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 pseudoMaxwellian molecules. Journal de Physique I, EDP Sciences, 1992, 2 (3), pp.229-232.
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Phys.
J.
France
I
(1992)
2
229-232
1992,
MARCH
PAGE
229
Classification
Physics
Abstracts
05.20D
05.60
47.45
Communication
Short
Derivation
of relaxional
transport equations
molecules
for a gas of pseudo-Maxwellian
Alexander
Dept.
V.
9,
127412,
Orlov(*)
High
for
Institute
(Received
1991,
October
16
In
Abstract.
this
revised
brief
The
a
+
go/4~
We
shall
=
method
iteration
is
fi
fi/fit
(*)
+
f
II /
=
gun
Address
+
was
ii
for
scheme
Vf
v
=
(go/4~)
=
const,
n
=
gun
fk+i
applied
(go/4~)
in
"
equations
transport
energy
of the
solution
1991)
December
20
of the
of
relaxational
equation
Boltzmann
for
correspondence:
f fi) ga(g, x)
sin x
dXded~vi
is
II1
f'f(
sin x
dXed~vi
(I)
f,
gun
d~v f.
=
(go/4~)
reference
/ / / ii
f'f(
[ii
molecules
equation (I) iteratively
fk+i/fit
Similar
+
ga(g, x)
solve
fi
V
v
pseudc-Maxwellian
fif/fit
where
Moscow
equation
Boltzmann
gas of
and
moment
iteration
derived
are
fif/fit
for
Sci.), 13/19 Izhorskaya St.,
Acad.
1991, accepted
December
9
the
note
by the proposed
molecules.
pseudo-Maxwellian
type
(USSR
Temperatures
Russia
with
the
II1
[2] to
fo)(
the
St.
following
of the
f( fk)(
sin x
Tashkentskaya
use
dXded~vi
sin x
Boltzmann
linearized
dxded~vi
10-2-39,
scheme:
v
Moscow
V
fo
"
109444,
v
V
fk
equation.
gun
fo
Russia.
v
The
V
fo,
first
(2)
JOURNAL
230
PHYSIQUE
DE
I
N°3
u)~/2kT)
(3)
where
fo
n,
and T here
u
we
~
"
afterwards
equation (2)
rewrite
can
and
i~
d/dt
alai
+
obtain
=
order
to
as
~'~
~
the
c
of
tensor
obtain
can
fi~a~ /fit
Taking
into
derive
can
we
left-hand
equations
The
left-hand
formal
a
as
integration
and
at
the
ii
fo
/
(fl
~~it~
~
~
2kT
for
over
Vln
~
~~
~
j
2
(4)
,
the
fluxes
multiply
we
equation (4) by the
velocities.
over
and
right-hand
The
spatial
derivatives
of the
contributions
multiplication
and
of
c.
/
cac~~a d~c
m
=
"
~~~
c
m
=
=
vav~~a
-(film) (p6a~ + pttatt~)
?fl (P"al
?a (PUfl)
d~v
V~ (pttatt~tt~
+
ptt~6a~)
(5)
equations
=
0, pdu/df
+
VP
0, dp/df
=
=
-pea~
neglected
gon~a~,
describes
ea~
receive
we
flows for
=
=
the
Vatt~
-~oe«~/ngo
which
the
+
+
(5/3)pT7
V~tta
well-known
u
=
0
=
(2/3)6a~V~tt~.
Navier-Stokes
linear
law
(6'j
-»e«~.
viscous
(6)
stresses
change quickly.
Equation (6)
solution
t
the
=
is
stresses
(pu)
~a~
Here
~a
equation (5) that
side is
side
fo
do not
contain
parameters
any ~a
from equation (4). This follows from
follows
~«~
has
of
u.
