Derivation of relaxional transport equations for a gas of pseudo-Maxwellian molecules

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Derivation of relaxional transport equations for a gas of pseudo-Maxwellian molecules

Alexander Orlov

To cite this version:

Alexander Orlov. Derivation of relaxional transport equations for a gas of pseudo-Maxwellian molecules. Journal de Physique I, EDP Sciences, 1992, 2 (3), pp.229-232. �10.1051/jp1:1992137�.

�jpa-00246476�

Classification Physics Abstracts

05.20D 05.60 47.45

Short Communication

Derivation of relaxional transport equations

for

_a

_gas of pseudo-Maxwellian molecules

Alexander V.

Orlov(*)

Dept. 9, Institute for High Temperatures

(USSR

Acad.

Sci.), 13/19

Izhorskaya St., Moscow 127412, Russia

(Received

16 October 1991, revised 9 December 1991, accepted 20 December

1991)

Abstract. In this brief note the moment and energy transport equations of relaxational type _are derived by the proposed iteration scheme of the solution of the Boltzmann equation for

pseudo-Maxwellian molecules.

The Boltzmann

equation

fif/fit

+ _v V

f

=

II / f'f( f fi) ga(g, x)

sin x

dXded~vi

for _{a gas} of

pseudc-Maxwellian

molecules

[ii

is

fif/fit

+ v

Vf

₌

(go/4~) II1 f'f(

sin x

dXed~vi

_gun

f, (I)

where

go/4~

₌

ga(g, x)

₌ const, n =

d~v f.

We shall solve

equation (I) iteratively

with the _use of the

following

scheme:

fi

fk+i/fit

+ _gun

fk+i

_"

(go/4~) II1 f( fk)(

sin _x

dXded~vi

_v V

fk

Similar method _was

applied

in reference [2] ^to the linearized Boltzmann

equation.

The first iteration is

fi

fi/fit

+ _gun

ii

=

(go/4~) / / / ii fo)(

sin _x

dxded~vi

_v V

fo

_{" gun}

fo

_v V

fo, (2)

(*)

Address for correspondence: Tashkentskaya St. 10-2-39, Moscow 109444, Russia.

230 JOURNAL DE PHYSIQUE I N°3

where

fo

_"

n(2~kT/m)~~/~

_exp

(-m(v u)~/2kT) (3)

n, u and T here and afterwards _are real parameters ^{of the} _gas. ^After introduction of

~a ⁼

ii fo

we can rewrite

equation (2)

as follows:

fi~a/fit

+ gon~a = -fi

fo/fit

_v V

fo

_"

"

~ i~ ~

~ ~'~ ~

i~ ~~

~

~~~

~

~~~ ^~~it~

^~

~~ ^~

^~

+d (cc

~I)

Vu _{+ c}

(fl

^~

Vln

j

,

(4)

kT 3 2kT 2

where

d/dt

=

alai

+ _u V, _{c = v u.}

In order to obtain the relaxation

equations

for the fluxes _we

multiply equation (4) by

the

corresponding polynonfial

of molecular

velocity

and

integrate

_over velocities. The

right-hand

side of the

resulting equation

is

expressed

in _terms of

temporal

and

spatial

derivatives of the parameters ^of

fo,

I-e- _{n, u,} T. These parameters do _not contain any ~a contributions and hence

obey

Euler

equations

as follows from

equation (4).

This follows from

multiplication

of

equation (4) by I,

_c, and c~ and

integration

_{over c.}

Since the tensor of viscous stresses is

~a~ = m cac~~a

d~c

= m

/

vav~~a

d~v

/

we can obtain in this _way

fi~a~ /fit

+ gon~a~ =

-(film) _(p6a~

₊

pttatt~) V~ (pttatt~tt~

+

ptt~6a~)

?fl (P"al

?a

(PUfl) (5)

Taking

into account the Euler

equations

tip/fit

_{+ T7}

(pu)

₌

0, pdu/df

+

VP

₌

0, dp/df

+

(5/3)pT7

u ₌ 0

we can derive from

equation (5)

that

fi~a~ /fit

_{= -pea~ gon~a~, ea~ =}

Vatt~

+

V~tta (2/3)6a~V~tt~. (6)

If the left-hand side is

neglected

_we receive the well-known linear Navier-Stokes law

~«~ =

-~oe«~/ngo

_{= -»e«~.}

(6'j

The left-hand side describes flows for which the viscous stresses

change quickly. Equation (6)

has _a formal solution

t t

~a~ =

~w

exp -go

~' n(r)dr (pea~),,

^dt'

(7)

Here the

subscript

t' shows that the

hydrodynamical

values in

parentheses

^should be calculated at the moment

t',

I.e. earlier than at t.

Equation (7)

is the sc-called transport

equation

with

delay (or

with

memory)

_[3], but contrary ^to [2, 3] in the

present

note the _memory kernel is

expressed explicitly.

We _can do similar calculations _to derive the relaxational

equation

for the heat flux

q =

(m/2) / cc~ f d~v.

In order to do this _we

multiply equation (4) by (m/2)vav~

and

integrate

_over

velocity. Taking

into account that

(m/2) /

vav~~a

d~v

= qa + u~~a~

we have the

equation

similar to the

preceding

_one:

(~

+

go) (q«

+

u~~a~)

₌

()pua

⁺

)pu~ua)

⁺

+$7~ ((~

^{~~ +}

_)u~) ^6a~

⁺

(jp

⁺

_)pu~) ^au~j

l'~

We

simplify

the

right-hand

side of this

equation using

the Euler

equations

_as earlier:

()

=

-~pV« $

+

?VfIP

gonq«.

(8)

The solution of this

equation

_can be

expressed

_as the transport ^law with memory similar _to

equation (7).

If _we

neglect

the left-hand side _we obtain in the _{case u}

= 0 the classical Fourier law

go =

(5p/2gon) Va(kT/m)

= AVaT.

(9)

Notice that

A/pcv

₌

5/3,

I-e- the Eucken relation [4]

A/pcv

=

5/2

is hence _not valid for

pseudc-Maxwellian

molecules.

Expressions

similar to our

equations (6), (8)

_were derived

by

another

approximate

method in reference [5], ^{but the}

equation

^for _qa ^obtained in that paper does not contain the term with

~a~.

Conclusion.

Relaxational

equations

for momentum and _energy fluxes

generalizing

well-known Navier-Stokes and Fourier laws _are obtained

using

the first iteration of the

proposed

^scheme of solution of the Boltzmann

equation.

This short note appears to be another

proof

of the fact that these

equations

_can be obtained from the Boltzmann

equation

with the _{same accuracy as} the classical

Ones.

Acknowledgements.

Valuable discussions with Dr. A-G. Bashkirov and Dr. A-D. Khonkin _are

gratefully appreciated.

The author thanks very much the referee for his

penetrating

criticism and very useful _notes that

helped

the author to understand the

problem

_more

deeply.

232 JOURNAL DE PHYSIQUE I N°3

References

[ii

I~OGAN M.N., Rarefied Gas Dynalrdcs

(Plenum,

N.Y.,

1969).

[2j ^{ZUBAREV D-N- and KHONKIN} A-D-, Theor. Math. Phys. II

(1972)

601.

[3j ^{ZUBAREV D-1i, and TISHCHENKO} S-V-, Physica 59

(1972)

285.

[4] ^{FERzIGER J.H. and I~APER} H-G-, Mathematical Theory of Transport Processes in Gases

(North-

Holland, Amsterdarr L.,

1972).

[5j ^KHONKIN A.D-, Fluid Mech. SOC. Res. 9

(1980)

93.