The method of elementary solutions for kinetic models with velocity-dependent collision frequency

BNNALS

The

OF

PHYSICS:

Method

40,

469-481

(1966)

of Elementary Velocity-Dependent

Solutions for Kinetic Models Collision Frequency*

CARLO

with

CERCIGNANI

The method of elementary solutions previousI> introdrlced for lrealillg problems in linearized kinet,ic theory of gases is extended to a gc~leral class of models wit,h velocity-dependent collision frequency. The previous treatment based on the Bhatnagar, (;ross, :tlld Krook model is :L particular rase of the present theory. The paper is devoted to steady shear flow problems. As BIL application the Framers problem is exactly solved in t.erms of quadratlues wit.hout specifyillp the model. ils a consequence a formula for the dependence of the slip cueflicietlt on the rollision frequency is given. I. INTI’LOl)UCTION

Models of the linearized Boltzmann equation have been repe:~tedly st’udictl t)y many authors. It is to be noted that a marked preferenw has been given to t~lw Bhatnagnr, Gross, and Krook (BGK) model (1) although other models have btwr proposed (2 --.$), and some of them also used (S-8) in prohlcms of sountl propagation, shear flow, and heat transfer. A common feature of the models used so far is that they are very close to the BGK model for what concerns the mathcm:~tic~:rl structure of the collision term. Essentially t’hcre is :I finite set, of relaxation frcquencies independent of t’he molecular velocity, and therefore the depcntlnwe of the collision frequency on the molecular VCkJcity does not appear in its ft~ll importance. It is felt that such models are good approximations for i\Iaxwell’s molet~~&s, and, perhaps, for any kind of molecules whose interaction (aan be described by :I potentfin with an angular cutoff of the kind suggested by Gr:kcl (9). However such a cutoff, which is very convenient mathematically, does ncJtSseem wry realistic* on physical grounds. Actually, a cutoff in physical space swms more suital+. The influence of the precise method of defining a cutof?’ potential on t,he various spect,ra arising in connection with t’he linearized BoltSzmann equation is wry peculiar. The changes in these spectra are very important for tinlct-dcl,cIltlcrlt * This research was sponsored under Cont,ract AF 61(052)-881 ITnited States Air Force.

by the United States through the European 469

Air Force Oflice of Scient,ific lIesearch Office of Aerospace Itesearch, (),\I:,

470

CERCIGNANI

problems since the spectrum of space transients dictates the asymptotic behavior far from physical boundaries in a very st’ringent way (10). If a physical cutoff is made, the linearized collision operator takes on the form I;h = Kh -

v(c)h

(1.1)

where Y(C) is a function of the molecular veIocity c such that V(C)/C goes to a con&ant when c + m, and h is the perturbation of a suitable Maxwellian fo . By analogy with the rigid spheres operator, one can conjecture that K is a completely continuous operator with respect to a Hilbert space H”, where the inner product is weighted by the Maxwellian f0 times Y(C). This means that K* 3 Y-1’2KY1’2 is self-adjoint and completely continuous with respect to a Hilbert space H, where the inner product is simply weighted by a Maxwellian. This result, in the special case of hard spheres, was proved by Hecke (II), more simply by Carleman (I$), and through a different procedure by Finkelstein (IS). It also follows from t.he complete continuity of K in H proved by Grad (3). Thanks to this complete continuity K can be expanded in a series of its square-summable cigenfunctions cpi (such that KP; = X~V(C)~~) :

Kh = v(c) gl Xdwi,

h)

(1.2)

where (f, g) denotes the inner product in H. Since the first five (oi are simply the five summantional invariants +;(i = 0, 2, 3, 4), and the corresponding eigenvalues are Xi = 1, a simple model of Eq. (1.1) is

(1.3) Here the #‘s are normalized in such a way that (v#i , $j) = 6ij . This model operator E shares with the original operator the properties: Lfh = 0;

(9, Ah) = (h, h)

(1.4)

and (h, Zh) S 0 Equations

(1.5)

(1.4) imply also that (#i,Eh)

= 0

(1.6)

We propose here to investigate the class of models defined by Eq. (1.3) in the case of steady problems in one-dimensional plane geomet’ry. The method to be used is that of finding separated variable solutions and then constructing the general solution by superpo&ion. Therefore the present study is an extension to t)he general model (1.3) of the previous research on the BGK model (14).

