On slip coefficients of polyatomic gases

Physica 143A (1987) 261-286 North-Holland. Amsterdam

ON SLIP COEFFICIENTS B.I.M.

TEN

Huygens Laboratorium

BOSCH

OF POLYATOMIC and J.J.M.

GASES

BEENAKKER

der Rijksuniversiteit, Leiden,

The Netherlands

I. KUSCER Department of Physics, University of Ljubljana,

Received

1 December

Yugoslavia

1986

A variational scheme is presented to evaluate field-dependent slip coefficients of polyatomic gases in a plane-parallel geometry. With the assumption of perfect accommodation at the surface and with bulk test functions only, fair agreement is obtained between theory and experiment. Also, it is shown that the present method is equivalent to a previously used variational technique, if bulk test functions are used in both cases. Finally, a comparison is made with Maxwell’s and Eddington’s methods. The equivalence of the latter method with the variational scheme is demonstrated.

1. Introduction Angular-momentum alignment or ‘polarization’ may occur in polyatomic gases out of equilibrium, because of non-spherical interaction of the molecules with each other or with the surfaces of the enclosing vessel. One of the ways to study this is to measure

changes

of transport

coefficients

caused

by a magnetic

field. These so-called Senftleben-Beenakker effects were first observed and analyzed in the hydrodynamic regime, where they are by now well understood’). More recently, experimental attention has also been paid to the near-hydrodynamic regime. Here the mean free path 1 is no longer negligibly small in comparison to the width of the channel b but the boundary layers may still be considered separate. Typically the Knudsen number Kn = l/b is about 0.1. Reliable data are by now available on field-dependent velocity slip*), thermal creep3), the viscomagnetic heat flux4) and the viscomagnetic diffusive flux5). In this paper we wish to perform a systematic study of these field-dependent rarefied-gas phenomena on the basis of kinetic theory. Comparison with 0378-4371 I87 /$03.50 0 (North-Holland Physics

Elsevier Science Publishers Publishing Division)

B .V.

262

B.I.M. TEN BOSCH et al

experimental

results

may then

answer

the question

to what extent

production

of polarization at the surface plays a role in these effects. We will restrict ourselves to pure gases consisting of diatomic diamagnetic molecules. Angular momentum surface

will be treated

classically,

and vibrational

of the walls shall be macroscopically

left-right

uniform,

excitation

ignored.

The

rotationally

invariant

and

symmetric.

The method to be employed is a variational scheme based on the linearized Boltzmann equation and boundary conditions. No fitting to experimental data is invoked, as is the case in previous attempts to describe field-dependent boundary-layer phenomena6 “). I n contrast to the method of ref. 10 (henceforth referred to as II), the variational functional to be used will not be related to flow coefficients but to slip coefficients, which occur naturally in a description of boundary layers. In section 2 the variational principle will be outlined, and Onsager’s reciprocity for slip coefficients shown to hold also in the variational approximation. Section 3 deals with applications of the technique to several field-dependent flow problems and in section 4 a link is established between the two variational methods. Finally, section 5 contains a discussion of Maxwell’s variational

and Eddington’s scheme.

2. A variational

methods,

and

a comparison

is made

with

the

principle

Bearing in mind that the boundary layers do not overlap, we shall idealize the problem by considering a gas filling the half-space above a wall at z = 0. The gas is subjected to a uniform magnetic field B, and constant gradients of temperature and flow velocity, VT and v’ = dvidz with v = (u,, u?), are present in the bulk. follows:

where

f” is a Maxwellian

T,(x> Y) = W) f”

For not too large z the distribution

=

+ q

at rest corresponding

function

to the surface

is linearized

as

temperature

- 7, T,

p(m/2~)“‘*(kT,)~“‘*(4~ZkT,)-’

exp(-EIkT,)

,

(2)

and 4” and 4” are correction terms. Most of the concepts and notation are taken from ref. 11, which is henceforth referred to as I. The symbol (( indicates ‘x-, y-components only’. The unknown functions +“, p = q, n-, satisfy the following equation and boundary condition:

ON SLIP COEFFICIENTS

OF POLYATOMIC

GASES

263

(W+i9+cZ&)+“=-PP,

(3a)

~“(O,c,J)=PO~~(O,c,J), c,>o,

(3b)

where

the microscopic

introduced,

fluxes

Q 4 = (E - zkT)c

as well as the Boltzmann

and time reversal operators 9, Chapman-Enskog-like solution,

collision,

and surface

W” = mc,c,, scattering,

have

been

precession

P,i9 and 0. For z + m, 4’ must approach

a

The 2 x 2 and 2 X 3 matrices of slip coefficients y”” and yV4 defined thereby account for velocity slip driven by either a velocity or a temperature gradient, jump in the boundary respectively. Similarly, yfP accounts for a temperature layer; yfP is a 1 X 2 or a 1 X 3 matrix according to whether p = rr or /3 = 4. All these coefficients are uniquely determined by the stated Boltzmann equation and boundary condition. Let us introduce the excess boundary-layer heat flux A4 according to

(5)

bulk heat flux, both where q,l and qf; are the true heat flux and the extrapolated parallel to the wall. The linear dependence of A” upon the gradients may be specified in terms of yet another set of slip coefficients yip, VT AY=y;;q-T+~;*~~‘r

(6)

with yif” a 2 x 3, and yi” a 2 x 2 matrix. The definition of the slip coefficients introduced above, which seems natural in the present context, differs from the one in I by viscosity and thermal conductivity

Y

TY

-

-

factors.

sty 3

The translation

r;;” = 5;;”

3

is

(7~ d)

(7e, f)

264

B.I.M.

where

the thermal

Here

and henceforth

[(h,(h,)]=

conductivity

BOSCH

and planar

the following

f jdz

TEN

inner

et al

viscosity

products

tensors

are given

by

are used:

jdI-h:/“h,.

the Hilbert

(i,ie)’ = -iZ

in X, and Z, ,

(10)

(c;P)’

in SC+ .

(11)

= cz P

spaces

X,,

CT+ and X%. The adjoints

defining, respectively, i_Y and c, P are

of %,

The heat fluxes in eq. (S), in case of a flow of type /3, contain the factors respectively. Definition (6) thus leads (!P~Ic$~) and -(!Pil(%! +L9?)‘Pa), to the following kinetic expressions for the riP,

We have to assume that integrations over z like in eqs. (5) and (12) do not lead to divergences. To obtain an approximation for some of the slip coefficients, we employ a variational method which has been used by several authors’2m’h) for monatomic gases. For the general background of the method the book by Finlayson”) may be consulted. The extension to polyatomic gases and inclusion of a magnetic field requires some modification. For each slip problem we

ON SLIP COEFFICIENTS

introduce

VP=

a tensor

of bilinear

[(O(f&, +(92 +

variational

OF POLYATOMIC

Yap, CX,p = q, rr,

functionals

-iif))‘t”)l(!S!

+iZ+

265

GASES

c, &)+*+

@)]

mb,Ic*(l- fw4)+I*=,

+ (0(9? The trial functions assumed to satisfy

- i.2)‘ly*(c,(&

+ (2

+ i,Fe)-‘*p))I,=O.

(13)

c#+ and 42 depend upon c, J and z, and in view of (4) are the following boundary conditions at infinity:

( 14b) where the parameters /.L, V, p and u stand in place of, but are not to be identified with, the slip coefficients. With some manipulation of the operators and a partial integration over z, one finds from eq. (13) the first variation of

It follows

that 6Y”’ = 0 if and only if C/J*satisfies

equations with opposite field and with P* instead the limiting behaviour of 4, for z--+ 00 must then +1

4

MC,,

The stationary

-

y”“(-B) value

+ (E - ikT)yq”(-B)

(3),

and +i the same

- (9 - iL!?)‘V

.

with (4),

(16)

of Yap is

Yzf = (O(%! - i.Z)‘Q”lc,(+P To establish 0(9X - i2)-‘Wa

eqs.

of W’. In analogy be

+ (6% + i~))‘!P’))I,=,

the meaning of Yzf, we take the with both sides of eq. (3a). After

.

(17)

[( ( )] inner product of some reshuffling of the

B.I.M.

266

operators

and a partial

integration

TEN BOSCH

et al

over z we obtain

the identity

The asymptotic form (4) of 4’ serves to relate the second term of this expression to slip coefficients, whereas the third term is linked to Yzf, eq. (17). So we get

+ [(

OW”l+” + (% +

iZ))‘W’)]

(18)

The second term on the right-hand side vanishes if Wf or 1v” are substituted for ?P”. Namely, if we take the ( ( ) mner product of E - i kT or mc,, with eq. (3a), three terms drop out and we see that (Py]+“) and (lu”l+“) are constants. Substitution of the asymptotic form (4) then shows that the constants are equal to -( V’f](% + i_Y)‘!Pa) and -(w~](% + iZ))‘~p), respectively. On the other hand, a comparison with eq. (12) shows that in case !P’ = !Pi the double-bracket inner product in eq. (18) is equal to -kTyiP/n. Hence, the following relations are obtained (with i, j, k = x, y and also k = z for p = q):

Y"" rk. rt

y@

rk.\t

=

(19c)

Approximate values for the slip coefficients may now be obtained in the standard way. First one chooses trial functions 4, and C#J*which satisfy the conditions (14) and depend upon a finite set of trial parameters. After the Y”’ are made stationary with respect to variations of the trial parameters, eqs. (19) to approximations +“’ for serve to relate the restricted stationary values, e:, the slip coefficients. We are now going to see that variational approximations for the yup do not violate the Onsager symmetries established in I. First, we write the functionals (13) in an alternative way,

261

ON SLIP COEFFICIENTS OF POLYATOMIC GASES

fwb1l@45)+I.=,

(c,(l-

+

+ (c,(&

+ (% - iZ)-‘ly”)lO(%

+ (O(Z

- iL.?)‘W”Ic,(%

+ iL!?)llyp)Iz_O ,

+ iZ)-‘!P’)

(20)

where the symmetry properties (9)-( 11) were used. The first three terms on the right constitute YP”‘(-B), the dagger denoting the transposition. When we request that

SP(B) = 0

SP-B)

)

=0 )

where the variations are performed within a family of trial functions 4, and &, the same solutions are obtained in both cases. Yet the stationary values of the two restricted functionals differ by a constant term, as seen from eq. (20), P$(B)

= ?fT+(-B)

+ (O(,$! - iZ)-‘ly~Ic,(%

With eqs. (19) the Onsager follow (i, j, k = x, y):

relations

y”gyB) + z-A,,(B)y,4,4(B) = y”;f(-B)

for the

approximate

+ TA,,(-B)y”,4y(-B)

r”;;(B) + TA;,(B)y”,4,9(B) = TA,,(-B)7;;(-B)

The tensor

of Burnett

A’ = - &

(O(%

To reach complete

heat flux coefficients - i~))‘lyyIc,(~

accord

has been

relations

(21)

slip coefficients

)

(224 (224

)

+ i_!Z)‘!PV)

with the Onsager

.

+ iZ)-‘!Pp)

introduced, . (1.62),

(23) (1.76) and (1.79),

268

B.I.M.

we must ignore discrepancy

products

with

approximation justification

TEN

BOSCH

of field-induced

I is not

was made

components

conceptual,

there

for neglecting

et al

however,

of A and the y”‘. since

as well (cf. the remarks

such products

and we will stick to this approximation

implicitly

the

This same

after eq. (1.69)).

is the smallness

The

of field effects”),

in the sequel.

3. Specific examples Three applications of the above variational principle will now be given, and a comparison with experimental results made. The effects to be analyzed are the transverse viscomagnetic slip, the transverse thermal creep, and the transverse viscomagnetic heat flux. Only the simplest possible trial functions will be employed, i.e. those which for any z have the bulk form (14),

c#+ = mc,, *p + (E

- zkT)v - (% - i5!?)‘!P”

+2 = mc,, . p + (E -

,

(244

$kT)a - (2 + i_Y)- ‘!PP .

(24b)

The approximations

for the slip coefficients resulting from ?zf contain inner quantities with the first products involving (9? + i2!?~ ’ and P. Whereas operator can in turn be re-expressed in terms of known matrix elements of 92, a guess must be made for the surface scattering operator to make a comparison with experiment possible. As in II, we assume that the surface is perfectly accommodating, Ph = (2nm/kT)“2(c,Ih)+ which results

3.1.

in considerable

,

(25)

simplification.

The transverse viscomagnetic slip

First we will summarize some experimental results. With B = (0,0, B), viscomagnetic slip has been measured for N, indirectly through an investigation of the pressure dependence of Ly,z IL”“. The tensor L”” relates the gapaveraged flow velocity V to the applied pressure gradient yip, (26) whereas

L””

denotes

the

field-free

coefficient.

In

the

near-hydrodynamic

ON SLIP COEFFICIENTS

regime

Breunese

et al.*) found

L;j: qP “” L --F(Kn)‘%&

OF POLYATOMIC

a scaling

behaviour

GASES

according

269

to

5 (

(27)

) ’

(28) with C, = 4(kTln-m)“*, and 71the field-free viscosity. The functions F and G are responsible for the pressure dependence of the magnitude and the position of the effect. The Lorentzian functions

(2% b) are used here and henceforth. Expressions for the non-sphericity parameter angle so*,, in terms of effective cross sections 9 “2Wand the effective precession G(;,z.z,) ‘) are given in the appendix. The scaling functions F and G were approximated by F=l+n,,Kn+B(Kn*),

G=l+n,Kn+Q(Kn*),

(3Oa, b)

with Knudsen correction parameters nP = 10.7 * 0.5, II, = 2.7 2 0.5. To extract from eq. (27) an expression for the velocity slip coefficient we note that up to order Kn this equation may alternatively be written L;.Z U” - - @“‘“g( L On the other

L”” = where

$

hand, b’(S

6 denotes U” L yx=_ “” L

A comparison

502m)(l

+

according

(n,

-

np)

+ f y”” + Q(Kn’))

+ E 7;:

-

to eq. (1.51)

the two-dimensional

%

Kn

- 7-l

2n,

Wl

-f(502?r))).

(31)

and eq. (7a) we have ,

unit tensor.

(32) Therefore.

+ Q(Kn’).

with eq. (31) confirms

y;T,“, as

(33) the bulk formula”)

(34)

270

B.I.M.

and also yields

an expression

1 6 (c,“%

y;;:”

TEN

BOSCH

et al.

for the slip coefficient,

5o*,)(n,

- n, + 250

(35)

- f( 502?r)))

Let us now turn to the variational scheme. With the bulk trial functions (24) the first term on the right-hand side of eq. (13) vanishes and the last one simplifies,

so that

IcAl - W(mc,,

‘P + (E - ;kT)a

- (% + iZ’)‘+~))+l,_,I

kT + y PI/ *P Putting the derivatives equations

(36)

with respect

to p, V, p and u equal to zero, we get the

(c,,Ic,(l- P@)(mc,,*p -

(3 + iT))‘VT))

(E - ;kTlc,(l

- $kT)o

- PO)((E

kT y ‘lit + (WC,, (O((E

- ikT)v

- (3 - i5!T)‘ly”)lcZ(l

account,

- iZ)-‘ly”lc,(l

X (mc,, .p + (E - $kT)a

- PO)mc,,), - PO)(E

When eqs. (19a) and (36) are taken into being omitted in the rest of this section) * (O(%

(374

- (%! + i_Y)‘TT))+

* P - ( 92 - iZ))‘lv”)[cZ(l

Y

r = 0 ,

- ;kT)) it follows

= 0,

WI

= 0,

(37c)

,_ = 0.

(37d)

that

tildes

(the

- PO)

- (% + i.Z)‘V”))

+ ,

(38)

with P =

(c,,IM - fwmc,,L’ *(c,,Ic,(l

- PO)(%!

+ iZ))‘Tm)

+ ,

(39)

and (E - ikTlc,(lu=

(E-

PO)(%

;kTlc,(l-PO)(E-

+ Z!?)‘W”) ;kT))+

+ (40)

ON SLIP COEFFICIENTS

With the assumption of field-effect

of perfect

coefficients,

OF POLYATOMIC

accommodation

271

GASES

(25) and neglecting

products

we get the result

(41) where

another

2 x 2 tensor

and its fieldless

value

An6 were introduced.

(42) In the appendix approximations are given for q, nYX, An and A;X in terms of effective cross sections. With eqs. (A.4), (A.6) and (A.7) eq. (41) becomes

(43) We see that scaling eq. (35), if 15 np = s

K

according

;+2

15 G(20) n, = 8 G(02T)

1

+p

to eq. (27) is recovered,

since eq. (43) reduces

S(20) E(02n)

to

(444

1’

(44b)

’

just as in eqs. (11.41). With the effective cross sections for N, listed in table I we obtain np = 9.4 and nY = 2.7, which is in satisfactory agreement with the previously quoted experimental values.

TABLEI Effective cross sections and gyromagnetic ratios for CO and N, used in this section. Sources: refs. 19 and 20.

G(20) 6(027r) s(p) 2( 1010) Z(1001) XX 5( 12q) ‘G(:;:, 5( :t’) V

10~” m’ 10~” m’ lo-z0 m.’ lo-*” m.’ loo*’ rn.’ lo-*” m2 lo-*” rn’ lo-‘” m’ 10~LOmL 10” (Ts‘)~’

co

N*

34.7 32.5 2.02 29 30 5.5 49 -0.91 4.1 -1.29

34.6 23.7 1.50 27 30 4.0 48 -0.67 4.4 -1.33

212

3.2.

B.I.M.

The transverse

Experimentally, Hermans et al.‘), of Eggermont and

thermal

TEN

BOSCH

et al

creep

this phenomenon has recently been reinvestigated in CO by who corrected systematic errors present in the original work

et a1.2’). The temperature

gradient

B = (0,B sin 0, B cos 0). The results

was taken

showed

in the z-direction,

a simple

orientational

dependence,

PVx -

PTY “;”

-L=-sin0(H+ZCOS20),

4,

(45)

A

where v, is the bulk creep velocity, q, the perpendicular heat flux and A the field-free thermal conductivity. The functions H and I depend upon B/p and are plotted in fig. 1. As we shall see, the variational representation simplifies if these functions are decomposed according to Vestner’), H=S”Y+A;Y,

I=2A;“y-A2”y,

A;‘(B/p)

= A;‘(2B/p).

(46)

in the kinetic theory for bulk Decompositions as in eqs. (46) occur naturally Fig. 2 shows 6”“ and A;i’ as fitted to the experimental transport coefficients. data. Since the derivation of the expression for the creep coefficient riz is

Fig. 1. Thermal creep in CO at room temperature. The coefficients H and I are explained by eq. (45). The symbols indicate experimental data from ref. 3, whereas the drawn curves show the variational results.

ON SLIP COEFFICIENTS

OF POLYATOMIC

273

GASES

05

pv, -9;

I

0

01

-

Blp

10

1

Fig. 2. Thermal creep in CO. The dashed (46) to the experimental data. The drawn

riz” = p - &

vi’

is quoted**),

* (O(CB - i2)-‘!Py”Icz(1

X(mc,,*p+(E-

100

curves show 6”“ and A;'as obtained from a fit of eqs. curves correspond to eq. (51) and (52).

as for ynm, only the result

analogous

mTlPa

- PO)

+ i2)-‘We))+,

ikT)a-(2

(47)

with P =

(c,,Ic,(l - P@)mc,,);’ - ( cIIIcz(l

PO)(%! + i2Z)‘ly~)

-

,

(48)

and

(E - ikTlcZ(l cr=

(E-

- PO)(%

+ i2))‘P’f)

;kTlc,(l-PO)(E-

;kT))+

+ ’

(49)

perfect Let us consider r,“,” and B = (0, B sin 8, B cos O), again assuming accommodation. By aid of the approximate relations (A.8) and (A.9) eq. (47) reduces to y;;(B)

=

&

A:,(B)

+& (n:,(-B)

- TA’;;-B)) ,

(50)

by eqs. (A. 10) and (A.ll). where the coefficients v,, and A:, are explained To make a comparison with experiment possible we employ approximations

B.I.M.

274

(A.14)-(A.23) cross sections. indeed

TEN BOSCH et al.

for hi,, 14:x and n,, in terms It is found that the variational

be cast into the form of eq. (45),

of quantities involving effective result for the ratio pu,Iqz may

with

(51)

substituted into eqs. (46). Cross sections for CO, needed for the evaluation of angle t,,,. the non-sphericity parameters $’ and $“‘, the effective precession and the quantities At/h and S, are listed in table I. The variational results for H, as curves in figs. 1 and 2. As in the case of I, aVq and A;” are presented viscomagnetic slip, the agreement with experimental data is satisfactory. 3.3.

The transverse viscomngnetic heat flux

This effect has been investigated in CO and N, by Eggermont et al.J). For the gap-averaged heat flux qV due to an applied pressure gradient Vyp = vu: and a field B = (0, B sin 0, B cos B), they found an orientational dependence of the form

Pq, M = cos 0 (J - K sin’ 0) hq TV:’

(53)

In figs. 3 and 4, J and K are plotted versus B/p. In addition, 6’” and AT” resulting from the decomposition J = 6””

+ AT-,

K = 24;”

To obtain

a variational

in subsection

3.1 to get

y 4” II = h

(O(S

-

expression

- i$P)-‘Iy~Jc,(l

ApT,

A;“(B/P)

figs. 5-8 show the

= Ay”(2Bip).

for YT:, we follow the procedure

- PO)(mc,, *p - (2 + iZ))‘lyp))

(54) outlined

+, (55)

with p from eq. (39). With surface, y y: is approximated

the specialization to a perfectly accommodating in terms of bulk transport coefficients,

(56)

ON SLIP COEFFICIENTS

OF POLYATOMIC

275

GASES

20 Id3

co

15

1.0

05,

Fig. 3. The transverse viscomagnetic heat flux in CO at room temperature. The coefficients J and K are explained by eq. (53). The symbols indicate experimental data from ref. 4, whereas the drawn curves show the variational results.

I 1 Fig. 4. The transverse

viscomagnetic

,1111111 10

heat flux in N, at room

mTI Pa temperature.

100 Notation

where use was made of relation (A.24). We then arrive at a formula averaged heat flux in Poiseuille flow in a gap (0 < z < b),

as in fig. 3.

for the

(57)

B.I.M.

276

TEN

BOSCH

et al.

15

Id3

10

05 6

‘?n

1 9

Fig. S. 6”” versus B/p for CO. The dashed curve is obtained from a fit of eqs. (S4) to the experimental data. The drawn curves show the variational result and the contribution from the Burnett heat flux, eq. (60).

03

I

I I I11111

Blp

Fig. 6. 3:”

versus

I 1

B/p

I

IIIIlll 10

for CO. Notation

mTlPo

100

as in fig. 5.

In the first equality the equations (1.50) and (6), and the relation u” = -2u’(O) lb were invoked. As in subsection 3.2 we relate the result to combinations of effective cross sections. With eqs. (A.25)-(A.27) from the appendix one finds that the variational expression (57) may be rewritten in the form of eq. (53), with J and K expressed via eqs. (54), where

ON SLIP COEFFICIENTS

OF POLYATOMIC

GASES

Fig. 7. 6 ” versus

E/p

for N,. Notation

as in fig. 5.

Fig. 8. A,‘a versus

B/p for N?. Notation

as in fig. 5.

211

(59) With the effective cross sections (59) yield the variational curves

for CO and N, listed in table I, eqs. (58) and drawn on figs. 3-8. For comparison, figs. 5-8

B.I.M. TEN BOSCH et al

278

also show the curves

and

it is seen

neglected. satisfactory,

that

Whereas

which would

the

result

contribution

on figs. 3, 5 and

in case of N, the deviation

from the Burnett

of 7;:

to the

heat flux alone,

total

6 the variational from experimental

effect

results

cannot

be

for CO look

data is more

than

15% for J (fig. 4). Figs. 7 and 8 indicate that the discrepancy must be attributed to ay”. Since it is known, however, that a quantitatively good description of thermal effects in N, invokes still more expansion functions than the ones taken into account in the evaluation of Sq” I’), it should come as no surprise that the disagreement is larger than for CO, where the employed moment approximations fully suffice. It is concluded that in all cases studied, the variational results compare reasonably well with experimental data, with the exception of 6’” in case of N,. In general, the results have so far shown no discrepancy with the simple assumption of perfect accommodation.

4. Comparison

of two variational

methods

One of the results of the previous section was that the variational method slip coefficients yields the same Knudsen corrections for viscosity as method of II. For the approximate results this correspondence is not evident, and we would like to establish this correspondence analytically. way of example we are going to refer to velocity slip. With some rephrasing other effects could be treated in the same way. We recall the stationary values of the functionals X and Y, eqs. (11.14)

for the selfBy the and

(lga),

(61) where the superscripts TV have been dropped. stationary values are related as

According

to eq. (32) the exact

where terms of higher order in Kn have been neglected. We are going to show that eq. (62) also holds in the variational approximation if only bulk trial functions are used.

ON SLIP COEFFICIENTS

The

functional

for the

problem

OF POLYATOMIC

of plane

Poiseuille

219

GASES

flow is given

by eq.

(II.12),

the outer (1.26),

brackets

(1.30)

denoting

a gap average.

As trial functions

we take (cf. eqs.

and (24))

(64a)

h=

b2n 12 mcll’

-6

+

6

SQ, - jj P

bn

(64b)

with Q,(Z) = P,(l - 2zib). The first expressions on the right-hand sides account for the Poiseuille profile with slip parameters p and p. Upon substitution of eqs. (64) into X two terms cancel, and we are left with

+ in2bqi’ Ic,(l

* (O(mc,, ap - (2 - iY))‘W’I)I

- PO)(mc,,

*p - (3 + i.JZ)‘*~))+

* vi’

.

(65)

In view

of eq. (36), the last two terms on the right-hand side constitute These terms only in2bqi’ ’ Ye qi’, if the terms with v and v are neglected. contribute to second-order field effects, and have been neglected in eqs. (64) anyway. As anticipated, we thus find

The ensuing restricted stationary values equation, and hereby the correspondence mations as well.

XS, and YS, clearly obey the same (62) is established for the approxi-

B.I.M.

280

5. Comparison

with Maxwell’s

TEN BOSCH

and Eddington’s

Apart from the variational method, considerable success in approximating viz.

Maxwell’s

and

et al

Eddington’s

methods

two other schemes have also met with slip coefficients for monatomic gases,

methods24m26).

In this

section

we offer

a

suitable generalization for polyatomic gases and compare the outcome with the variational results obtained earlier. Maxwell’s procedure is based on the assumption that the impinging distribution function may be approximated by its bulk form (4) down to z = 0. Subsequently, the scattering law (3b) is invoked to obtain the outgoing part at the surface. Explicitly, at z = 0 one takes +

mc,,.y”“+(E-~kT)y~“-(~+i~)~‘~“,

c,
(67a)

To obtain approximations for the ynp and yfP, use is made of the exact relations expressing the constancy of shear stress and perpendicular heat flux throughout the boundary layer, (68a) (68b) Substituting eqs. (67) into eqs. (68), one obtains equations which may be easily solved for y@ and yYP. In case of Couette flow (p = V) one obtains

(6% b) with p and u from eqs. (39) and (40), deal with thermal

creep

and take

respectively.

On the other

hand,

if we

$2 for $,“, then

(7% b) where p and u are given by eqs. (48) and (49). In general, it may be concluded that Maxwell’s method yields slip coefficients which equal the corresponding tensors of trial parameters as obtained from the variational scheme. With the assumption of perfect accommodation, eq. (69a) implies

(71)

ON SLIP COEFFICIENTS

OF POLYATOMIC

281

GASES

and 15lr

z5.9,

*p = 8 which

should

scheme

be compared

does

not

even

(7% b)

n, =o,

with eqs. (41) and (44).

give qualitative

agreement

‘shift of the maximum’ is obtained (G = 1 + B(Kn2)). creep is equally bad. From eq. (70a) one obtains

It is seen that Maxwell’s with

experiment

The situation

since

no

for thermal

(73)

s=q = A”4 n

=

; ;

+%a2,)

+

%Tcx,,,)) ,

(744

0,

(74b)

which may be compared with eqs. (50)-(52), and also with fig. 2. It is seen again that Maxwell’s method is by no means satisfactory in dealing with effects related to angular momentum polarizations. Finally, it is noticed that the transverse viscomagnetic heat flux cannot be treated with Maxwell’s method. This is due to the fact that the corresponding slip coefficient y q” does not appear in the distribution function (67). Let us now turn to the Eddington method, which is slightly more elaborate than Maxwell’s In addition to eqs. (68) use is also made of relations between the constant quantities (O(!B - i_!Z)‘t,k(I”[~,(+~ + (% + iZ)-‘+‘)) and the various slip coefficients. These relations are obtained by equating the righthand sides of eqS. (17) and (19). In analogy to Maxwell’s procedure the inner products

are evaluated

+‘=mc

,,-p+(E-

at z = 0, with the distribution ;kT)

a-(

%$+iZ))‘*‘,

function c,
(75a)

WI the p and u being reminiscent of the trial parameters in eq. (14b). Finally, the resulting equations may be solved for p, u and the slip coefficients. As an example, let us treat the transverse viscomagnetic heat flux by Eddington’s method. The relations (67) with j3 = r are augmented with rfi”=

j$

(O(%

- iZ)-‘*cl;flc,(+”

and eqs. (75) with p = 7~ are employed

+ (% + iZ)-1+L7i))lZ=0

,

to solve for p, u and 71”.

(76) It is found

B.I.M.

282

that ~4” is given

by eq. (55),

the a-term

may be neglected).

variational

method

ence has already

TEN BOSCH

with p from eq. (39) (just as in subsection 3.3) Hence, Eddington’s method is equivalent to the

in this particular

been noted

et al

case.

by Loyalka”).

For monatomic Further

gases such equival-

inspection

schemes give identical results for all the slip coefficients, equally successful in a description of slip coefficients

shows that both

and consequently are for polyatomic gases.

Ultimately, this correspondence is due to the fact that in both methods slip coefficients are evaluated through equivalent approximations for the quantities (O(9 - i9))‘+“lc,(+” + (9 + Sir?) ‘JIP))Iz _,,.

Appendix The appendix lists some known relations between bulk transport coefficients and effective cross sections, supplemented by a few new derivations. The non-sphericity parameter $0’~ used in eq. (27) is given byiX) -_(

‘::,“)’

(A.11

@‘” = G(20)G(027r) The labels

correspond

to the basis functions

needed

to describe

viscous

flow’),

G4.2) with W= (mi2kT)“‘c sion angle

and P(f an unknown

scalar factor.

The effective

preces-

.$r,z,i is

(A.3)

with y the gyromagnetic ratio. In the derivation of eq. (43) invoked’“):

from

eq.

(41),

the

following

formulae

are

(A.4)

in standard To evaluate A” and AFX, we proceed (.% 2 iT))‘W” in terms of the relevant basis functions

fashion by (A.2) I*),

expanding

ON SLIP COEFFICIENTS

OF POLYATOMIC

+d502*P;?)

283

GASES

(A.5)

)

and similarly for (6%.+ iZ)-‘@:q. Inserting these evaluating the inner products, we finally get

relations

into

eq. (42) and

(A.61

g!y;)d&12,)(1-f(a,J).

fqx = To derive

(0(.92-

eq. (50) from eqs.

iZ))‘V*“lc,(E

(c,Ic,(% + i~))‘W~)+ where tensor

the viscosity A’ are given

77zx

A’= Relation

=

j+

j+

($mc*

(A.8)

+

n,,

i.Z)‘?P~)

= g

the approximate

n,,(-B)

,

relations

(A.8)

A:, ,

(A.9)

and the translational

thermal

conductivity

(A.lO)

, + iL!?-‘ly4)

via the moment T,,(--B)(cz

we need

= g

= g

- gkT)cl(%!

is derived

(CJS!- iZ))‘W~

- iAT))+

coefficient by

(mc:I(%!

(47)-(49)

(A.7)

.

(A.ll)

approximation”)

- cz) + . . . .

(A.12)

The deleted terms (indicated by dots) contain components of @*O and @Ozr which do not contribute to (A.8). Upon insertion of the expansion (A.12) into the left-hand side of eq. (A.8), the stated relation ensues. Relation (A.9) is obtained in analogous fashion. We expand (3 + iZ))‘?Pz as follows23): (22 + iZ)-‘qf

= A:, -j$$j ( $mc2 - $kT)c,

+. . . .

(A.13)

B.I.M. TEN BOSCH et al.

284

The omitted

terms include

the microscopic rotational which both drop @I”,

heat flux W(J’12ZkT - 1) out from the stated inner

and the Kagan polarization product. With only the quoted term taken into account, eq. (A.9) follows. To reduce eq. (50) to eqs. (51) and (52), we need the approximations given below.

First we have’.“)

Aiz = sin ~h’$‘(g(S,7y) with the field-free

translational

thermal

G(lOO1) + ($)“%( *I=*

conductivity $6)

G(1001)+2(~)“2S(~~‘:~I)+

an effective

precession

(A. 14)

T

+ 2g(25,,,))

(A.15)

@(lOlO)

’

angle (A.16)

and a non-sphericity

-S(

parameter

~~~)G(lOlO)

x 5( 12q)(G( Next,

the Burnett AzX(-B)

+ G( :;;‘)G(

lOlO)G(

coefficient

:;;A)

(A. 17)

1001) - G( :;;;)‘)

is given

by7.23)

= sin O(8 + A2 + cos’ 8(2A,

(A.18)

- A,)) ,

with (A.19)

(A.20) q

12q)

(A.21)

s=-qiGj’ *‘”

=

$ j-q;:‘b”, + q;;“,‘)

‘q;“,;;) + ( ~)“26(1010) G(1001)

+ ( ~)“%(

G(ii”) ;;;;) > G(O2%-)G( 12q) (A.22)

ON SLIP COEFFICIENTS

Finally,

the expression

OF POLYATOMIC

for the viscosity

coefficient

285

GASES

(A.lO)

is”)

= sin1377+02n(g(250211) + cos2 0 (2d5o2,) - @to,,))) .

G-B)

(A.23)

The relation

(A.24) needed in the derivation of eq. (56), may be obtained in the same way as eq. (A.9). To obtain eqs. (58) and (59) from eq. (57), we use the relations7’23) -/‘TX= cos 8 (6 + A, + sin2 8 (A, - 24,))

G,(W)

=

cm

0

oJtM5*2q)

+

WY,,,))

(A.25)

,

(A.26)

,

and”) rl yx

=

ax

f3 r1$02=(

d

50271)

+

sin2

8

( d2502n

> -

2g(

to2?r)))

.

(A.27)

References 1) E. Mazur, J.J.M. Beenakker and I. KuScer, Physica 121A (1983) 430. 2) J.N. Breunese, F.W. Godecke, L.J.F. Hermans and J.J.M. Beenakker, Physica 126A (1984) 82. 3) L.J.F. Hermans, R. Horne and E.L.M.J. van Wonderen, Physica 134A (1985) 209. 4) G.E.J. Eggermont, P.W. Hermans, L.J.F. Hermans, H.F.P. Knaap and J.J.M. Beenakker. Z. Naturforsch. 33a (1978) 749. 5) J.N. Breunese, L.J.F. Hermans and J.J.M. Beenakker, Physica 126A (1984) 66. 6) H. Vestner, Z. Naturforsch. 28a (1973) 869. 7) H. Vestner, 2. Naturforsch. 31a (1976) 540. 8) V.D. Borman, S.Yu. Krylov and B.I. Nikolayev, Zh. Eksp. Teor. Fiz. 76 (1979) 1551; Sov. Phys. JETP 49 (1979) 787. 9) E.A. Mason and E. Mazur, Physica 130A (1985) 437. 10) B.I.M. ten Bosch, J.J.M. Beenakker and I. KuSEer, Physica 134A (1986) 522, to be referred to as II. 11) B.I.M. ten Bosch, J.J.M. Beenakker and I. KuSEer, Physica 123A (1984) 443, to be referred to as I. 12) C. Cercignani and C.D. Pagani, Phys. Fluids 9 (1966) 1167. 13) S.K. Loyalka, Z. Naturforsch. 26a (1971) 964. 14) T. Kline and I. KuSEer, Phys. Fluids 15 (1972) 1018. 15) M.H. de Wit, Thesis, Technical University of Eindhoven (1975). 16) C. Cercignani, Theory and Application of the Boltzmann Equation (Scottish Academic Press, Edinburgh, 1975).

286

B.I.M.

TEN

BOSCH

et al.

The Method of Weighted Residuals and Variational Principles (Academic 17) B.A. Finlayson, Press, New York, London, 1972). 18) F.R. McCourt and R.F. Snider, J. Chem. Phys. 47 (1967) 4117. W.A.P. Denissen, L.J.F. Hermans. H.F.P. Knaap and J.J.M. Beenakker, 19) B.J. Thijsse, Physica 97A (1979) 467. L.J.F. Hermans and J.J.M. Beenakker, Physica 130A (1985) 505. 20) H. van Houten, P. Oudeman, L.J.F. Hermans and J.J.M. Beenakker, Physica 91A (1978) 21) G.E.J. Eggermont, 345. 22) B.I.M. ten Bosch, Chem. Phys. Lett. 122 (1985) 230. 23) A.C. Levi, F.R. McCourt and J.J.M. Beenakker, Physica 42 (1969) 363. The Internal Constitution of the Stars (Cambridge Univ. Press, Cambridge, 24) A.S. Eddington, 1926) p. 333. Phys. Fluids 14 (1971) 2291. 25) S.K. Loyalka, Z. Naturforsch. 26a (1971) 1708. 26) S.K. Loyalka,