Comprehensive Kinetic Model For Weakly Rarefied Gases
Hakuro Oguchi
Institute of Space and Astronautical Science
Sagamihara C., Kanagawa-Ken ,Japan
Abstract. This paper concerns with the construction of a kinetic model equation which closely resembles the Boltzmann
collision integrals for weakly rarefied gases of the monatomic molecules obeying the inverse power-law potentials. The
collision integrals are dealt with under the assumption that the velocity distribution functions involved take the form of
the thirteen-moment expansion. Through analytic manipulations we shall derive a comprehensive kinetic model equation
which is applicable to analyses of the molecular motions under a wide range of positive-power interaction. The present
formulation of the kinetic model equation suggests further possible extension to cases of the molecules obeying much
practical potential laws.
INTRODUCTION
As is well known, the direct solution of the full Boltzmann equation is still difficult even by means of the recent
advent of the computational technology. Apparently this is mainly due to the complexity of the collision integrals.
So far there have been evolved a large variety of approximate approaches, one of which is to apply the kinetic
model equation instead of the full Boltzmann equation.1,2 In the preceding paper,3 we presented a kinetic model
equation, in which the velocity distribution function involved in the gain term of the collision integral was assumed
to take an expression based upon the thirteen-moment approximation proposed by Grad.4 Then the gain term of the
collision integral was obtained in an analytic formula, so that the kinetic model thus derived was of a form similar
to the one of the existing model equations.２ In this paper, however the total collision integral is dealt with under the
same approximation as the previous one for the velocity distribution functions. Furthermore, we shall much
straightforwardly derive a kinetic model equation, which is comprehensively applicable to the cases of molecular
gases obeying a wide range of the inverse power-law potentials and also may provide its possible extension for cases
of other physical potentials. As an additional remark, the relation of the present kinetic model with the thirteenmoment system of equations is examined.
BOLTZMANN EQUATION WITH APPROXIMATE COLLISION TERM
Let us consider simple dilute monatomic gases which are free of any external force. As is well known, the state
of gases is described by the full Boltzmann equation written with the familiar symbols, as follows:
D F ( t , x ,v )
=
Dt
∞
π
2π
0
0
∫ dv ∫ dχ ∫ dε
1
−∞
g 1σ sin χ [ F (v 1∗ ) F (v ∗ ) − F (v 1 ) F (v )]
(1)
where F is the velocity distribution function, ｖ the particle velocity, σ is the differential collision cross section , χ
the deflection angle in molecular collisions, εthe azimuth, and g1 the relative speed given by
g1 = V − V1 = V * − V1*
where V is the peculiar (or thermal) velocity related to the mass velocity u by V = ｖ – u. For a specified pair of the
pre-collision molecules with the velocities V and V1, the inverse collisions between pairs of molecules having the
velocities V* and V1 * contribute to the gain of the number of V molecules. It means that the double integrals with
respect to χ and ε must be carried out over the surface of a sphere of the radius r (= g1/2 ) having its center at the
CP663, Rarefied Gas Dynamics: 23rd International Symposium, edited by A. D. Ketsdever and E. P. Muntz
© 2003 American Institute of Physics 0-7354-0124-1/03/$20.00
296
center-of-mass velocity in the velocity space. The center-of-mass velocity G1 for a pair of pre-collision molecules (V,
V1) as well as pairs of the allowable inverse collision molecules (V*, V1 *) are given, from the conservation of
momentum, by
G 1 = (V +V 1 ) / 2 = (V ∗ +V 1∗ ) / 2
With a half of the relative velocity r (= g/2), we have the following relations from the elementary collision kinetics
V = r1 + G 1 , V 1 = −r1 + G 1 , V ∗ = r + G 1 , V 1∗ = −r + G 1
r = r 1 = r = g1 / 2
(2)
The above equations are derived from the energy conservation in molecular collisions. As will be noted later, in
order to ensure the conservation of the lower-order five moments (mass, momentum, and energy), the set of the
above relations must hold for the variables pertinent to the peculiar velocities. In this paper, we mainly concerns
with the molecules obeying the inverse power – law interaction potential U; i.e.
U = K /{( s − 1)rT
s −1
}
where rT is the distance of the reduced mass molecule from the fixed scattering center, K is the potential constant
and (s-1) the power of the potential.
In the preceding paper,3) only the gain term in the collision integral was dealt with in somewhat schematic way
in the discreet velocity space. In the present paper, however the total collision integral of the Boltzmann equation (1)
is dealt with much analytically and straightforwardly, though the same approximation that in the preceding one for
the velocity distribution function is made. For weakly rarefied gases of our primary concern, the velocity distribution
functions involved in the collision term are assumed to take the form of the thirteen-moment approximation,
primarily proposed by Grad. 4-7
F (V ) = F0 (V )[1 +
p mn
q
V2
VmVn − m Vm (5 −
)]
RT
2 RTp
5 RTp
(3)
where V is the peculiar velocity of the molecules and F0 is the local equilibrium distribution function given by
F0 (V ) = (2πRT )
−3 / 2
(
n exp − V 2 / 2 RT
)
(4)
In Eqs. (3) and (4), n denotes the number density, T the temperature, p the pressure, pmn the non-divergence tensor
pertinent to the stress tensor, qm the heat flux vector, R the gas constant. Here we have the following relation
between the non-divergence tensor pmn and the stress tensor Pmn; that is,
p mn = Pmn − p δ mn
where δmn is the Kronecker delta. The subscript m or n denotes an axis among the Cartesian coordinates (x1, x2, x3)
adopted in the physical space concerned. The summation convention is adopted for the repeated subscripts; i.e.,
3
p mnVmVn ≡ ∑
m =1
3
∑p
n =1
3
mnVmVn , q mVm ≡ ∑ q mVm
m =1
From the definition of the stress tensor Pmn we have
3
3
m =1
m =1
p mm ≡ ∑ p mm = ∑ Pmm − 3 p = 0
(5)
Thus the thirteen independent moments n, u, T, p, pmn, qm are relevant to the velocity distribution function., together
with the law of state ; p=k n T where k is the Boltzmann constant. In terms of the velocity distribution function F,
the thirteen moments concerned are given as follows:
n = ∫ F dV ,
u=
1
v FdV ,
n∫
p mn = m ∫ VmVn FdV − 3 pδ mn ,
3
1 mV 2
kT = ∫
FdV ,
2
n
2
m
q m = ∫ V 2Vm F dV ,
2
where the integrations are performed over a whole velocity space. Since, as mentioned before, we primarily
concerns with the weakly rarefied gases, the velocity distribution functions involved in the collision term are
assumed to take the form given by the thirteen- moment approximation (3). If this is allowed, the following relation
is derived with neglect of the higher-order cross product terms,
297
:
F (V ∗ ) F(V 1* ) = F0 (V
*
)F0 (V 1 )[1 +
:*
p mn
q V∗
q V∗
V ∗2
V ∗2
(Vm∗Vn∗ + V1∗mV1∗n ) − m m (5 −
) − m 1m (5 − 1 )]
2 RTp
5 RTp
5 RTp
RT
RT
With the aid of the variable relations (2), the above equation becomes in terms of rm and G1,m
G1,m (r 2 + G12 )
p mn
2q m
F (V * ) F (V 1* )
2
(rm rn + G1,m G1,n ) −
{G1,m −
rm rn G1,n }
= 1+
−
*
*
RTp
RTp
5 RT
5 RT
F0 (V ) F0 (V 1 )
(6)
Quite similarly, we also have
G1,m (r 2 + G12 )
p
2q
F (V ) F (V 1 )
2
= 1 + mn (r1,m r1,n + G1,m G1,n ) − m {G1,m −
−
r1,m r1,n G1,n }
F0 (V ) F0 (V 1 )
RTp
RTp
5 RT
5 RT
(7)
In view of the energy conservation in binary collisions of the molecules, we have
[
F0 (V * ) F0 (V1* ) = F0 (V ) F0 (V1 ) = n 2 (2πRT ) −3 exp − (V 2 + V1 ) / 2 RT
2
]
The substitution of Eqs. (6) and (7) into Eq.(1) yields
∞
p π 2π
DF (t , x,v )
= ∫ dV 1 F0 (V ) F0 (V 1 )(2r1 )[ mn ∫ dχ ∫ dε σ sin χ (rm rn − r1,m r1,n ) +
Dt
RTp 0
0
−∞
+
4q m G1,n
π
5( RT ) 2 p ∫0
2π
(8)
dχ ∫ dε σ sin χ (rm rn − r1,m r1,n )]
0
In what follows, we shall perform the above five-fold integrations to derive a closed analytic formula or our
kinetic model. First, the double integrals regarding χ and εin Eq. (8) are dealt with for a fixed pair of V and V１.
Next, by performing the remaining ternary integrals regarding V１, the collision integrals in Eq. (8) are obtained in
an analytic form which is computable for actual applications.
COLLISION INTEGRALS FOR A FIXED PAIR OF V-AND V１MOLECULES
As regards the integrals on the right hand side of Eq. (8), we first deal with the integral regarding the azimuth ε
and then the integral regarding the deflection angle χ. We introduce the new Cartesian coordinates (ξ1, ξ2, ξ3 )
which are generated by the rotation of the spatial coordinates (x1 , x2 , x3 ) around the origin located at G1. Then the
ξ3 –axis is chosen such that its direction coincides with r1-direction, theξ1 –axis lies on the plane containing both
x3 – andξ3 –axes, and theξ2 -axis is normal to that plane . We note that the coordinates thus generated are pertinent
to r1.
If we choose the polar axis to coincide with theξ3 –axis, then a half of the relative velocity r is expressed as (r
sinχ cosε, r sinχ sinε, r cosχ) component wise in the ξ-coordinates. Let us denote the angle between x3 –
andξ3-axes by θ and the angle of the x1-axis, making against the plane containing the both x3-and ξ3 –axes, by φ.
Then , the (x-ξ) transformation T1 is given by the following relation,
 r1   cosθ cos φ
  
 r2  =  cosθ sin φ
 r   − sin θ
 3 
Since we have
(r ,
1 1
− sin φ
cos φ
0
sin θ cos φ 

sin θ sin φ 

cosθ

 r sin χ cos ε 


 r sin χ sin ε 
 r cos χ



(9)
)
/ r , r1, 2 / r , r1,3 / r = (r sin θ cos φ , r sin θ sin φ , r cosθ )
Eq. (9) is also rewritten as
∗
∗
 r1   r1,1 r1,3 / rr − r1, 2 / r
 
∗
r1,1 / r ∗
 r2  =  r1, 2 r1,3 / rr

r   ∗
0
 3  − r / r
r1,1 / r
r1, 2 / r
r1,3 / r
298






 r sin χ cos ε 


 r sin χ sin ε 
 r cos χ



(10)
where
(
r ∗ = r12,1 + r12, 2
)
1/ 2
r 2 = r1,m r1,m = rm rm
As regards the integrals involved in Eq. (8), the following lemma derived by the transformation of Eq. (10) is useful,
2π
∫r
3
r dε = 2πr1,m r1,n (1 − sin 2 χ ) + δ mnπr 2 sin 2 χ
2
(11)
m n
0
Thus we obtain
π
2π
3
r2
sin
(
)
d
d
r
r
r
r
r
r
χ
σ
χ
ε
−
=
−
σ
+
δ
σ
m n
µ 1, m 1, n
mn µ
1, m 1, n
∫0
∫0
2
2
(12)
The σμ is the viscosity cross- section defined by
π
σ µ = 2π ∫ σ sin 3 χ dχ = A* ( s )(2r ) − 4 /( s −1)
(13)
0
where
A* = 2π (2 K / m) 2 /( s−1) A2 ( s )
The function A２ (s) is tabulated in the contexts5, 6 for a variety of the potential exponent s. By the use of Eq. (13)
with the relation pmm = 0, we have from Eq. (8)
∞
3 p mn
2q m
DF (t , x,v )
= ∫ dV1 A* (2r ) ( s −5) /( s −1) F0 (V ) F0 (V1 )[−
r1,m r1,n +
(r 2 G1,m − 3r1,m r1,n G1,n )]
2
2
pRT
Dt
5
(
)
p
RT
−∞
With the aid of the variable transformation (2) the above equation becomes
∞
DF (t , x,v )
A* n 2
(2r ) ( s −5) /( s −1) exp[−(2r 2 − 2r1,mVm + V 2 ) / RT ]
= ∫ dV1
3
Dt
(2πRT )
−∞
3p
2q m
× [− mn r1,m r1,n +
(r 2Vm + 2r 2 r1,m − 3r1,m r1,nVn )]
2 RTp
5( RT ) 2 p
(14)
DERIVATION OF KINETIC MODEL
In order to conclude our kinetic modeling of the Boltzmann equation, we must perform the integration of Eq.(14).
Since the integrand in Eq. (14) is expressed in terms of r1, the integral variable V1 must be transformed into r1 for
fixed V as
dV 1 = −dg 1 = −d (2r1 )
Introducing the polar coordinates (r, κ, ω) having theξ3 –axis in V-direction as the polar axis and V as its origin ,
then we have
∞
∫ dV1 [
−∞
∞
π
0
0
] = − 8∫ r 2 dr ∫ sin κ
2π
dκ ∫ dω [
]
0
In what follows, without any confusion we employ the non-dimensional variables defined by
r = r /( RT )1 / 2 ,
rm = r1,m /( RT )1 / 2 ,
C = V /( RT )1 / 2 , C m = Vm /( RT )1 / 2
(15)
Then Eq. (14) is rewritten as
∞
π
2π
DF A* n 2 (4 RT ) ( s −5) / 2 ( s −1)
=
exp(−C 2 ) ∫ dr ∫ sin κ dκ ∫ dω ×
Dt
π 3 ( RT ) 3 / 2
0
0
0
×r
2 + ( s −5 ) /( s −1)
3p
2q m
exp(−2r + 2rm C m )[ mn rm rn −
(r 2 C m + 2r 2 rm − 3rm rn C n )]
1/ 2
2p
5 p ( RT )
2
299
(16)
In Eq. (15) the variables r and r1,m are pertinent to V1 for a fixed V. In order to perform the integration, we introduce
the similar transformation to T1 of Eq. (9). In this case theξ3 is chosen in V-direction, and theξ1-axis chosen to lie
on the x3 -ξ3 plane. The angle ofξ3 against x3 is denoted by θ, and the angle of ξ1-axis making for the x3 -ξ3
plane is denoted by φ . Quite similarly to the derivation of the transformation T1, we have the following
transformation T2 expressed in terms of non-dimensional variables r and rm ,
 r1   cosθ cos φ
  
 r2  =  cosθ sin φ
 r   − sin θ
 3 
,
where
− sin φ
cos φ
0
sin θ cos φ 

sin θ sin φ 

cosθ

 r sin κ cos ω 


 r sin κ sin ω 

 r cos κ


(17)
(V1 / V , V2 / V , V3 / V ) = (C1 / C , C 2 / C , C 3 / C ) = (sin θ cos φ , sin θ sin φ , cosθ )
In performing the integration in Eq. (16), we use the following relations derived by use of Eq. (17)
r m C m = rn C n = rC cos κ
2π
∫r
m
dω = 2π
0
2π
∫r
Cm
r cos κ
C
r dω = 2π
m n
0
CmCn 2
3
r (1 − sin 2 κ ) + δ mn πr 2 sin 2 κ
2
2
C
It should be noted that the last equation is quite similar to Eq. (11). The resulting formula after integration of
Eq.(16)is expressed as follows:
3p C C
4q m
2
DF (t , x,v ) A* n 2 (4 RT ) ( s −5) / 2( s −1)
=
exp(−C 2 ) [− mn m 2 n P −
C m ( P − Q)] (18)
2
3/ 2
1/ 2
C
Dt
2p C
π ( RT )
5 p ( RT )
where P and Q are the functions of C alone given by
∞
P(C ) = ∫ r 2 ( s −3) /( s −1) exp(−2r 2 )[(
0
r2
3r
3
−
+
) exp(2Cr ) −
2
C 2C
4C 3
r2
3r
3
−( +
+
) exp(−2Cr )]d r
2
C 2C
4C 3
∞
r3
r2
−
Q(C ) = ∫ r 2 ( s −3) /( s −1) exp(−2r 2 )[(
) exp(2Cr ) +
2C 4C 2
0
r3
r2
+(
+
) exp(−2Cr )]d r
2C 4C 2
(19)
(20)
Equation (18) with P and Q, respectively, given by Eqs. (19) and (20) is the kinetic model equation which contains
the thirteen moments of the velocity distribution function. The parameter involved, A* or A ２ , pertinent to the
viscosity cross- section σμ is presumed to be related to the viscosity given by 5, 6
µ=
5m( RT / π )1 / 2 (2mRT / K ) 2 /( s −1)
,
8 A2 ( s )Γ{4 − 2 /( s − 1)}
∞
Γ( s ) = ∫ x s −1 exp(− x)dx
(21)
0
where Γ(s) is the gamma function. From Eqs. (13) and (21) we derive the following relation ,
A*n =
5 π 2 2( 3− s ) /( s −1)
p
( RT ) (5− s ) / 2 ( s −1)
Γ{4 − 2 /( s − 1)}
µ
(22)
Apparently the present kinetic model constitutes a closed system for computation. It should be noted that the model
equation (18) contains the measurable thirteen moments alone and the right hand side pertinent to the collision
processes does no differentials explicitly.
300
Special Molecular Models
Maxwell Molecules
For Maxwell molecules (s=5) the integrals (19) and (20) simply reduce to the followings
(2π )1 / 2 2
C2
P=
C exp( )
16
2
(2π )1 / 2
C2
2
Q=
C (5 + C ) exp( )
64
2
Consequently, we have the kinetic model equation for the Maxwell molecules as follows:
DF (t , x,v )
A* n 2
V2
3 p mn VmVn q mVm V 2
=
exp(
−
)[
−
−
(
− 1)]
Dt
2 RT
8 p RT
2 pRT 5 RT
(2π RT ) 3 / 2
where from Eq. (22) we have
(23)
A∗ n = (4 / 3) p / µ
which represents the collision frequency. We note the present model is qualitatively similar to the model proposed
by Shakov 2 except that his model remains an indefinite parameter.
Hard Sphere Molecules
For hard sphere molecules (s →∞) we also obtain the following kinetic model equation,
p V V
q V
DF (t , x,v ) A* n 2 ( RT )1 / 2
V2
=
exp(
−
)[− P1 mn m n − Q1 m m ]
3/ 2
Dt
p RT
2 RT
pRT
(2π RT )
(24)
where
P1 =
Q1 =
3
3
3
3
2
3
3
C2
C
−
−
)
exp(
) + (C + − 3 + 5 ) erf( )
2
4
2
8
C C
C
C
C
2
2
9
1
15 9
C
C
(C 2 − 4 − 2 ) exp(−
) + (C 3 − 3C − + 3 ) erf( )
2
10
C C
C
2
(1 +
4(2π )1 / 2
1
5(2π )1 / 2
with the definition of the error function:
erf(a) =
a
2
π
∫ exp(−t
2
) dt
0
The parameter A* is given from Eq.(22) as follows:
A∗ n =
5 π p
( RT ) −1 / 2
24 µ
Another Specific Molecules (s = 3)
For another specific molecules (s=3), we have a closed analytic expression for the collision integral; i.e.
p V V
q V
DF (t , x,v ) A∗ n 2 ( RT ) −1 / 2
C2
=
exp(
−
)[− P3 mn m n − Q3 m m ]
3/ 2
Dt
p RT
2
pRT
(2πRT )
where
P3 =
3
4(2π )1 / 2
(
1
3
3 1
2
3
C2
C
−
−
)
exp(
) + ( − 3 + 5 ) erf( )
2
4
2
8 C C
C
C
C
2
301
(25)
Q3 =
1
5(2π )1 / 2
(1 −
7
1
6
7
C2
C
−
)
exp(
) + (C − + 3 ) erf( )
2
2
10
C C
C
2
and
A∗ n =
5 π p
( RT )1 / 2
2 µ
As shown above, the analytic expressions of the collision term have been derived for the specific molecular models
with the potential power s=3, 5 (Maxwell molecules) and infinity (hard sphere molecules) (see Eqs. (23-25)).
Relation With Thirteen –Moment System of Equations
Referring to the context by Kogan7, we have the following moment equations which are similar to the one
primarily derived by Grad (so- called thirteen-moment system of equations);4
∂ρ
∂
+
(ρ ur ) = 0
∂t ∂x r
∞
∂u i
∂u
1 ∂Pir 1
+ ur i +
= ∫ Vi Fcol dV ≡ I 1
ρ −∞
∂t
∂x r ρ ∂x r
∂u
∂p
∂
2
2 ∂ qr
+
(u r p ) + Pir i +
=0
∂t ∂x r
3 ∂x r 3 ∂x r
∂u j
∂ pij
∂u 2
∂u
∂
2 ∂q ∂q j 2 ∂q r
+ p jr i − δ ij p rs r
− δ ij
) + pir
+
(u r pij ) + ( i +
∂x r
∂x r 3
∂x s
∂x r
∂t
5 ∂x j ∂xi 3 ∂x r
∞
∂u i ∂u j 2 ∂u r
+ p(
+
− δ ij
) = m ∫ vi v j Fcol dV ≡ I 2.
∂x j ∂xi 3 ∂x r
−∞
(26)
∂qi
p ∂P
∂
7 ∂u
2 ∂u
2 ∂u
kT ∂pir 7
∂ kT
+
(u r qi ) + q r i + q r r + qi r +
( ) − ir rs
+ pir
∂t ∂x r
5 ∂x r 5 ∂xi 5 ∂x r
m ∂x r 2
∂x r m
ρ ∂x s
∞
+
m
5 ∂ kT
p
( ) = ∫ vi v 2 Fcol dV ≡ I 3
2 ∂xi m
2 −∞
(27)
where Fcol. denotes the collision term given by the right hand side of Eq. (18). The collision integrals I’s for the
special molecular models mentioned before are evaluated. First,. it is analytically confirmed that I１= 0. For any
other molecular models of positive s-potential it was also confirmed numerically. Consequently, for the present
kinetic model the conservation law pertinent to the basic lower moments was ensured. For the remaining integrals,
we have
∞
I 2 = m ