The temperature-jump problem for a variable ... - Semantic Scholar

PHYSICS OF FLUIDS

VOLUME 14, NUMBER 1

JANUARY 2002

The temperature-jump problem for a variable collision frequency model L. B. Barichello Instituto de Matema´tica, Universidade Federal do Rio Grande do Sul, 91509-900 Porto Alegre, RS, Brazil

A. C. R. Bartz Programa de Po´s-Graduac¸˜ao em Matema´tica Aplicada, Universidade Federal do Rio Grande do Sul, 91509-900 Porto Alegre, RS, Brazil

M. Camargoa) Programa de Po´s-Graduac¸˜ao em Engenharia Mecaˆnica, Universidade Federal do Rio Grande do Sul, 90050-170 Porto Alegre, RS, Brazil

C. E. Siewert Mathematics Department, North Carolina State University, Raleigh, North Carolina 27695-8205

共Received 3 April 2001; accepted 14 August 2001兲 An analytical version of the discrete-ordinates method is used here in the field of rarefied-gas dynamics to solve a version of the temperature-jump problem that is based on a linearized, variable collision frequency model of the Boltzmann equation. In addition to a complete development of the discrete-ordinates method for the application considered, the computational algorithm is implemented to yield accurate numerical results for three specific cases: the classical BGK model, the Williams model 共the collision frequency is proportional to the magnitude of the velocity兲, and the rigid-sphere model. © 2002 American Institute of Physics. 关DOI: 10.1063/1.1416192兴

I. INTRODUCTION

the variable collision frequency model was used to solve the classical Kramers’ problem,3 and so in this work we extend our use13 of the discrete-ordinates method14 to solve the temperature-jump problem15 for a general version of a linearized, variable collision frequency model of the Boltzmann equation. Here we base our notation on Williams’ book;3 however, the papers of Cercignani7 and Loyalka and Ferziger8 are the ones we consider to be the defining works on this subject of the variable collision frequency model. It therefore seems reasonable to refer to the general model equation used in this work as the CLF equation and to consider the BGK model 共constant collision frequency兲, the Williams model 共the collision frequency is proportional to the particle speed兲, and the rigid-sphere model as special cases that correspond to certain choices of the collision frequency. To introduce the mathematical statement of the problem to be solved, we follow Williams3 and consider the defining balance equation to be

The state of a gas can be described mathematically by a distribution function that satisfies the nonlinear Boltzmann equation.1–3 While, for example, Monte Carlo methods and computationally intensive iterative methods are ways of attempting to extract some physical information from the nonlinear Boltzmann equation, another approach that can be used when the density of particles is small 共rarefied-gas dynamics兲 is to approximate the nonlinear Boltzmann equation by a so-called model equation.4 In recent years, we have seen an increased interest in the general area of rarefied-gas dynamics essentially because of applications to small-scale problems 共for example, as related to micro-machines and high-speed disk drives兲 where the Boltzmann equation or a model equation is required in order to describe well gas-flow and heat-flow mechanisms. In this work, we take advantage of some recent mathematical and numerical improvements in the discrete-ordinates method in order to establish a series of high-quality results for the temperature-jump problem as based on a generalization of the standard BGK model. Although the so-called BGK model5 introduced by Bhatnagar, Gross, and Krook has been the focus of the vast majority of mathematical studies in the general area of rarefiedgas dynamics, there exist numerous models that have been used to try to improve on the simplest form of the BGK model. One such approach6 –11 is based on the variable collision frequency model 共sometimes referred to as the generalized BGK model兲 since it has been shown9 better able to support some experimental observations. In a recent work12

c␮

⳵ h 共 x,c兲 ⫹V 共 c 兲 h 共 x,c兲 ⳵x ⫽

冕冕 冕 ⬁

0

1

⫺1

2␲

0

c ⬘ 2 e ⫺c ⬘ K 共 c:c⬘ 兲 h 共 x,c⬘ 兲 d ␹ ⬘ d ␮ ⬘ dc ⬘ . 2

共1兲

Here h 共 x,c兲 ⇒h 共 x,c, ␮ , ␹ 兲

共2兲

a兲

Permanent address: Universidade Regional Integrada do Alto Uruguai e ˜ es. das Misso

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and 382

© 2002 American Institute of Physics

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Phys. Fluids, Vol. 14, No. 1, January 2002

K 共 c:c⬘ 兲 ⫽

Temperature-jump problem

II. QUANTITIES OF INTEREST

1 V 共 c 兲 V 共 c ⬘ 兲关 ␥ 0 ⫹ ␥ 1 c•c⬘ ⫹ ␥ 2 共 c 2 ⫺ ␻ 兲 4␲ ⫻ 共 c ⬘ 2 ⫺ ␻ 兲兴 ,

共3兲

where 1 , V0

共4a兲

3 ␥ 1⫽ , V2

共4b兲

␥ 0⫽

␥ 2⫽

V0 V 0 V 4 ⫺V 22

While our problem is defined in terms of the basic unknown h(x,c), we require only two elementary integrals of h(x,c). To be clear, we note that here we seek the temperature and density perturbations8 defined by N共 x 兲⫽

1 ␲ 3/2

冕冕 冕 ⬁

1

⫺1

0

2␲

0

2

c 2 e ⫺c h 共 x,c, ␮ , ␹ 兲 d ␹ d ␮ dc 共8兲

and 2 3 ␲ 3/2

T共 x 兲⫽

共4c兲

,

383

冕冕 冕 ⬁

1

⫺1

0

2␲

0

2

c 2 e ⫺c 共 c 2 ⫺3/2兲

⫻h 共 x,c, ␮ , ␹ 兲 d ␹ d ␮ dc,

and

共9兲

or V2 ␻⫽ V0

共4d兲

with V n⫽

冕

冕冕 ⬁

N共 x 兲⫽

2 ␲ 1/2

T共 x 兲⫽

4 3 ␲ 1/2

1

2

⫺1

0

c 2 e ⫺c ␾ 共 x,c, ␮ 兲 d ␮ dc

共10兲

and ⬁

0

2

V 共 c 兲 c n⫹2 e ⫺c dc.

