(12)
(13)
where −νn2 are the eigenvalues of the partial Laplace operator on L2 (y) space. Taking into account (13), the general solution of Eq. (11) reduces to f n (y) = an cos(νn y),
(14)
where an is a set of constants to be defined. Then using Eq. (12), we obtain the following equation: cotan(βn ) = 2A1 K nβn
(15)
βn = νn h/2.
(16)
with
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In expression (15), the local Knudsen number Kn, defined by relation (3), depends on z via the molecular mean free path (2). The solution of Eq. (15) gives the eigenvalues νn2 (or β n with notation (16)) of the partial Laplace operator on L2 (y) space. Using the Hermitian scalar product we normalize the fn (y) functions, thus obtaining the set of corresponding constants an , √ 2 2βn / h an = √ . (17) 2βn + sin(2βn ) Next, we will obtain the explicit expressions for the gn (x) functions. Let us remember that the second member of Eq. (7) depends only on the z space variable and so may be noted as P(z) =
1 dp . μ dz
(18)
Using the u* velocity expansion (10) in Eq. (7) and taking into account the basis properties of the fn (y) functions, we obtain the differential equation governing the gn (x) functions d 2 gn (x) − νn2 gn (x) − sn = 0 (19) dx2 valid for all natural n, where sn are the coefficients of the expansion of P(z) (see Eqs. (7) and (18)) on the fn (y) basis functions P(z) =
∞
sn f n (y).
(20)
n=0
The solution of the homogeneous equation associated with Eq. (19) involves the sum of two exponential functions. Moreover, the complete solution of the non-homogeneous equation (19) includes a constant obvious solution. Finally, the gn (x) expression leads to gn (x) = An eνn x + Bn e−νn x − sn /νn2 .
(21)
Using expression (10), we obtain the velocity u* in the following form: u∗ =
∞
An eνn x + Bn e−νn x − sn /νn2 an cos(vn y).
(22)
n=0
The constants An and Bn are determined using the boundary conditions (9) on the x space variable at 0 and w/2. Thus, we find first that An = Bn , and hence we obtain An =
sn h 2 (cosh(βn w/ h) + 2A1 K nβn sinh(βn w/ h))−1 , 8βn2
(23)
where the infinite set of β n (or ν n ) values is obtained by solving Eq. (15). Finally, expanding P(z) we find h/2 h an f n (y)dy = P(z) √ sin(βn ). (24) sn = P(z) 2 βn 0 Then, from Eq. (22), using expressions (16)–(18) and (24), we find u∗ =
∞ h 2 dp φn (x, y), μ dz n=0
where the following new notations are used: φn (x, y) =
sin(βn ) cos(2βn y/ h) × βn2 (2βn + sin(2βn ))
cosh(2βn x/ h) −1 , cosh(βn w/ h) + βn ε sinh(βn w/ h)
ε = 2A1 K n,
ε 50, the implementation of expression (38), instead of (31), gives an error of the order of 1% (see Table II). The second comparison with the experimental data is based on the mass flow rate measured in a silica (SiO2 ) microchannel of rectangular cross-section with h = 24.3 μm, w = 50.1 μm, L = 13.68 mm. The width to height ratio w/ h for this case is equal to 2.06. For this w/ h ratio, the difference between the mass flow rate between two parallel plates (38) and the mass flow rate
Dimensionless mass flow rate
0
0.1
0.2
0.3
3
3
2
2
1
1
0
0.1
0.2
0.3
Knm FIG. 2. Fitting curve (solid line) using expression (31) and the experimental data for nitrogen (circles);24 the fitting curve (dashed line) using expression (38); and the experimental data for nitrogen (squares);24 the fitting curve with the coefficient ex p B0 set equal to 1 (dashed dotted line).
