Plane Poiseuille-Couette problem in micro-electro-mechanical systemsapplications with gas-rarefaction effectsCarlo Cercignani, Maria Lampis, and Silvia Lorenzani Citation: Phys. Fluids 18, 087102 (2006); doi: 10.1063/1.2335847 View online: http://dx.doi.org/10.1063/1.2335847 View Table of Contents: http://pof.aip.org/resource/1/PHFLE6/v18/i8 Published by the American Institute of Physics. Related ArticlesHigh frequency microbubble-switched oscillations modulated by microfluidic transistors Appl. Phys. Lett. 101, 073509 (2012) Convenient quantification of methanol concentration detection utilizing an integrated microfluidic chip Biomicrofluidics 6, 034111 (2012) Giant augmentations in electro-hydro-dynamic energy conversion efficiencies of nanofluidic devices usingviscoelastic fluids Appl. Phys. Lett. 101, 043905 (2012) Liquid flow retardation in nanospaces due to electroviscosity: Electrical double layer overlap, hydrodynamicslippage, and ambient atmospheric CO2 dissolution Phys. Fluids 24, 072001 (2012) Thermocoalescence of microdroplets in a microfluidic chamber Appl. Phys. Lett. 100, 254105 (2012) Additional information on Phys. FluidsJournal Homepage: http://pof.aip.org/ Journal Information: http://pof.aip.org/about/about_the_journal Top downloads: http://pof.aip.org/features/most_downloaded Information for Authors: http://pof.aip.org/authors

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PHYSICS OF FLUIDS 18, 087102 �2006�

Plane Poiseuille-Couette problem in micro-electro-mechanical systemsapplications with gas-rarefaction effects

Carlo Cercignani, Maria Lampis, and Silvia LorenzaniDipartimento di Matematica, Politecnico di Milano, Milano 20133, Italy

�Received 27 February 2006; accepted 13 July 2006; published online 23 August 2006�

Rarefied gas flows in micro-electro-mechanical systems �MEMS� devices, calculated from thelinearized Bhatnagar-Gross-Krook model equation �P. L. Bhatnagar, E. P. Gross, and M. Krook,Phys. Rev. 94, 511 �1954��, are studied in a wide range of Knudsen numbers. Both plane Poiseuilleand Couette flows are investigated numerically by extending a finite difference technique firstintroduced by Cercignani and Daneri �J. Appl. Phys. 34, 3509 �1963��. Moreover, a variationalapproach, applied to the integrodifferential form of the linearized Boltzmann equation �C.Cercignani, J. Stat. Phys. 1, 297 �1969��, is used to solve in a unified manner the planePoiseuille-Couette problem by means of the computation of only one functional. General boundaryconditions of Maxwell’s type have been considered, assuming both symmetric and nonsymmetricmolecular interaction between gas-solid interfaces, in order to take into account possible differencesin the accommodation coefficients on the walls of MEMS devices. Based on the analysis presentedin this paper, an accurate database valid in the entire Knudsen regime can be created for thePoiseuille-Couette problem, to be used in micromechanical applications. © 2006 American Instituteof Physics. �DOI: 10.1063/1.2335847�

I. INTRODUCTION

In gas film lubrication problems, operating under a sub-micron or less clearance conditions, the gas cannot be treatedas a continuous medium since the molecular mean free pathis not negligible compared with the film thickness. Accord-ingly, the kinetic theory of rarefied gas flows in narrow chan-nels must be applied.1–3 Typical examples are the start/stopoperations of hydrodynamic gas-lubricated bearings or flyinghead sliders employed in magnetic disk storage devices.4,5

Since the conventional Reynolds equation based on the no-slip boundary conditions cannot be used to describe gasflows in noncontinuum regimes, Fukui and Kaneko first de-rived a generalized Reynolds equation from a linearizedBoltzmann equation.6,7 The generalized Reynolds equation isa flow rate-based model and is obtained by calculating thefundamental flows in the lubrication film: a pressure flow�Poiseuille flow� and a shear flow �Couette flow�, when iso-thermal conditions are established.

Beyond the lubrication problems, microstructures thatmove in the direction parallel or perpendicular to their sur-faces are used in surface-micromachined inertial sensors,resonating filter structures for signal processing and in mi-cromachined capacitive accelerometers.8,9 The damping, dueto the internal friction of the flowing gas, namely the viscos-ity, in the small gaps between these moving structures, is animportant design parameter since it determines, e.g., thefrequency-domain behavior of the sensor or the quality fac-tor of the vibrating filter structure.10–12 At low pressures or inultrathin films, the gas rarefaction effect and the molecularinteraction with the surfaces effectively change the viscosity.In this flow regime, the continuum equations are no longervalid and the Boltzmann equation must be considered to un-

derstand and compute the rarefied flows related with these

1070-6631/2006/18�8�/087102/14/$23.00 18, 08710

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devices. The gas rarefaction effects including the influence ofthe gas-surface interactions have been taken into account byusing an effective viscosity instead of the static gas viscositycoefficient. To model the squeezed-film damping, due to amotion of the micromechanical structures perpendicular totheir surfaces, the effective viscosity can be expressed as afunction of the inverse Knudsen number ��=�� / �2Kn�,where the Knudsen number Kn is the ratio between the gasmean free path � and a characteristic length scale of the gasfilm� and the Poiseuille flow-rate coefficient.13,14 Similarly,the viscous damping coefficient, due to laterally movingstructures, is related to the shear stress whose computation isbased on the solution of the linearized Boltzmann equationfor the Couette-flow problem.15 Therefore, the investigationon rarefied gas flows between parallel plates, in spite of itssimple geometry, deserves particular attention since ultrathingas films are encountered in many industrial applications,especially in the nanoscale technology field.

In this paper a particular attention has been also paid tothe role of boundary conditions by allowing for boundingsurfaces with different physical structures. Recently, in lubri-cation theory, it has been pointed out16,17 that in modernmagnetic hard disk drives the lubricated disk surface issmoother than the carbon-coated slider surface, so that non-symmetric gas-wall interactions have to be considered. Be-yond the lubrication problems, such a kind of investigationdeserves a particular attention also with reference to micro-mechanical accelerometers. In fact, e.g., in the actuator con-sidered by Veijola et al.,13,14 the single-crystal silicon surfaceof the accelerometer’s proof mass is a potential candidate fornoticeable reflections close to specular, since it has a smooth,pure surface having an ordered molecular structure, whereas

the other surface �a metallized electrode� is rough.

© 2006 American Institute of Physics2-1

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087102-2 Cercignani, Lampis, and Lorenzani Phys. Fluids 18, 087102 �2006�

II. THE POISEUILLE-COUETTE PROBLEM: BASICEQUATIONS

Let us consider two plates separated by a distance d anda gas flowing parallel to them, in the z direction, due to apressure gradient. The lower boundary �placed at x=−d /2�moves to the right with velocity U, while the upper boundary�placed at x=d /2� is fixed. Both boundaries are held at aconstant temperature T0. The basic geometry of the two-dimensional gas layer is outlined in Fig. 1.

If the pressure gradient is taken to be small as well as thevelocity U, it can be assumed that the velocity distribution ofthe flow is nearly the same as that occurring in an equilib-rium state. This means that the Boltzmann equation can belinearized about a Maxwellian f0 by putting2

f = f0�1 + h� , �1�

where f�x ,z ,c� is the distribution function for the molecularvelocity c expressed in units of �2RT0�1/2 �R being the gasconstant�, x is the coordinate normal to the plates, and h�x ,c�is the small perturbation upon the basic equilibrium state.The above mentioned Maxwellian in equilibrium with thewalls is given by

f0�z,c� = �1 + kz��0�−3/2 exp�− c2� ,

where �0 is the density on the boundaries and

k =1

p

�p

�z=

1

�

��

�z,

with p and � being the gas pressure and density, respectively.Using Eq. �1�, the linearized Boltzmann equation reads as18

kcz + cx�h

�x= Lh. �2�

If one assumes the linearized Bhatnagar-Gross-Krook �BGK�model for the collision operator Lh,19 Eq. �2� becomes

kcz + cx�h

�x= ��−3/2/���� e−c1

2h1 dc1

+ 2c ·� c1e−c12h1 dc1 +

2

3�c2 −

3

2

�� �c12 − 3/2�e−c1

2h1 dc1 − h/� , �3�

where � is the collision time and h1�h�x ,c1�. Integrations

FIG. 1. Geometry of the Poiseuille-Couette problem.

are extended to the whole velocity space. Multiplying Eq. �3�

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by �cz /��exp�−�cy2+cz

2�� and integrating with respect to cy

and cz, it turns out that18

1

2k + cx

�Z

�x=

1

���−1/2�−�

+�

e−cx1

2Z�x,cx1

�dcx1− Z�x,cx� ,

�4�

where, by definition,

Z�x,cx� = �−1�−�

+� �−�

+�

e−cy2−cz

2czh�x,c�dcy dcz. �5�

Hence, the bulk velocity of the gas can be written as follows:

q�x� = �−1/2�−�

+�

e−cx1

2Z�x,cx1

�dcx1. �6�

Appropriate boundary conditions on the two plates must besupplied for the Boltzmann equation �4� to be solved.