共5兲

In addition, c is used, with dimensionless units, to denote the magnitude of the particle velocity vector c, x⭓0 is the spatial variable that measures 共in dimensionless units兲 the distance from the wall, V(c) is the collision frequency, and ␮ and ␹ are the two angular variables that define the direction 共relative to the positive x axis兲 of the velocity. In addition to Eq. 共1兲 we consider the boundary condition at the wall written as h 共 0,c, ␮ , ␹ 兲 ⫺ 共 1⫺ ␣ 兲 h 共 0,c,⫺ ␮ , ␹ ⫹ ␲ 兲 ⫺ 共 Ih 兲共 0 兲 ⫽0 共6a兲 for ␮苸共0,1兴, c苸 关 0,⬁), and ␹苸关0,␲兴 and

for ␮苸共0,1兴, c苸 关 0,⬁), and ␹苸关␲,2␲兴. Here 2␣ 共 Ih 兲共 0 兲 ⫽ ␲

冕冕冕 ⬁

0

1

0

2␲

0

1

2

⫺1

0

c 2 e ⫺c 共 c 2 ⫺3/2兲 ␾ 共 x,c, ␮ 兲 d ␮ dc, 共11兲

␾ 共 x,c, ␮ 兲 ⫽

1 2␲

冕

2␲

0

h 共 x,c, ␮ , ␹ 兲 d ␹

共12兲

is an azimuthal average. We can integrate Eqs. 共1兲 and 共6兲 over ␹ to find c␮

⳵ ␾ 共 x,c, ␮ 兲 ⫹V 共 c 兲 ␾ 共 x,c, ␮ 兲 ⳵x

冕冕 ⬁

1

⫺1

0

c ⬘ 2 e ⫺c ⬘ K 共 c, ␮ :c ⬘ , ␮ ⬘ 兲 ␾ 共 x,c ⬘ , ␮ ⬘ 兲 d ␮ ⬘ dc ⬘ , 2

共13兲

for x⬎0, ␮苸关⫺1,1兴 and c苸 关 0,⬁), and

␾ 共 0,c, ␮ 兲 ⫺ 共 1⫺ ␣ 兲 ␾ 共 0,c,⫺ ␮ 兲

c ⬘ 3 e ⫺c ⬘ h 共 0,c ⬘ ,⫺ ␮ ⬘ , ␹ ⬘ 兲

⫻ ␮ ⬘ d ␹ ⬘ d ␮ ⬘ dc ⬘

⬁

where

⫽

h 共 0,c, ␮ , ␹ 兲 ⫺ 共 1⫺ ␣ 兲 h 共 0,c,⫺ ␮ , ␹ ⫺ ␲ 兲 ⫺ 共 Ih 兲共 0 兲 ⫽0 共6b兲

冕冕

2

⫺4 ␣ 共7兲

and ␣苸共0,1兴 is the accommodation coefficient. Our basic unknown h(x,c) is the perturbation from an initial Maxwellian distribution that, due to the presence of the wall, is a component of the particle distribution function. In regard to Eqs. 共6兲, we note that some fraction 1⫺␣ of the particles is reflected specularly and that the remaining fraction ␣ is reflected diffusely. In other words, the wall acts somewhat like a mirror and at the same time appears to absorb some of the particles and then re-emit them isotropically. Because there is no loss or supply of particles due to the presence of the wall, the boundary condition can be thought of as conservative. In addition to the boundary condition given by Eqs. 共6兲, we note that, as will be discussed later, we must also impose a condition on h(x,c) as x tends to infinity.

冕冕 ⬁

0

1

0

c ⬘ 3 e ⫺c ⬘ ␾ 共 0,c ⬘ ,⫺ ␮ ⬘ 兲 ␮ ⬘ d ␮ ⬘ dc ⬘ ⫽0, 共14兲 2

for ␮苸共0,1兴 and c苸 关 0,⬁). Here K 共 c, ␮ :c ⬘ , ␮ ⬘ 兲 ⫽ 21 V 共 c 兲 V 共 c ⬘ 兲关 ␥ 0 ⫹ ␥ 1 c ␮ c ⬘ ␮ ⬘ ⫹ ␥ 2 共 c 2 ⫺ ␻ 兲共 c ⬘ 2 ⫺ ␻ 兲兴 .

共15兲

As Eqs. 共1兲 and 共6兲 are homogeneous, we must specify a driving term for the temperature-jump problem. We do this implicitly by requiring that h(x,c, ␮ , ␰ ) diverge as x tends to infinity. More specifically, we impose the condition that the temperature perturbation satisfies the Welander condition16 lim x→⬁

d T 共 x 兲 ⫽K, dx

共16兲

where K is considered specified. Now let V共 c 兲⫽␴␩共 c 兲,

共17兲

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384

Barichello et al.

Phys. Fluids, Vol. 14, No. 1, January 2002

where ␴ is a scale factor to be defined later and where ␩ (c) is a ‘‘shape factor’’ used to define the variable collision frequency. We also introduce

␶⫽␴x

共18a兲

Y 共 ␶ ,c, ␮ 兲 ⫽ ␾ 共 ␶ / ␴ ,c, ␮ 兲

共18b兲

and

K d T 共 ␶ 兲⫽ . d␶ * ␴

lim ␶ →⬁

共27兲

At this point we note that

and rewrite our problem as

⫽

冕冕 ⬁

1

⫺1

0

c ⬘ 2 e ⫺c ⬘ F 共 c, ␮ :c ⬘ , ␮ ⬘ 兲 Y 共 ␶ ,c ⬘ , ␮ ⬘ 兲 d ␮ ⬘ dc ⬘ , 2

共19兲

for ␶⬎0, ␮苸关⫺1,1兴 and c苸 关 0,⬁), and Y 共 0,c, ␮ 兲 ⫺ 共 1⫺ ␣ 兲 Y 共 0,c,⫺ ␮ 兲 ⫺4 ␣

冕冕 ⬁

0

Z a 共 ␶ ,c, ␮ 兲 ⫽ 共 c 2 ⫺5/2兲关 ␶ ⫺c ␮ / ␩ 共 c 兲兴

1

0

Y 共 ␶ ,c, ␮ 兲 ⫽

c ⬘ 3 e ⫺c ⬘ Y 共 0,c ⬘ ,⫺ ␮ ⬘ 兲 ␮ ⬘ d ␮ ⬘ dc ⬘ ⫽0, 共20兲

冋

2

N 共 ␶ 兲 ⫽ 共 K/ ␴ 兲 ⫺ ␶ ⫹

*

F 共 c, ␮ :c ⬘ , ␮ ⬘ 兲 ⫽ 21 ␩ 共 c 兲 ␩ 共 c ⬘ 兲关 ␤ 0 ⫹ ␤ 1 c ␮ c ⬘ ␮ ⬘ ⫹ ␤ 2 共 c 2 ⫺ ␻ 兲共 c ⬘ 2 ⫺ ␻ 兲兴 ,

冕冕 ⬁

⫻ 共21兲

and

␤ 1⫽

␩2 3

␩4

,

共22a兲

,

共22b兲

␩2 ␤ 2⫽ ␩ 2 ␩ 6 ⫺ ␩ 24

共22c兲

␻⫽

␩4 ␩2

共22d兲

with

␩ n⫽

冕␩ ⬁

0

共29兲

冋

*

2 ␲ 1/2 2

c 2 e ⫺c Z 共 ␶ ,c, ␮ 兲 d ␮ dc

T 共 ␶ 兲 ⫽ 共 K/ ␴ 兲 ␶ ⫹

4 3 ␲ 1/2

冕冕 ⬁

0

册

1

⫺1

册

共30兲

2

c 2 e ⫺c 共 c 2 ⫺3/2兲

⫻Z 共 ␶ ,c, ␮ 兲 d ␮ dc .