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TABLE VI. Fitting coefficients for nitrogen. The first line corresponds to the implementation of expression (31); the second line corresponds to the use of expression (38). Gas N2
ex p
B0
1 0.692 ± 0.000
ex p
B1
7.883 ± 0.120 5.536 ± 0.002
ex p
B2
6.084 ± 0.644 4.038 ± 0.303
σp
α
1.070 ± 0.016 0.818 ± 0.010
0.972 ± 0.008 1.115 ± 0.006
taking the influence of the lateral walls into account is of the order of 30% (see Table II). The results of implementing relation (31) to extract the slip and accommodation coefficients for Ar and N2 are presented in Table V, where the fitting coefficients according to (41) are given. The corresponding fitting curve for nitrogen (solid line) and the dimensionless mass flow rate (circles) are shown in Fig. 2. Now, we will show what happens when we try to apply the analytical expression for the mass flow rate between two parallel plates (38) to extract the slip and accommodation coefficients. Expressions (31) and (38) differ by the coefficient Vn which takes the influence of the channel cross-section into account (34). If we use expression (38), then in order to fit the dimensionless data of the mass flow rate, as described before, we divide the dimensional experimental data by the Poiseuille mass flow rate M˙ P and then fit these normalized data according to expression (41). But by doing this, we underestimate the dimensionless mass flow rate by a factor 1/Vn (Vn = 0.695) when w/ h = 2.06. The corresponding experimental data normalized by M˙ P are represented by squares in Fig. 2. These dimensionless experimental data are fitted according to expression (41) but without setting the ex p coefficient B0 to 1. The fitting curve is plotted by the dashed line on Fig. 2. The fitting coefficients are given in Table VI. It can be seen that the two kinds of fitting coefficients differ by ∼30%, but the difference in the accommodation coefficients is of the order of 15% due to the fact that the coefficient before the term with Knm is approximately 10% higher (see Table III) when expression (31) is used. Therefore, the implementation of the expression for the mass flow rate between two parallel plates in the case of a rectangular channel with a small aspect ratio may introduce a large error in the determination of the slip and the accommodation coefficients. In addition, when the experimental data are fitted ex p with the first fitting coefficient B0 set to 1, according to analytical results, the errors on the two coefficients become of the order of 100%. The corresponding fitting curve is plotted on Fig. 2 with a dashed-dotted line.
IX. CONCLUSIONS AND PERSPECTIVES
The gas flow through a long rectangular cross-section microchannel was modeled in the hydrodynamic and slip flow regime under the isothermal flow assumptions. An original approach was implemented here based on the spectral properties of the Laplace operator. The use of this approach allowed us to obtain an explicit analytical expression for the tangential component of the bulk velocity and mass flow rate valid in the hydrodynamic and slip flow regimes. On the other hand, by using the analytical expression obtained for the bulk velocity and the property of mass conservation along the channel, the pressure gradient expression along the channel may be obtained. The expression proposed for the mass flow rate may be used directly for the mass flow rate estimation at first or second order according to the Knudsen number. More precisely the relation given in Sec. VIII allows us to predict the influence of the aspect ratio of the micro tube section (height/width ratio) on the mass flow rate data. For example, the present model easily brings out significant influences of the two-dimensional cross-section for usual sizes of microchannels: for the case w/ h = 2 the one-dimensional calculation overestimates the mass flow rate by approximately 29% of its value: and it is to note that the novel analytical mass flow rate expression gives results that are completely consistent with the existing semi-numerical calculations for the zero order and first order terms.