.4 MODEL

IN

KINETIC

1’71

THEORY

As was done in t,he simplest case we shall preliminarily split I<(~. ( 1.3 ) int80 simpkr equations by decoupling shear from heat transfer effects. Then pursuing the analogy wit’h ref. 14, we shall restrict to shear flow prt~l)lerns, leaving aside for the moment heat transfer effe&. In fact, although rnuc~h of t#he present theory can be cxt#ended to heat transfer situations, the not:ttion:Ll complexit,y and the impossibility of obtaining the explicit expression of so111e quantities in closed form make it preferable to defer the treatment of such prob lems. In the particular case of the BGK model t.his treat,mcntj has hem dcsuihd (15) t)ogcther with the analogous theory for the more general models propos(4 by Gross and Jackson (2) and Sirovich (3). By a proper choice of variables the equations to he solved are put’ in tjhcxU~IP form as the corresponding ones of the BGK model; the situat,ions which arisc~ are not essentially new and the generalization, though not, trivial, is straightsforward. The same cannot be said about time-dependent problems: thescl \V(W t’reatecl with the BGB model (16, l?) by means of a Laplace transform with respect to time. The same thing can be done with the present model, but th(a L’F sult,ing st,eady equation presents the new situation that, the c*ontinuous spcct rlml connected with space transients fills a twc)-dimensional region of the ~ornJ)lcs plane instead of lying along a line. This new fact changes considerably th(s whole matt,er in a very interesting way. Therefore a separak treatment of these problems seems worthwhile deferring to a subsequent. paper. In the present paper, after the general trent,ment of the met~hod for shcrar flow problems, the particular case of the Iiramers problem ( 14, 18) is complcttly worked out. In particular, the evaluation of the slip coeflkient corresponding to the general model (1.3) is reduced t,o quadrat~nres. II. SPLITTING

OF THE

ROI)I’:L

I’C/UATlOX

The Boltzmnnn equation with the model &lision t,erm int.roduc*cd in Sec+tiotl I can be written as follows when the problem exhibits a one-dimensional pl;nrc~ symmetry :

where L is given by Eq. (1.3) and cZis the s-component of the molecular WIGI(‘ity c. If we adopt a polar coordinate system in the velo&y space with (.r ;IS polar axis, me have:

where ,U = cos 0, while 0 and ‘p are the colatitutle

and

azimut’h of c. :1s a VOIIW

472

CERCIGNANI

quence, if we write h = hl (2,

C, p)

+

~-~‘~h~(z,

c, p)

cosp

+

n-li2h3(x,

c, P)

sin9

(2.3) + hoc& 6 CL,P> where hl , hp , h, are so chosen that Js* ho& = Sf”h, cos cp&p = SiThO sin (O C&P= 0, ho , h, , hz , h, arc lying in four orthogonal subspaces, which are invariant with respect to z. Therefore Eq. (2.1) split’s as follows:

CP2 = v(c)k/o(vgo, hl) + !/I(w, hl) + qdvos, hd - h,l WE

= v(c)[qi(vqi,

dho YG=

-

(i

h;) - hi]

=

(2.4)

2, 3)

(2.5)

v(c)ho

Equation (2.6) is easily treated, since it is simply a differential equation where c and p play the role of parameters. Equation (2.4) describes heat transfer effects, while Eqs. (2.5) are connected with the shear effects due to motions in the y- and x-direction respectively. We shall consider in detail one of the two identical equations (2.5), as mentioned in the Introduction. III.

