On The Reynolds Equation For Linearized Models Of ...

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On The Reynolds Equation For Linearized Models Of The Boltzmann Operator a

a

C. Cercignani , M. Lampis & S. Lorenzani a

a

Dipartimento di Matematica, Politecnico di Milano, Milano, Italy

Version of record first published: 15 Dec 2010.

To cite this article: C. Cercignani, M. Lampis & S. Lorenzani (2007): On The Reynolds Equation For Linearized Models Of The Boltzmann Operator, Transport Theory and Statistical Physics, 36:4-6, 257-280 To link to this article: http://dx.doi.org/10.1080/00411450701465643

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Transport Theory and Statistical Physics, 36:257–280, 2007 Copyright # Taylor & Francis Inc. ISSN: 0041-1450 print/1532-2424 online DOI: 10.1080/00411450701465643

ON THE REYNOLDS EQUATION FOR LINEARIZED MODELS OF THE BOLTZMANN OPERATOR C. CERCIGNANI, M. LAMPIS, and S. LORENZANI

Downloaded by [Politecnico di Milano Bibl] at 03:16 19 October 2012

Dipartimento di Matematica, Politecnico di Milano, Milano, Italy

Rarefied gas flows in ultra-thin film slider bearings are studied in a wide range of Knudsen numbers. The generalized Reynolds equation, first derived by Fukui and Kaneko (1987, 1988, 1990) on the basis of the linearized Bhatnagar-Gross-Krook (BGK) Boltzmann equation (Bhatnagar et al. 1954), has been extended by considering a more refined kinetic model of the collisional Boltzmann operator, i.e., the linearized ellipsoidal statistical (ES) model, which allows the Prandtl number to assume its proper value (Cercignani and Tironi 1966). Since the generalized Reynolds equation is a flow rate – based model and is obtained by calculating the fundamental flows in the lubrication film (i.e., the Poiseuille and Couette flows), the plane Poiseuille-Couette flow problem between parallel plates has been preliminarily investigated by means of the linearized ES model. General boundary conditions of Maxwell’s type have been considered by allowing for bounding surfaces with different physical properties. Keywords: Rarefied gas flows, Boltzmann operator, Reynolds equation

1. Introduction In gas film lubrication problems, operating under submicron or less clearance conditions, the gas cannot be treated as a continuous medium since the molecular mean free path is not negligible when compared with the film thickness. Accordingly, the kinetic theory of rarefied gas flow in narrow channels must be applied (Cercignani 1969, 1988, 2000). Typical examples are the start/ stop operations of hydrodynamic gas-lubricated bearings or flying head sliders employed in magnetic disk storage devices Received 14 October 2005, Revised 3 July 2006, Accepted 17 July 2006 Address correspondence to S. Lorenzani, Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, Milano 20133, Italy. E-mail: silva.lorenzani@ polimi.it

257


258

C. Cercignani et al.

(Deckert 1990; Tagawa 1993). Traditionally, the classical Reynolds lubrication equation based on the no-slip boundary conditions has been used to model slider bearings. In noncontinuum regimes, Fukui and Kaneko (1987, 1988, 1990) first derived a generalized Reynolds equation starting from the linearized Boltzmann equation based on the Bhatnagar-GrossKrook (BGK) model (Bhatnagar et al. 1954). The generalized Reynolds equation is a flow rate –based model and is obtained by calculating the fundamental flows in the lubrication film: a pressure flow (Poiseuille flow) and a shear flow (Couette flow), when isothermal conditions are established. Because of its simplicity compared to the Boltzmann equation, the BGK model is widely used in the kinetic theory of gases, although one of the best-known shortcomings of this model is that the Prandtl number Pr turns out to be unity. Since the classical value for a monoatomic gas is known to be Pr ¼ 2/3, one cannot make both viscosity and thermal conductivity agree with the Chapman-Enskog values for a Maxwell gas. This circumstance is easily avoided in linearized problems since viscosity and temperature effects can be decoupled. However, even in the frame of a linearized analysis, one is induced to suspect that the incorrect Prandtl number can be influent in the transitional regime. Therefore, it appears worthwhile to investigate the slider bearing problem through the generalized Reynolds equation based on a model more refined than the BGK one i.e., the ellipsoidal statistical (ES) model, which allows the Prandtl number to take on its correct value (Cercignani and Tironi 1966). In the present paper a particular attention has also been paid to the role of boundary conditions by allowing for bounding surfaces with different physical structures. Recently, it has been pointed out (Kang et al. 1999; Huang and Bogy 2000) that in modern magnetic hard disk drives the lubricated disk surface is smoother than the carbon-coated slider surface, so that nonsymmetric gas-wall interactions have to be considered. 2. The Poiseuille-Couette Problem According to the ES Model Let us consider two plates separated by a distance h and a gas flowing parallel to them in the x direction due to a pressure

Reynolds Equation for Linearized Models

259


FIGURE 1 Geometry of the Poiseuille-Couette problem.