共31兲

It follows now that we seek a solution Z( ␶ ,c, ␮ ) that is bounded as ␶ tends to infinity and that satisfies c␮

and

1

⫺1

0

where

␤ 0⫽

K 关 Z 共 ␶ ,c, ␮ 兲 ⫹Z a 共 ␶ ,c, ␮ 兲兴 ␴

and find from Eqs. 共25兲 and 共26兲, after using Eqs. 共28兲 and 共29兲, results for the density and temperature perturbations expressed in terms of the bounded component Z( ␶ ,c, ␮ ), viz.

for ␮苸共0,1兴 and c苸 关 0,⬁). Here

1

共28兲

is a solution of Eq. 共19兲 that is linear in ␶, and so we choose to decompose the required solution into a part that has the desired behavior as ␶ tends to infinity and a part that is bounded. We therefore write

⳵ Y 共 ␶ ,c, ␮ 兲 ⫹ ␩ 共 c 兲 Y 共 ␶ ,c, ␮ 兲 ⳵␶

c␮

Finally to complete the definition of our problem, we rewrite Eq. 共16兲 as

⳵ Z 共 ␶ ,c, ␮ 兲 ⫹ ␩ 共 c 兲 Z 共 ␶ ,c, ␮ 兲 ⳵␶ ⫽

冕冕 ⬁

1

c ⬘ 2 e ⫺c ⬘ F 共 c, ␮ :c ⬘ , ␮ ⬘ 兲 Z 共 ␶ ,c ⬘ , ␮ ⬘ 兲 d ␮ ⬘ dc ⬘ , 2

⫺1

0

共32兲

for ␶⬎0, ␮苸关⫺1,1兴 and c苸 关 0,⬁), and n ⫺c 2

共 c 兲c e

共23兲

dc.

Now we let

⫺4 ␣

T 共 ␶ 兲 ⫽T 共 ␶ / ␴ 兲 and N 共 ␶ 兲 ⫽N 共 ␶ / ␴ 兲 ,

* and so we can write 2 N 共 ␶ 兲 ⫽ 1/2 * ␲

Z 共 0,c, ␮ 兲 ⫺ 共 1⫺ ␣ 兲 Z 共 0,c,⫺ ␮ 兲

*

冕冕 ⬁

1

2 ⫺c 2

⫺1

0

c e

共24兲

冕冕 ⬁

0

冕冕 ⬁

0

1

⫺1

0

c ⬘ 3 e ⫺c ⬘ Z 共 0,c ⬘ ,⫺ ␮ ⬘ 兲 ␮ ⬘ d ␮ ⬘ dc ⬘ 2

⫽R 共 c, ␮ 兲 ,

共33兲

for ␮苸共0,1兴 and c苸 关 0,⬁). Here Y 共 ␶ ,c, ␮ 兲 d ␮ dc

共25兲

R 共 c, ␮ 兲 ⫽ 共 2⫺ ␣ 兲共 c 2 ⫺5/2兲

and 4 T 共 ␶ 兲⫽ * 3 ␲ 1/2

1

c␮ 4␣ ⌫, ⫹ ␩共 c 兲 3

共34兲

where 2 ⫺c 2

c e

共 c ⫺3/2兲 Y 共 ␶ ,c, ␮ 兲 d ␮ dc. 2

共26兲

⌫⫽

冕␩ ⬁

0

c 4 ⫺c 2 2 e 共 c ⫺5/2兲 dc. 共c兲

共35兲

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Phys. Fluids, Vol. 14, No. 1, January 2002

Temperature-jump problem

Our basic statement of the problem to be solved is now complete, and so we proceed with our solution; however, before making use of our version of the discrete-ordinates method, we introduce some elementary transformations that will facilitate the development of the final results.

where c苸M ␰

␩共 c 兲兩 ␰ 兩

if

c

⭐1.

共40兲

In regard to boundary conditions, we substitute Eq. 共37兲 into Eq. 共33兲 to obtain G 1 共 0,␰ 兲 ⫺ 共 1⫺ ␣ 兲 G 1 共 0,⫺ ␰ 兲 ⫺⌬

III. BASIC TRANSFORMATIONS

Rather than deal explicitly with Eqs. 共32兲 and 共33兲, we choose to follow Busbridge17 and to introduce the convenient change of variables c␮ ␰⫽ ␩共 c 兲

共36a兲

4␣ ⌫⫹ 共 2⫺ ␣ 兲共 ␻ ⫺5/2兲 ␰ , 3

共41a兲

G 2 共 0,␰ 兲 ⫹ 共 1⫺ ␣ 兲 G 2 共 0,⫺ ␰ 兲 ⫽0,

共41b兲

⫽

and G 3 共 0,␰ 兲 ⫺ 共 1⫺ ␣ 兲 G 3 共 0,⫺ ␰ 兲 ⫽ 共 2⫺ ␣ 兲 ␰

and

␥ ⫽sup兵 c/ ␩ 共 c 兲 其 .

共36b兲

⌬⫽

Z 关 ␶ ,c, ␰ ␩ 共 c 兲 /c 兴 ⫽G 1 共 ␶ , ␰ 兲 ⫹ ␰ ␩ 共 c 兲 G 2 共 ␶ , ␰ 兲 ⫹ 共 c 2⫺ ␻ 兲 G 3共 ␶ , ␰ 兲

共37兲

we find, after an interchange of orders of integration,

⳵ G 共 ␶ , ␰ 兲 ⫹G i 共 ␶ , ␰ 兲 ⫽ ⳵␶ i

冕

␥

⫺␥

⫹ ␺ i,3共 ␰ ⬘ 兲 G 3 共 ␶ , ␰ ⬘ 兲兴 d ␰ ⬘ for i⫽1,2,3. Here

␺ 1,1共 ␰ 兲 ⫽ ␺ 1,2共 ␰ 兲 ⫽ ␺ 1,3共 ␰ 兲 ⫽ ␺ 2,1共 ␰ 兲 ⫽ ␺ 2,2共 ␰ 兲 ⫽ ␺ 2,3共 ␰ 兲 ⫽ ␺ 3,1共 ␰ 兲 ⫽ ␺ 3,2共 ␰ 兲 ⫽

