Gas flow rates in microchannels have been rigorously evaluated by means of a variational ...... A. Bird, Molecular Gas Dynamics and the Direct Simulation of Gas.
Variational approach to gas flows in microchannels Carlo Cercignani, Maria Lampis, and Silvia Lorenzani Citation: Phys. Fluids 16, 3426 (2004); doi: 10.1063/1.1764700 View online: http://dx.doi.org/10.1063/1.1764700 View Table of Contents: http://pof.aip.org/resource/1/PHFLE6/v16/i9 Published by the American Institute of Physics.
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PHYSICS OF FLUIDS
VOLUME 16, NUMBER 9
SEPTEMBER 2004
Variational approach to gas flows in microchannels Carlo Cercignani, Maria Lampis, and Silvia Lorenzani Dipartimento di Matematica, Politecnico di Milano, Milano, Italy 20133
共Received 17 February 2004; accepted 28 April 2004; published online 2 August 2004兲 Gas flow rates in microchannels have been rigorously evaluated by means of a variational technique which applies to the integrodifferential form of the Boltzmann equation based on the Bhatnagar, Gross, and Krook model. The Maxwell scattering kernel has been used to describe the gas-wall interactions in the most general case of two surfaces with different accommodation coefficients. © 2004 American Institute of Physics. 关DOI: 10.1063/1.1764700兴
deal with this equation.4 – 6 The Boltzmann equation must be used when the Knudsen number (Kn⫽/L) is no longer small enough to be negligible. The flow between parallel plates has a simple geometry and has been studied by researchers analytically, experimentally, and numerically using different techniques.7–11 In the numerical investigations, the direct simulation Monte Carlo 共DSMC兲 method of Bird12 has proven to be a valuable tool, and various microchannel flows have been simulated successfully.13 Experimental data for flows inside channels much larger than the microdevices are presented by Taguchi et al.7 for a wide range of Reynolds numbers covering most of the transitional regime between collisionless flow and continuum limits. Molecular based numerical schemes, such as the DSMC method,14 are physically appropriate for this kind of gas flows where noncontinuum, rarefied gas effects become important. In the DSMC method, macroscopic observable quantities, such as velocity and temperature, are obtained through averaging appropriate microscopic properties over a small space region. The simulated results are therefore inherently accompanied by statistical noise due to finite sample size. Some researchers9,15,16 applied the DSMC method to microscale gas flows like those in microchannels, and found it very difficult to obtain statistically convergent results under experimental conditions.3 According to Ref. 13 the signal-to-noise ratio in a dilute gas may be written as 共see Ref. 17 for a more detailed discussion兲
I. INTRODUCTION
Rarefied gas flows occur in many microelectromechanical systems 共MEMS兲,1,2 such as actuators, microturbines, gas chromatographs, and microair vehicles. A correct prediction of these flows is important to design and develop MEMS. Nanoscale design occurs for computer components as well and is no longer limited to chip technology but extends to mechanical devices as well. In a modern disk drive, the read/ write head floats at distances of the order of 50 nm above the surface of the spinning platter. The prediction of the vertical force on the head 共as obtained from the pressure distribution in the gas兲 is a crucial design calculation since the head will not accurately read or write if it flies too high. If the head flies too low, it can catastrophically collide against the platter. Microchannels may have further computer applications because they are supposed to dissipate the heat generated in microchips more effectively than fans, and may be used as a more practical cooling system in integrated circuit chips.3 Since, as these examples indicate, microdevices are gaining popularity both in commercial applications and in scientific research, there exists a rapidly growing interest in improving the conventional design techniques related with these devices. Microdevices are often operated in gaseous environments 共typically air兲, and thus their performances are affected by the gas around them. The numerical simulation of all these flows cannot be performed with the NavierStokes equations 共or the related Reynolds equation for a slider air bearing兲 because the smallest characteristic length of MEMS 共to be denoted by L) or of the thin air film occurring in a computer drive is comparable with 共or smaller than兲 the mean free path of the gas molecules. For this reason the continuum equations are no longer valid and the Boltzmann equation must be considered4 – 6 to understand and compute the rarefied flows related with these devices. The Boltzmann equation describes the time evolution of the distribution function of the gas molecules. Numerical methods based on this equation are generally numerically expensive especially when the flow to be considered progresses from free molecular, through transitional, to continuum regions. Since these flows, contrary to the flow past space vehicles, are usually at low Mach number, the use of the linearized Boltzmann equation is permissible and this revives old methods developed in the 1960s and 1970s to 1070-6631/2004/16(9)/3426/12/$22.00
u/ ␦ u⫽Ma冑␥ N, where u is the characteristic flow velocity, ␦ u is the statistical fluctuation, Ma is the Mach number, ␥ is the specific heat ratio, and N is the sample size. A typical flow velocity in microchannel experiments3 is about 0.2 m/s that corresponds to Ma of 10⫺4 . The above relation indicates that if we require the signal to be ten times larger than the noise, N must be 1010. Such an enormous sample size is extremely time consuming and beyond the capabilities of current computers.18 According to Ref. 13 when Ma is small enough for compressibility effects to be negligible, favorable relative statistical errors may be obtained by performing simulations at a Ma increased to a level where compressibility is still negligible. This idea is consistent with the intuitive feeling 3426
© 2004 American Institute of Physics
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Phys. Fluids, Vol. 16, No. 9, September 2004
that, for small Ma, the deviation from a Maxwellian is proportional to the Mach number, no matter what Kn; this reasonable assumption justifies linearization but may be doubted, as indicated in Ref. 19. While it is true that for low Mach number flows the number of statistical samples needs to be quite large in order to measure the flow velocity accurately, in most cases this can still be done with DSMC, given the speed of modern computers. Furthermore, the properties of interest, such as flow rate and skin friction, are integrated quantities, which reduce the statistical scatter. Another method, the so-called information preservation method20–25 has been used to analyze subsonic, microscale gas flows. The statistically convergent results obtained in test problems for characteristic velocities ranging from 0.01 m/s to 1 m/s, corresponding to Ma of 10⫺5 – 10⫺3 and a normal sample size of 103 – 104 were in excellent agreement with the exact solutions in the continuum and free molecular regimes, with those of the linearized Boltzmann equation24,26 and also with the experimental data25 in the transition regime. Here we shall adopt the linearized equation, since all the available mathematical results27 indicate that this is correct for sufficiently low Mach numbers; how small these must is an open and interesting question. An important aspect of the matter is to have an approximate closed form solution for the flow rate of plane Poiseuille flow in order to use it in applications. A typical use occurs in the correction of the Reynolds equation in lubrication theory proposed by Fukui and Kaneko,28 who in a subsequent paper provided a database29 for this flow rate. The database was interpolated by Robert with a formula given in Ref. 17; this matter was further investigated by Veijola et al.30,31 In order to develop a more accurate formula directly from kinetic theory there is a particularly useful technique, the variational method proposed by one of the authors.32 We stress that this variational technique32 applies directly to the integrodifferential form of the Boltzmann equation and can be used for any linearized Boltzmann equation, while the variational methods used by other authors apply to the integral form of the Boltzmann equation, which is available only for special models, such as the Bhatnagar, Gross, and Krook 共BGK兲 model.33 In this paper the variational technique32 is used to compute the flow rate of plane Poiseuille flow as a function of the Knudsen number by using, for the sake of simplicity, the BGK model.33 The variational results will be compared with accurate numerical solutions of the same equation. We shall pay particular attention to the role of boundary conditions. The boundaries will be permitted to have different gas-surface interaction properties, an aspect which, in these days, is no longer academic and requires a particular attention. In fact, e.g., in the actuator considered by Veijola et al.,30,31 the single-crystal silicon surface of the accelerometer’s proof mass is a potential candidate for noticeable reflections close to specular, since it has a smooth, pure surface having an ordered molecular structure, whereas the other surface 共a metallized electrode兲 is rough.