gradient. The lower boundary (placed at z ¼ 2h/2) moves to the right with velocity U, while the upper boundary (placed at z ¼ h/2) is fixed. Both boundaries are held at a constant temperature T0. The basic geometry of the two-dimensional gas layer is outlined in Figure 1. If the pressure gradient as well as the velocity U are taken to be 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 f0 by putting (Cercignani 1988, 2006) ~ f ¼ f0 ð1 þ hÞ

ð1Þ

where f(x, z, c) is the distribution function for the molecular velocity c expressed in units of (2RT0)1/2 (R being the gas constant), z is the coordinate normal to be plates, and ˜h(z, c) is the small perturbation upon the basic equilibrium state. The above-mentioned maxwellian in equilibrium with the walls is given by f0 ðx; cÞ ¼ ð1 þ kxÞr0 p3=2 expðc2 Þ

ð2Þ

where r0 is the density on the boundaries, located at z ¼ +h/2, and k¼

1 @p 1 @r ¼ ; p @x r @x

with p and r being the gas pressure and density respectively. Using (1), which defines the unknown perturbation function h˜(z, c), the linearized Boltzmann equation reads (Cercignani and Daneri 1963) kcx þ cz

@h~ ~ ¼ Lh: @z

ð3Þ

260



Let us assume, as collision operator Lh˜, the linearized ES model (Cercignani and Tironi 1966) 1 2 l Lh~ ¼ 0 n þ 2c q þ t ðc2 3=2Þ lci c j Pij þ ðn þ tÞc2 h~ 3 u 2 ð4Þ where u 0 is a suitable mean free time; n and t are the perturbations of the density r0 and temperature T0, respectively, ð 2 3=2 ~ cÞdc ec hðz; n¼p

t¼p

3=2

ð

2 ~ cÞdc; ðc2 3=2Þec hðz;

q is the perturbation of the bulk velocity (in (2RT0)1/2 units) ð 2 3=2 ~ cÞdc; cec hðz; q¼p and Pij is the stress tensor (in 2r0RT0 units) ð 2 3=2 ~ cÞdc: Pij ¼ p ci cj ec hðz; Integrations are extended to the whole velocity space. In (4), l is a constant to be chosen in such a way as to have the correct Prandtl number, defined by Pr ¼ cp

m k

where cp is the specific heat at constant pressure, m the viscosity, and k the heat conductivity. l is equal to 0 for the BGK model (Pr ¼ 1) and 1 for a Maxwell gas (Pr ¼ 2/3). For a general monoatomic gas the relation between l and Pr is Pr ¼

2 ; 2þl

i.e.,

l¼

2 2: Pr


261

Multiplying (3), with Lh˜ given by (4), by (cx/p) exp[2(c2x þ c2y )] and integrating with respect to cx and cy, it turns out that ð þ1 2 1 @Z 1 1=2 ecz1 Zðz; cz1 Þdcz1 lcz p1=2 k þ cz ¼ 0 p 2 @z u 1 ð þ1 2 ecz1 cz1 Zðz; cz1 Þdcz1 Zðz; cz Þ ð5Þ Downloaded by [Politecnico di Milano Bibl] at 03:16 19 October 2012

1

where, by definition, Zðz; cz Þ ¼ p

1

ð þ1 ð þ1 1

2 2 cÞdcx dcy : eðcx þcy Þ cx hðz;

1

From (5) one can obtain the momentum conservation equation k @ þ Pxz ¼ 0 2 @z

ð6Þ

with 1=2

ð þ1

Pxz ðzÞ ¼ p

2

ecz cz Zðz; cz Þdcz :

ð7Þ

1

The integration of (6) gives k Pxz ðzÞ ¼ z þ P 2

ð8Þ

where the integration constant P will be determined in the following. Therefore (5) can be rewritten as ð 1 @Z 1 1=2 þ1 c2z k e 1 Zðz;cz1 Þdcz1 þ lcz z lcz P Zðz;cz Þ : k þ cz ¼ 0 p 2 @z u 2 1 ð9Þ Appropriate boundary conditions on the two plates must be supplied for the Boltzmann Equation (9) to be solved. If one assumes that the bulk velocity of the gas, defined by ð þ1 2 1=2 ecz Zðz; cz Þdcz ; ð10Þ qðzÞ ¼ p 1

262


is a known quantity, the integrodifferential Boltzmann Equation (9) can be formally handled as an ordinary inhomogeneous differential equation whose solution reads h h 0 Zðz; cz Þ ¼ exp z þ sgncz ðcz u Þ Z sgncz ; cz 2 2 ðz jz tj exp þ jcz ju 0 ðh=2sgncz Þ Downloaded by [Politecnico di Milano Bibl] at 03:16 19 October 2012

=

½qðtÞ ku 0 =2 lcz P þ lkcz t=2=ðcz u 0 Þdt:

ð11Þ

The values of the Z function at the boundary, Z(2h/2sgncz, cz), depend on the model of boundary conditions chosen. In the following, we will consider the Maxwell boundary conditions and specialize the analysis to walls having different physical properties so that two accommodation coefficients (a1, a2) must be used. In this case, the boundary conditions can be written as (Cercignani et al., 2004a, b, 2005) Zþ ðh=2; cz Þ ¼ ð1 a1 ÞZ ðh=2; cz Þ

ð12Þ

þ

ð13Þ

Z ðh=2; cz Þ ¼ a2 U þ ð1 a2 ÞZ ðh=2; cz Þ

where U is expressed in units of (2RT0)1/2; Z2 (2h/2, cz), Z2 (h/2, cz), are the distribution functions of the molecules impinging upon the walls and Zþ (2h/2, cz), Zþ (h/2, cz) the distribution functions of the molecules reemerging from them. Enforcing (11) at the boundaries, one obtains two equations: ð h=2 ðh=2 þ tÞ 0 þ exp Z ðh=2;cz Þ ¼ expðh=ðcz u ÞÞZ ðh=2;cz Þ þ jcz ju 0 h=2 ½qðtÞ ku 0 =2 lcz P þ lkcz t=2 dt ð14Þ jcz ju 0 ð h=2 ðh=2 tÞ 0 þ exp Z ðh=2;cz Þ ¼ expðh=ðcz u ÞÞZ ðh=2;cz Þ þ jcz ju 0 h=2

½qðtÞ ku 0 =2 lcz P þ lkcz t=2 dt: jcz ju 0

ð15Þ

After substituting (12) and (13) into (14) and (15), the following system of two equations for the distribution functions of the


263

impinging molecules can be derived: Aðcz Þ ¼ Zðh=2; cz Þ ¼ ð1 a1 Þ expðh=ðc3 u 0 ÞÞBðcz Þ þ Xðcz Þ þ Kðcz Þ

ð16Þ

Bðcz Þ ¼ Zðh=2; cz Þ ¼ a2 U expðh=ðcz u 0 ÞÞ


þ ð1 a2 Þ expðh=ðcz u 0 ÞÞAðcz Þ þ fðcz Þ þ Kðcz Þ

ð17Þ

with ðh=2 þ tÞ ½qðtÞ ku 0 =2 þ lcz P X ðcz Þ ¼ exp dt jcz ju 0 jcz ju 0 h=2 ð h=2 ðh=2 tÞ ½qðtÞ ku 0 =2 lcz P fðcz Þ ¼ exp dt jcz ju 0 jcz ju 0 h=2 ð h=2

Kðcz Þ ¼

kl hcz u 0 =2 ðcz u 0 Þ2 þ expðh=ðcz u 0 ÞÞ 2u 0 ðhcz u 0 =2 þ ðcz u 0 Þ2 Þ :

ð18Þ ð19Þ

ð20Þ

The functions A(cz) and B(cz) are defined on the halfspace cz . 0. The system of Equations (16) and (17) admits the solution Aðcz Þ ¼ Dðcz Þ1 ½a2 ð1 a1 ÞU expð2h=ðcz u 0 ÞÞ þ ð1 a1 Þ expðh=ðcz u 0 ÞÞðfðcz Þ þ Kðcz ÞÞ þ X ðcz Þ þ Kðcz Þ

ð21Þ

Bðcz Þ ¼ Dðcz Þ1 ½a2 U expðh=ðcz u 0 ÞÞ þ ð1 a2 Þ expðh=ðcz u 0 ÞÞ ðX ðcz Þ þ Kðcz ÞÞ þ fðcz Þ þ Kðcz Þ

ð22Þ

with Dðcz Þ ¼ 1 ð1 a1 Þð1 a2 Þ expð2h=ðcz u 0 ÞÞ:

ð23Þ

After substituting the integral solution Z(z, cz) given by (11) in the definition (10) of q(z) by standard manipulation, the bulk

264


velocity of the gas reads


qðzÞ ¼

ð ð1 a1 Þ 1 z h pffiffiffiffi exp c2z þ Bðcz Þdcz cz u 0 2cz u 0 p 0 ð a2 U 1 z h ð1 a2 Þ 2 pffiffiffiffi þ pffiffiffiffi exp cz 0 0 dcz þ cz u 2cz u p 0 p ð h=2 ð1 z h 1 exp c2z T1 0 0 Aðcz Þdcz þ pffiffiffiffi 0 cz u 2cz u pu h=2 0 ð1 jz tj kl 0 ffiffiffi ffi p u =2 dt þ expðc2z Þ ½qðtÞ k u0 2 pu 0 0 ðh=2 zÞ ðh=2 þ zÞ exp þ exp cz u 0 cz u 0 ð klu 0 lP 1 hcz u 0 =2 þ ðcz u 0 Þ2 dcz þ pffiffiffiffi cz expðc2z Þ 4 p 0 ðh=2 þ zÞ ðh=2 zÞ exp dcz exp ð24Þ cz u 0 cz u 0

where Tn(x) is the Abramowitz function defined by ð1 sn expðs2 x=sÞ ds: Tn ðxÞ ¼

ð25Þ

0

It is convenient now to rescale all variables appearing in (24) as follows:

d 0 ¼ h=u 0 ;

w0 ¼ t=u 0 ;

u0 ¼ z=u 0 ;

so that q(z) can be rewritten in terms of the nondimensional functions cp(u0 ) and cc(u0 ), giving the Poiseuille and Couette contributions, respectively, as follows: 1 ql ðzÞ ¼ ku 0 ½1 c p ðu0 Þ þ Ucc ðu0 Þ: 2

ð26Þ

To determine the integration constant P, (8) has been used, where Pxz(z) has been computed by inserting in the definition (7) the integral solution Z(z, cz), given by (11), and fixing the value of


265

z at the boundary. After some simple manipulations, the constant P may then be written in nondimensional form as

l pffiffiffiffi P ¼ 1 þ pffiffiffiffi p=4 T2 ðd0 Þ ð1 a1 ÞðS2 ðd0 Þ S2 ð2d0 ÞÞ p


ð1 a2 ÞðS2 ð0Þ S2 ðd0 ÞÞ io1 þ ð1 a1 Þð1 a2 ÞðS2 ðd0 Þ S2 ð3d0 ÞÞ

pffiffiffiffi ku 0 l k a2 U u 0 d0 þ pffiffiffiffi a22 ð1 a1 Þ US1 ð2d0 Þ= p þ pffiffiffiffi 4 2 p 2 p 0 d d0 0 0 0 0 ð1 a1 Þ S2 ðd Þ þ S2 ð2d Þ S3 ðd Þ þ S3 ð2d Þ 2 2 0 d d0 0 0 þ ð1 a2 Þ S2 ð0Þ þ S2 ðd Þ S3 ð0Þ þ S3 ðd Þ 2 2 0 d d0 þ ð1 a1 Þð1 a2 Þ S2 ðd0 Þ S2 ð3d0 Þ 2 2 d0 d0 S3 ðd0 Þ þ 2S3 ð2d0 Þ S3 ð3d0 Þ T2 ð0Þ T2 ðd0 Þ 2 2 ð 0 1 d =2 T3 ðd0 Þ þ T3 ð0Þ pffiffiffiffi dw0 ½ku 0 =2c p ðw0 Þ p d0 =2 0 þ Ucc ðw Þ T0 ðd0 =2 þ w0 Þ þ ð1 a1 ÞS0 ð3d0 =2 w0 Þ ð1 a2 ÞS0 ðd0 =2 þ w0 Þ þ ð1 a1 Þð1 a2 ÞðS0 ð5d0 =2 þ w0 Þ 0 0 S0 ð3d =2 w ÞÞ ð27Þ where Sn(x) is a generalized Abramowitz function defined by ð þ1 sn expðs2 x=sÞ 0 Sn ðx; d ; a1 ; a2 Þ ¼ ds: 0 0 1 ð1 a1 Þð1 a2 Þ expð2d =sÞ ð28Þ Let us now go back to (24). Using the explicit expressions (21) – (22) for A(cz) and B(cz) along with (26) and (27), the

266


nondimensional functions cp(u0 ) and cc(u0 ) can be written as 1 c p ðu0 Þ ¼ 1 þ pffiffiffiffi p

ð d0 =2 d0 =2

h dw0 c p ðw0 Þ ð1 a1 ÞS1 ðd0 u0 w0 Þ


þ ð1 a2 ÞS1 ðd0 þ u0 þ w0 Þ þ ð1 a1 Þð1 a2 Þ ðS1 ð2d0 u0 þ w0 Þ þ S1 ð2d0 þ u0 w0 ÞÞ

l l d0 0 0 ðT1 ðd0 =2 u0 Þ þ T1 ðju w jÞ þ pffiffiffiffi 2 p 2 þ T1 ðd0 =2 þ u0 ÞÞ þ T2 ðd0 =2 u0 Þ þ T2 ðd0 =2 þ u0 Þ þ ð1 a1 Þ 0 d ðS1 ðd0 =2 u0 Þ þ S1 ð3d0 =2 u0 ÞÞ S2 ðd0 =2 u0 Þ 2 0 d þ S2 ð3d0 =2 u0 Þ þ ð1 a2 Þ ðS1 ðd0 =2 þ u0 Þ 2 0 0 0 0 0 0 þ S1 ð3d =2 þ u ÞÞ S2 ðd =2 þ u Þ þ S2 ð3d =2 þ u Þ þ ð1 a1 Þð1 a2 Þ

d0 ðS1 ð3d0 =2 u0 Þ þ S1 ð3d0 =2 þ u0 Þ 2

þ S1 ð5d0 =2 u0 Þ þ S1 ð5d0 =2 þ u0 ÞÞ S2 ð3d0 =2 u0 Þ 0 0 0 0 0 0 S2 ð3d =2 þ u Þ þ S2 ð5d =2 u Þ þ S2 ð5d =2 þ u Þ

l l pffiffiffiffi pffiffiffiffi 1 þ pffiffiffiffi p=4 T2 ðd0 Þ ð1 a1 ÞðS2 ðd0 Þ p p S2 ð2d0 ÞÞ ð1 a2 ÞðS2 ð0Þ S2 ðd0 ÞÞ þ ð1 a1 Þð1 a2 Þ 1 0 0 ðS2 ðd Þ S2 ð3d ÞÞ ½T1 ðd0 =2 þ u0 Þ T1 ðd0 =2 u0 Þ