冕 ␤ ␰ 冕 ␤ 冕 ␤ ␰ 冕 ␤ ␰ 冕 ␤ ␰ 冕 ␤ 冕 ␤ ␰ 冕 ␤0 2

M␰

2

M␰

2

共39b兲

ce ⫺c ␩ 2 共 c 兲共 c 2 ⫺ ␻ 兲 dc,

共39c兲

M␰

2

2

ce ⫺c ␩ 3 共 c 兲 dc,

2

M␰

2

ce ⫺c ␩ 4 共 c 兲 dc, 2

ce

2

M␰

共39e兲

ce ⫺c ␩ 3 共 c 兲共 c 2 ⫺ ␻ 兲 dc, ⫺c 2

M␰

2

冕

M␰

共42兲

冕

␥

⫺␥

⌿共 ␰ ⬘ 兲 G共 ␶ , ␰ ⬘ 兲 d ␰ ⬘ ,

共43兲

共39g兲

2

共39h兲

2

ce ⫺c ␩ 3 共 c 兲共 c 2 ⫺ ␻ 兲 dc

␥

0

⌼共 ␰ ⬘ 兲 G共 0,⫺ ␰ ⬘ 兲 ␰ ⬘ d ␰ ⬘ ⫽R共 ␰ 兲

S⫽diag兵 1,⫺1,1其 , 4 ⌼共 ␰ 兲 ⫽ ␤0 and R共 ␰ 兲 ⫽

共39f兲

␩ 共 c 兲共 c ⫺ ␻ 兲 dc, 2

冕

共44兲

for ␰苸共0,␥兴. Here

冋

冋

␺ 1,1共 ␰ 兲

⫺ ␺ 1,2共 ␰ 兲

␺ 1,3共 ␰ 兲

0

0

0

0

0

0

册

ce

␩ 共 c 兲共 c ⫺ ␻ 兲 dc, 2

2

2

共39i兲

共46兲

册

共 2⫺ ␣ 兲共 ␻ ⫺5/2兲 ␰ ⫹ 共 4/3兲 ␣ ⌫ 0 . 共 2⫺ ␣ 兲 ␰

再

N 共 ␶ 兲 ⫽ 共 K/ ␴ 兲 ⫺ ␶ ⫹

*

1 ␲ 1/2

冕

␥

⫺␥

共47兲

关 n 1共 ␰ 兲 G 1共 ␶ , ␰ 兲

⫹n 2 共 ␰ 兲 G 2 共 ␶ , ␰ 兲 ⫹n 3 共 ␰ 兲 G 3 共 ␶ , ␰ 兲兴 d ␰ ⫺c 2

共45兲

And so we seek a bounded 共as ␶→⬁兲 solution of Eq. 共43兲 that satisfies Eq. 共44兲. Of course, once we have solved the G problem, we can use Eq. 共37兲 to rewrite Eqs. 共30兲 and 共31兲 as

and

␤2 ␺ 3,3共 ␰ 兲 ⫽ 2

⳵ G共 ␶ , ␰ 兲 ⫹G共 ␶ , ␰ 兲 ⫽ ⳵␶

⫺2 ␣

共39d兲

2

M␰

1

2

ce

2

2

关 ␺ 1,1共 ␰ 兲 G 1 共 0,⫺ ␰ 兲 ⫺ ␺ 1,2共 ␰ 兲 G 2 共 0,⫺ ␰ 兲

where the 3⫻3 matrix ⌿共␰兲 has components ␺ i, j ( ␰ ). To have our boundary conditions in vector form, we rewrite Eqs. 共41兲 as

共39a兲

␩ 3 共 c 兲 dc,

1

1

⫺c 2

M␰

0

0

G共 0,␰ 兲 ⫺ 共 1⫺ ␣ 兲 SG共 0,⫺ ␰ 兲 2

ce ⫺c ␩ 2 共 c 兲 dc,

0

2

␥

We now introduce the vector-valued function G共␶,␰兲, with components G i ( ␶ , ␰ ), i⫽1,2,3, and write Eq. 共38兲 as

␰

共38兲

冕

8␣ ␤0

⫹ ␺ 1,3共 ␰ 兲 G 3 共 0,⫺ ␰ 兲兴 ␰ d ␰ .

关 ␺ i,1共 ␰ ⬘ 兲 G 1 共 ␶ , ␰ ⬘ 兲

⫹ ␺ i,2共 ␰ ⬘ 兲 G 2 共 ␶ , ␰ ⬘ 兲

共41c兲

for ␰苸共0,␥兴. Here the diffuse term in Eq. 共41a兲 is given by

And so now if we go back to Eq. 共32兲 and introduce the decomposition

␰

385

冎

共48兲

and

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386

Barichello et al.

Phys. Fluids, Vol. 14, No. 1, January 2002

再

2 3 ␲ 1/2

T 共 ␶ 兲 ⫽ 共 K/ ␴ 兲 ␶ ⫹

*

冕

␥

⫺␥

range’’ quadrature scheme that we partially attribute the especially good accuracy we have obtained from the solution reported here. Continuing, we substitute

关 t 1共 ␰ 兲 G 1共 ␶ , ␰ 兲

冎

⫹t 2 共 ␰ 兲 G 2 共 ␶ , ␰ 兲 ⫹t 3 共 ␰ 兲 G 3 共 ␶ , ␰ 兲兴 d ␰ ,

n 2 共 ␰ 兲 ⫽2

N

冕 ␰冕

ce

⫺c 2

M␰

M␰

␩ 共 c 兲 dc,

共50a兲

2

共50b兲

ce ⫺c ␩ 2 共 c 兲 dc,

冕

2

M␰

ce ⫺c ␩ 共 c 兲共 c 2 ⫺ ␻ 兲 dc,

共50c兲

t 2 共 ␰ 兲 ⫽2

冕 ␰冕

M␰

2

ce ⫺c ␩ 共 c 兲共 c 2 ⫺3/2兲 dc, 2

M␰

冕

ce ⫺c ␩ 2 共 c 兲共 c 2 ⫺3/2兲 dc,

共51a兲 共51b兲

ce

M␰

␩ 共 c 兲共 c ⫺ ␻ 兲共 c ⫺3/2兲 dc. 2

2

M⫽diag兵 ␰ 1 I, ␰ 2 I,..., ␰ N I其 ,

16 ⫺1/2 ␲ 15

冕␩ ⬁

0

⫺1

2

共 c 兲 c 4 e ⫺c 共 c 2 ⫺5/2兲 2 dc

w 2 ⌿共 ␰ 2 兲

R⫺ ⫽ 关 w 1 ⌿共 ⫺ ␰ 1 兲

¯

w N ⌿共 ␰ N 兲兴

共58兲

¯

w N ⌿共 ⫺ ␰ N 兲兴 共59兲

␯ ⌽⫹ 共 ␯ 兲 ⫺M⌽⫹ 共 ␯ 兲 ⫽ ␯ 关 W⫹ ⌽⫹ 共 ␯ 兲 ⫹W⫺ ⌽⫺ 共 ␯ 兲兴

共60a兲

and

␯ ⌽⫺ 共 ␯ 兲 ⫹M⌽⫺ 共 ␯ 兲 ⫽ ␯ 关 W⫹ ⌽⫹ 共 ␯ 兲 ⫹W⫺ ⌽⫺ 共 ␯ 兲兴 .