III. KINETIC MODELS OF BOUNDARY CONDITIONS

According to the scattering kernel theory, the boundaryconditions for the perturbation h, defined by Eq. �1�, turn outto be2,3

h+ = h0 + Kh−, �7�

where h+ and h− concern, respectively, the reemitted and theimpinging molecules. The boundary source term, h0, is givenby

h0 = �f0�c��cn��−1�cn��0

R�c� → c�f0�c���cn��dc� − 1 �cn 0�

�8�

and K denotes the following operator:

Kh− = �f0�c��cn��−1�cn��0

R�c� → c�f0�c��

��cn��h−�c��dc� �cn 0� , �9�

where f0 is the basic Maxwellian with zero velocity andtemperature To, R�c�→c� is the scattering kernel, cn and cn�are the velocity components of the reemitted and impingingmolecules, respectively, normal to the plates. Let fw�c� de-note a Maxwellian with velocity and temperature equal to thewall velocity and temperature, respectively. Then, using thecondition of preservation of equilibrium for the Maxwellianfw, Eq. �8� reads as

h0 = �cn�−1�cn��0

R�c� → c�� f0�c��f0�c�

−fw�c��fw�c�

�cn��dc�.

�10�

If the gas molecules are specularly reflected by theboundaries, the scattering kernel is given by

R�c� → c� = �c − c� + 2n�n · c��� �11�

� is the Dirac delta�, and

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087102-3 Plane Poiseuille-Couette problem in MEMS applications Phys. Fluids 18, 087102 �2006�

h0 = 0, �12�

since in our case there is no component of the wall velocitynormal to the boundaries.

If the gas molecules are reemitted diffusely from thewall with the Maxwellian distribution fw�c�, the scatteringkernel is given by

R�c� → c� = �cn�fw�c� . �13�

Neglecting terms of order higher than first in the speed of thewall, and assuming Tw=T0, the problem can be linearized byputting

fw = f0�1 + �� , �14�

where

� = 2c · U �15�

and U is the wall velocity expressed in units of �2RTw�1/2.Inserting Eq. �14� in �10� and neglecting again terms of orderhigher than first in the perturbation parameters, the sourceterm reads as

h0 = �+ − K�−, �16�

where the operator K�− is obtained by Eq. �9� replacing thetemperature and velocity of the wall by those of the Max-wellian f0, and h− by �−.

Depending on the models for the scattering kernel, wecan obtain the boundary conditions for the Z�x ,cx� function.Inserting the general form of the boundary conditions for theperturbation h �Eq. �7�� in the definition �5�, we have

Z+�cx� = �−1�−�

+� �−�

+�

exp�− cy2 − cz

2�czh0�c�dcy dcz

+ �−1�−�

+� �−�

+�

exp�− cy2 − cz

2�czKh−�c�dcy dcz.

�17�

For Maxwell diffusion, the second term on the right-handside of Eq. �17� vanishes, while the first term gives �usingEq. �16��

Z+ = U. �18�

In the case of specular reflection, the first term on the right-hand side of Eq. �17� vanishes, so that Eq. �17� becomes

Z+�cx� = KZ−�cx� = Z−�− cx� , �19�

where K is the reflection operator with respect to the molecu-lar velocity component normal to the walls.

IV. ANALYTICAL FORMULATION OF THEBOUNDARY-VALUE PROBLEM

If one assumes that the bulk velocity of the gas, given byEq. �6�, is a known quantity, the integrodifferential Boltz-mann equation �4� can be formally handled as an ordinaryinhomogeneous differential equation whose solution reads

20,21

as

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Z�x,cx� = exp�− �x +d

2sgn cx �cx��

�Z�−d

2sgn cx,cx + �

−d/2 sgn cx

x

exp�− �x − t��cx��

��q�t� − k�/2�/�cx��dt . �20�

The values of the Z function at the boundary,Z�−d /2 sgn cx ,cx�, depend on the model of boundary condi-tions chosen and can be specified as follows. LetZ−�−d /2 ,cx�, Z−�d /2 ,cx�, be the distribution functions of themolecules impinging upon the walls and Z+�−d /2 ,cx�,Z+�d /2 ,cx� the distribution functions of the molecules re-emerging from them. Enforcing Eq. �20� at the boundaries,one obtains two equations:

Z−�− d/2,cx� = exp�d/�cx���Z+�d/2,cx�

+ �−d/2

d/2

exp�− �d

2+ t

�cx��� �q�t� − k�/2�

�cx��dt ,

�21�

Z−�d/2,cx� = exp�− d/�cx���Z+�− d/2,cx�

+ �−d/2

d/2

exp�− �d

2− t

�cx��� �q�t� − k�/2�

�cx��dt .

�22�

These equations can be simply reexpressed in terms of theincoming stream adding the appropriate boundary conditionson the two plates. Assuming the diffuse-specular reflectioncondition of Maxwell’s type and specializing the analysis towalls having different physical properties so that two accom-modation coefficients ��1 ,�2� can be defined, the boundaryconditions read as20,21

Z+�d/2,cx� = �1 − �1�Z−�d/2,− cx� , �23�

Z+�− d/2,cx� = �2U + �1 − �2�Z−�− d/2,− cx� , �24�

where the wall velocity U is expressed in units of �2RT0�1/2.Substituting Eqs. �23� and �24� into Eqs. �21� and �22�, thefollowing system of two equations for the distribution func-tions of the impinging molecules can be derived:22

A�cx� = Z�− d/2,− cx�

= �1 − �1�exp�− d/�cx���B�cx� + �cx� , �25�

B�cx� = Z�d/2,cx�

= �2U exp�− d/�cx���

+ �1 − �2�exp�− d/�cx���A�cx� + ��cx� , �26�

with

�cx� = �d/2

exp�− �d

2+ t

�cx��� �q�t� − k�/2�

�cx��dt , �27�

−d/2

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087102-4 Cercignani, Lampis, and Lorenzani Phys. Fluids 18, 087102 �2006�

��cx� = �−d/2

d/2

exp�− �d

2− t

�cx��� �q�t� − k�/2�

�cx��dt . �28�

The functions A�cx� and B�cx� are defined on the half-spacecx�0. The system of Eqs. �25� and �26� admits the solution

A�cx� = D�cx�−1 � ��2�1 − �1�U exp�− 2d/�cx���

+ �1 − �1�exp�− d/�cx�����cx� + �cx�� , �29�

B�cx� = D�cx�−1 � ��2U exp�− d/�cx���

+ �1 − �2�exp�− d/�cx��� �cx� + ��cx�� , �30�

with

D�cx� = 1 − �1 − �1��1 − �2�exp�− 2d/�cx��� . �31�

Inserting in the definition �6� of q�x� the Z function �20�,the bulk velocity of the gas reads, after standard manipula-tion, as

q�x� =�1 − �1�

���

0

�

exp�− cx2 +

x

cx�−

d

2cx�B�cx�dcx

+�2U��

�0

�

exp�− cx2 −

x

cx�−

d

2cx�dcx

+�1 − �2�

���

0

�

exp�− cx2 −

x

cx�−

d

2cx�A�cx�dcx

+1

����

−d/2

d/2

T−1� �x − t��

�q�t� − k�/2�dt , �32�

where Tn�x� is the Abramowitz function, defined by

Tn�x� = �0

�

sn exp�− s2 − x/s�ds . �33�

It is convenient now to rescale all variables appearing inEq. �32� as follows:

� = d/�; w = t/�; u = x/� ,

so that q�x� can be rewritten in terms of the nondimensionalfunctions �p�u� and �c�u�:22

q�x� = 12k��1 − �p�u�� + U�c�u� . �34�

Equation �34� shows that the gas velocity is induced by thesuperposition of two distinct effects. The gas moves by animposed pressure gradient �Poiseuille flow� and by the sheardriven flow due to the motion of the bottom surface �Couetteflow�. Using the explicit expressions �29� and �30� for A�cx�and B�cx� in Eq. �32�, the nondimensional functions �p�u�and �c�u�, giving the Poiseuille and Couette contributions,

respectively, can be written as

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�p�u� = 1 +1

���

−�/2

�/2

dw�p�w���1 − �1�S−1�� − u − w�

+ �1 − �2�S−1�� + u + w� + �1 − �1��1 − �2�

��S−1�2� − u + w� + S−1�2� + u − w��

+ T−1��u − w��� , �35�

�c�u� =�2

���T0��/2 + u� + �1 − �1�S0�3/2� − u� + �1 − �1�

��1 − �2�S0�5/2� + u�� +1

���

−�/2

�/2

dw �c�w�

���1 − �1�S−1�� − u − w�

+ �1 − �2�S−1�� + u + w� + �1 − �1��1 − �2�

��S−1�2� − u + w� + S−1�2� + u − w��

+ T−1��u − w��� , �36�

where Sn�x� is a generalized Abramowitz function, definedby

Sn�x,�,�1,�2� = �0

+� sn exp�− s2 − x/s�1 − �1 − �1��1 − �2�exp�− 2�/s�

ds .

�37�

Taking into account Eq. �34�, the flow rate �per unit timethrough unit thickness�, defined by

F = ��−d/2

d/2

q�x�dx , �38�

can be expressed as the sum of the Poiseuille flow �Fp� andthe Couette flow �Fc� as follows:

F = Fp + Fc = −�p

�zd2Qp��,�1,�2� +

�Ud

2Qc��,�1,�2� ,

�39�

where

Qp��,�1,�2� = −1

�+

1

�2�−�/2

�/2

�p�u�du , �40�

Qc��,�1,�2� =2

��

−�/2

�/2

�c�u�du , �41�

are the nondimensional volume flow rates and � is the rar-efaction parameter �inverse Knudsen number�. For the prob-lem of Couette flow, another macroscopic quantity of somevalue is the xz component of the stress tensor,

Pxz = ��−1/2�−�

+�

e−cx1

2cx1

Z�x,cx1�dcx1

, �42�

which can be rewritten in terms of the nondimensional bulk

velocity of the gas, �c as follows:

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087102-5 Plane Poiseuille-Couette problem in MEMS applications Phys. Fluids 18, 087102 �2006�

�xz =Pxz

�U

= −�2

���1 − �1�S1��� +

�2

���1 − �1��1 − �2�S1�3��

+�2

��T1��� −

�1 − �1���

�−�/2

�/2

dw �c�w�S0��/2 − w�

+�1

���1 − �2� � �

−�/2

�/2

dw�c�w�S0�3�/2 + w�

+�1 − �1�

���1 − �2��

−�/2

�/2

dw�c�w� � S0�5�/2 − w�

+1

���

−�/2

�/2

dw�c�w�T0��/2 − w� . �43�

Thus, the problem has been reduced to the task of solving theintegral equations �35� and �36�.

V. NUMERICAL METHOD

In order to solve numerically the integral equations �35�and �36�, we extend a finite difference technique first intro-duced in a paper by Cercignani and Daneri,18 where theplane Poiseuille flow is analyzed under the diffuse reflectioncondition at the boundaries. The one-dimensional computa-tional domain is divided into n mesh points �for simplicity,only constant mesh steps are considered� and ��u� �whichstands for either the Poiseuille or the Couette nondimen-sional bulk velocity� is approximated by a stepwise function.The general form of the numerical scheme is given by

�k=0

n−1

�hk�k = �h �h = 0, . . . ,n − 1� . �44�

Following the idea reported in Ref. 18, the constant valueassigned to the function ��u� on every interval can be inter-preted as either �a� the value in the midpoint or �b� the meanvalue on the whole interval, so that two methods of differ-encing can be defined with two possible choices for the co-efficients �hk and �h. Though with a resolution of n=200mesh points �used in the present computations to reach veryhigh accuracy�, the two schemes approach so closely eachother that they can be considered equivalent, the second fi-nite difference method is, in general, more accurate. There-fore, in the following, we report only the coefficients relatedto the method of differencing �b�:

�hk = −n

����T1��h − k��/n − �/n� − 2T1��k − h��/n�

+ T1��k − h��/n + �/n�� +n

����1 − �1�

��2S1�2� − �h + k + 1��/n�

− S1�2� − �h + k + 2��/n� − S1�2� − �h + k��/n��

+n

�1 − �2��2S1��h + k + 1��/n�

���

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− S1��h + k + 2��/n� − S1��h + k��/n��

+n

����1 − �1��1 − �2� � ��hk

+ + �hk− � , �45�

where

�hk± = 2S1�2� ± �h − k��/n� − S1�2� ± �h − k + 1��/n�

− S1�2� ± �h − k − 1��/n� .

The coefficients �hk are the same for both functions �p�u�and �c�u�, while

�h = 1 �46�

for the Poiseuille velocity profile, and

�h =n

����2�S1�h�/n� − S1��2h + 2��/2n��

+n

����2�1 − �1� � �S1�2� − �2h + 2��/2n�

− S1�2� − h�/n�� +n

����2�1 − �1��1 − �2�

� �S1�2� + h�/n� − S1�2� + �2h + 2��/2n�� , �47�

for the Couette velocity profile. Since it has been pointed outin Ref. 23 that an accurate estimate of the flow rate of the gasis strongly related to the high accuracy in the computation ofthe generalized Abramowitz functions, the Sn functions ap-pearing in Eqs. �45� and �47�, have been numerically evalu-ated using the SLATEC libraries. In the following, we willomit the results on the Poiseuille flow rate coefficient:Qp�� ,�1 ,�2�, which have been analyzed in great detail inRefs. 20, 21, and 23.

If the two bounding walls are identical ��1=�2=��, theCouette flow rate is independent of the Knudsen numberregardless of the value of the accommodation coefficient �:

Qc��,�� = 1, " �, " � �48�

as already observed in literature.24,25

VI. THE VARIATIONAL APPROACH TO PLANEPOISEUILLE-COUETTE FLOW

The steady linearized Boltzmann equation, given by Eq.�4�, can be rewritten in symbolic form as follows:

�D − L�Z = S , �49�

where DZ=cx�Z /�x, S=−k /2, and

LZ =1

���−1/2�−�

+�

e−cx1

2Z�x,cx1

�dcx1− Z�x,cx� .

The boundary conditions to be matched to Eq. �49� have thegeneral expression

Z+ = Z0 + KZ−, �50�

where the explicit form of the source term Z0 and the opera-tor K depends on the scattering kernel used. In the case ofMaxwell’s boundary conditions, Eq. �50� reduces to Eqs.

�23� and �24�. Using the variational principle described in

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087102-6 Cercignani, Lampis, and Lorenzani Phys. Fluids 18, 087102 �2006�

Ref. 26, we introduce the following functional J of the test

function Z:

J�Z� = ��Z,P�DZ − LZ��� − 2��PS,Z��

+ �Z+ − KZ− − 2Zo,PZ−�B, �51�

where P is the parity operator in velocity space and ��,�� �,�B

denote two scalar products defined as follows:

��h,g�� = �−1/2�−d/2

+d/2 �−�

+�

exp�− cx2�h�x,cx�g�x,cx�dcx dx ,

�52�

�h±,g±�B = �−1/2����

cx�0cx exp�− cx

2�h±�cx�g±�cx�dcx d� .

�53�

Here g±, h± are the restrictions of the functions g and h,defined on the boundary, to positive, respectively negative,values of cx. In the one-dimensional case, the integrationover the boundary �� reduces to the sum of the terms at x= ±d /2.