þ ð1 a1 ÞðS1 ð3d0 =2 u0 Þ S1 ðd0 =2 u0 ÞÞ þ ð1 a2 ÞðS1 ðd0 =2 þ u0 Þ S1 ð3d0 =2 þ u0 ÞÞ þ ð1 a1 Þð1 a2 ÞðS1 ð3d0 =2 u0 Þ S1 ð3d0 =2 þ u0 Þ þ S1 ð5d0 =2 þ u0 Þ S1 ð5d0 =2 u0 ÞÞ


267


n d0 l d0 0 0 0 0 p ffiffiffi ffi ð1 a1 Þ ðS2 ðd Þ þ S2 ð2d ÞÞ S3 ðd Þ þ S3 ð2d Þ 2 p 2 l d0 0 0 þ ð1 a2 Þ pffiffiffiffi ðS2 ð0Þ þ S2 ðd ÞÞ S3 ð0Þ þ S3 ðd Þ p 2 l d0 þ ð1 a1 Þð1 a2 Þ pffiffiffiffi ðS2 ðd0 Þ S2 ð3d0 ÞÞ p 2 S3 ðd0 Þ þ 2S3 ð2d0 Þ 0 l d 0 0 0 S3 ð3d Þ þ pffiffiffiffi ðT2 ð0Þ þ T2 ðd ÞÞ þ T3 ð0Þ T3 ðd Þ p 2 ð d0 =2 1 0 0 þ pffiffiffiffi dw c p ðw Þ T0 ðd0 =2 þ w0 Þ þ ð1 a1 ÞS0 ð3d0 =2 w0 Þ p d0 =2 ð1 a2 Þ S0 ðd0 =2 þ w0 Þ þ ð1 a1 Þð1 a2 Þ 0 0 0 0 ðS0 ð5d =2 þ w Þ S0 ð3d =2 w ÞÞ

ð29Þ

a2 p ffiffiffi ffi cc ðu Þ ¼ T0 ðd0 =2 þ u0 Þ þ ð1 a1 ÞS0 ð3=2d0 u0 Þ p ð 0 1 d =2 þ ð1 a1 Þð1 a2 ÞS0 ð5=2d0 þ u0 Þ þ pffiffiffiffi dw0 cc ðw0 Þ p d0 =2 ð1 a1 ÞS1 ðd0 u0 w0 Þ þ ð1 a2 ÞS1 ðd0 þ u0 þ w0 Þ 0

þ ð1 a1 Þð1 a2 ÞðS1 ð2d0 u0 þ w0 Þ þ S1 ð2d0 þ u0 w0 ÞÞ

l l pffiffiffiffi 0 0 þ T1 ðju w jÞ þ pffiffiffiffi 1 þ pffiffiffiffi p=4 T2 ðd0 Þ p p ð1 a1 ÞðS2 ðd0 Þ S2 ð2d0 ÞÞ ð1 a2 ÞðS2 ð0Þ S2 ðd0 ÞÞ þ ð1 a1 Þð1 a2 Þ 1 0 0 ðS2 ðd Þ S2 ð3d ÞÞ ½T1 ðd0 =2 þ u0 Þ T1 ðd0 =2 u0 Þ þ ð1 a1 ÞðS1 ð3d0 =2 u0 Þ S1 ðd0 =2 u0 ÞÞ

268



þ ð1 a2 ÞðS1 ðd0 =2 þ u0 Þ S1 ð3d0 =2 þ u0 ÞÞ þ ð1 a1 Þð1 a2 Þ ðS1 ð3d0 =2 u0 Þ S1 ð3d0 =2 þ u0 Þ þ S1 ð5d0 =2 þ u0 Þ

a2 0 0 S1 ð5d =2 u ÞÞ pffiffiffiffi 1=2 ð1 a1 ÞS1 ð2d0 Þ þ ð1 a1 Þ p ð 0 1 d =2 0 0 0 ð1 a2 ÞS1 ð2d Þ pffiffiffiffi dw cc ðw Þ T0 ðd0 =2 þ w0 Þ p d0 =2 þ ð1 a1 ÞS0 ð3d0 =2 w0 Þ ð1 a2 ÞS0 ðd0 =2 þ w0 Þ