Variational approach to gas flows in microchannels
3427
II. BOUNDARY-VALUE PROBLEM FOR POISEUILLE FLOW
Plane Poiseuille flow is the flow of a gas between two parallel walls due to a pressure gradient parallel to the bounding surfaces.5,6,34 In the (x,z) plane, the z axis coincides with the direction of the fluid motion, while the two plates, whose temperature is assumed to be constant, are fixed at x⫽⫾d/2. If the pressure gradient is small, it can be assumed that the velocity distribution of the flow is nearly the same as that occurring in an equilibrium state. This means that the Boltzmann equation can be linearized about a Maxwellian f 0 by putting f ⫽ f 0 共 1⫹h 兲 ,
共1兲
where f (x,z,c) is the distribution function for the molecular velocity c expressed in units of (2RT) 1/2 (R being the gas constant, T the temperature兲, and h(x,c) is the small perturbation upon the basic equilibrium state. The above mentioned Maxwellian is given by f 0 共 z,c兲 ⫽ 共 1⫹kz 兲 0 ⫺ 3/2 exp共 ⫺c 2 兲 ,
共2兲
where 0 is the density on the boundaries and k⫽
1 p 1 ⫽ , p z z
with p and being the gas pressure and density, respectively. If one assumes the linearized BGK model for the collision operator,33 the Boltzmann equation reads:34
冋
1 Z 1 ⫺ 共 1/2兲 k⫹c x ⫽ 2 x
冕
⫹⬁
⫺⬁
册
2
e ⫺c x 1 Z 共 x,c x 1 兲 dc x 1 ⫺Z 共 x,c x 兲 , 共3兲
where by definition Z 共 x,c x 兲 ⫽ ⫺1
冕 冕 ⫹⬁
⫺⬁
⫹⬁
⫺⬁
2
2
e ⫺c y ⫺c z c z h 共 x,c兲 dc y dc z
and is the collision time. Hence, the bulk velocity of the gas can be written as follows: q 共 x 兲 ⫽ ⫺ 共 1/2兲
冕
⫹⬁
⫺⬁
2
e ⫺c x 1 Z 共 x,c x 1 兲 dc x 1
共4兲
and the flow rate F 共per unit time through unit thickness兲 as F⫽
冕
d/2
⫺d/2
共5兲
q 共 x 兲 dx.
The solution of Eq. 共3兲 in integral form is35,36
冋冉 冉 冉
d Z 共 x,c x 兲 ⫽exp ⫺ x⫹ sgn c x 2
冊冒 册
冊 冕
d ⫻Z ⫺ sgn c x ,c x ⫹ 2 ⫻exp
冊
共 c x 兲
x
⫺ 共 d/2兲 sgn c x
⫺ 兩 x⫺t 兩 关 q 共 t 兲 ⫺k /2兴 / 共 c x 兲 dt, 兩 c x兩
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共6兲
3428
Phys. Fluids, Vol. 16, No. 9, September 2004
Cercignani, Lampis, and Lorenzani
with the values at the boundary Z(⫺ (d/2)sgn cx ,cx) depending on the model of boundary conditions chosen. Let Z ⫺ (⫺d/2,c x ), Z ⫺ (d/2,c x ) be the distribution functions of the molecules impinging upon the walls and Z ⫹ (⫺d/2,c x ), Z ⫹ (d/2,c x ) the distribution functions of the molecules reemerging from them. Enforcing Eq. 共6兲 at the boundaries, one obtains two equations Z ⫺ 共 ⫺d/2,c x 兲 ⫽exp关 d/ 共 c x 兲兴 Z ⫹ 共 d/2,c x 兲
冉
⫺
冉 冊冊 d ⫹t 2 兩 c x兩
⫹
冕
⫻
关 q 共 t 兲 ⫺k /2兴 dt, 兩 c x兩
d/2
⫺ 共 d/2兲
exp
共7兲
Z ⫺ 共 d/2,c x 兲 ⫽exp关 ⫺d/ 共 c x 兲兴 Z ⫹ 共 ⫺d/2,c x 兲
⫹
冕
d/2
⫺ d/2
exp
冉
⫺
冉 冊冊 d ⫺t 2 兩 c x兩
关 q 共 t 兲 ⫺k /2兴 dt. 兩 c x兩
共8兲 These equations can be simply reexpressed in terms of the incoming stream adding the appropriate boundary conditions on the two plates. In the following, we will focus upon Maxwell’s scattering kernel and specialize the analysis to walls having different physical properties so that two accommodation coefficients ( ␣ 1 , ␣ 2 ) can be defined. In this case, the boundary conditions can be written as35,36 ⫹
⫺
Z 共 d/2,c x 兲 ⫽ 共 1⫺ ␣ 1 兲 Z 共 d/2,⫺c x 兲 , ⫹
⫺
Z 共 ⫺d/2,c x 兲 ⫽ 共 1⫺ ␣ 2 兲 Z 共 ⫺d/2,⫺c x 兲 .
共9兲 共10兲
Substituting Eqs. 共9兲 and 共10兲 into Eqs. 共7兲 and 共8兲 one finally obtains a system of two equations for the distribution functions of the impinging molecules. If we introduce two functions S(c x ) and T(c x ) defined on the half space c x ⬎0 as S共 c x 兲 ⫽Z 共 ⫺d/2,⫺c x 兲 ,
T共 c x 兲 ⫽Z 共 d/2,c x 兲 ,
共11兲
S共 c x 兲 ⫽ 共 1⫺ ␣ 1 兲 exp关 ⫺d/ 共 c x 兲兴 T共 c x 兲 ⫹ 共 c x 兲 ,
共12兲
T共 c x 兲 ⫽ 共 1⫺ ␣ 2 兲 exp关 ⫺d/ 共 c x 兲兴 S共 c x 兲 ⫹ 共 c x 兲 ,
共13兲
冉 冉 冊冊 冉 冉 冊冊
关 q 共 t 兲 ⫺k /2兴 dt, 兩 c x兩
d ⫺t 2 兩 c x兩
关 q 共 t 兲 ⫺k /2兴 dt. 兩 c x兩
共 cx兲⫽
冕
⫺
d/2
⫺ d/2
exp
共14兲
III. THE VARIATIONAL APPROACH
Let us rewrite the Boltzmann equation given by Eq. 共3兲 in a symbolic way35 共19兲
共 D⫺L 兲 Z⫽S,
where DZ⫽c x ( Z/ x) , S⫽⫺k/2, and LZ⫽
冋
1 ⫺ 共 1/2兲
冕
⫹⬁
⫺⬁
2
册
e ⫺c x 1 Z 共 x,c x 1 兲 dc x 1 ⫺Z 共 x,c x 兲 .
The boundary conditions to be matched to Eq. 共19兲 have the following general expression: Z ⫹ ⫽KZ ⫺ , where K is a generic operator whose explicit form depends on the scattering kernel used 关see Eqs. 共9兲 and 共10兲兴. Using the general technique described in Refs. 5, 6, and 37 we introduce the following functional J of the test function Zˆ
⫹ 共 Zˆ ⫹ ⫺KZˆ ⫺ , PZˆ ⫺ 兲 B ,
共15兲
D 共 c x 兲 ⫽1⫺ 共 1⫺ ␣ 1 兲共 1⫺ ␣ 2 兲 exp关 ⫺2d/ 共 c x 兲兴
共16兲
共20兲
where P is the parity operator in velocity space and 关共,兲兴, (,) B denote two scalar products defined as follows:
冕 冕 ⫹d/2
⫺d/2
⫹⬁
⫺⬁
exp共 ⫺c 2x 兲
⫻h 共 x,c x 兲 g 共 x,c x 兲 dc x dx, 共 h ⫾ ,g ⫾ 兲 B ⫽ ⫺1/2
The system formed by Eqs. 共12兲 and 共13兲, whose determinant is
admits the solution
If the two walls are identical ( ␣ 1 ⫽ ␣ 2 ⫽ ␣ ), the problem is symmetric and the following relations hold: q(x)⫽q(⫺x), (c x )⫽ (c x ), S(c x )⫽T(c x ). In the limiting cases of perfectly reflecting surfaces ( ␣ →0) and of free molecular flow (d/ →0), the determinant D(c x ) vanishes and the distribution functions S(c x ), T(c x ) are divergent. This divergence does not arise only if ␣ 1 ⫽ ␣ 2 . This is because at least one of the accommodation coefficients must be necessarily different from zero, which is to say, at least one of the walls slows down the flow. Armed with the integral solution of the Boltzmann equation 关Eq. 共6兲兴 subject to the boundary conditions explicitly given by Eqs. 共17兲 and 共18兲, the Poiseuille flow rate can be evaluated by means of a semianalytical procedure, based on a variational principle.5,6,37
共 h,g 兲 ⫽ ⫺1/2
d ⫺ ⫹t d/2 2 共 cx兲⫽ exp 兩 c x兩 ⫺ d/2
冕
T 共 c x 兲 ⫽D 共 c x 兲 ⫺1 †共 1⫺ ␣ 2 兲 exp关 ⫺d/ 共 c x 兲兴 共 c x 兲 ⫹ 共 c x 兲 ‡. 共18兲
J 共 Zˆ 兲 ⫽ 关 „Zˆ , P 共 DZˆ ⫺LZˆ 兲 …兴 ⫺2†共 PS,Zˆ 兲 ‡
such a system becomes
with
S共 c x 兲 ⫽D 共 c x 兲 ⫺1 †共 1⫺ ␣ 1 兲 exp关 ⫺d/ 共 c x 兲兴 共 c x 兲 ⫹ 共 c x 兲 ‡, 共17兲
冕冕 ⍀
c x ⬎0
共21兲
c x exp共 ⫺c 2x 兲
⫻h ⫾ 共 c x 兲 g ⫾ 共 c x 兲 dc x d .