The functional J�Z� attains its minimum value when Z=Z�x ,cx� solves Eq. �49� with appropriate boundary condi-

tions. If we let Z=Z, Eq. �51� gives

J�Z� =k

2��1,Z�� − �Z0,PZ−�B. �54�

It can be easily shown that the stationary value of J has adirect connection with three quantities of physical interest, aswill be specified in the following.

In the general formulation of the variational principle

�expressed by Eq. �51�� Z does not need to satisfy the sameboundary conditions as Z. Nevertheless, taking advantage ofthe mathematical formulation presented in Sec. IV, we canfind a more refined trial function that satisfies the boundaryconditions, in order to attain very accurate results. Thus, to

obtain an expression for the test function Z�x ,cx�, the solu-tion of the Boltzmann equation in integral form has beenconsidered,

Z�x,cx� = exp�− �x +d

2sgn cx �cx��

�Z�−d

2sgn cx,cx + �

−d/2 sgn cx

x

exp�− �x − t��cx��

��q�t� − k�/2�/�cx��dt . �55�

Assuming Maxwell’s boundary conditions imposed on wallswith different accommodation coefficients, the values at the

boundary, Z�−d /2 sgn cx ,cx�, defined on the half-space cx

�0, are given by

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Z�− d/2,− cx� = D�cx�−1��2�1 − �1�U exp�− 2d/�cx���

+ �−d/2

+d/2

�q�t� − k�/2�/�cx��

� �exp�− �d/2 + t�/�cx���

+ �1 − �1�exp�− �3d/2 − t�/�cx����dt ,

�56�

Z�d/2,cx� = D�cx�−1��2U exp�− d/�cx���

+ �−d/2

+d/2

�q�t� − k�/2�/�cx��

� �exp�− �d/2 − t�/�cx��� + �1 − �2�

�exp�− �3d/2 + t�/�cx����dt , �57�

as for Z �Eqs. �29� and �30��.The analytical formulation presented in Sec. IV has

shown that the bulk velocity of the gas for the problemsketched in Fig. 1 can be expressed as the sum of the Poi-seuille bulk velocity, qp�x�, and the Couette bulk velocity,qc�x�. Furthermore, the analysis developed in Ref. 21 for thePoiseuille flow problem has shown that for nonsymmetricgas-wall interactions the bulk velocity profile can be ap-proximated by a parabola asymmetric with respect to thechannel axis,

qp�x� = Apx2 + Bpx + Cp �58�

�Ap, Bp, and Cp being constants�, while as a trial function forthe Couette velocity profile, we can consider the continuumsolution, i.e., a straight profile, with a quadratic correction totake into account the presence of the Knudsen layers and thenonsymmetric gas-wall interactions,

qc�x� = Acx2 + Bcx + Cc �59�

�Ac, Bc, and Cc being constants�. By using Eqs. �58� and �59�,the total bulk velocity of the gas reads as

q�x� = Ax2 + Bx + C , �60�

where

A = Ap + Ac, B = Bp + Bc, C = Cp + Cc, �61�

are adjustable constants to be varied in order to obtain the

best value of J�Z�. Inserting in Eq. �55� the expressions of

Z�−d /2 sgn cx ,cx� and q�x�, given by �56�, �57�, and �60�,after some simple manipulations, the test function Z can be

explicitly written as

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087102-7 Plane Poiseuille-Couette problem in MEMS applications Phys. Fluids 18, 087102 �2006�

Z�x,cx� = G+��2�U sgn cx exp�− �x +d

2sgn cx �cx�� + D�cx�−1 exp�− �x +

d

2sgn cx �cx��

� �− G−��2��1 − F��1,�2��U sgn cxexp�− d sgn cx/�cx��� + G+��2��1 − �1��1 − �2�U sgn cx

�exp�− 2d sgn cx/�cx���� + D�cx�−1 � exp�− �x +d

2sgn cx �cx�� � ��1 − F��1,�2�� � ��C − k�/2�

+ A/4�d2 − 4dcx� sgn cx + 8cx2�2� + B/2�− d sgn cx + 2cx�� − exp�− d sgn cx/�cx��� � ��C − k�/2�

+ A/4�d2 + 4dcx� sgn cx + 8cx2�2� + B/2�d sgn cx + 2cx���� + �1 − �1��1 − �2� � �exp�− d sgn cx/�cx��� � ��C

− k�/2� + A/4 � �d2 − 4dcx� sgn cx + 8cx2�2� − B/2�− d sgn cx + 2cx��� − exp�− 2d sgn cx/�cx��� � ��C − k�/2�

+ A/4�d2 + 4dcx� sgn cx + 8cx2�2� − B/2�d sgn cx + 2cx����� + �C − k�/2� + A�x2 − 2xcx� + 2cx

2�2� + B�x − cx��

− exp�− �x +d

2sgn cx �cx�� � ��C − k�/2� + A/4�d2 + 4dcx� sgn cx + 8cx

2�2� − B/2�d sgn cx + 2cx��� , �62�

where G+��2�=�2 if cx�0 and 0 otherwise, G−��2�=�2 ifcx�0 and 0 otherwise, and

F��1,�2� = ��2, cx � 0,

�1, cx � 0.�

It is convenient to rescale all variables appearing in Eq.�62� as follows:

� = d/�; w = t/�; u = x/�; A = A/�2; B = B/� .

If one substitutes Z, given by �62�, in Eq. �51� and splits eachconstant as in �61�, with the following normalization

Ac = Ac/U; Bc = Bc/U; Cc = Cc/U ,

Ap = Ap/�k��; Bp = Bp/�k��; Cp = Cp/�k�� ,

the functional J�Z� is reduced to the sum of three functionals:

J�1��Z�, J�2��Z�, J�3��Z�, simply grouping together the termsproportional to �k��2, �Uk��, U2, respectively. Each func-tional is a polynomial of the second order with respect to theconstants Ap, Ac, Bp, Bc, Cp, Cc, that are to be determined:

J�1��Z�/�k��2 = ����−1 � � c11

2Ap

2 +c22

2Bp

2 +c33

2Cp

2

+ c12ApBp + c13ApCp + c23BpCp − c1Ap

− c B − c C +1

�c − c /4� , �63�

2 p 3 p 2 3 33

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J�2��Z�/�k�U� = ����−1 � �c11ApAc + c22BpBc + c33CpCc

+ c12�ApBc + AcBp� + c13

� �ApCc + AcCp� + c23�BpCc + BcCp�

− d1Ap − c1Ac − d2Bp − c2Bc − d3Cp

− c3Cc + d5� , �64�

J�3��Z�/U2 = ����−1 � � c11

2Ac

2 +c22

2Bc

2 +c33

2Cc

2 + c12AcBc

+ c13AcCc + c23BcCc − d1Ac − d2Bc − d3Cc

+ d4 . �65�

The nondimensional coefficients c11, c12, c13, c1, c22, c23, c2,c33, c3 are the same as reported in Ref. 23, and

d1 = �2��1 − �1����2

2+ 2�S1��� − S1�2���

− 2��S2��� + S2�2��� + 4�S3��� − S3�2��� + �1 − �1�

��1 − �2����2

2+ 2�S1�2�� − S1�3���

− 2��S2�2�� + S2�3��� + 4�S3�2�� − S3�3���− ��2

2+ 2T1��� − 2�T2��� − 4T3��� +

�2

4

−��

� + 3� +2�2T1, �66�

2 ��

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087102-8 Cercignani, Lampis, and Lorenzani Phys. Fluids 18, 087102 �2006�

d2 = �2��1 − �1����S1��� + S1�2��� − 2�S2��� − S2�2����

+ �1 − �1��1 − �2� � �− ��S1�2�� + S1�3���

+ 2�S2�2�� − S2�3���� − �T1��� − 2T2��� −�

2+

��

2

+2�2

��T2, �67�

d3 = �2„�1 − �1��2�S1��� − S1�2���� + �1 − �1��1 − �2�

��2�S1�2�� − S1�3���� − 2T1��� + 1… +2�2

��T3, �68�

d4 = − �22�1 − �1�S1�2�� −

�22

� T4, �69�

�

hand side of Eq. �54� is related to the perturbation part of the

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d5 = �2„�1 − �1��2�S1��� − S1�2���� + �1 − �1��1 − �2�

��2�S1�2�� − S1�3���� − 2T1��� + 1… +�2

��T3, �70�

where the Abramowitz functions Tn�x�, Sn�x� are defined byEqs. �33� and �37�. The symbols T1, T2, T3, T4, stand forintegral expressions involving the Tn�x� and Sn�x� functions,which are negligible in the free molecular flow regime. Theexplicit form of these integrals is given in the Appendix.

The derivatives of J�1��Z� / �k��2, J�2��Z� / �k�U�,J�3��Z� /U2, with respect to Ap, Ac, Bp, Bc, Cp, Cc vanish incorrespondence of the optimal values of these constants. The

resulting expressions for the minimum of J�2��Z� / �k�U� and

J�3��Z� /U2 are

minJ�2��Z��k�U�

= ����−1 · �c132 c22 − 2c12c13c23 + c12

2 c33 + c11�c232 − c22c33��−1 � �c12c23c3d1 − c12c2c33d1 − c11c23c3d2 + c11c2c33d2

− c11c2c23d3 − c122 c3d3 + c11c22c3d3 + c1�c22c33d1 − c23

2 d1 − c12c33d2 + c12c23d3� + c11c232 d5 + c12

2 c33d5

− c11c22c33d5 + c132 �c22d5 − c2d2� + c13�c2c23d1 − c22c3d1 + c1c23d2 + c12c3d2 + c12c2d3 − c1c22d3 − 2c12c23d5�� ,

�71�

minJ�3��Z�

U2 = �2���−1 · �c132 c22 − 2c12c13c23 + c12

2 c33 + c11�c232 − c22c33��−1 � �− 2c12c33d1d2 − c13

2 d22 + c11c33d2

2 + 2c12c13d2d3

− c122 d3

2 + 2c122 c33d4 − c23

2 �d12 − 2c11d4� + 2c23�c13d1d2 + c12d1d3 − c11d2d3 − 2c12c13d4� + c22�c33d1

2 − 2c13d1d3

+ c11d32 + 2c13

2 d4 − 2c11c33d4�� , �72�

while the minimum of J�1��Z� / �k��2 has been reported inRef. 23.

Let us now go back to Eq. �54�. If one looks at thedefinitions �6� and �38�, taking into account the fact that thegas flow can be expressed as the sum of the Poiseuille flowrate �Fp� and the Couette flow rate �Fc�, along with Eq. �18�and the definition �42�, it is easy to see that the first term onthe right-hand side of Eq. �54� is related to the Poiseuille andCouette flow rates, through the relations

Fp

�− 1/2�kd2�= − 4

J�1��Z��2 , �73�

where J�1��Z�=J�1��Z� / �k��2,

Fc

�1/2�Ud�= 2

J�2��Z��

, �74�

where J�2��Z�=J�2��Z� / �k�U�. The second term on the right-

stress component Pxz that is constant and gives the drag ex-erted by the gas on each plate:

Pxz

��U�= J�3��Z� +

�2

�2���, �75�

where J�3��Z�=J�3��Z� /U2.Thus, the computation of the optimal value of the func-

tionals J�1��Z�, J�2��Z�, J�3��Z� �Eqs. �71� and �72� and Ref.23� will lead to an accurate estimate of the Poiseuille flowrate,

Qp��� =Fp

�− 1/2�kd2�, �76�

the Couette flow rate,

Qc��� =Fc

�1/2�Ud�, �77�

and the �xz� component of the stress tensor,

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087102-9 Plane Poiseuille-Couette problem in MEMS applications Phys. Fluids 18, 087102 �2006�

�xz =Pxz

�U. �78�

It is worth noting that the advantage of the variationalprinciple expressed by Eq. �51�, with respect to the previ-ously known methods,27–30 is that general molecular modelsfor the collision term can be used, beyond the BGK model.Moreover, the particular formulation adopted here enables usto estimate the three basic quantities Qp, Qc, and �xz, con-sidered separately in literature, by means of the computationof only one functional J�Z�.

VII. RESULTS AND DISCUSSION

The analysis developed in previous sections can be ap-plied to model gas flows in very small gaps between themoving surfaces of micromechanical structures. The varioustypes of motions experienced by micro-electro-mechanicalsystems �MEMS� components can be summarized as one ora combination of the following:25

�a� lateral motion of the lower boundary with slight incli-nation of the upper plate �slider bearing�;

�b� normal oscillations between two surfaces �squeezedfilm�;

�c� lateral motion between two surfaces.

All of these situations, with the exception of the lastcase, can be analyzed using the Reynolds equation for a thingas lubricated film.7,13,14,24,31 For slider type applications,Fukui and Kaneko derived a generalized Reynolds equationbased on the linearized Boltzmann equation, which holds for

TABLE I. The shear stress Pxz / �U /2� for Couette flow versus � with �=1. A32–34.

�Num.

�Ref. 32�Var.

�Ref. 27�Var.

�Ref. 26�

1.0�10−3 --- --- ---

0.01 5.592 81�−1� 5.593 37�−1� ---

0.1 5.223 26�−1� 5.223 26�−1� 5.211 98�−1�1.0 3.389 65�−1� 3.389 65�−1� 3.398 67�−1�2.0 2.505 00�−1� 2.503 87�−1� 2.517 41�−1�3.0 1.996 66�−1� 1.995 53�−1� 2.006 25�−1�5.0 1.425 14�−1� 1.424 01�−1� 1.429 65�−1�7.0 1.108 06�−1� 1.108 06�−1� 1.110 88�−1�10.0 8.316 15�−2� 8.316 15�−2� 8.327 44�−2�20.0 4.553 01�−2� 4.541 73�−2� 4.541 73�−2�

FIG. 2. Geometry of a slider bearing.

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arbitrary Knudsen numbers, in order to take into account thegas rarefaction effects.6,7,24 This equation, extended in Refs.16, 17, and 22 by allowing for bounding surfaces with dif-ferent physical structures, can be written in nondimensionalform as follows:

d

dZ�Qp��oPD,�1,�2�PD3dP

dZ− Qc��oPD,�1,�2��PD

= 0, �79�

where Qp is the relative flow rate of pressure-driven channelflows normalized by the no-slip Poiseuille flow rate, Qc is theCouette flow rate, P= p / p0 is the pressure field, D=d /d0 isthe film thickness, Z=z / l is the longitudinal coordinate, �0 isthe characteristic inverse Knudsen number defined by theminimum film thickness, d0, and the ambient pressure p0, �is the bearing number:

� =6�Ul

p0d02 ,

with � being the dynamic viscosity of the gas �see Fig. 2�.Beyond the lubrication problems, the Reynolds equation

in linearized form has been applied in modeling the squeezedfilm damping effects in various microsystems:13,14

d2

12�effp0

�2

�z2� p

p0 −

�

�t� p

p0 =

�

�t� x

d , �80�

where p0 is the static pressure, p�z , t� is the small variation inthe pressure distribution, d is the static plate separation, x isthe small variation in the plate separation. In Eq. �80� apressure-dependent effective viscosity �eff replaces the staticclassical gas viscosity � in order to incorporate rarefactioneffects:

�eff =�

Qp����

6. �81�

In the continuum limit Qp���→� /6, so that �eff→�.In the last case �c�, since the plate motion is purely in the

lateral direction, streamwise pressure variations do not ariseand the Reynolds equation cannot be used. Unlike thesqueezed film and the slider bearing problems, there will be

parison between our results �CLL� and those presented in Refs. 26, 27, and

Num.�Ref. 33�

Num.�Ref. 34�

Num.�CLL�

Var.�CLL�

.636 92�−1� 5.636 92�−1� 5.636 92�−1� 5.630 06�−1�

.593 36�−1� --- 5.593 36�−1� 5.593 36�−1�

.223 25�−1� 5.223 25�−1� 5.223 25�−1� 5.223 25�−1�

.389 25�−1� 3.389 25�−1� 3.389 25�−1� 3.389 29�−1�

.503 01�−1� --- 2.503 02�−1� 2.503 16�−1�

.994 60�−1� --- 1.994 62�−1� 1.994 88�−1�

.423 15�−1� --- 1.423 23�−1� 1.423 55�−1�

.107 37�−1� --- 1.107 54�−1� 1.107 78�−1�

.311 22�−2� 8.311 22�−2� 8.314 91�−2� 8.314 64�−2�

.538 77�−2� --- 4.554 35�−2� 4.540 28�−2�

com

5

5

5

3

2

1

1

1

8

4

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087102-10 Cercignani, Lampis, and Lorenzani Phys. Fluids 18, 087102 �2006�

no lift. The viscous damping problem, considering again gasrarefaction, can be analyzed defining an effective viscosity:15

�eff = ��xz2� , �82�

where �xz= Pxz / ��U� is the normalized shear stress for theCouette-flow problem. In the continuum limit, �xz→1/ �2��,so that �eff→�. We stress that Eqs. �81� and �82� have beenderived assuming slowly moving surfaces. Hence, onlyfrequency-independent damping models have been pre-sented, which assume fully established Poiseuille and Cou-ette flows, respectively, ignoring the inertia effect of the gas.The rigorous computation of the Poiseuille flow rate Qp re-ported in Ref. 23 and the Couette flow rate Qc along with the

TABLE II. The shear stress Pxz / �U /2� for Couette flow versus � for Maxw�CLL� and those presented in Ref. 34.

�

�=0.1Num.

�Ref. 34�Num.�CLL�

V�CL

1.0�10−3 2.969 28�−2� 2.969 28�−2� 2.969

0.1 2.956 18�−2� 2.956 18�−2� 2.956

1.0 2.859 30�−2� 2.859 30�−2� 2.859

10.0 2.262 21�−2� 2.262 39�−2� 2.263

TABLE III. The shear stress �xz for Couette flow versus � for Maxwell’s b

� �2

�1=0.1Num. �CLL� Var. �CLL�

1.0�10−4 0.1 1.484 70�−2� 1.489 10�−2�0.5 2.564 48�−2� 2.566 16�−2�0.8 2.752 12�−2� 2.754 50�−2�1.0 2.820 92�−2� 2.825 64�−2�

1.0�10−3 0.1 1.484 64�−2� 1.484 50�−2�0.5 2.564 29�−2� 2.560 30�−2�0.8 2.751 91�−2� 2.752 08�−2�1.0 2.820 70�−2� 2.820 70�−2�

0.01 0.1 1.484 02�−2� 1.484 02�−2�0.5 2.562 45�−2� 2.562 45�−2�0.8 2.749 79�−2� 2.749 78�−2�1.0 2.818 47�−2� 2.818 47�−2�

0.05 0.1 1.481 33�−2� 1.481 33�−2�0.5 2.554 49�−2� 2.554 49�−2�0.8 2.740 66�−2� 2.740 66�−2�1.0 2.808 90�−2� 2.808 90�−2�

0.1 0.1 1.478 09�−2� 1.478 09�−2�0.5 2.544 96�−2� 2.544 96�−2�0.8 2.729 77�−2� 2.729 77�−2�1.0 2.797 51�−2� 2.797 51�−2�

0.5 0.1 1.454 87�−2� 1.454 87�−2�0.5 2.478 12�−2� 2.478 12�−2�0.8 2.653 94�−2� 2.653 94�−2�1.0 2.718 53�−2� 2.718 53�−2�

1.0 0.1 1.429 65�−2� 1.429 66�−2�0.5 2.407 23�−2� 2.407 25�−2�0.8 2.573 92�−2� 2.573 93�−2�1.0 2.635 37�−2� 2.635 38�−2�

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�xz� component of the stress tensor �xz presented in this pa-per, is therefore motivated by its relevance in several MEMSapplications when low working pressures impose correctionsdue to gas rarefaction effects. To validate our codes, the re-sults about the shear stress for the Couette-flow problem ob-tained by means of the numerical method described in Sec. Vand the variational approach presented in Sec. VI, have beencompared in Tables I and II, with the outputs published byother authors in the case of Maxwell’s boundary conditionson two physically identical walls ��1=�2=��.26,27,32–34 Allthese authors analyzed the classical plane Couette flow be-tween two parallel plates moving with velocities ±U /2 andnormalized the �xz component of the stress tensor Pxz using

boundary conditions on identical walls. A comparison between our results

�=0.5Num.

�Ref. 34�Num.�CLL�

Var.�CLL�

2� 1.880 08�−1� 1.880 08�−1� 1.879 25�−1�2� 1.829 84�−1� 1.829 84�−1� 1.829 84�−1�2� 1.523 54�−1� 1.523 54�−1� 1.523 56�−1�2� 6.360 78�−2� 6.362 49�−2� 6.366 04�−2�

ary conditions on different walls.

=0.5. �CLL� Var. �CLL�

�1=0.8Num. �CLL� Var. �CLL�

48�−2� 2.564 54�−2� 2.752 12�−2� 2.752 14�−2�88�−2� 9.403 15�−2� 1.253 70�−1� 1.254 51�−1�70�−1� 1.255 85�−1� 1.880 52�−1� 1.880 68�−1�41�−1� 1.411 94�−1� 2.256 60�−1� 2.256 76�−1�29�−2� 2.564 13�−2� 2.751 91�−2� 2.751 91�−2�39�−2� 9.396 88�−2� 1.253 26�−1� 1.253 26�−1�26�−1� 1.253 26�−1� 1.879 52�−1� 1.878 02�−1�85�−1� 1.409 85�−1� 2.255 16�−1� 2.255 17�−1�45�−2� 2.562 45�−2� 2.749 79�−2� 2.749 79�−2�76�−2� 9.375 76�−2� 1.248 90�−1� 1.248 90�−1�90�−1� 1.248 90�−1� 1.869 76�−1� 1.869 76�−1�34�−1� 1.404 34�−1� 2.241 16�−1� 2.241 16�−1�49�−2� 2.554 49�−2� 2.740 66�−2� 2.740 66�−2�31�−2� 9.271 31�−2� 1.230 56�−1� 1.230 56�−1�56�−1� 1.230 56�−1� 1.829 37�−1� 1.829 37�−1�29�−1� 1.381 29�−1� 2.183 74�−1� 2.183 74�−1�96�−2� 2.544 96�−2� 2.729 77�−2� 2.729 77�−2�19�−2� 9.149 19�−2� 1.209 38�−1� 1.209 38�−1�38�−1� 1.209 38�−1� 1.783 66�−1� 1.783 66�−1�84�−1� 1.354 84�−1� 2.119 54�−1� 2.119 54�−1�12�−2� 2.478 12�−2� 2.653 94�−2� 2.653 94�−2�89�−2� 8.357 91�−2� 1.076 83�−1� 1.076 83�−1�83�−1� 1.076 83�−1� 1.514 18�−1� 1.514 18�−1�16�−1� 1.192 16�−1� 1.753 42�−1� 1.753 43�−1�23�−2� 2.407 25�−2� 2.573 92�−2� 2.573 93�−2�68�−2� 7.617 79�−2� 9.586 80�−2� 9.586 93�−2�80�−2� 9.586 93�−2� 1.293 70�−1� 1.293 72�−1�30�−1� 1.050 31�−1� 1.467 01�−1� 1.467 03�−1�

ell’s

ar.L�

04�−18�−32�−40�−

ound

�1

Num

2.564

9.402

1.253

1.410

2.564

9.400

1.253

1.409

2.562

9.375

1.248

1.404

2.554

9.271

1.230

1.381

2.544

9.149

1.209

1.354

2.478

8.357

1.076

1.192

2.407

7.617

9.586

1.050

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087102-11 Plane Poiseuille-Couette problem in MEMS applications Phys. Fluids 18, 087102 �2006�

the velocity of one wall. On the contrary, we have assumedthat the lower boundary moves with velocity U, while theupper boundary is fixed and we have normalized Pxz by thevelocity of the moving plate. Since the shear stress is pro-portional to the relative velocity between the boundingsurfaces,35 our outputs can be directly compared with thosepresented in Refs. 26, 27, and 32–34, simply multiplying by2 the results obtained from Eqs. �43� and �75�. As revealedby Tables I and II, the agreement between the outcomes ofour numerical and variational methods and the findings ob-tained in previous numerical works is very good �the relativeerror is less than 0.1%�. Moreover, a direct comparison withthe data reported in Ref. 26, where the same variational prin-ciple has been considered, reveals that the more refined testfunction assumed here provides highly accurate results, evenin the transitional region for every value of �. Looking atTables I and II, one sees that the shear stress decreases withdecreasing �, progressing from free molecular, through tran-sitional, to continuum regions.

Since walls with different physical structures are fullyused in designing micromechanical systems,13,14,25 Tables IIIand IV show the �xz� component of the stress tensor �xz,computed by our variational and numerical methods, versus

TABLE IV. The shear stress �xz for Couette flow versus � for Maxwell’s b

� �2

�1=0.1Num. �CLL� Var. �CLL� Nu

3.0 0.1 1.346 18�−2� 1.346 32�−2� 2.1

0.5 2.182 18�−2� 2.182 45�−2� 5.7

0.8 2.320 31�−2� 2.320 56�−2� 6.8

1.0 2.371 52�−2� 2.371 75�−2� 7.3

5.0 0.1 1.275 95�−2� 1.276 30�−2� 2.0

0.5 2.004 13�−2� 2.004 77�−2� 4.6

0.8 2.120 61�−2� 2.121 20�−2� 5.3

1.0 2.163 68�−2� 2.164 23�−2� 5.6

7.0 0.1 1.213 56�−2� 1.214 10�−2� 1.8

0.5 1.854 62�−2� 1.855 54�−2� 3.9

0.8 1.954 10�−2� 1.954 95�−2� 4.4

1.0 1.990 72�−2� 1.991 51�−2� 4.5

10.0 0.1 1.131 00�−2� 1.131 70�−2� 1.6

0.5 1.668 60�−2� 1.669 72�−2� 3.1

0.8 1.748 76�−2� 1.749 79�−2� 3.4

1.0 1.778 07�−2� 1.779 03�−2� 3.6

20.0 0.1 9.221 47�−3� 9.229 72�−3� 1.2

0.5 1.250 90�−2� 1.251 93�−2� 1.9

0.8 1.295 43�−2� 1.296 38�−2� 2.0

1.0 1.311 46�−2� 1.312 35�−2� 2.0

50.0 0.1 5.933 61�−3� 5.940 55�−3� 7.1

0.5 7.145 35�−3� 7.149 61�−3� 8.9

0.8 7.288 79�−3� 7.292 35�−3� 9.2

1.0 7.339 33�−3� 7.342 58�−3� 9.2

100.0 0.1 3.725 23�−3� 3.726 73�−3� 4.1

0.5 4.173 80�−3� 4.169 01�−3� 4.7

0.8 4.222 70�−3� 4.217 14�−3� 4.8

1.0 4.239 71�−3� 4.233 89�−3� 4.8

the rarefaction parameter � in the general, nonsymmetrical

Downloaded 19 Oct 2012 to 131.175.56.27. Redistribution subject to AIP lice

case, �1��2. If we keep the accommodation coefficient ofone of the surfaces fixed and vary the other one in the inter-val �0, 1�, �xz increases in going from the entirely specularreflection to the complete Maxwell diffusion for every valueof �. It is to be noted that exchanging �1 with �2, we obtainthe same values for the shear stress, since it is constant withrespect to x.

In lubrication problems, if the two bounding plates areallowed to reemit the impinging gas molecules differently, arigorous evaluation of the Couette flow rate Qc is required. Inthe late 1990s, Kang et al.16 computed the Couette flow ratein the case of Maxwell’s boundary conditions imposed ontwo walls having different physical properties, but the twoaccommodation coefficients were allowed to independentlytake only the values varying in the interval �0.7, 1�. More-over, the Couette flow rate Qc has been obtained by Kangonly for the cases with �1��2. In this paper we have evalu-ated Qc over the whole range of the Knudsen number and foreach value of the accommodation coefficients. Table V re-ports the values of Qc versus the inverse of the Knudsennumber �, evaluated by means of our numerical and varia-tional methods. As revealed by this table, the deviation fromthe condition expressed by Eq. �48�, when the two bounding

ary conditions on different walls.

.5CLL� Var. �CLL�

�1=0.8Num. �CLL� Var. �CLL�

7�−2� 2.182 45�−2� 2.320 30�−2� 2.320 56�−2�0�−2� 5.760 69�−2� 6.834 63�−2� 6.836 05�−2�5�−2� 6.836 05�−2� 8.405 32�−2� 8.406 90�−2�3�−2� 7.301 39�−2� 9.121 79�−2� 9.123 31�−2�1�−2� 2.004 77�−2� 2.120 58�−2� 2.121 20�−2�8�−2� 4.671 09�−2� 5.354 06�−2� 5.356 55�−2�9�−2� 5.356 55�−2� 6.275 73�−2� 6.278 26�−2�9�−2� 5.640 01�−2� 6.669 25�−2� 6.671 59�−2�8�−2� 1.855 54�−2� 1.954 04�−2� 1.954 95�−2�6�−2� 3.934 22�−2� 4.406 99�−2� 4.409 82�−2�2�−2� 4.409 82�−2� 5.013 68�−2� 5.016 40�−2�5�−2� 4.600 45�−2� 5.262 22�−2� 5.264 68�−2�4�−2� 1.669 72�−2� 1.748 67�−2� 1.749 79�−2�6�−2� 3.183 02�−2� 3.484 64�−2� 3.487 28�−2�9�−2� 3.487 28�−2� 3.853 53�−2� 3.855 93�−2�6�−2� 3.605 43�−2� 3.998 82�−2� 4.000 94�−2�7�−2� 1.251 93�−2� 1.295 25�−2� 1.296 38�−2�2�−2� 1.945 26�−2� 2.053 33�−2� 2.054 75�−2�2�−2� 2.054 75�−2� 2.176 15�−2� 2.177 31�−2�8�−2� 2.095 16�−2� 2.221 76�−2� 2.222 75�−2�4�−3� 7.149 61�−3� 7.285 70�−3� 7.292 35�−3�6�−3� 8.976 61�−3� 9.202 79�−3� 9.202 80�−3�7�−3� 9.202 80�−3� 9.442 22�−3� 9.440 69�−3�3�−3� 9.282 95�−3� 9.527 24�−3� 9.525 07�−3�2�−3� 4.169 01�−3� 4.220 82�−3� 4.217 14�−3�1�−3� 4.730 40�−3� 4.806 35�−3� 4.792 46�−3�0�−3� 4.792 46�−3� 4.871 35�−3� 4.856 18�−3�6�−3� 4.814 11�−3� 4.894 01�−3� 4.878 40�−3�

ound

�1=0m. �

82 1

59 4

34 6

00 0

04 1

68 6

54 0

37 6

54 5

31 3

07 0

97 8

68 5

80 2

84 6

03 0

50 7

43 7

53 4

93 9

43 2

76 3

03 9

84 7

72 4

43 6

06 9

28 9

walls are identical, is significant particularly in the free mo-

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087102-12 Cercignani, Lampis, and Lorenzani Phys. Fluids 18, 087102 �2006�

lecular flow limit and in the transitional regime. In order toquantify these deviations, Fig. 3 shows the variation of theCouette flow rate Qc as a function of the accommodationcoefficient of one of the walls, keeping the other one fixed.The circles indicate that, when the accommodation coeffi-cient of the upper wall ��1� is fixed and the other one ��2�varies in the range �0, 1�, the gas flow rate increases bydecreasing the fraction of gas molecules specularly reflectedby the moving wall. On the contrary, the square symbolsshow that, keeping the accommodation coefficient of themoving wall ��2� fixed and varying the other one ��1�, Qc

decreases with increasing �1. In Fig. 3 the rarefaction param-eter � is fixed to 1, but the same qualitative profiles can beobtained for every value of �. This trend of the Couette flowrate profiles is particularly significant in applications since it

TABLE V. Couette flow rate Qc versus � for Maxwell’s boundary condition

� �2

�1=0.1Num. �CLL� Var. �CLL�

1.0�10−4 0.1 1.000 00 1.000 00

0.5 1.727 24 1.727 26

0.8 1.853 62 1.854 07

1.0 1.899 96 1.899 41

1.0�10−3 0.1 1.000 00 1.000 00

0.5 1.727 06 1.727 07

0.8 1.853 38 1.853 38

1.0 1.899 68 1.899 69

0.1 0.1 1.000 00 1.000 00

0.5 1.719 46 1.719 46

0.8 1.843 09 1.84309

1.0 1.888 14 1.888 14

1.0 0.1 1.000 00 1.000 00

0.5 1.680 87 1.680 87

0.8 1.795 18 1.795 19

1.0 1.836 73 1.836 74

3.0 0.1 1.000 00 1.000 00

0.5 1.618 95 1.619 00

0.8 1.719 88 1.719 93

1.0 1.756 82 1.756 86

5.0 0.1 1.000 00 1.000 00

0.5 1.569 24 1.569 35

0.8 1.659 33 1.659 42

1.0 1.692 31 1.692 37

10.0 0.1 1.000 00 1.000 00

0.5 1.853 38 1.474 77

0.8 1.544 98 1.545 00

1.0 1.570 52 1.570 50

50.0 0.1 1.000 00 1.000 00

0.5 1.204 95 1.203 48

0.8 1.229 17 1.227 47

1.0 1.237 68 1.235 90

100.0 0.1 1.000 00 1.000 00

0.5 1.122 39 1.118 66

0.8 1.135 71 1.131 56

1.0 1.140 33 1.136 05

s on different walls.