þ ð1 a1 Þð1 a2 ÞðS0 ð5d =2 þ w Þ S0 ð3d =2 w ÞÞ : 0

0

0

0

ð30Þ

Until now we have not mentioned the relation between u0 and the collision time u defined in the BGK model (Cercignani and Daneri 1963; Cercignani 1988). In order to get the same viscosity coefficient from the BGK model and the present one, we must put

u0 ¼

ðl þ 2Þ u: 2

ð31Þ

Therefore, the rarefaction parameter d0 can be rewritten in term of the inverse Knudsen number d, appearing in the BGK solution of the Poiseuille-Couette problem, as follows:

d0 ¼

2d : ð2 þ lÞ

ð32Þ

Using (26), the flow rate (per unit time through unit thickness) defined by ð h=2 qðzÞdz ð33Þ F¼r h=2

can be expressed as the sum of the Poiseuille flow (Fp) and the Couette flow (Fc) as follows: Fl ¼ Flp þ Flc ¼

@p 2 l rUh l h Q p ðd; a1 ; a2 Þ þ Q c ðd; a1 ; a2 Þ ð34Þ @x 2


269

where ð 0 1 1 d =2 þ c ðu0 Þ du0 d0 d02 d0 =2 p ð 0 2 d =2 l Q c ðd; a1 ; a2 Þ ¼ 0 c ðu0 Þ du0 d d0 =2 c


Q lp ðd; a1 ; a2 Þ ¼

are the nondimensional volume flow rates. In order to solve numerically the integral Equations (29) and (30), we have extended a finite difference technique first introduced by Cercignani and Daneri (1963). The one-dimensional computational domain is divided into n mesh points ( for simplicity, only constant mesh steps are considered) and c(u0 ) (which stands for either the Poiseuille or the Couette nondimensional bulk velocity) is approximated by a stepwise function. The general form of the numerical scheme is given by n1 X

ahk ck ¼ bh

ðh ¼ 0; . . . ; n 1Þ:

ð35Þ

k¼0

Following the idea reported in Cercignani and Daneri (1963), the constant value assigned to the function c(u0 ) on every interval can be interpreted as either (a) the value in the midpoint or (b) the mean value on the whole interval, so that two methods of differencing can be defined with two possible choices for the coefficients ahk and bh. Obviously the second finite difference method is more accurate. However, with a resolution of n ¼ 200 mesh points (used in the present computations to reach very high accuracy), the two schemes approach each other so closely that they can be considered equivalent. Technical details of the algorithm for l ¼ 0 are given in Cercignani et al. (2006) the generalization to the case l = 0 is straightforward. The correction to the results given by the BGK solution of the Poiseuille problem is apparent from Figures 2 and 3, where the Poiseuille flow rate Qlp is plotted versus the inverse of the Knudsen number d for different values of the accommodation coefficients a1, a2. In the continuum flow limit the differences between the two models are extremely small (the relative error is less than 0.5%), while in the free molecular flow limit and in the transitional region the influence of the model is more


270


FIGURE 2 Poiseuille flow rate Qlp versus the inverse Knudsen number d for l ¼ 0 (BGK model, solid line) and l ¼ 1 (ES model, dashed line). The accommodation coefficients are a1 ¼ a2 ¼ 1 (left) and a1 ¼ a2 ¼ 0.5 (right).

evident (the relative error is within (8%– 9%) for each value of the accommodation coefficient. In the near-free molecular flow regime, the discrepancy between the results given by the BGK and ES models increases if one of the walls is allowed to reemit

FIGURE 3 Poiseuille flow rate Qlp versus the inverse Knudsen number d for l ¼ 0 (BGK model, solid line) l ¼ 1 (ES model, dashed line). The accommodation coefficients are a1 ¼ 0.1, a2 ¼ 0.8 (left); and a1 ¼ 0.5, a2 ¼ 0.3 (right).



271

FIGURE 4 Couette bulk velocity profile cc versus X ¼ z/h for l ¼ 0 (BGK model, solid line) and l ¼ 1 (ES model, dashed line). For all panels d ¼ 0.1. The accommodation coefficients are a1 ¼ 0.1, a2 ¼ 0.5 (top panel, left); a1 ¼ 0.3, a2 ¼ 0.5 (top panel, right); a1 ¼ 0.5, a2 ¼ 0.5 (bottom panel, left); and a1 ¼ 1.0, a2 ¼ 0.5 (bottom panel, right).

part of the molecules specularly, but it decreases faster than in the case of complete Maxwell diffusion, in the transitional region. Concerning the Couette flow rate Qlc , the differences between the two models are always extremely small progressing from free molecular, through transitional, to continuum regions, regardless of the value of the accommodation coefficients. However, the influence of the model is not completely negligible if one looks at the bulk velocity profiles in the near-free molecular flow regime (see Figure 4). It is worth noting that the differences between the velocity profile results for the BGK model and the ES model of the Couette flow problem increase, raising the accommodation coefficient of the fixed wall (a1) (see Figure 4). 3. Generalized Reynolds Equation Based on the ES Model The analysis developed in the previous section can be applied to the slider bearing problem in lubrication theory.

272



FIGURE 5 Geometry of a slider bearing.

The basic geometry of the two-dimensional gas film is outlined in Figure 5. Unlike the configuration considered in the previous section, the upper plate is slightly inclined at a small angle g. Since the pitch angle g is typically less than 18, the pressure p in the gas is taken to be constant in the vertical direction, while at the left (x ¼ 0) and right (x ¼ L) boundaries it is fixed at ambient pressure po. Assuming that the heat generation in the gas is very small, so that an isothermal process can be considered, the generalized Reynolds equation, modified to take into account the gas rarefaction effects, can be derived by applying the mass flow conservation equation across the film thickness with the flow rate Fl given by (34) (Cercignani et. al. 2005): d dp 2 l rUh l h Qp ðd; a1 ; a2 Þ Q ðd; a1 ; a2 Þ ¼ 0: dx dx 2 c

ð36Þ

In terms of the dimensionless quantities x X¼ ; l

P¼

p ; po

H¼

h ; ho

the nondimensional generalized Reynolds equation reads d ~l 3 dP l Q p ðdo PH; a1 ; a2 ÞPH Qc ðdo PH; a1 ; a2 ÞLPH ¼ 0 dX dX ð37Þ ¯ lp ¼ Qlp /Qcon, with ¯ lp is the Poiseuille relative flow rate Q where Q Qcon ¼ d/6 being the continuum flow limit; L is the bearing


273

number defined as L¼

6mUl po h20


with m being the dynamic viscosity of the gas. Furthermore, the rarefaction parameter d has been expressed as d ¼ doPH, with do being the characteristic inverse Knudsen number defined by the minimum film thickness, ho, and the ambient pressure po: po ho do ¼ pffiffiffiffiffiffiffiffiffiffiffi : m 2RTo If the continuum limit (d ! 1) is taken for any fixed a1 ¼ a2, then the limiting solution for the Poiseuille and Couette flow rates is given by l Q~ p !1;

Qcl !1

ð38Þ

so that (37) reduces to the classical Reynolds equation used in standard hydrodynamic lubrication theory (Fukui and Kaneko 1987, 1988, 1990). Writing the nondimensional film thickness H in terms of the longitudinal coordinate X, h2 l h2 1 X ð39Þ H¼ ho L ho such that dP l h2 dP ¼ : 1 dX L ho dH Equation (37) can be analytically integrated to give l h2 dP l 1 Q~ p ðdo PH; a1 ; a2 ÞPH3 þ Qcl ðdo PH; a1 ; a2 Þ L ho dH LPH ¼ K1

ð40Þ

where K1 is a constant of integration. The substitution of PH ¼ z

ð41Þ

274


in (40) gives dz z ½Qcl ðdo z; a1 ; a2 ÞLz K1 ¼ : dH H l=Lðh2 =ho 1ÞQ~ l ðdo z; a1 ; a2 ÞHz

ð42Þ

p

The boundary conditions to be matched to (42) are given by


z ¼ h2 =ho at H ¼ h2 =ho z ¼ 1 at H ¼ 1 Equation (42) has been solved numerically using the relaxation methods. To apply this numerical scheme, the differential equations have to be replaced by finite-difference equations on a point mesh. The solution of the resulting set of equations is determined by starting with a guess and improving it iteratively using Newton’s method. The Poiseuille and Couette flow rate coefficients. Qlp (d, a1, a2) and Qlc (d, a1, a2), respectively, have been evaluated by means of the numerical method described in Section 2. 4. Results and Discussion Once z(H) has been numerically evaluated on a grid that spans the domain of interest, (39) and (41) give the pressure field in the gas film as a function of the longitudinal coordinate X. Furthermore, a prediction of the vertical force acting on the upper surface of the slider bearing may be obtained from the loadcarrying capacity W, defined as ð 1 L=l ðP 1ÞdX: ð43Þ W¼ L 0 A comparison between the Reynolds equation solutions, obtained using the ES and BGK models, and the numerical findings obtained from direct simulation Monte Carlo (DSMC) (Alexander et al. 1994) and the information preservation IP method (Jiang et al. 2005) in the case of Maxwell’s boundary conditions on two physically identical walls, is shown in Figures 6 and 7. The parameters describing the gas film geometric configuration were fixed at the following values: h2/ho ¼ 2, L/ho ¼ 100.



275

FIGURE 6 Pressure profile versus X. Comparison between the Reynolds-BGK results (solid line), the Reynolds-ES results (dashed line), and DSMC data (Alexander et al. 1994) (open circles). The parameters are do ¼ 0.7, L ¼ 61.6, a1 ¼ a2 ¼ 1 (left); and do ¼ 0.2, L ¼ 1264, a1 ¼ a2 ¼ 1 (right).

Looking at the figures, one sees that the pressure distribution in the gas film increases with increasing L. Furthermore, at a fixed bearing number, the pressure field reduces by increasing the fraction of gas molecules specularly reflected by the walls.

FIGURE 7 Pressure profile versus X. Comparison between Reynolds-BGK results (solid line), the Reynolds-ES results (dashed line), DSMC data (Alexander et al. 1994) (open circles), and IP data (Jiang et al. 2005) (open squares). The parameters are do ¼ 0.7, L ¼ 61.6, a1 ¼ a2 ¼ 0.7.


276


Figures 6 and 7 show that the present Reynolds equation solutions, obtained using the ES and BGK models, are in good agreement with the DSMC data presented by Alexander et al. (1994) and the IP results reported by Jiang et al. (2005). It is worth noting that in Figure 7, the results of the IP method given by Jiang et al. (2005) are closer to the Reynolds equation numerical solutions than the DSMC data obtained previously by Alexander et al. (1994). Furthermore, the solution of the Reynolds equation based on the ES model slightly underestimates the pressure profiles given by the DSMC and IP simulations, suggesting that in isothermal conditions and at low Mach numbers the corrections introduced by a more refined kinetic model of the collisional Boltzmann operator are extremely small. The load capacity values, corresponding to the set of parameters listed in Figures 6 and 7, are summarized in Table 1. In order to investigate the effects of the rarefaction parameter do and the accommodation coefficients on the basic lubrication characteristics, Figure 8 shows the pressure profiles obtained through Reynolds equation based on the ES model, in the near-free molecular flow and near-continuum flow limits for different values of a1 and a2. The figure reveals that, for small do, if one keeps the accommodation coefficient of the slider (a1) fixed and varies the other one (a2), the pressure distribution in the gas film, at a fixed bearing number, increases with increasing a2, as always happens in the continuum region; while at fixed a2, the pressure distribution decreases by increasing a1. Such inverted pressure profiles, which appear in studying the slider air bearing Table 1 Summary of load capacity values

do L a1 a2 Load capacity Reynolds (BGK) Reynolds (ES) DSMC (Alexander et al. 1994) ( from pressure) DSMC (Alexander et al. 1994) ( from force)

0.7 61.6 1. 1.

0.2 1264 1. 1.

0.7 61.6 0.7 0.7

0.174 0.167 0.175

0.347 0.345 0.357

0.122 0.117 0.129

0.174

0.329

0.132



277

FIGURE 8 Pressure profiles, from the Reynolds-ES equation, versus X for L ¼ 50. The line styles indicate a1 ¼ 0.1, a2 ¼ 0.8 (dashed line); a1 ¼ 0.8, a2 ¼ 0.8 (solid line); and a1 ¼ 0.8, a2 ¼ 0.1 (dot-dashed line). The inverse Knudsen number do is 1023 (top panels) and 10 (bottom panels).

problem in the free molecular flow regime, are triggered by the Couette contribution to the lubrication flow rate (Cercignani et al. 2006), irrespective of which of the two kinetic models (BGK or ES) have been considered to derive the Reynolds equation. The values of load capacities are presented in Table 2. 5. Concluding Remarks In the present paper, the plane Poiseuille-Couette flow of a rarefied gas between two parallel plates has been investigated by means of the linearized ES model of the Boltzmann equation, as an issue of relevance for applications. General boundary conditions of Maxwell’s type have been imposed on the two walls, allowing for possible differences in the accommodation coefficients of the top and bottom plates. While the well-known BGK model leads to the wrong Prandtl number, the ES model can be adjusted to give its proper value, since one can make both viscosity and thermal conductivity agree with the Chapman-Enskog values for a

278



Table 2 Summary of load capacity values

do 1023 1023 1023 L 50 50 50 a1 0.1 0.8 0.8 a2 0.8 0.8 0.1 Load capacity Reynolds 6.316 . 1025 5.755 . 1025 — (BGK) Reynolds 5.942 . 1025 5.438 . 1025 4.710 . 1026 (ES)

10 50 0.1 0.8

10 50 0.8 0.8

10 50 0.8 0.1

0.295

0.333

—

0.295

0.333

0.239

Maxwell gas. This circumstance should turn out to be important in nonlinear problems. However, our study shows that, even in the frame of a linearized analysis, the corrections introduced by the ES model are not completely negligible. Focusing on issues of practical usefulness, the analysis of the Poiseuille-Couette flow problem has been applied to the slider bearing problem in lubrication theory. The results obtained through the generalized Reynolds equation, using both the BGK and ES models, are in good agreement with the calculations of the DSMC method published by Alexander et al. (1994) and the IP method introduced by Jiang et al (2005). Since the ES model is a more refined model than the BGK one, the pressure profiles given by the Reynolds equation based on the ES model, which slightly underestimate the DSMC and IP data, should be regarded as the lower limit to which improved DSMC-IP simulations should tend, as Figure 7 suggests. Finally, it is worth stressing that the Reynolds equation solutions, obtained using the ES and BGK models, reveal the existence of inverted pressure profiles in the free molecular flow regime, when two different accommodation coefficients (a1, a2) for the bounding surfaces are considered. The origin of this kind of inverted pressure profiles can be traced back of the Couette contribution to the lubrication flow rate. Acknowledgments This research was supported by the Italian MIUR (PRIN Project “Mathematical Problems of Kinetic Theories”) and by the


279

INDAM project “Kinetic Innovative Models for the Study of the Behavior of Fluids in Micro/Nano Electromechanical Systems.” The authors wish to thank Dr. Alberto Maurizi for helpful suggestions.


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