共22兲
Here g ⫾ , h ⫾ are the restrictions of the functions g and h, defined on the boundary, to positive, respectively negative, values of c x . In the one-dimensional case, the integration over the boundary ⍀ reduces to the sum of the terms at x ⫽⫾d/2. The first variation ␦ J of J vanishes if and only if
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Phys. Fluids, Vol. 16, No. 9, September 2004
Variational approach to gas flows in microchannels
Zˆ ⫽Z, where Z(x,c x ) is the solution of Eq. 共3兲 with appropriate boundary conditions. Let us now consider the value attained by J when Z⫽Zˆ . Equation 共20兲 gives k J 共 Z 兲 ⫽⫺ 关共 PS,Z 兲兴 ⫽ 关共 1,Z 兲兴 . 2
共23兲
with different accommodation coefficients, the values at the boundary, Zˆ (⫺ (d/2)sgn cx ,cx), defined on the half space c x ⬎0, are given by Zˆ 共 ⫺d/2,⫺c x 兲 ⫽D 共 c x 兲 ⫺1
Looking at the definitions 共4兲 and 共5兲, it can be easily shown that the stationary value of J is related to a quantity of physical interest, the flow rate of the gas, through the relation F⫽
2 J共 Z 兲. k
共24兲
It must be stressed that in the general formulation of the variational principle 关expressed by Eq. 共20兲兴 Zˆ does not need to satisfy the same boundary conditions as Z. Nevertheless, taking advantage of the mathematical formulation presented in Sec. II, where the values of Z at the boundary have been explicitly evaluated, we can simplify the problem. Thus, in order to obtain an expression for the trial function Zˆ (x,c x ), the solution of the Boltzmann equation in integral form has been considered
冋冉 冉
d Zˆ 共 x,c x 兲 ⫽exp ⫺ x⫹ sgn c x 2
冊冒 册 共 c x 兲
冊 冕
d ⫻Zˆ ⫺ sgn c x ,c x ⫹ 2
x
⫺ 共 d/2兲 sgn c x
exp
冉
⫺ 兩 x⫺t 兩 兩 c x兩
⫻ 关 qˆ 共 t 兲 ⫺k /2兴 / 共 c x 兲 dt.
冊
共25兲
Assuming Maxwell’s boundary conditions imposed on walls
冋冉
d Zˆ 共 x,c x 兲 ⫽D 共 c x 兲 ⫺1 exp ⫺ x⫹ sgn c x 2
3429
冕
⫹d/2
⫺d/2
关 qˆ 共 t 兲 ⫺k /2兴 / 共 c x 兲
⫻ 兵 exp关 ⫺ 共 d/2⫹t 兲 / 共 c x 兲兴 ⫹ 共 1⫺ ␣ 1 兲 ⫻exp关 ⫺ 共 3d/2⫺t 兲 / 共 c x 兲兴 其 dt, Zˆ 共 d/2,c x 兲 ⫽D 共 c x 兲 ⫺1
冕
⫹d/2
⫺d/2
共26兲
关 qˆ 共 t 兲 ⫺k /2兴 / 共 c x 兲
⫻ 兵 exp关 ⫺ 共 d/2⫺t 兲 / 共 c x 兲兴 ⫹ 共 1⫺ ␣ 2 兲 exp关 ⫺ 共 3d/2⫹t 兲 / 共 c x 兲兴 其 dt, 共27兲 as for Z 关Eqs. 共17兲 and 共18兲兴. The numerical results presented in Ref. 36 have shown that for nonsymmetric gas-wall interactions the bulk velocity profile can be approximated by a parabola asymmetric with respect to the channel axis. This enables us to assume the following test function for the bulk velocity of the gas qˆ 共 x 兲 ⫽Ax 2 ⫹Bx⫹C,
共28兲
where A, B, and C are adjustable constants to be varied in order to obtain the best value of J(Zˆ ). Putting in Eq. 共25兲 the expressions of Zˆ (⫺ (d/2)sgn cx ,cx) and qˆ (x), given by 共26兲– 共28兲, by means of analytical calculations, the trial function Zˆ can be explicitly written as follows:
冊冒 册
共 c x 兲 兵 关 1⫺F共 ␣ 1 , ␣ 2 兲兴 †共 C⫺k /2兲 ⫹A/4共 d 2 ⫺4dc x sgn c x ⫹8c 2x 2 兲
⫹B/2共 ⫺d sgn c x ⫹2c x 兲 ⫺exp关 ⫺d sgn c x / 共 c x 兲兴关共 C⫺k /2兲 ⫹A/4共 d 2 ⫹4dc x sgn c x ⫹8c 2x 2 兲 ⫹B/2共 d sgn c x ⫹2c x 兲兴 ‡⫹ 共 1⫺ ␣ 1 兲共 1⫺ ␣ 2 兲 †exp关 ⫺d sgn c x / 共 c x 兲兴 ⫻ 关共 C⫺k /2兲 ⫹A/4共 d 2 ⫺4dc x sgn c x ⫹8c 2x 2 兲 ⫺B/2共 ⫺d sgn c x ⫹2c x 兲兴 ⫺exp关 ⫺2d sgn c x / 共 c x 兲兴关共 C⫺k /2兲 ⫹A/4共 d 2 ⫹4dc x sgn c x ⫹8c 2x 2 兲 ⫺B/2共 d sgn c x ⫹2c x 兲兴 ‡其 ⫹ 共 C⫺k /2兲
冋冉
d ⫹A 共 x 2 ⫺2xc x ⫹2c 2x 2 兲 ⫹B 共 x⫺c x 兲 ⫺exp ⫺ x⫹ sgn c x 2
冊冒 册
共 c x 兲 关共 C⫺k /2兲 ⫹A/4共 d 2 ⫹4dc x sgn c x
⫹8c 2x 2 兲 ⫺B/2共 d sgn c x ⫹2c x 兲兴 ,
much simpler variational formulation. It is convenient now to rescale all variables appearing in Eq. 共29兲 as follows:
where
F共 ␣ 1 , ␣ 2 兲 ⫽
再
共29兲
␣2 ,
c x ⬎0
␣1 ,
c x ⬍0.
Since this test function satisfies the same boundary conditions as Z, the last term on the right-hand side of Eq. 共20兲 can be dropped because it is identically zero, leading to a
␦ ⫽d/ ;
w⫽t/ ;
u⫽x/ ;
A⫽A/ 2 ;
B⫽B/ .