ELEMENTARY

SOLUTIONS

Let us rewrite I:‘2

= V(C)

OF

THE

STEADY

RHEAK

FLOW

EQUATION

Eq. (2.5) as follows: c(l

- py

;

1

m *s0

+1

& s -1

cl/.~lv(~&?-~~~(l

-

C114)1’@.hi(c1,/.LI) - hi

1

(3.1)

where the dependence on some of the variables has not been explicity indicated when obvious. In Eq. (3.1) we have put 2RT = 1 if T is the temperature of the gas and R the gas constant. We have also introduced the quantity:

jj=- 4 m v( c)c4e-c2 clc 3 s0 In what follows v(c) will be supposed to be a positive increasing function for 00. Let us now introduce, in place of the variables hi, P, c, the variables Y(x, U, c), U, c, where Y and u are defined as follows:

OgCs

hi = 27?c(l u = c/h/v(c)

- u’)~‘~ Y(z, U, c)

(3.3)

A MODEL

IN

where the index i has been omitted, written as follows:

where

k and c = c(u)

KINETIC

173

THEORY

as is clearly possible. Then Eq. (3.1) can be

are defined by

-=c “(Cl

k=lii&; and we have introduced

the following

auxiliary

Then multiplying Eq. (3.6) by e-c2[v(~)]2 {c2 respect to c from c(u) to 00 , we have

I 7L(

(3.5 )

function

u~[v(c)]~]

and integrating

u g + 2 = 4(u)s: d?Al 2(x,ZCl 1

with

(

Z4.i

i

where

It is useful to note that + (u) >= 0 (since integration), u +‘(zc) 5 0, and

cl/v(cl)

2 / ?hl/ in the whole range of

The latter result is easily obtained by a partial integration. Equation (3.7 ) is very similar to equations which have already been studied (14, 19, 20). The variable u has a noticeable physical significance, being the projection on t’he z-direct.ion of t’he free path of a molecule. It degenerates into a quant,ity proport,ional to CP = c, if the collision frequency is constant (BGK model; ref. 14), or to Jo if the free path averaged over the different directions (C/V(C) 1) is constant (one-velocity equation of neutrons; ref. 19). It is well know (20) that Eq. (3.7) has separated variable solutions given by &(z,

u) = e-““f”(U)

c#J(u)

(3.10

j

where f,(u)

= P -f!-v-u

+ ph)d(v

- u)

(3.11 )

474 and

CERCIGNANI

P means Cauchy principal part, while p(u) is given by

(3.12) In such a way: (3.13) Since also 4(u) and (Z - u)$(u) general solution of this equation:

Z(z, u) = +(u>

satisfy

Eq. (3.7), we can write

AO + Al(z - u) + [:

A(v)e-“‘“fdu)

du

down

>

the

(3.14)

No discrete values of v are allowed, since if v = vi were a discrete value of the v-spectrum, we would have: k udu) du = 0 1--I; vi - u i.e., since 4(u)

(3.15)

is an even function

k u2h4 & = 0 s0 vi7 - IL2 Now if vi is a solution gives :

(eventually Im (vi’) l

complex), k

(3.16)

the imaginary

part of Eq. (3.15)

&h> , vi2 _ u2 ,2 ch = 0

Since the integral cannot be zero, we must have Im (vi”) = 0, i.e., either Re vi = 0 or Im vi = 0. In the first case Eq. (3.16) cannot be satisfied, since the function under the sign of the integral is negative. In the second case the same function should always be positive since 1vi 1 > k. The conclusion is that no finite v of the complex plane with a cut on the real axis from -k to k can belong to the discrete spectrum. The general solution of Eq. (3.4) can now be easily written down since it is a matter of solving an ordinary differential equation with constant coefficients and given source term. We obtain

k du Y(z,u,c)= $i A0+ A1(x- u>+ s-*A(u)e-“‘*y&L)

+ B(u, c)ewziu\

(3.17)

-4 MODEL

where B(zL, c) is an arbitrary

In-

KINETIC

function,

4i.T

THEOHI

such t,hat, (3.1s)

In the new variables

the mass velocity is given by: m q( J. ) = zr--l,“’ fllll rzc1 v(cl)c1((‘12 - u,“[v(r,,]‘]r r -1s c(u1)

.-I? l’i x’, u, Cl )

ci',.l!i )

and consequent,ly : ( :cN )

( :L! 1 1 ( :i.l!‘L I

( :;.“:; ) Analogously 7r,?

=

‘)ipa

the z-component k 00

-l/2

dll

s-k

s c(u)

of t’he stress is given by:

dcl[v(cl)]‘cl~rl~

-

U2[Y(C1

f]c’

c11 I’(

.c, ‘U, 1’1 I

h

= 2plr = “pa

-11’1 I,

(111uZ(s,

?A)

i :<.,4 I

-, :L’ k IL“+ ( 11) du I,

where p is the density of the gas. Equation (3.24) shows that the stressis I’OII&ant and proportional to A1 . The constancy is correct since it reflcctJs the conservation of the x-component of momentum. The integral which appcws in Eq. (3.24 ) is proportional to the viscosity coefficient p asswiatrd with the 111~ NIPI under invcstigat.ion. As a matter of fact we find P -= P

-l,? 4na

k I-I;

u”+(u) du

i ::.,.-I I

where p is the pressure related to p and T by p = pRT ( = p/2 in tlw c*hoscw unit,s) .

476

CERCIGNANI IV.

COMPLETENESS

PROPERTIES

The completeness properties of the elementary solutions of Eq. (3.4) can be easily deduced from the treatment of a slightly more general kind of equation studies by Aamodt and Case (20). Here the main results will be recalled and explicit formulas given. Generally speaking one can establish the following THEOREM. The set of solutions (f%(u)} (CY=< v =< p; -k < a! < p < k) form a complete set on the class of functions which satisfy a Halder condition in cv < u < p, and are such that lim Z(u) U’Y

5 C j u - y I-’

(4.1)

where y = CL,/3. In the casethat CY= -k or (respectively and) p = k, thefunctions f*(u) or (respectively and) fm’(u) = u are to be added to the set in order to have a completeset. If k = 00 and either cx = -k or tr = k, then the following condititrn is also to be satisjied: 0 &(u) du < cfz (4.2) scl Two casesare particularly interesting: the full-range completeness (a = -k, /? = k) and the half-range completeness (a = 0, /3 = k). For these casesit is convenient to give explicit formulas. To be precise, if Z(u)

= 4(u) po

+ Al u + I: A(v)j,(u)

du],

i-k

< u < k)

(4.3)

then the coefficients A,,, Al , A (v) are given by Ao = [I’

&J(U)

du]-’ 1’ u2Z(u) du

-k

(4.4)

k

A1= [I1 t&“&u)du]-l I: uZ(u) du A(V) = [v+(v)]-‘([p(v>J”

+ T~v*]-~

(4.5)

’ ufv(u>Z(u) I-k

du

Analogously, if Z(u) = 4(u)

Ao + I^ A(v).L(u)

do]

(0 < u < k)

(4.7)

0

we have

k A0 =

A(v)

u[X( -u>l-‘c$(u>

= [v#J(v)l-‘{[p(v)]’

+ 7%)

u[X( -u>]-%‘(u)

s0

k 4X( -u)l-tfv(u)Z(u)

du du

(4.8) (4.9)

A MODEL

IN

KIZiETIC

477

THEORY

Here for any complex x: (4.10)

G(t)

i-t.1 I)

= log pS$

the branch of t,he logarit,hm being such that G(t) 3 0 when t + k; with t*his choice, G(t) -+ -2ni, when t -+ 0. The function X(x) defined by Eq. (4.10) is far from being an elementary function. However it satisfies some important identities, which make t8hc I~KLnipulation of integrals involving X(x) much easier than would be esp(~ct8cd. These identiCes can be obtained by standard methods i 14, l/i, 18) and NV:

S(r)S(

= M(z)

-2)

1

Sk z&b(,lL) f/U --’ [ -k

Here M(z) is :L function

= 1; ;gz

rl1c

that’, when Im z -+ 0, goes into M*(t)

= rjqt)fp(t)

Comparison of t#he limiting behavior as x t and Eq. (4.12) gives the following result

*

nit] 30 of X( z ) :as given by Eq. ( 1.10)

[s’ &(u)q =-1 Ioig$)clt --I; Ry using this result, Eq. (4.6) can be writ,ten

as follows:

(4.17)

375

CERCIGNANI

Equations (4.9) and (4.17) can be interpreted by saying that the eigensolutions J”~(zL) (0 < v < k) andf- = 1 are orthogonal on the segment (0, /c) with respect to the weight’: 1.(u) = ur$(u)[-X( 1’. APPLICATION

OF

THE

GENERAL

-u)]-’

METHOD

TO

(4.19) THE

KRAMERS

PROBLEM

In hhis section we shall apply the above results to a typical problem of kinetic theory investigated by many authors, and usually referred to as the Garners problem (see refs. 16 and 18) for a list of previous papers on the subject). It consists in finding the molecule distribution function of a gas in the following situation: t.he gas fills the half-space J: > 0 bounded by a physical wall in the plane J = 0, and is nonuniform because of a gradient along the x-axis of the Zcomponent of the mass velocity; this gradient goes to a constant N when x goes to infinity. It is seen that this problem is a limiting case of plane Couette flow (when one of the plates is pushed to infinity). The classical no-slip solution of this problem would give simply a x-component of the mass velocity equal to HZ. It is, therefore, convenient to linearize the problem about a Naxwellian endowed with a translational velocity in the xdirection equal to XX (21) . Because of the nonuniformity of this Maxwellian distribution, linearization gives an inhomogeneous linearized Boltzmann equation Zxc, c, + c, f$ = Lh where c is the molecular velocity referred to a local frame endowed with a translational velocity in the z-direction equal to XX. Equation (5.1) can be reduced to the homogeneous linearized Boltzmann equation by subtracting a particular solution. A particular solution not depending on x is given by the ChapmanEnskog theory for any molecular model. If we choose a collision term as given by Eq. (3) we have h = -ZKC,U where zc = c,/Y(c) given by:

+ Zc,Y(x,

u, c)

and Y (z, U, c) satisfies Eq. (3.4). The mass velocity

q(x) = xx + 27r-1’2

(5.2) is now

(5.3)

the first term being the contribution from the maxwellian. Concerning the boundary conditions we shall assume that the molecules are re-emitted according to a Maxwellian distribution completely accomodated to the state of the wall (for different assumptions, see ref. 18). Therefore the boundary

A

MODEL

IN

KIXETIC

17!)

THEORY

conditjion for h rends as follows h(0, cj = 0

cc, > 0)

(*5.-1-i

and this in t8crms of 2’ becomes (;i.., )

Y(0, u, c) = MU Aacording t,o t,he previous discussion (Section III), thca general solution ( 3.4 ) sat,isfying t,he condition of boundedness at infinity is l-ix,

dv + I?(& (‘)Ci( 4

24,cl = + il” + Ik A4(V)e-“~vfv(uj -0 1

,;

of I:((.

(.-,.ci 1

where B(u, c) satisfies Eq. (3.18). The condition t,o be satisfied at t,he plat#e gives: imc. = Ao +

I0

k A(v)f*(u)

dv + R!u, f’)

( .-b.T’

It easily follows that B(u, c) = 0 and consequently solving Eq. (.5.7) means expanding the function ML according to the half-range completeness theorem. Therefore A, and A(V) are given by Eqs. (4.18) and (4.9) respectively, if FW is subst#ituted for Z(U). The result is as follows:

.k

A” = -ct-l

I0

tan-“[at/p(

f )I o’t

( .i.s

I

k

.-I(V)

= M~x( -v)[+(vj]-'{[p(v)]"

+ ;Y-1

&(,

s v-1.

(1) (111

( -FL!))

where use has been made of Eq. (4.12) and the identity I, fLs-( I’) lk t’&)[x( -t,1-’ df s

c1t = 0p(t) - 7rif

zz- 1 L tan-‘[7rt/p(t)] r s0

rlf

which is obt,ained from Eqs. (4.10) and (4.13) by asymptotical1.y expanding for large Z. Substituting Eqs. (5.8) and (5.9) into Eq. (5.7) (with B(u, c) = 0) gives the solution of the Kramers problem. The mass velocity is readily obtained from Eqs. ~~5.3) and (5.6): h-

tlv.\ (ii.11 ) i where n(v) is given by Eq. (3.2), A, by Eq. (5.8), and A(v) by Eq. (5.9). From Eq. (5.11) we recognize that AOF-’ ’1sthe macroscopic slip on the plate: q(x) = JCZ++

i

Ao+

10

il(V)n(v)e-“”

430

CERCIGNANI

it has the form [H where [ is the slip coefficient

which

turns out to be

k [

=

-T-l s0

ban?[7rt/p

( t )]

(5.12)

clt

In such a way the evaluation of the slip coefficient has been reported to quadratures for an arbitrary V(C). The evaluation of the microscopic slip, i.e., the velocity of the gas at the wall, can be reduced to even simpler quadratures. As a matter of fact, we have: k q(0) = -x i tan-‘[&/p(t)] clt b”lr 0

s k

-

0

XC-V)) 4(v)

1

h

([io(v)l”

+

g2v2)

s --k 112

=

-K

[x(o)]-’

=

1

7t

(5.13)

=H

where use of Eqs. (4.13), (4.14), and (3.25) has been made. Analogously we can evaluate analytically (in terms of the function X(z) which thus receives a physical interpretation) the distribution function of the molecules arriving at the plate. We have: Y(0, u) = x(u -

[x(-u)]-‘)

where Eq. (4.13) has been used. Then h (the perturbation distribution) is given by: h(0, c) = - 2,&(X VI. CONCLUDING

[-&]F’

(5.14) of the maxwellian (5.15)

REMARKS

We have shown that the technique of separate variables solutions can be extended to models of the linearized Boltzmann equation which allow a dependence of the collision frequency on the molecular speed. We have also shown that the present method can give useful information on the boundary value problems for the linearized Boltzmann equation. In particular we have found an analytical solution to the Kramers problem. III other cases,as e.g. problems between parallel plates, the method could be used to discuss the solution, as was done in refs. 22 and 23 with the BGI< model. The technique turns out to be also suitable for treating the fundamental problem of completing the Chapman-Enskog and Hilbert theories wit)h proper boundary conditions. This problem is much more difficult than the analogous problem of supplementing the mentioned theories with proper initial conditions, which was solved by Grad in a general fashion (24). A sketch of the procedure to

A MODEL

IiV

KINETIC

THEORY

4Sl

he used to attack the problem of the “Knudsen layers” was given by the author (R5,15) together with an illust’ration of the possible applications of the technique of elementary solutions to this fundamental question. The present extension of the previous work appears interesting ancl by no means trivial, but, one must recognize that no essentially new situations arise wit,h respect to t>he BGK model. The same thing cannot be said about problems depending 011 time as well as on a space coordinate. This further extension appea,rs very int,eresting and promising, bot’h from the mathematical and thr physical point of view. A paper on this subject’ will be presented shortly. RECEIVED:

h&y

%,

1966 REFERENCES

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