�1=0.5Num. �CLL� Var. �CLL�

�1=0.8Num. �CLL� Var. �CLL�

2.727 56�−1� 2.727 52�−1� 1.463 80�−1� 1.463 23�−1�1.000 00 1.000 00 6.667 36�−1� 6.670 53�−1�1.333 26 1.332 76 1.000 00 1.000 00

1.499 88 1.498 90 1.199 92 1.197 49

2.729 35�−1� 2.729 34�−1� 1.466 23�−1� 1.466 22�−1�1.000 00 1.000 00 6.671 75�−1� 6.671 68�−1�1.332 82 1.332 83 1.000 00 1.000 00

1.499 13 1.499 13 1.199 44 1.199 44

2.805 36�−1� 2.805 36�−1� 1.569 09�−1� 1.569 09�−1�1.000 00 1.000 00 6.849 71�−1� 6.849 71�−1�1.315 03 1.315 03 1.000 00 1.000 00

1.468 72 1.468 72 1.180 72 1.180 72

3.191 32�−1� 3.191 28�−1� 2.048 17�−1� 2.048 13�−1�1.000 00 1.000 00 7.500 06�−1� 7.500 03�−1�1.249 99 1.250 00 1.000 00 1.000 00

1.363 62 1.363 63 1.125 27 1.125 27

3.810 54�−1� 3.809 99�−1� 2.801 23�−1� 2.800 74�−1�1.000 00 1.000 00 8.181 10�−1� 8.180 95�−1�1.181 89 1.181 90 1.000 00 1.000 00

1.259 08 1.259 09 1.080 98 1.080 98

4.307 64�−1� 4.306 49�−1� 3.406 68�−1� 3.405 78�−1�1.000 00 1.000 00 8.560 51�−1� 8.560 49�−1�1.143 95 1.143 95 1.000 00 1.000 00

1.202 61 1.202 59 1.060 37 1.060 35

5.253 30�−1� 5.252 28�−1� 4.550 24�−1� 4.549 97�−1�1.000 00 1.000 00 9.053 28�−1� 9.054 03�−1�1.094 67 1.094 60 1.000 00 1.000 00

1.131 15 1.131 01 1.036 91 1.036 85

7.950 49�−1� 7.965 23�−1� 7.708 30�−1� 7.725 34�−1�1.000 00 1.000 00 9.745 94�−1� 9.748 53�−1�1.025 40 1.025 15 1.000 00 1.000 00

1.034 40 1.034 04 1.009 00 1.008 90

8.776 13�−1� 8.813 38�−1� 8.642 88�−1� 8.684 35�−1�1.000 00 1.000 00 9.864 51�−1� 9.868 93�−1�1.013 55 1.013 11 1.000 00 1.000 00

1.018 26 1.017 67 1.004 71 1.004 57

is responsible for a kind of inverted pressure profiles that

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FIG. 3. Couette flow rate Qc versus the accommodation coefficient of one ofthe walls, keeping the other one fixed, for �=1. The squares are obtained

varying �1 ��2=0.5�, while the circles are obtained varying �2 ��1=0.5�.

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087102-13 Plane Poiseuille-Couette problem in MEMS applications Phys. Fluids 18, 087102 �2006�

appear in studying the slider air bearing problem in the free-molecular flow regime. In fact, for small �, if one keeps theaccommodation coefficient of the slider ��1� fixed and variesthe other one ��2�, the pressure distribution in the gas film, ata fixed bearing number, increases with increasing �2, as italways occurs in the continuum region, while at fixed �2, thepressure distribution decreases by increasing �1.

VIII. CONCLUDING REMARKS

In the present study, we have investigated the planePoiseuille-Couette flow of a rarefied gas between two paral-lel plates as an issue of relevance in MEMS applications.General boundary conditions of Maxwell’s type have beenimposed on the two walls, allowing for possible differencesin the accommodation coefficients of the top and bottomplates. Based on the numerical and variational analysis pre-sented, a general database of practical usefulness can be cre-ated, exploiting the ability of our methods to give highlyaccurate results in the entire Knudsen regime. Of course, the

variational approach has the enormous advantage of reducing

��Sn�3�/2 + u� − Sn�3�/2 − u�� ± �1 − �1��1 − �2

Downloaded 19 Oct 2012 to 131.175.56.27. Redistribution subject to AIP lice

drastically the computational CPU time, giving at one timethe same accuracy for each value of the rarefaction param-eter �.

To conclude the paper, we discuss the profiles of theCouette flow rate Qc, shown in Fig. 3, from a qualitativephysical point of view. The lower boundary, which moveswith velocity U, imparts the tangential component of themomentum to the diffusely reflected molecules. When �2

increases, the rate of the transfer of the tangential momen-tum, which is supplied continuously, also increases and thegas flow enhances. On the contrary, if one keeps �2 fixed andincreases the accommodation coefficient of the upper bound-ary, the gas is not continuously accelerated, but the shearstress �just as the viscosity, according to Eq. �82�� raises.This produces a braking effect on the gas flow that is conse-quently reduced.

APPENDIX: DETAILED FORM OF THE INTEGRALEXPRESSIONS APPEARING IN THE VARIATIONALCOEFFICIENTS

In the following we report the explicit form of the inte-

grals T1, T2, T3, T4 appearing in Eqs. �66�–�70�:

T1 = �−�/2

�/2

du��T0��/2 + u� + �1 − �1�S0�3�/2 − u� + �1 − �1��1 − �2�S0�5�/2 + u�� � ���2

4L0

− − �L1+ + 2L2

−��1,�2�

−�2

4�T0��/2 + u� + T0��/2 − u�� − ��T1��/2 + u� + T1��/2 − u�� − 2�T2��/2 + u� + T2��/2 − u��� , �A1�

T2 = �−�/2

�/2

du��T0��/2 + u� + �1 − �1�S0�3�/2 − u� + �1 − �1��1 − �2�S0�5�/2 + u�� � ���

2A0

+ − A1−��1,�2�

+�

2�T0��/2 + u� − T0��/2 − u�� + T1��/2 + u� − T1��/2 − u�� , �A2�

T3 = �−�/2

�/2

du��T0��/2 + u� + �1 − �1�S0�3�/2 − u� + �1 − �1��1 − �2�S0�5�/2 + u��

� �L0−��1,�2� − T0��/2 + u� − T0��/2 − u��� , �A3�

T4 = �−�/2

�/2

du�T0��/2 + u� + �1 − �1�S0�3�/2 − u� + �1 − �1��1 − �2�S0�5�/2 + u��2, �A4�

where

Ln±��1,�2� = �1 − �1�Sn��/2 − u� + �1 − �2�Sn��/2 + u� ± ��1 − �1�Sn�3�/2 − u� + �1 − �2�Sn�3�/2 + u�� + �1 − �1��1 − �2�

��Sn�3�/2 − u� + Sn�3�/2 + u�� ± �1 − �1��1 − �2��Sn�5�/2 − u� + Sn�5�/2 + u�� ,

An±��1,�2� = �1 − �1�Sn��/2 − u� − �1 − �2�Sn��/2 + u� ± ��1 − �1�Sn�3�/2 − u� − �1 − �2�Sn�3�/2 + u�� + �1 − �1��1 − �2�

��Sn�5�/2 + u� − Sn�5�/2 − u�� .

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087102-14 Cercignani, Lampis, and Lorenzani Phys. Fluids 18, 087102 �2006�

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