These
Euler
the
+ T7
fi~a~ /fit
If the
~
V
v
velocity and integrate
expressed in terms of temporal
T.
gon~a~
account
from
introduction
way
+
tip/fit
v
is
viscous
this
in
After
molecular
~a~
we
~
3
=
of
the
~~~
Vu +
relaxation
fo, I-e- n, u,
hence obey Euler equations
equation (4) by I, c, and c~
Since
fo/fit
-fi
=
~I)
corresponding polynonfial of
side of the resulting equation
parameters
+ gon~a
(cc
V,
u
of the gas.
parameters
i~ ~~
~
kT
In
real
are
(-m(v
exp
follows:
fi~a/fit
~
+d
where
n(2~kT/m)~~/~
"
=
~w
exp
-go
~'
t
n(r)dr
(pea~),,
dt'
(7)
subscript t' shows that the hydrodynamical values in parentheses should be calculated
t', I.e. earlier than at t. Equation (7) is the sc-called transport equation with
moment
EQUATION
N°3
FOR
delay (or with memory) [3],
expressed explicitly.
We
do
can
similar
A
GAS
contrary
but
calculations
order
into
do
to
this
the
have
equation
similar
(~
go)
((~
+$7~
(q«
+
+
simplify the right-hand
=
The
6a~
-~pV«
$
If
(jp +
+
+
velocity. Taking
over
)pu~ua)
au~j
+
equations
Euler
in the
obtain
earlier:
as
(8)
gonq«.
law
transport
the
as
+
)pu~)
?VfIP
equation can be expressed
neglect the left-hand side we
we
flux
u~~a~
()pua
=
of this
solution
qa +
=
integrate
and
equation using the
side of this
()
equation (7).
)u~)
heat
is
one:
u~~a~)
~~
l'~
We
+
kernel
memory
the
for
231
cc~ f d~v.
vav~~a d~v
preceding
the
to
the
note
equation
relaxational
/
(m/2)
=
/
(m/2)
we
the
MOLECULES
present
[2, 3] in the
multiply equation (4) by (m/2)vav~
we
that
account
to
derive
to
q
In
PSEUDO-MAXWELLIAN
OF
case
u
with
=
similar
memory
classical
0 the
to
Fourier
law
go
(5p/2gon) Va(kT/m)
=
5/3, I-e- the Eucken relation
that A/pcv
pseudc-Maxwellian
molecules.
Expressions similar to our equations (6), (8) were
in reference [5], but the equation for qa obtained in
Notice
[4]
=
A/pcv
5/2
=
by
derived
that
(9)
AVaT.
=
another
does
paper
not
is
hence
valid
not
approximate
contain
the
for
method
with
term
~a~.
Conclusion.
equations
Relaxational
and
Fourier
the
Boltzmann
equations
laws
can
for
using
equation.
This
obtained
from
be
and
momentum
obtained
are
short
the
the
energy fluxes generalizing well-known
first
iteration
of the proposed scheme
note
appears
Boltzmann
be
to
equation
proof of the
another
with
the
accuracy
same
Navier-Stokes
of
fact
as
solution
that
the
of
these
classical
Ones.
Acknowledgements.
Valuable
The
that
author
discussions
thanks
helped the
with
very
author
Dr.
much
to
A-G.
the
understand
Bashkirov
referee
the
and
Dr.
A-D.
Khonkin
penetrating
problem more deeply.
for
his
are
criticism
gratefully appreciated.
and
very
useful
notes
JOURNAL
232
DE
PHYSIQUE I
N°3
References
[ii I~OGAN M.N., Rarefied Gas Dynalrdcs (Plenum, N.Y., 1969).
Math. Phys. II (1972) 601.
[2j ZUBAREV D-N- and KHONKIN A-D-, Theor.
S-V-, Physica 59 (1972) 285.
TISHCHENKO
[3j ZUBAREV D-1i, and
[4]
FERzIGER
Holland,
[5j KHONKIN
J.H.
and
I~APER
Amsterdarr
A.D-,
Fluid
H-G-,
Mathematical
Theory
L., 1972).
Mech.
SOC.
Res.
9
(1980)
93.
of
Transport
Processes
in
Gases
(North-