共60b兲

At this point we find, after noting some basic properties of ⌿共␰兲, that we can write W⫺ ⫽DW⫹ D,

共61兲

where the 3N⫻3N diagonal matrix D can be written as D⫽diag兵 S,S,...,S其

共62兲

with S as given by Eq. 共45兲. We now multiply Eq. 共60b兲 by D and rewrite Eqs. 共60兲 as

␯ ⌽⫹ 共 ␯ 兲 ⫺M⌽⫹ 共 ␯ 兲 ⫽ ␯ 关 W⫹ ⌽⫹ 共 ␯ 兲 ⫹DW⫹ D⌽⫺ 共 ␯ 兲兴 共63a兲 and

␯ D⌽⫺ 共 ␯ 兲 ⫹MD⌽⫺ 共 ␯ 兲 ⫽ ␯ 关 DW⫹ ⌽⫹ 共 ␯ 兲 ⫹W⫹ D⌽⫺ 共 ␯ 兲兴 . 共63b兲

N

w k 关 ⌿共 ␰ k 兲 G共 ␶ , ␰ k 兲 ⫹⌿共 ⫺ ␰ k 兲 G共 ␶ ,⫺ ␰ k 兲兴

w 2 ⌿共 ⫺ ␰ 2 兲

so that we can write Eqs. 共55兲 as

共52兲

d G共 ␶ ,⫾ ␰ i 兲 ⫹G共 ␶ ,⫾ ␰ i 兲 d␶

兺

共57兲

where I is the 3⫻3 identity matrix. In addition we let W⫹ and W⫺ denote 3N⫻3N matrices each 3⫻3N row of which is, respectively,

共51c兲

We note first of all that the characteristic matrix ⌿共␰兲, as defined by Eqs. 共39兲, is not symmetric. We note also that ⌿共␰兲⫽⌿共⫺␰兲, and so we write our discrete-ordinates version of Eq. 共43兲 as

k⫽1

共56a兲

⌽T 共 ␯ ,⫺ ␰ 2 兲 ¯ ⌽T 共 ␯ ,⫺ ␰ N 兲兴 T , 共56b兲

⌽⫺ 共 ␯ 兲 ⫽ 关 ⌽T 共 ␯ ,⫺ ␰ 1 兲

IV. THE DISCRETE-ORDINATES SOLUTION

⫽

⌽T 共 ␯ , ␰ 2 兲 ¯ ⌽T 共 ␯ , ␰ N 兲兴 T ,

and ⫺c 2

for all models we consider. As noted by Loyalka,9 the use of ␴ ⫽ ⑀ t corresponds to measuring our spatial variable x in terms of a mean-free path l t that is defined in terms of the thermal conductivity.

⫾␰i

共55兲

for i⫽1,2,...,N. Now let

R⫹ ⫽ 关 w 1 ⌿共 ␰ 1 兲

It is clear that the scale factor ␴ will have a fundamental effect on our reported numerical results, and since there already exist various possibilities in the literature concerning the definition of an appropriate mean-free path, we elect here to use one of Loyalka’s choices9 for scaling our results. We therefore define

␴ ⫽ ⑀ t⫽

w k 关 ⌿共 ␰ k 兲 ⌽共 ␯ , ␰ k 兲

⫹⌿共 ⫺ ␰ k 兲 ⌽共 ␯ ,⫺ ␰ k 兲兴

and t 3 共 ␰ 兲 ⫽2

兺

k⫽1

and

and also where t 1 共 ␰ 兲 ⫽2

共 ␯ ⫿ ␰ i 兲 ⌽共 ␯ ,⫾ ␰ i 兲 ⫽ ␯

⌽⫹ 共 ␯ 兲 ⫽ 关 ⌽T 共 ␯ , ␰ 1 兲

and n 3 共 ␰ 兲 ⫽2

共54兲

into Eqs. 共53兲 to find

where n 1 共 ␰ 兲 ⫽2

G共 ␶ ,⫾ ␰ i 兲 ⫽⌽共 ␯ ,⫾ ␰ i 兲 e ⫺ ␶ / ␯

共49兲

共53兲

for i⫽1,2,...,N. In writing Eqs. 共53兲 as we have, we clearly are considering that the N quadrature points 兵 ␰ k 其 and the N weights 兵 w k 其 are defined for use on the integration interval 关0,␥兴. We note that it is to this feature of using a ‘‘half-

Now let U⫽⌽⫹ 共 ␯ 兲 ⫹D⌽⫺ 共 ␯ 兲

共64a兲

V⫽⌽⫹ 共 ␯ 兲 ⫺D⌽⫺ 共 ␯ 兲

共64b兲

and

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Phys. Fluids, Vol. 14, No. 1, January 2002

Temperature-jump problem

冋 册

and add Eqs. 共63兲 to obtain

␯ 关 I⫺ 共 I⫹D兲 W⫹ 兴 U⫽MV,

␻ ⫺5/2 0 G4 共 ␶ , ␰ 兲 ⫽ 共 ␶ ⫺ ␰ 兲 . 1

共65兲

where now I is the 3N⫻3N identity matrix. We also compute the difference between Eqs. 共63兲 to find

␯ 关 I⫺ 共 I⫺D兲 W⫹ 兴 V⫽MU.

共66兲

We now can eliminate V共␯兲 between Eqs. 共65兲 and 共66兲 to find the eigenvalue problem 共67兲

AU⫽␭U,

共71d兲

If we now let GT 共 ␶ ,⫾ ␰ 2 兲 ¯ GT 共 ␶ ,⫾ ␰ N 兲兴 T , 共72兲

G⫾ 共 ␶ 兲 ⫽ 关 GT 共 ␶ ,⫾ ␰ 1 兲

then our discrete-ordinates solution can be written 共after we exclude all solutions that are not bounded as ␶ tends to infinity兲 as

where ␭⫽1/␯ and

3N

2

A⫽M⫺1 关 I⫺ 共 I⫺D兲 W⫹ 兴 M⫺1 关 I⫺ 共 I⫹D兲 W⫹ 兴 .

d␰ ⌿共 ␰ 兲 ⌳共 z 兲 ⫽I⫹z ␰ ⫺z ⫺␥ ␥

共69a兲

and note that we consider ⌳(z) to be the exact version of the discrete-ordinates quantity N

⍀共 z 兲 ⫽I⫹z

兺

k⫽1

冋

册

1 1 ⫺⌿共 ⫺ ␰ k 兲 w k ⌿共 ␰ k 兲 . ␰ k ⫺z ␰ k ⫹z 共69b兲

Since we know that the separation constants ␯ j defined by the zeros of det ⍀(z) are the same as those we compute from the eigenvalues of the matrix A, we base our discussion about the eigenvalues of A 共that accumulate at zero as N tends to infinity兲 on the zeros of det ⌳(z) as z tends to infinity. We find that M det ⌳共 z 兲 ⬃ 4 , z

M ⫽0,

共70兲

as z→⬁, and so we conclude that, as N tends to infinity, A should have ␭⫽0 as a 共two-fold兲 repeated eigenvalue. And so instead of using Eq. 共54兲 for the two smallest eigenvalues of A we use instead the following four exact solutions of Eq. 共43兲:

冋册

0 G1 ⫽ 0 1 and

冋册

0 G2 ⫽ 1 0

共71a兲

共71b兲

along with

冋册

1 G3 ⫽ 0 0 and

G⫾ 共 ␶ 兲 ⫽A 1 ⌽1 ⫹A 2 ⌽2 ⫹B 1 ⌽3 ⫹

共68兲

And so our first computational job is to find the 3N eigenvalues of A. However we first wish to address the issue of infinite values of the separation constant ␯ 共or equivalently, the eigenvalues of A that approach zero as N tends to infinity兲. We first introduce

冕

387

A j ⌽⫾ 共 ␯ j 兲 e ⫺ ␶ / ␯ j , 共73兲

where B 1 and A j , for j⫽1,2,...,3N, are arbitrary constants. In addition ⌽ j ⫽ 关 GTj

¯

GTj

GTj 兴 T ,

j⫽1,2,3,

共74兲

and the ⌽⫾ ( ␯ j ) are available from Eqs. 共64兲, 共65兲, and 共66兲. We find ⌽⫹ 共 ␯ j 兲 ⫽ 21 兵 I⫹ ␯ j M⫺1 关 I⫺ 共 I⫹D兲 W⫹ 兴 其 U j

共75兲

⌽⫺ 共 ␯ j 兲 ⫽ 21 D兵 I⫺ ␯ j M⫺1 关 I⫺ 共 I⫹D兲 W⫹ 兴 其 U j ,

共76兲

and

where U j is the eigenvector of A that corresponds to the eigenvalue ␭ j . Looking back now to Eq. 共44兲, we find, for this formulation, that the boundary condition can be written as G⫹ 共 0 兲 ⫺ 共 1⫺ ␣ 兲 Rs G⫺ 共 0 兲 ⫺2 ␣ Rd G⫺ 共 0 兲 ⫽R,

共77兲

where the known right-hand side is given by R⫽ 关 RT 共 ␰ 1 兲

RT 共 ␰ 2 兲

¯

RT 共 ␰ N 兲兴 T .

共78兲

Note that R共␰兲 is given by Eq. 共47兲. In addition, we find we can write the specular matrix as Rs ⫽diag兵 S,S,...,S其 ,

共79兲

where S is given by Eq. 共45兲. Finally, to account for diffuse reflection, Rd is a 3N⫻3N matrix each 3⫻3N row of which is given by Rr ⫽ 关 w 1 ␰ 1 ⌼共 ␰ 1 兲

␻ 2 ␰ 2 ⌼共 ␰ 2 兲

¯

␻ N ␰ N ⌼共 ␰ N 兲兴 , 共80兲

where ⌼共␰兲 is given by Eq. 共46兲. It is clear that Eq. 共77兲 is a general result, but when the exact terms ⌽1 , ⌽2 , and ⌽3 are used in Eq. 共44兲 the integrals resulting from the diffuse reflection can be done exactly. Finally, we note that since ⌽3 satisfies the homogeneous version of Eq. 共77兲 the constant B 1 cannot be determined from that equation. However, we can impose on our solution the additional 共arbitrary兲 normalization condition lim 关 N 共 ␶ 兲 ⫹T 共 ␶ 兲兴 ⫽⌫ .

␶ →⬁

共71c兲

兺

j⫽3

*

*

*

共81兲

At this point, we follow other works15,18,19 and use ⌫ ⫽0 * which, when we consider Eqs. 共48兲, 共49兲, and 共73兲, yields B 1 ⫽ 共 ␻ ⫺5/2兲 A 1 .

共82兲

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388

Barichello et al.

Phys. Fluids, Vol. 14, No. 1, January 2002

Considering now the quantities we wish to evaluate, we substitute Eq. 共73兲 into Eqs. 共48兲 and 共49兲 to find, after using Eq. 共82兲,

再

where we have imposed the normalization K⫽1. If now we let T asy共 x 兲 ⫽x⫹A 1 / ␴

3N

1 A 关N ⌽ 共␯ 兲 N 共 ␶ 兲 ⫽ 共 K/ ␴ 兲 ⫺ ␶ ⫺A 1 ⫹ 1/2 * ␲ j⫽3 j ⫹ ⫹ j ⫹N⫺ ⌽⫺ 共 ␯ j 兲兴 e ⫺ ␶ / ␯ j

兺

冎

共89兲

and define the temperature-jump coefficient ␨ by 共83兲

T asy共 0 兲 ⫽ ␨

d T 共 x 兲 兩 x⫽0 , dx asy

共90兲

and then clearly

再

3N

T 共 ␶ 兲 ⫽ 共 K/ ␴ 兲 ␶ ⫹A 1 ⫹

*

2 A 关T ⌽ 共␯ 兲 3 ␲ 1/2 j⫽3 j ⫹ ⫹ j

冎

兺

⫹T⫺ ⌽⫺ 共 ␯ j 兲兴 e ⫺ ␶ / ␯ j ,

␨ ⫽A 1 / ␴ . 共84兲

where N⫾ ⫽ 关 w 1 N共 ⫾ ␰ 1 兲

w 2 N共 ⫾ ␰ 2 兲

¯

共91兲

To be very clear, we note that the constant ⌫ introduced in * Eq. 共81兲 does not affect the temperature-jump coefficient or the temperature perturbation T(x). In fact, another choice of ⌫ would change only the density perturbations N(x) by the * addition of a constant factor.

w N N共 ⫾ ␰ N 兲兴 共85a兲

and V. SPECIAL CASES

T⫾ ⫽ 关 w 1 T共 ⫾ ␰ 1 兲

w 2 T共 ⫾ ␰ 2 兲

¯

w N T共 ⫾ ␰ N 兲兴 共85b兲

with N共 ␰ 兲 ⫽ 关 n 1 共 ␰ 兲

n 2共 ␰ 兲

n 3 共 ␰ 兲兴

共86a兲

Having developed our general solution to the temperature-jump problem for the CLF model of the linearized Boltzmann equation, we are ready to list the specific forms of certain basic quantities for the three special cases we consider in this work. A. Constant collision frequency

and

For this case, the classical BGK model, we write T共 ␰ 兲 ⫽ 关 t 1 共 ␰ 兲

t 2共 ␰ 兲

t 3 共 ␰ 兲兴 .

共86b兲

Note that the components of the vectors introduced in Eqs. 共86兲 are defined by Eqs. 共50兲 and 共51兲. We note also that in obtaining Eqs. 共83兲 and 共84兲 from Eqs. 共48兲 and 共49兲, we have analytically integrated the first three terms of Eq. 共73兲, but we have used our defined quadrature scheme to integrate the remaining terms. Now putting Eqs. 共83兲 and 共84兲 back in terms of the x variable, we find

␩ 共 c 兲 ⫽1, and so we find

3N

1 A 关N ⌽ 共␯ 兲 N 共 x 兲 ⫽⫺x⫺A 1 / ␴ ⫹ ␴␲ 1/2 j⫽3 j ⫹ ⫹ j

兺

⫹N⫺ ⌽⫺ 共 ␯ j 兲兴 e ⫺ ␴ x/ ␯ j

共87兲

and

共92兲

␥ ⫽⬁,

共93a兲

⌫⫽0,

共93b兲

␻ ⫽3/2,

共93c兲

␤ 0 ⫽4/␲ 1/2,

共93d兲

␴ ⫽1.

共93e兲

and 3N

T 共 x 兲 ⫽x⫹A 1 / ␴ ⫹

2 A 关T ⌽ 共␯ 兲 3 ␴␲ 1/2 j⫽3 j ⫹ ⫹ j

兺

⫹T⫺ ⌽⫺ 共 ␯ j 兲兴 e ⫺ ␴ x/ ␯ j ,

共88兲

In addition, we find from Eqs. 共39兲

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Phys. Fluids, Vol. 14, No. 1, January 2002

⫺␰2

⌿共 ␰ 兲 ⫽

e ␲ 1/2

冋

Temperature-jump problem

1

␰

␰ 2 ⫺1/2

2␰

2␰2

2 ␰ 共 ␰ 2 ⫺1/2兲

共 2/3兲共 ␰ 2 ⫺1/2兲

共 2/3兲 ␰ 共 ␰ 2 ⫺1/2兲

共 2/3兲共 ␰ 4 ⫺ ␰ 2 ⫹5/4兲

We can also use Eqs. 共50兲 and 共51兲 to write Eqs. 共86兲, for this case, as 2

N共 ␰ 兲 ⫽e ⫺ ␰ 关 1

␰

␰ 2 ⫺1/2兴

2

共95a兲

␰ 共 ␰ 2 ⫺1/2兲

␰ 4 ⫺ ␰ 2 ⫹5/4兴 . 共95b兲

B. The Williams model

For this case we write

␩ 共 c 兲 ⫽c,

共96兲

⌫⫽⫺1/4,

共97b兲

␻ ⫽2,

共97c兲

␤ 0 ⫽2,

共97d兲

␴ ⫽ 共 6/5兲 ␲ ⫺1/2.

共97e兲

共101c兲

⌫⫽⫺0.06063367084623,

共101d兲

␴ ⫽0.2753345876233.

共101e兲

In regard to Eqs. 共39兲, 共50兲, and 共51兲, we have used numerical methods to evaluate the various functions required to establish ⌿共␰兲, N共␰兲, and T共␰兲. As discussed in a previous work12 concerning Kramers’ problem, we let c ␩共 c 兲

冋

1 1 共 9/8兲 ␲ 1/2␰ ⌿共 ␰ 兲 ⫽ 2 0

共 3/4兲 ␲ ␰ 1/2

3␰

2

共 3/16兲 ␲ 1/2␰

0

m 共 ␰ 兲 ⫽ f ⫺1 共 兩 ␰ 兩 兲 ,

册

1

共98兲

Again we can use Eqs. 共50兲 and 共51兲 to write Eqs. 共86兲, for this case, as N共 ␰ 兲 ⫽ 关共 1/2兲 ␲ 1/2

␰

⫺ 共 1/4兲 ␲ 1/2兴

共99a兲

and T共 ␰ 兲 ⫽ 关 0

␰ /2

共 3/4兲 ␲ 1/2兴 .

共99b兲

C. The rigid-sphere model

For the rigid-sphere model, we follow Loyalka and Hickey11 and write

冉

␩ 共 c 兲 ⫽ 2c⫹

冊

1 ␲ 2 erf共 c 兲 ⫹e ⫺c , c 2

␰苸关⫺␥,␥兴,

共103兲

exists, and thus we can write the required functions 共written symbolically as兲 P共 ␰ 兲⫽

冕

P共 ␰ 兲⫽

冕

M␰

p 共 c 兲 dc

共104兲

as

共 9/16兲 ␲ ␰ . 1/2

共102兲

and note that we can show, for the considered case, that f ⬘ (c)⬎0, for c⭓0 and so the inverse function

and In addition, we find from Eqs. 共39兲

to find the

␤ 0 ⫽0.7978845608029,

f 共 c 兲⫽ 共97a兲

MAPLE V

and

and so we find

␥ ⫽1,

共94兲

.

and we have used the software package numerical values

and T共 ␰ 兲 ⫽e ⫺ ␰ 关 ␰ 2 ⫺1/2

册

389

1/2

共100兲

where erf(c) is the error function. Here we find the exact results

␥ ⫽1/␲ 1/2

共101a兲

␻ ⫽7/4,

共101b兲

and

⬁

m共 ␰ 兲

p 共 c 兲 dc,

共105兲

which can be evaluated numerically once m( ␰ ) is available; as before12 we use Newton’s method to establish the required numerical values of m( ␰ ). VI. NUMERICAL RESULTS

The first thing we must do is to define the quadrature scheme to be used in our discrete-ordinates solution, and, since we have considered three different cases, to which we refer as case 1, case 2, and case 3 while meaning, respectively, the BGK model, the Williams model, and the rigidsphere model, we have used three different maps. For case 1, we used the 共nonlinear兲 transformation u 共 ␰ 兲 ⫽exp兵 ⫺ ␰ 其

共106兲

to map ␰苸关0,⬁兲 into u苸 关 0,1兴 , and we then used the Gauss– Legendre scheme mapped 共linearly兲 onto the interval 关0,1兴. For cases 2 and 3 we simply mapped the Gauss–Legendre scheme onto, respectively, the intervals 关0,1兴 and 关 0,␲ ⫺1/2兴 . Having defined our quadrature scheme, we used the driver program RG from the EISPACK collection20 to find the eigenvalues and eigenvectors defined by Eq. 共67兲. And so, after using the subroutines DGECO and DGESL from the LINPACK package21 to solve the linear system derived from Eq. 共77兲 to

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390

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Phys. Fluids, Vol. 14, No. 1, January 2002

TABLE I. The temperature-jump coefficient ␨. Model

␣⫽0.1

␣⫽0.3

␣⫽0.5

␣⫽0.6

␣⫽0.7

␣⫽0.9

␣⫽1.0

Case 1 Case 2 Case 3

21.45012 21.19359 21.24657

6.630514 6.406417 6.452894

3.629125 3.435960 3.476180

2.867615 2.689383 2.726563

2.317534 2.153897 2.188095

1.570264 1.434848 1.463247

1.302716 1.180947 1.206526

find the constants A j , for j⫽1,2,...,3N, we consider our solution complete. Finally, but importantly, we have found, that some elements of the matrix-valued function ⌿共␰兲 as defined by Eqs. 共39兲 can be essentially zero 共from a computational point-of-view兲. In such cases, we found that by defining an element to be precisely zero when that element is less than, say, ⑀ ⫽10⫺50, we greatly increased the ability of the linearalgebra package to yield the required number of independent eigenvectors when there is a 共nearly兲 repeated eigenvalue. To complete our work we list in Tables I and II some results obtained from our FORTRAN implementation of the developed solution of the temperature-jump problem for the three explicitly considered cases. We note that our results are given with what we believe to be seven, in Table I, and six, in Table II, figures of accuracy. While we have no proof of the accuracy achieved in this work, we have done some things to support the confidence we have. First of all our results for case 1 agree perfectly with some 共quasi兲 independent calculations15 done previously. In addition we found agreement, to three figures, with the value of ␨ for case 2, with ␣⫽1.0, that was reported by Cassell and Williams.22 We also found apparent convergence in our numerical results as we increased N, the number of quadrature points used, and to reduce the possibility of FORTRAN errors, we have implemented two independent versions of the algorithm. Finally, we note that for case 2, the Williams model, the three-vector G problem can, as discussed by Williams and Cassell,22 be

solved as three consecutive scalar problems. This approach has been used by Bartz23 to confirm all of the results given in Tables I and II that refer to case 2. We have typically used N⫽50 to generate results for the temperature-jump coefficient ␨ and the temperature and density perturbations good to, say, five or six significant figures, and so we note that our FORTRAN implementation 共no special effort was made to make the code especially efficient兲 of our discrete-ordinates solution 共with N⫽50兲 runs in a few seconds on a 400 MHz Pentium-based PC. To have some idea of the merits of the CLF model, as we have used it here, we note that Loyalka9 and Sone, Ohwada, and Aoki10 give, respectively, for the case ␣⫽1, the results ␨⫽1.2486 and ␨⫽1.2482 for the case of the linearized Boltzmann equation relevant to hard-sphere collisions. If we consider these results to be the best available for the problem as defined in this work, then our use of the CLF model for the rigid-sphere case 共␨⫽1.206526兲 provides a modest improvement 共in regard to the temperature-jump coefficient for the case ␣⫽1兲 over the classical BGK model 共␨⫽1.302716兲. VII. FINAL COMMENTS

In concluding this work, we note that we have been able to extend the use of our analytical version of the discreteordinates method to solve the temperature-jump problem for a general version of the variable collision frequency model.

TABLE II. The temperature and density perturbations for the case ␣⫽0.5. Case 1

Case 2

Case 3

x

T(x)

N(x)

T(x)

N(x)

T(x)

N(x)

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 20.0

2.91597 3.18042 3.36278 3.52167 3.66754 3.80489 3.93615 4.06283 4.18593 4.30614 4.42400 5.52928 6.57466 7.59758 8.61013 9.61737 10.6217 11.6243 12.6260 13.6271 23.6291

⫺3.07437 ⫺3.31664 ⫺3.48323 ⫺3.62947 ⫺3.76478 ⫺3.89310 ⫺4.01653 ⫺4.13633 ⫺4.25334 ⫺4.36814 ⫺4.48113 ⫺5.55674 ⫺6.58912 ⫺7.60560 ⫺8.61476 ⫺9.62011 ⫺10.6234 ⫺11.6254 ⫺12.6267 ⫺13.6275 ⫺23.6291

3.10167 3.30560 3.45447 3.58758 3.71209 3.83110 3.94628 4.05866 4.16891 4.27749 4.38475 5.42014 6.43030 7.43378 8.43508 9.43559 10.4358 11.4359 12.4359 13.4359 23.4360

⫺3.27399 ⫺3.42542 ⫺3.54940 ⫺3.66556 ⫺3.77746 ⫺3.88665 ⫺3.99397 ⫺4.09990 ⫺4.20480 ⫺4.30888 ⫺4.41233 ⫺5.42888 ⫺6.43349 ⫺7.43503 ⫺8.43559 ⫺9.43581 ⫺10.4359 ⫺11.4359 ⫺12.4359 ⫺13.4360 ⫺23.4360

3.00508 3.23434 3.39748 3.54146 3.67483 3.80132 3.92295 4.04097 4.15620 4.26921 4.38044 5.44006 6.46062 7.46901 8.47273 9.47447 10.4753 11.4757 12.4759 13.4761 23.4762

⫺3.16704 ⫺3.35822 ⫺3.50093 ⫺3.63010 ⫺3.75193 ⫺3.86906 ⫺3.98291 ⫺4.09435 ⫺4.20394 ⫺4.31208 ⫺4.41906 ⫺5.45530 ⫺6.46735 ⫺7.47216 ⫺8.47426 ⫺9.47523 ⫺10.4757 ⫺11.4759 ⫺12.4761 ⫺13.4761 ⫺23.4762

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Phys. Fluids, Vol. 14, No. 1, January 2002

In addition to a formulation valid for a general form of the collision frequency, the algorithm was implemented to yield high-quality numerical results for three well-regarded forms. Since the reported solution is easy to evaluate and yields excellent numerical results, and since the developed code runs typically in a few seconds on a 400 MHz Pentium-based PC, we consider the solution ready for additional applications. It is clear that the formalism reported here can readily be used to solve other classical problems, in semi-infinite media and for plane-parallel channel flow, in rarefied-gas dynamics, when one of the three explicitly developed variants of the CLF model equation is required to give 共perhaps兲 more realistic results than the standard BGK model can provide. And finally, since the development reported here is also general, we believe that our solution can immediately be used with forms of the collision frequency additional to the three special cases considered. ACKNOWLEDGMENTS

The authors take this opportunity to thank S. K. Loyalka, F. Sharipov, and M. M. R. Williams for some helpful discussions concerning this 共and other兲 work. In addition, it is noted that the work of L.B.B. was supported in part by CNPq of Brazil. C. Cercignani, Mathematical Methods in Kinetic Theory 共Plenum, New York, 1969兲. 2 C. Cercignani, The Boltzmann Equation and its Applications 共SpringerVerlag, New York, 1988兲. 3 M. M. R. Williams, Mathematical Methods in Particle Transport Theory 共Butterworth, London, 1971兲. 4 F. Sharipov and V. Seleznev, ‘‘Data on internal rarefied gas flows,’’ J. Phys. Chem. Ref. Data 27, 657 共1998兲. 5 P. L. Bhatnagar, E. P. Gross, and M. Krook, ‘‘A model for collision processes in gases. I. Small amplitude processes in charged and neutral onecomponent systems,’’ Phys. Rev. 94, 511 共1954兲. 6 M. M. R. Williams, ‘‘Boundary-value problems in the kinetic theory of gases Part I. Slip flow,’’ J. Fluid Mech. 36, 145 共1969兲. 1

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