Substituting Zˆ , given by Eq. 共29兲, in the reduced expression of J(Zˆ ), we obtain the following polynomial of the second order with respect to the constants A, B, and C, that are to be determined
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3430
Phys. Fluids, Vol. 16, No. 9, September 2004
J 共 Zˆ 兲 ⫽ 共 冑 兲 ⫺1
再
Cercignani, Lampis, and Lorenzani
c 11 2 c 22 2 c 33 2 A ⫹ B ⫹ C ⫹c 12AB⫹c 13AC 2 2 2
冎
1 ⫹c 23BC⫺c 1 A⫺c 2 B⫺c 3 C⫹ 共 c 3 ⫺c 33/4兲 , 2
⫺2 ␦ S 2 共 2 ␦ 兲 ⫺4S 3 共 0 兲 ⫹8S 3 共 ␦ 兲 ⫺4S 3 共 2 ␦ 兲 ⫹2 共 1⫺ ␣ 1 兲共 1⫺ ␣ 2 兲
共30兲
冋冉 冊
⫻ ⫺
where the same symbol J(Zˆ ) has been used to denote the dimensionless quantity J(Zˆ )/(k ) 2 . The coefficients in nondimensional form are given by c 11⫽
␦4 8
冑
⫺
6
␦ 3 ⫹ ␦ 2 ⫹24⫺
冉
⫹8 ␦ 兲 T 2 共 ␦ 兲 ⫺ 共 8 ␦ 2 ⫹16兲 T 3 共 ␦ 兲 ⫺16␦ T 4 共 ␦ 兲 ⫺16T 5 共 ␦ 兲
冋冉
⫹ 共 2⫺ ␣ 1 ⫺ ␣ 2 兲 ⫺ ⫺
冉
冊
␦4
␦4 8
冊
⫹ ␦ 2 S 1共 0 兲 ⫹
冉
␦4 4
冊
⫹2 ␦ 2 S 1 共 ␦ 兲
⫹16S 3 共 ␦ 兲 ⫺ 共 4 ␦ 2 ⫹8 兲 S 3 共 2 ␦ 兲 ⫹8 ␦ S 4 共 0 兲 ⫺8 ␦ S 4 共 2 ␦ 兲
冋冉 冊 冉 冊
⫹2 共 1⫺ ␣ 1 兲共 1⫺ ␣ 2 兲 ⫺ ⫺
冉
␦
冊
4
⫹ ␦ 2 S 1共 ␦ 兲 ⫹
8
册
␦
␦
8
⫹␦
2
S 1共 3 ␦ 兲
12
␦ 3⫹
␦2 2
冋冉 冊 冉 冊 ␦2 2
␦2 2
⫹1 S 1 共 2 ␦ 兲 ⫺2 ␦ S 2 共 0 兲 ⫹2 ␦ S 2 共 2 ␦ 兲 ⫹4S 3 共 0 兲
册
⫺8S 3 共 ␦ 兲 ⫹4S 3 共 2 ␦ 兲 ⫺2 共 1⫺ ␣ 1 兲共 1⫺ ␣ 2 兲
冋冉 冊 ␦2 2
⫹1 S 1 共 3 ␦ 兲 ⫹
冉 冊
册
⫺ 共 4 ␦ 2 ⫹8 兲 S 3 共 3 ␦ 兲 ⫺8 ␦ S 4 共 3 ␦ 兲 ⫺8S 5 共 3 ␦ 兲 ⫹
2
冑
c 22⫽ I1 , 共31兲
4
⫹ ␦ S 1共 0 兲 ⫹
⫹4 ␦ S 3 共 2 ␦ 兲 ⫹4S 4 共 0 兲 ⫹4S 4 共 2 ␦ 兲 ⫹ 共 ␦ 2 ⫺4 兲 S 2 共 ␦ 兲
册
⫺8S 4 共 ␦ 兲 ⫹
c 13⫽
␦
2
2
冑
冉 冊 ␦2 2
2
1
冑
共34兲
I3 ,
⫺2⫹ ␦ 2 T 1 共 ␦ 兲 ⫹4 ␦ T 2 共 ␦ 兲 ⫹4T 3 共 ␦ 兲 ⫺ 共 2⫺ ␣ 1
⫺␣2兲
冋
␦2 2
S 1共 0 兲 ⫹ ␦ 2S 1共 ␦ 兲 ⫹
␦2 2
S 1 共 2 ␦ 兲 ⫺2 ␦ S 2 共 0 兲
册
⫺ ␣ 1 兲共 1⫺ ␣ 2 兲
冋
␦2 2
S 1共 3 ␦ 兲 ⫹
␦2 2
S 1共 ␦ 兲 ⫹ ␦ 2S 1共 2 ␦ 兲
⫹2S 3 共 ␦ 兲 ⫺4S 3 共 2 ␦ 兲 ⫺2 ␦ S 2 共 ␦ 兲 ⫹2 ␦ S 2 共 3 ␦ 兲
册
⫹2S 3 共 3 ␦ 兲 ⫹
2
冑
共35兲
I4 ,
c 23⫽ 共 ␣ 2 ⫺ ␣ 1 兲关 ⫺ ␦ S 1 共 0 兲 ⫹ ␦ S 1 共 2 ␦ 兲 ⫹2S 2 共 0 兲
⫺ 冑 ␦ ⫹6⫺ 共 ␦ 2 ⫹4 兲 T 1 共 ␦ 兲 ⫺4 ␦ T 2 共 ␦ 兲 ⫺8T 3 共 ␦ 兲
冋冉 冊
⫹ 共 2⫺ ␣ 1 ⫺ ␣ 2 兲 ⫺ ⫺
共32兲
I2 ,
␦2
⫹ ␦ S 1共 2 ␦ 兲
3 3 ⫹ ␦ 2 ⫹2 S 2 共 0 兲 ⫹ ␦ 2 ⫹2 S 2 共 2 ␦ 兲 ⫺4 ␦ S 3 共 0 兲 2 2
2
⫹1 S 1 共 ␦ 兲 ⫺ 共 ␦ 2
⫹2 ␦ S 2 共 2 ␦ 兲 ⫹2S 3 共 0 兲 ⫺4S 3 共 ␦ 兲 ⫹2S 3 共 2 ␦ 兲 ⫹2 共 1
冉 冊 冊 4
2
册
⫹ 共 ␦ 3 ⫹4 ␦ 兲 S 2 共 ␦ 兲 ⫹8 ␦ S 4 共 ␦ 兲 ⫺ 共 ␦ 3 ⫹4 ␦ 兲 S 2 共 3 ␦ 兲
冋冉 冊 冉 冊 冉
␦2
⫹2 ␦ S 2 共 3 ␦ 兲 ⫹4S 3 共 3 ␦ 兲 ⫹
⫺ 共 4 ␦ ⫹8 兲 S 3 共 ␦ 兲 ⫹16S 3 共 2 ␦ 兲 ⫺8S 5 共 ␦ 兲 ⫹16S 5 共 2 ␦ 兲
c 12⫽ 共 ␣ 2 ⫺ ␣ 1 兲 ⫺
I 3 , 共33兲
⫹2 兲 S 1 共 2 ␦ 兲 ⫹4S 3 共 ␦ 兲 ⫺8S 3 共 2 ␦ 兲 ⫺2 ␦ S 2 共 ␦ 兲
⫹2 ␦ 2 S 1 共 2 ␦ 兲
␦3
冑
⫹1 S 1 共 0 兲 ⫺ 共 ␦ 2 ⫹2 兲 S 1 共 ␦ 兲
2
␦3
2
⫺ 冑 ␦ ⫹5⫺ 共 ␦ 2 ⫹2 兲 T 1 共 ␦ 兲
4
4
⫹2 S 1 共 ␦ 兲
2
⫺4 ␦ T 2 共 ␦ 兲 ⫺8T 3 共 ␦ 兲 ⫺ 共 2⫺ ␣ 1 ⫺ ␣ 2 兲
⫻
4
冑
c 1 ⫽⫺
⫹
⫺ 共 ␦ 3 ⫹4 ␦ 兲 S 2 共 2 ␦ 兲 ⫺ 共 4 ␦ 2 ⫹8 兲 S 3 共 0 兲
⫺8S 5 共 0 兲 ⫹16S 5 共 ␦ 兲 ⫺8S 5 共 2 ␦ 兲
冉 冊
册
⫻
⫹ ␦ 2 S 1 共 2 ␦ 兲 ⫹ 共 ␦ 3 ⫹4 ␦ 兲 S 2 共 0 兲
8
⫹2 S 1 共 3 ␦ 兲 ⫺
⫹2 ␦ S 2 共 ␦ 兲 ⫺2 ␦ S 2 共 3 ␦ 兲 ⫺4S 3 共 3 ␦ 兲 ⫹
⫹2 ␦ 2 T 1 共 ␦ 兲 ⫺ 共 2 ␦ 3
4
2
␦2
⫹ 共 ␦ 2 ⫹4 兲 S 1 共 2 ␦ 兲 ⫺4S 3 共 ␦ 兲 ⫹8S 3 共 2 ␦ 兲
冊
␦4
␦2
册
␦2 2
⫹2 S 1 共 0 兲 ⫹ 共 ␦ 2 ⫹4 兲 S 1 共 ␦ 兲
⫹2 S 1 共 2 ␦ 兲 ⫹2 ␦ S 2 共 0 兲
⫹2S 2 共 2 ␦ 兲 ⫺4S 2 共 ␦ 兲兴 ⫹
2
冑
I5 ,
共36兲
c 2 ⫽ 共 ␣ 2 ⫺ ␣ 1 兲关 ⫺ ␦ S 1 共 0 兲 ⫹ ␦ S 1 共 2 ␦ 兲 ⫹2S 2 共 0 兲 ⫹2S 2 共 2 ␦ 兲 ⫺4S 2 共 ␦ 兲兴 ⫹
1
冑
I5 ,
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共37兲
Phys. Fluids, Vol. 16, No. 9, September 2004
Variational approach to gas flows in microchannels
c 33⫽2⫺4T 1 共 ␦ 兲 ⫺ 共 2⫺ ␣ 1 ⫺ ␣ 2 兲关 2S 1 共 0 兲 ⫺4S 1 共 ␦ 兲
3431
and S n (x) is a generalized Abramowitz function defined by
⫹2S 1 共 2 ␦ 兲兴 ⫺2 共 1⫺ ␣ 1 兲共 1⫺ ␣ 2 兲关 2S 1 共 3 ␦ 兲 ⫹2S 1 共 ␦ 兲 ⫺4S 1 共 2 ␦ 兲兴 ⫹
2
冑
共38兲
I6 ,
S n 共 x, ␦ , ␣ 1 , ␣ 2 兲 ⫽
冕
⫹⬁
0
t n exp共 ⫺t 2 ⫺x/t 兲 dt. 1⫺ 共 1⫺ ␣ 1 兲共 1⫺ ␣ 2 兲 exp共 ⫺2 ␦ /t 兲
c 3 ⫽⫺ 冑 ␦ ⫹2⫺4T 1 共 ␦ 兲 ⫺ 共 2⫺ ␣ 1 ⫺ ␣ 2 兲关 2S 1 共 0 兲 ⫺4S 1 共 ␦ 兲 ⫹2S 1 共 2 ␦ 兲兴 ⫺2 共 1⫺ ␣ 1 兲共 1⫺ ␣ 2 兲 ⫻ 关 2S 1 共 3 ␦ 兲 ⫹2S 1 共 ␦ 兲 ⫺4S 1 共 2 ␦ 兲兴 ⫹
1
冑
I6 ,
共39兲
where T n (x) is an Abramowitz function defined by T n共 x 兲 ⫽
冕
⫹⬁
0
The symbols I 1 , I 2 , I 3 , I 4 , I 5 , I 6 stand for integral expressions involving the T n (x) and S n (x) functions, which are negligible in the free molecular flow regime. The explicit form of these integrals is given in the Appendix. The derivatives of J(Zˆ ) with respect to A, B, C vanish in correspondence of the optimal values of these constants. The resulting expression for the minimum of J(Zˆ ) is