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Moreover, as shown in this section, using this analytical expression for the mass flow rate and the corresponding fitting of measured data, “experimental” consistent values of the slip and of the accommodation coefficients were obtained. Considering the relative simplicity of the new two-dimensional approach, we plan to use the method in the near future to give an evaluation of the corrections that need to be made to previously obtained results, where the weak two-dimensional effects were neglected. We also plan to present and analyze new experimental results which have recently been obtained for channels with various rectangular cross-sections and where all the surfaces are homogeneously machined with the same material. Then the second order boundary conditions presented in this article will be usable. From the comparison of second order terms quantitative information could be extract about the theoretical expression of the A2 coefficient given in Sec. III of Ref. 22. ACKNOWLEDGMENTS
The research leading to these results has received funding from the European Community’s Seventh Framework Programme (ITN - FP7/2007-2013) under Grant Agreement No. 215504. The researchers thank Femto-ST Laboratory (Besanon-France) for providing the microchannels. APPENDIX: EXPANSION OF u* FUNCTION
Expression (25) of u* is expanded up to the first order according to the small parameter ε defined in Eq. (27) which is proportional to the local Knudsen number. Since u* depends on ε not only explicitly through (27), but also implicitly via β n which is related with the local Knudsen number in Eq. (15). Using the ε parameter, Eq. (15) is now rewritten as follows: cotan(βn ) = βn ε. Then the basic form of the expansion u* will be ∗
u =
u ∗0
where u ∗0 = u ∗
ε=0
du ∗ +ε dε
,
du ∗ dε
(A1)
,
(A2)
du ∗ , dε ε=0
(A3)
0
= 0
and
du ∗ dε
= 0
∞ ∂u ∗ dβn n=0
∂βn dε
+
0
∂u ∗ ∂ε
.
(A4)
0
To make the calculation easier to handle, it is important to note now that, at zero order following ε, Eq. (A1) reduces to cotan(β n ) = 0, so that, = (βn )0 = π/2(2n + 1). (A5) βn ε=0
We start with the calculation of the first term in Eq. (A2), which is easily obtained, using Eqs. (26) and (A5) equal to u ∗0 =
∞ h 2 dp (φn )0 μ dz n=0
(A6)
with (φn )0 =
4(−1)n cosh(kn x/ h) − cosh(ωn ) , cos(kn y/ h) 3 kn cosh(ωn )
where kn = π (2n + 1) and ωn = kn w/(2h).
(A7)
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Now we calculate the derivatives in Eq. (A4). First, this equation may be written in the following form using the function φ n (26): ∗ ∗ ∞ ∂u du h 2 dp ∂φn dβn = + . (A8) dε 0 μ dz n=0 ∂βn dε 0 ∂ε 0
n The derivative ∂φ is easily calculated from Eq. (26). Otherwise, the derivative (dβ n /dε)0 is ∂βn 0 calculated from the implicit function defined by Eq. (A1), so, taking into account (A5), we obtain dβn = −π/2(2n + 1). (A9) dε 0 Then, we finally obtain from the calculation required in Eq. (A8), ∂φn dβn (ϕn )0 = , ∂βn dε 0
(A10)
where
2(−1)n+1 4 2y (cosh(ωn ) − cosh(kn x/ h)) (ϕn )0 = cos(kn y/ h)+ sin(kn y/ h) − cos(kn y/ h) kn cosh(ωn ) kn h w w 2x sinh(kn x/ h)− sinh(ωn ) − tanh(ωn ) cos(kn y/ h) (cosh(kn x/ h)− cosh(ωn )) . (A11) × h h h
The second term on the right-hand side of Eq. (A8) has the following form deduced from (25) and (26), ∗ ∞ h 2 dp ∂u = (χn )0 (A12) ∂ε 0 μ dz n=0 with (χn )0 =
2(−1)n+1 tanh(ωn ) cos(kn y/ h) cosh(kn x/ h). kn2 cosh(ωn )
Finally, from Eqs. (A2) and (A6) and using Eqs. (A11) and (A12), we obtain
∞ ∞ h 2 dp ∗ u = (φn )0 + 2A1 K n (ψn )0 μ dz n=0 n=0
(A13)
(A14)
with (ψn )0 = (ϕn )0 + (χn )0 . Then uz is deduced from (A14), recalling Eq. (5) rewritten as u z = u ∗ − A2 K n 2
1 C.
h 2 dp . μ dz
(A15)
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