Fourier transform of the Boltzmann collision integral for a bimodal distribution function

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Fourier transform of the Boltzmann collision integral for a bimodal distribution function

Alexander Orlov

To cite this version:

Alexander Orlov. Fourier transform of the Boltzmann collision integral for a bimodal distribution function. Journal de Physique II, EDP Sciences, 1993, 3 (3), pp.339-342. �10.1051/jp2:1993136�.

�jpa-00247836�

Classification

Physics

Abstracts

05.20D 47.45 47.40N

Fourier transform of the Boltzmann collision integral for

_a

bimodal distribution function

Alexander Orlov

(*)

Dept.

9, Institute for

High Temperatures

(Russian Acad. Sci.), 13/19

Izhorskaya

Street, Moscow 127412, Russia

(Received 3 September J992,

accepted

J3 November J992)

Rdsum£. Pour le modble d'onde de choc de Mott-Smith

correspondant

h des blteractions Maxwelliermes, la transfornlde de Fourier du terme de collision est calcul£e. La m£thode utilis£e est celle

qui

a dtd donn£e par

Bobylev

_pour

l'dquation

de Boltzmann.

Abstract. The Fourier transform of the Boltzmann collision

integral

for _a gas of Maxwell molecules is calculated

explicitly

for the bimodal distribution

proposed by

Mott-Smith for the shock _wave structure. The results of

Bobylev's approach

of Fourier

transforming

of the Boltzmann equation were used _as the

starting point.

1. Introduction.

The Boltzmann collision

integral

is

usually

calculated

approximately

with the use of

expansion

in Sonine _or Hermite

polynomials

of

velocity [I].

This

expansion

is correct in the

vicinity

of

equilibrium.

Mott-Smith's bimodal distribution function

[2]

is _a linear combination of Maxwellians with different parameters. ^{It is} an

example

of _a

non-equilibrium

distribution. That is

why

it _was

frequently

used for the

problem

of shock wave structure

(see

_e.g.

[3]

and references

therein).

However, the _users of the bimodal distribution function

usually

added

some

closing

moment

equation

to obtain the shock

profile.

As far _{as we} known, the Boltzmann collision

integral

for _a combination of Maxwellians _was

explicitly

calculated

only

^for

rigid spheres by Deshpanded

and Narasimha

[4]

and for

quasi-Maxwell

molecules

by Segal [8] (the

result without derivation _was cited

by Segal

and

Ferziger [7]).

The results in both _{cases are} rather awkward and

expressed by

the confluent

hypergeometric

function

,F

i

(in

the first

case)

or

by

^the

hyperbolic

sine sinh

(in

the second

case)

of such

non-analytical

function of the molecular

velocity

_as

R

[(Cl

₊

C( _{)~/4 (C~ Cp}

_)~]~'~

,

(*) Address

for

correspondence:

Tashkentskaya

Street10-2-39, Moscow 109444, Russia.

340 JOURNAL DE PHYSIQUE II N° 3

where

C~,

_{p =}

(2 RT~,

p

)~

^~'~

(v

_u~,

p

).

It _seems rather hard to calculate the moment of this

integral with,

_{e.g., some}

integer

_power of the molecular

velocity vi.

Rode and Tanenbaum

[7]

have done this for _a Maxwell molecule gas

(notice

that the derivation in

[7]

^is

again absent),

but their result in the form of

8-uple

series appears ^not so easy ^{to treat.}

In this short paper we calculate the Fourier transform of the collision

integral

for _a gas of Maxwell molecules

using Bobylev's

results

[5].

This

approach

reduces the

problem

of the calculation of the moments of the collision

integral

from

integration

of the awkward functions to

simple

differentiation with

respect

to the Fourier variable k and

putting

k

= 0 afterwards.

Section 2 contains _an

analysis

and section 3 is devoted to conclusions.

2.

Analysis.

Bobylev [5]

showed that the _use of the Fourier transform of the

velocity

distribution function for _a gas of Maxwell molecules leads to the

following

form of the Boltzmann collision

integral

J=

jd~ng(k.n/k) [~(~~~" ~(~~~" -~(k)~(0)j, (1)

2 2

where _~ is the Fourier transform of the distribution

function,

~

(k )

=

d~v f (v )

e~ ^{'~ ~}

,

(2)

g is the

product

of the absolute value of the relative

velocity

and the collisional _cross

section,

and the

integration

is _over the surface of the unit

sphere.

For

plane,

one-dimensional

problems (e,g.

shock

wave)

the Boltzmann

equation

is thus of the form _:

I

a~~lazifk~

= J

(3)

We shall further _use the linear combination of _two Maxwellians

f(v)

_{= c~} ^{e~ ~~}

^~°~~~

+ cp e~ ^"

~fl'~

,

(4)

where c, = n,

(2 gra( )~

~'~,

a(

=

RT,,

so that the Fourier-transformed distribution function is

~

(k)

_{= c~}

~~

+ cp

#ip

,

(5)

where

w,(k)

₌

e-aS~~-ik.Ua, ₍₆₎

and the collision

integral

is J

=

J~p

+

Jp~,

^where e,g.

J~p

^is

k~(a$ +

a)

y4 _>k;(u~ _{+ up Y2 2} k

jai al

_)k

n + ik(u

p u~)nj2

J~p

~~

=

(g/4

_{gr c~}

_cp

_e d _{n e}

)

and _we assumed g ₌

#/4

_gr

(isotropic scattering)

to be able to lead

explicit analysis

to the end.

Therefore,

all _we need _to do is _to evaluate the

integral ^d~n

eP'~

Usually

_{one puts} the

n~ direction

along

the vector p and then the matter is _{easy :}

iv

^2w

_I-

^I

d~n

eP'~

=

do sin 0 de eP~°~~

= 2 _gr dx eP~

=

0 0

= 2 _gr

(eP e~P)/p

=

4 _gr sinh

p/p, (8)

the function _can be viewed _{as a} series _over

p~.

But in _{our case}

Im~p~)

_~

0, p~

=

[k(aj a$ )/4]~ _(k~ kj)

₊

[k(a( a$ ) k/4

₊

ik~(up

u~

)/2

,

(9)

so we cannot

put

n~

along

_p. The

integral id~n ^e~'~~~~~~

^{should be calculated}

thoroughly.

We _set n~ =

sin 0 _{cos ~,} n~ = sin 0 sin _~, n~ = cos 0 and let a~ =

0. Then

~ ^#

i~2~

^{~a n +ibn~} #

^~

^{~ ~}E

)~

^~~ e^{a, sin sin p + az cos + ib cos} Sin ^{~ ~~}

0 0

The

integral

_over

s is easy ^:

Ids _e5

~~~ ~ _"~ ~

=

2

grJo(ia~

^sin

0)

= 2

grIo(a~

^sin

^{0), (11)}

~

where

lo

is the modified Bessel

function,

_so

A = 2 _gr

~

do sin 0

e~'~~~~~°~

^~

Io(a~

^sin 0

)

= 2

gr

l~

^dt

^e~"

^~'~~~

^Io(a~ /l).

(12)

-1

To

integrate

this

expression

_we rewrite the Bessel function in the form of _a series

j

^W

^{a)~(l t~)~}

Io(a~

^I t

=

~j ⁽¹³⁾

q=o

2~~ (ql)~

and _use the well-known formula

([6], 3.387)

l~

^{dx i}

^x~

^p

^{_1~-~~ ~~}

^~~~

^{~ ~~~}

^{~P~~~ ~}

^Ip

ij~~p

).

_~~~~

As

Iq

+ i12(A

)

=

2

~ ^{q + i} i d _q +

~~ ~ ~~ ~~~~ ~

~

(15)

the result is

l~

^dt

^{ezt Io(A} fi)

=

2

Si@

= 2

io (Wm)

,

(16)

z + A

io being

^the

spherical

Bessel function. Thus _{we see} that the r-h- _s, of the Fourier transform of the BoltzmanJJ

equation

is

j

~

j~

_~

j~ ~

^ii~

~

^ii~ _~ ~

~ j ~i~~

Here

~

~~~~4~~~~~~~~2~ ^~~' ~~~)~_~~'

~~~~

a fl

342 JOURNAL DE PHYSIQUE II N° 3

3. Conclusions.

In this brief _{note we} have calculated the Fouder transform of the Boltzmann collision

integral

for a bimodal distribution function in _a gas of Maxwell molecules with

isotropic scattering.

The Fourier-transform

approach permits

_us to calculate the _moments of the collision

integral by

differentiation with respect to k _at the

point

k

= 0 instead of

performing explicit

or

numerical

quadratures

of

hypergeometric

_or

hyperbolic

functions of

non-analytical

_arguments,

see

[4, 7-9].

For

example,

the coefficients of the

kj

and

k(

terms in the

Taylor expansion

of _our result _at k

=

0

give

the well-known

v)

and

v(

moments of the collision

integral corresponding

to the

original

Mott-Smith choice

[2].

Acknowledgements.

Valuable conversations with Dr. A.

V.Bobylev

and Dr. A. S. Goncharov _are

gratefully

appreciated.

References

[ii FERzIGER J. H. and KAPER H. G., Mathematical

Theory

of

Transport

Processes in Gases (North- Holland, Amsterdam-L., 1972).

[2] Mom-SMITH H. M., Phys. Rev. 82 (1951) 885.

[3] BASHKIROV A. G. and ORLOV A. V., J. Statist. Phys. 64 (1991) 429.

[4] DESHPANDE S. M. and NARASIMHA R., J. Fluid Mech. 36 (1969) 545.

[5]

BOBYLEV A. V., Theor. Math.

Phys.

60

(1984)

^820.

[6] GRADSHTEYN I. S. and RYzHIK I. M., Tables of

Integrals,

Series, and Products (Academic, N-Y-, 1965).

[7] SEGAL B. M. and FERzIGER J. H.,

Phys.

Fluids 15 (1972) 1233.

[8] SEGAL B. M., Ph. D. thesis, Stanford

University

(1971).

[9] RODE D. L. and TANENBAUM B. S.,

Phys.

Fluids 10 (1967) 1352.

Proofnot

corrected

by

the author.