t n exp共 ⫺t 2 ⫺x/t 兲 dt
2 2 2 2 2 2 min J 共 Zˆ 兲 ⫽ 共 8 冑 兲 ⫺1 关 c 13 c 22⫺2c 12c 13c 23⫹c 12 c 33⫹c 11共 c 23 ⫺c 22c 33兲兴 ⫺1 †⫺8c 11c 2 c 23c 3 ⫹4c 11c 23 c 3 ⫺4c 12 c 3 ⫹4c 11c 22c 23 2 2 ⫹4c 11c 22 c 33⫺c 11c 23 c 33⫹4c 12 c 3 c 33 2 2 2 2 ⫺4c 11c 22c 3 c 33⫺c 12 c 33⫹c 11c 22c 33 ⫺8c 1 c 12共 c 2 c 33⫺c 23c 3 兲 ⫺4c 21 共 c 23 ⫺c 22c 33兲 ⫹2c 13共 4c 1 c 2 c 23⫹4c 12c 2 c 3 2 ⫺4c 1 c 22c 3 ⫺4c 12c 23c 3 ⫹c 12c 23c 33兲 ⫺c 13 关 4c 22 ⫹c 22共 c 33⫺4c 3 兲兴 ‡.
Thus, the computation of the optimal value of the functional J(Z) 共Eq. 40兲 will lead to an accurate estimate of the flow rate of the gas, which in nondimensional form reads Q共 ␦ 兲⫽
F 4 ⫽⫺ 2 J 共 Z 兲 , ␦ ⫺ kd 2 2
共41兲
where ␦ is the rarefaction parameter 共inverse Knudsen number兲.
IV. RESULTS AND DISCUSSION
The rigorous computation of the Poiseuille flow rate Q, presented in this paper, is mainly motivated by its relevance to several applications. In the 1980s Fukui and Kaneko derived a generalized Reynolds equation for lubrication problems,28 based on the linearized Boltzmann equation, which holds for arbitrary Knudsen number gas films and arbitrary accommodation coefficients. The solution of this equation requires that the Poiseuille flow rate has to be accurately calculated in advance. To this end, Fukui and Kaneko gave a database,29 created by both numerical and variational calculations38 共based on an integral approach to the BGK model兲, for the values of the gas flow rate Q versus the inverse of the Knudsen number ␦, with ␦ ranging from 0.01 to 100.0 and the accommodation coefficient ␣ of the bounding surfaces 共supposed physically identical兲 varying in the interval 关0.7, 1兴. This database is reported in Tables I and
共40兲
II along with our results obtained by means of the variational approach presented in Sec. III and the numerical method described in Ref. 36, for a comparison. As revealed by these tables, the agreement between our numerical and variational outputs is very good 共the relative error is less than 0.1%兲. It is to be noted that our numerical findings have been compared in Ref. 36 with those published by other authors,39– 41 in the case of Maxwell’s boundary conditions on two physically identical walls, showing a basic good agreement. Of course, the variational approach carried out here has the enormous advantage of reducing drastically the computational CPU time, giving at one time the same accuracy for each value of the rarefaction parameter ␦. For this reason, only our variational datas are listed in Tables I and II for ␦ ⭓10.0. If one wanted to reach the same high accuracy using the finite difference numerical technique described in Refs. 34 and 36, the number of mesh points should be considerably increased as ␦ increases, with the loss in the efficiency of the calculation. Some considerations are in order here concerning the results obtained by Fukui and Kaneko which compare poorly with our outputs. Their variational findings are inaccurate for every value of ␦ as ␣ decreases, while the numerical outputs are in fair agreement only in a narrow ␦ range which reduces decreasing the accommodation coefficient. This is demonstrated in Fig. 1 which shows as a function of the rarefaction parameter ␦ the flow rate Q evaluated by Fukui and Kaneko. For comparison, our variational results are drawn in the same figure. The inaccuracy of the tabulated values in Ref. 29 is likely related to the high accuracy required in the computation of the gen-
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3432
Phys. Fluids, Vol. 16, No. 9, September 2004
Cercignani, Lampis, and Lorenzani
TABLE I. Poiseuille flow rate Q vs ␦ for Maxwell’s boundary conditions on identical walls. Comparison between our results 共CLL兲 and Fukui and Kaneko’s results Ref. 29.
␣ ⫽0.9
␦ ⫺5
1.⫻10 1.⫻10⫺3 0.01 0.03 0.06 0.09 0.1 0.3 0.5 0.7 1.0 2.0 3.0 5.0 7.0 10.0 20.0 30.0 60.0 80.0 100.0
␣ ⫽1
Num. 共Ref. 29兲
Var. 共Ref. 29兲
Num. 共CLL兲
Var. 共CLL兲
Num. 共Ref. 29兲
Num. 共CLL兲
Var. 共CLL兲
¯ ¯ 3.564 2.923 2.570 2.384 2.338 1.947 1.826 1.774 1.746 1.789 1.886 2.141 2.431 2.898 4.494 6.102 10.737 13.725 16.779
¯ ¯ 3.566 2.942 2.578 2.391 2.345 1.946 1.821 1.729 1.750 1.795 1.909 2.189 2.494 2.969 4.598 6.250 ¯ ¯ ¯
8.194 5.042 3.557 2.924 2.570 2.383 2.338 1.946 1.826 1.773 1.745 1.793 1.908 2.190 2.495 2.970 ¯ ¯ ¯ ¯ ¯
8.194 5.042 3.557 2.924 2.570 2.383 2.338 1.946 1.826 1.773 1.745 1.793 1.908 2.189 2.494 2.969 4.600 6.253 11.237 14.567 17.898
¯ ¯ 3.060 2.522 2.228 2.071 2.033 1.703 1.602 1.559 1.539 1.595 1.711 1.991 2.292 2.768 4.398 6.049 11.033 14.363 17.693
6.855 4.274 3.050 2.524 2.227 2.071 2.033 1.702 1.602 1.559 1.539 1.595 1.710 1.990 2.294 2.767 ¯ ¯ ¯ ¯ ¯
6.855 4.274 3.050 2.524 2.227 2.071 2.033 1.702 1.602 1.559 1.539 1.595 1.710 1.990 2.294 2.767 4.394 6.045 11.029 14.358 17.689
eralized Abramowitz functions S n especially for small values of ␦. In our numerical method and variational analysis solution, the S n functions have been numerically evaluated using the SLATEC libraries. We note in passing that a likely reason for the slight deviation registered between the DSMC datas obtained by Alexander et al.17 and the Fukui and Kaneko modified Reynolds equation for ␣ ⫽0.7 can be related to the inaccuracy of the tabulated values in Ref. 29.
Beyond the lubrication problems, the Reynolds equation in linearized form has been applied in modeling the gas damping effects acting on oscillating microstructures in a micromechanical accelerometer.30,31 In ultrathin films the gas rarefaction effects have been taken into account by replacing the gas viscosity with an effective viscosity which has been expressed as a function of the Poiseuille flow rate coefficient Q. 31 Again a rigorous evaluation of Q is required. In order to
TABLE II. Poiseuille flow rate Q vs ␦ for Maxwell’s boundary conditions on identical walls. Comparison between our results 共CLL兲 and Fukui and Kaneko’s results 共Ref. 29兲.
␣ ⫽0.7
␣ ⫽0.8
␦
Num. 共Ref. 29兲
Var. 共Ref. 29兲
Num. 共CLL兲
Var. 共CLL兲
Num. 共Ref. 29兲
Var. 共Ref. 29兲
Num. 共CLL兲
Var. 共CLL兲
1.⫻10⫺5 1.⫻10⫺3 0.01 0.03 0.06 0.09 0.1 0.3 0.5 0.7 1.0 2.0 3.0 5.0 7.0 10.0 20.0 30.0 60.0 80.0 100.0
¯ ¯ 4.932 4.004 3.497 3.232 3.168 2.620 2.452 2.376 2.330 2.324 2.357 2.540 2.794 3.246 4.803 6.432 11.024 13.935 17.051
¯ ¯ 5.868 4.599 3.922 3.572 3.487 2.781 2.570 2.475 2.417 2.450 2.572 2.864 3.175 3.654 5.290 6.945 ¯ ¯ ¯
11.923 7.147 4.929 4.005 3.496 3.232 3.168 2.620 2.452 2.376 2.330 2.360 2.472 2.757 3.066 3.546 ¯ ¯ ¯ ¯ ¯
11.923 7.147 4.929 4.005 3.496 3.232 3.168 2.620 2.452 2.376 2.330 2.360 2.472 2.756 3.065 3.544 5.181 6.836 11.823 15.153 18.485
¯ ¯ 4.174 3.406 2.984 2.762 2.708 2.245 2.102 2.039 2.002 2.026 2.096 2.320 2.595 3.054 4.633 6.251 10.869 13.823 16.906
¯ ¯ 4.388 3.540 3.084 2.842 2.783 2.279 2.124 2.054 2.009 2.059 2.175 2.458 2.765 3.242 4.875 6.528 ¯ ¯ ¯
9.841 5.978 4.169 3.407 2.983 2.762 2.708 2.245 2.102 2.039 2.002 2.041 2.155 2.438 2.746 3.223 ¯ ¯ ¯ ¯ ¯
9.841 5.978 4.169 3.407 2.983 2.762 2.708 2.245 2.102 2.039 2.002 2.041 2.155 2.437 2.744 3.221 4.855 6.509 11.495 14.825 18.156
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Phys. Fluids, Vol. 16, No. 9, September 2004
Variational approach to gas flows in microchannels
3433
FIG. 1. Poiseuille flow rate Q vs ␦ for Maxwell’s boundary conditions on identical walls. Comparison between the numerical 共circles兲 and variational 共squares兲 data tabulated by Fukui and Kaneko 共Ref. 29兲 and our variational results 共triangles兲.
model the damping produced by a squeezed gas film in an accelerometer where one surface is rough 共metallized electrode兲 while the other surface is smooth 共single-crystal silicon兲, Veijola et al.31 solved the linearized Boltzmann equation imposing different boundary conditions on the two walls. In particular, they suggested that the metallized surface reflects the molecules diffusely ( ␣ 2 ⫽1) while the other wall reflects specularly a noticeable part of the particles ( ␣ 1 ⭓0). In Table III we list the Veijola et al. results31 obtained by combining a variational analysis and a numerical approach, along with our variational outputs. As shown by the table, the agreement is very good in the transitional regime as well as in the continuum flow limit. Since walls with different physical structures are fully used in designing micromachines and other devices with very tiny gas gaps, Table IV shows the Poiseuille flow rate Q, computed by our variational analysis, versus the rarefaction parameter ␦ in the general, nonsymmetrical case, ␣ 1 ⫽ ␣ 2 . A comparison with the numerical results presented in Ref. 36 reveals that the agreement is of the same quality as
found for ␣ 1 ⫽ ␣ 2 ⫽ ␣ 共see Tables I and II兲. Therefore a complete and general database of practical usefulness can be created exploiting the ability of our variational method to give highly accurate results in the free molecular flow limit as well as in the continuum regime. V. CONCLUDING REMARKS
In the present paper a variational approach has been used to solve the plane Poiseuille problem between two parallel plates as an issue of relevance for applications. The variational method turns out to be quite precise in evaluating the flow rate Q progressing from free molecular, through transitional, to continuum regions. Since the implementation of our variational solution is computationally straightforward, an accurate database can be created for the gas flow rates at different Knudsen numbers. Knowing that walls with different physical structures are fully used in designing micromachines, different accommodation coefficients ( ␣ 1 ⫽ ␣ 2 ) have been considered in the
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3434
Phys. Fluids, Vol. 16, No. 9, September 2004
Cercignani, Lampis, and Lorenzani
TABLE III. Poiseuille flow rate Q vs ␦ for Maxwell’s boundary conditions on different walls. ␣ 2 ⫽1. Comparison between our results 共CLL兲 and Veijola et al.’s results 共Ref. 31兲.
␣ 1 ⫽0.0
␦ ⫺5
1.⫻10 1.⫻10⫺3 0.01 0.03 0.07 0.1 0.2 0.5 0.7 1.0 3.0 7.0 10.0 20.0 50.0 70.0 100.0
␣ 1 ⫽0.25
␣ 1 ⫽0.5
␣ 1 ⫽0.75
CLL
Reference 31
CLL
Reference 31
CLL
Reference 31
CLL
Reference 31
12.929 7.790 5.422 4.455 3.835 3.616 3.282 3.077 3.095 3.190 4.281 6.826 8.790 15.409 35.378 48.705 68.701
¯ ¯ 5.422 4.455 3.835 3.616 3.282 3.077 3.095 3.189 4.279 6.821 8.784 15.402 35.370 48.697 68.692
11.410 6.908 4.819 3.953 3.386 3.181 2.854 2.596 2.564 2.576 3.014 3.969 4.617 6.556 11.813 15.207 20.257
¯ ¯ 4.819 3.953 3.386 3.182 2.854 2.596 2.564 2.576 3.013 3.966 4.613 6.551 11.806 15.201 20.250
9.892 6.028 4.222 3.464 2.959 2.774 2.470 2.204 2.153 2.134 2.368 3.039 3.549 5.233 10.249 13.586 18.589
¯ ¯ 4.222 3.464 2.959 2.774 2.470 2.203 2.153 2.134 2.367 3.037 3.545 5.228 10.243 13.580 18.583
8.373 5.150 3.633 2.988 2.553 2.392 2.123 1.877 1.826 1.800 1.975 2.575 3.055 4.692 9.672 13.001 17.997
¯ ¯ 3.633 2.988 2.553 2.392 2.123 1.877 1.826 1.800 1.974 2.572 3.051 4.688 9.667 12.996 17.992
TABLE IV. Poiseuille flow rate Q vs ␦ for Maxwell’s boundary conditions on different walls.
␦
␣2
␣ 1 ⫽0.1
␣ 1 ⫽0.3
␣ 1 ⫽0.5
␣ 1 ⫽0.7
␣ 1 ⫽0.8
␣ 1 ⫽0.9
1.⫻10⫺5
0.3 0.5 0.8
48.084 30.125 17.256
32.598 23.412 14.918
23.412 18.371 12.763
17.285 14.432 10.778
14.918 12.763 9.841
12.882 11.259 9.303
0.01
0.3 0.5 0.8
16.804 11.161 6.896
11.921 8.918 6.036
8.918 7.200 5.246
6.852 5.832 4.515
6.036 5.246 4.169
5.325 4.711 3.837
0.05
0.3 0.5 0.8
12.137 8.073 5.070
8.582 6.446 4.431
6.446 5.223 3.856
4.998 4.264 3.333
4.431 3.856 3.090
3.939 3.486 2.857
0.1
0.3 0.5 0.8
10.733 7.096 4.466
7.520 5.634 3.887
5.634 4.556 3.375
4.375 3.724 2.918
3.887 3.375 2.708
3.466 3.060 2.508
0.5
0.3 0.5 0.8
8.906 5.786 3.660
6.013 4.446 3.090
4.446 3.544 2.635
3.459 2.896 2.263
3.090 2.635 2.102
2.778 2.406 1.954
1.0
0.3 0.5 0.8
8.683 5.673 3.655
5.763 4.257 3.000
4.257 3.368 2.520
3.337 2.758 2.154
3.000 2.520 2.002
2.717 2.314 1.865
5.0
0.3 0.5 0.8
9.493 6.825 5.075
6.141 4.742 3.700
4.742 3.773 3.009
3.970 3.210 2.590
3.700 3.009 2.437
3.479 2.841 2.309
10.0
0.3 0.5 0.8
10.694 8.297 6.754
6.945 5.600 4.666
5.600 4.569 3.831
4.903 4.020 3.379
4.666 3.831 3.221
4.473 3.677 3.093
60.0
0.3 0.5 0.8
20.315 18.707 17.743
15.247 14.011 13.265
14.011 12.860 12.163
13.448 12.333 11.658
13.265 12.163 11.495
13.121 12.028 11.366
100.0
0.3 0.5 0.8
27.258 25.793 24.928
21.910 20.690 19.967
20.690 19.522 18.830
20.143 18.998 18.320
19.967 18.830 18.156
19.828 18.697 18.027
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Phys. Fluids, Vol. 16, No. 9, September 2004
Variational approach to gas flows in microchannels
3435
out that, for Maxwell reemission, the flow rate diverges as
frame of Maxwell’s model for boundary conditions. This aspect deserves a particular attention since in ultrathin films, when gas rarefaction effects dominate, the molecular interactions with the surfaces affect strongly the velocity profile and the gas flow rate. If two identical walls are considered, the Poiseuille flow rate increases significantly when going from the complete Maxwell diffusion ( ␣ ⫽1) to the entirely specular reflection ( ␣ ⫽0). Previous works already pointed
␦ →0 共free molecular flow limit兲. In addition, our results
show that this divergence is enhanced in the limit of entirely specular reflection: for free molecular flow, Q( ␦ ) diverges faster as ␣ decreases. If the symmetry of the problem is broken by assuming two different accommodation coefficients ( ␣ 1 , ␣ 2 ) for the bounding surfaces, the wall which reflects specularly less molecules slows down the flow.
APPENDIX
I 1⫽
冕
␦ /2
0
⫹
冋
再 冋
du ⫺2
␦2 2
␦2 4
共 T 0 共 ␦ /2⫺u 兲 ⫹T 0 共 ␦ /2⫹u 兲兲 ⫹ ␦ 关 T 1 共 ␦ /2⫺u 兲 ⫹T 1 共 ␦ /2⫹u 兲兴 ⫹2 关 T 2 共 ␦ /2⫺u 兲 ⫹T 2 共 ␦ /2⫹u 兲兴
关 T 0 共 ␦ /2⫺u 兲 ⫹T 0 共 ␦ /2⫹u 兲兴 ⫹2 ␦ ⫻ 关 T 1 共 ␦ /2⫺u 兲 ⫹T 1 共 ␦ /2⫹u 兲兴 ⫹4 关 T 2 共 ␦ /2⫺u 兲 ⫹T 2 共 ␦ /2⫹u 兲兴
冊
⫹2L ⫺ 2 共 ␣1 ,␣2兲⫹ ⫹
I 2⫽
冕
冉
␦2 4
␦ /2
du
冋冉 冋冉 冉 ␦
2
⫻ ⫺ ⫻
I 3⫽
冕
␦2 4
␦2 4
再冋
␦2 4
冊
2
共␣2 ,␣1兲
4
册冎
冊
冉 冊
␦ 2
0
再冋
␦2 4
␦2
4
⫹ ⫺ L⫺ 0 ⫺ ␦ L 1 ⫹2L 2
冊
4
⫹ L⫺ 0 ⫺␦L 1
2
共␣1 ,␣2兲
共A1兲
⫹ ⫺ L⫺ 0 ⫺ ␦ L 1 ⫹2L 2 共 ␣ 1 , ␣ 2 兲 ⫹
冊
du
2
␦2
2
,
冊
册冋
⫺ A⫹ 0 ⫺A 1 共 ␣ 2 , ␣ 1 兲 ⫹
⫹ ⫺ L⫺ 0 ⫺ ␦ L 1 ⫹2L 2 共 ␣ 1 , ␣ 2 兲 ⫺
␦ /2
册 冑 冋冉
关 T 0 共 ␦ /2⫺u 兲 ⫹T 0 共 ␦ /2⫹u 兲兴 ⫹ ␦ 关 T 1 共 ␦ /2⫺u 兲 ⫹T 1 共 ␦ /2⫹u 兲兴 ⫹2 关 T 2 共 ␦ /2⫺u 兲 ⫹T 2 共 ␦ /2⫹u 兲兴
⫺ A⫹ 0 ⫺A 1 共 ␣ 1 , ␣ 2 兲 ⫺
␦2
冊
⫹ ⫺ L⫺ 0 ⫺ ␦ L 1 ⫹2L 2 共 ␣ 2 , ␣ 1 兲 ⫹
⫹ ⫺ L⫺ 0 ⫺ ␦ L 1 ⫹2L 2
0
⫻
冉
册冋 冉
册
冉
␦ 2
冉
␦2 4
␦ 2
册
关 T 0 共 ␦ /2⫹u 兲 ⫺T 0 共 ␦ /2⫺u 兲兴 ⫹ 关 T 1 共 ␦ /2⫹u 兲 ⫺T 1 共 ␦ /2⫺u 兲兴
冊 冉
册 冑 冋冉
⫹ ⫺ L⫺ 0 ⫺ ␦ L 1 ⫹2L 2 共 ␣ 2 , ␣ 1 兲 ⫹
冊
⫺ A⫹ 0 ⫺A 1 共 ␣ 2 , ␣ 1 兲 ⫻
␦2 4
2
␦
2
冊 册冎
册
⫺ A⫹ 0 ⫺A 1 共 ␣ 1 , ␣ 2 兲
冊
⫹ ⫺ L⫺ 0 ⫺ ␦ L 1 ⫹2L 2 共 ␣ 2 , ␣ 1 兲
共A2兲
,
关 T 0 共 ␦ /2⫺u 兲 ⫹T 0 共 ␦ /2⫹u 兲兴 ⫹ ␦ 关 T 1 共 ␦ /2⫺u 兲 ⫹T 1 共 ␦ /2⫹u 兲兴 ⫹2⫻ 关 T 2 共 ␦ /2⫺u 兲 ⫹T 2 共 ␦ /2⫹u 兲兴
册
⫺ ⫻ 关 ⫺2 关 T 0 共 ␦ /2⫺u 兲 ⫹T 0 共 ␦ /2⫹u 兲兴 ⫹L ⫺ 0 共 ␣ 1 , ␣ 2 兲 ⫹L 0 共 ␣ 2 , ␣ 1 兲兴 ⫹ 关 T 0 共 ␦ /2⫺u 兲 ⫹T 0 共 ␦ /2⫹u 兲兴
⫻ ⫻
I 4⫽
冕
冋冉 冉
␦2 4
␦2 4
␦ /2
0
冊
⫹ ⫺ L⫺ 0 ⫺ ␦ L 1 ⫹2L 2 共 ␣ 1 , ␣ 2 兲 ⫹
␦2 4
冊
冊
再
册 冑冋 冊 册冎
⫹ ⫺ L⫺ 0 ⫺ ␦ L 1 ⫹2L 2 共 ␣ 2 , ␣ 1 兲 ⫹
⫹ ⫺ ⫺ L⫺ 0 ⫺ ␦ L 1 ⫹2L 2 共 ␣ 1 , ␣ 2 兲 ⫹L 0 共 ␣ 2 , ␣ 1 兲
冉
␦2 4
2
L⫺ 0 共␣1 ,␣2兲
⫹ ⫺ L⫺ 0 ⫺ ␦ L 1 ⫹2L 2 共 ␣ 2 , ␣ 1 兲
,
共A3兲
1 du ⫺ †␦ 关 T 0 共 ␦ /2⫹u 兲 ⫺T 0 共 ␦ /2⫺u 兲兴 ⫹2 关 T 1 共 ␦ /2⫹u 兲 ⫺T 1 共 ␦ /2⫺u 兲兴 ‡2 ⫹†␦ 关 T 0 共 ␦ /2⫹u 兲 ⫺T 0 共 ␦ /2⫺u 兲兴 2
冋冉 冉
⫹2 关 T 1 共 ␦ /2⫹u 兲 ⫺T 1 共 ␦ /2⫺u 兲兴 ‡ ⫺ ⫹
冉
2
冑
冋冉
␦ 2
⫺ A⫹ 0 ⫺A 1
冊
2
共 ␣1 ,␣2兲⫹
␦
2
␦ 2
冊
⫺ A⫹ 0 ⫺A 1 共 ␣ 1 , ␣ 2 兲 ⫺
⫺ A⫹ 0 ⫺A 1
冊
2
共␣2 ,␣1兲
册冎
冉
␦ 2
冊
⫺ A⫹ 0 ⫺A 1 共 ␣ 2 , ␣ 1 兲
册
,
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共A4兲
3436
I 5⫽
Phys. Fluids, Vol. 16, No. 9, September 2004
冕
0
⫹ ⫻ I 6⫽
␦ /2
冕
冋 冉
⫹
␦
冋冉
␦ 2
冊
⫺ A⫹ 0 ⫺A 1 共 ␣ 1 , ␣ 2 兲 ⫺
冉
␦ 2
册 册冎
冊
⫺ A⫹ 0 ⫺A 1 共 ␣ 2 , ␣ 1 兲
册
⫺ 关 T 0 共 ␦ /2⫹u 兲 ⫺T 0 共 ␦ /2⫺u 兲兴 ⫹ 关 T 1 共 ␦ /2⫹u 兲 ⫺T 1 共 ␦ /2⫺u 兲兴 关 ⫺L ⫺ 0 共 ␣ 1 , ␣ 2 兲 ⫹L 0 共 ␣ 2 , ␣ 1 兲兴 ⫹
2
␦ 2
␦ /2
0
再
du 关 T 0 共 ␦ /2⫺u 兲 ⫹T 0 共 ␦ /2⫹u 兲兴
Cercignani, Lampis, and Lorenzani
冊
⫺ ⫺ A⫹ 0 ⫺A 1 共 ␣ 1 , ␣ 2 兲 ⫺L 0 共 ␣ 2 , ␣ 1 兲
再
冉
␦ 2
冊
⫺ A⫹ 0 ⫺A 1 共 ␣ 2 , ␣ 1 兲
2
冑
冋
L⫺ 0 共␣1 ,␣2兲
,
共A5兲
⫺ du ⫺2 关 T 0 共 ␦ /2⫺u 兲 ⫹T 0 共 ␦ /2⫹u 兲兴 2 ⫹2 关 T 0 共 ␦ /2⫺u 兲 ⫹T 0 共 ␦ /2⫹u 兲兴关 L ⫺ 0 共 ␣ 1 , ␣ 2 兲 ⫹L 0 共 ␣ 2 , ␣ 1 兲兴
2
冑
冎
⫺ 2 2 关共 L ⫺ 0 兲 共 ␣ 1 , ␣ 2 兲 ⫹ 共 L 0 兲 共 ␣ 2 , ␣ 1 兲兴 ,
共A6兲
where L⫾ n 共 ␣ 1 , ␣ 2 兲 ⫽ 共 1⫺ ␣ 1 兲 S n 共 ␦ /2⫺u 兲 ⫹ 共 1⫺ ␣ 2 兲 S n 共 ␦ /2⫹u 兲 ⫾ 关共 1⫺ ␣ 1 兲 S n 共 3 ␦ /2⫺u 兲 ⫹ 共 1⫺ ␣ 2 兲 S n 共 3 ␦ /2⫹u 兲兴 ⫹ 共 1⫺ ␣ 1 兲共 1⫺ ␣ 2 兲关 S n 共 3 ␦ /2⫺u 兲 ⫹S n 共 3 ␦ /2⫹u 兲兴 ⫾ 共 1⫺ ␣ 1 兲共 1⫺ ␣ 2 兲关 S n 共 5 ␦ /2⫺u 兲 ⫹S n 共 5 ␦ /2⫹u 兲兴 , A⫾ n 共 ␣ 1 , ␣ 2 兲 ⫽ 共 1⫺ ␣ 1 兲 S n 共 ␦ /2⫺u 兲 ⫺ 共 1⫺ ␣ 2 兲 S n 共 ␦ /2⫹u 兲 ⫾ 关共 1⫺ ␣ 1 兲 S n 共 3 ␦ /2⫺u 兲 ⫺ 共 1⫺ ␣ 2 兲 S n 共 3 ␦ /2⫹u 兲兴 ⫹ 共 1⫺ ␣ 1 兲共 1⫺ ␣ 2 兲关 S n 共 3 ␦ /2⫹u 兲 ⫺S n 共 3 ␦ /2⫺u 兲兴 ⫾ 共 1⫺ ␣ 1 兲共 1⫺ ␣ 2 兲关 S n 共 5 ␦ /2⫹u 兲 ⫺S n 共 5 ␦ /2⫺u 兲兴 .
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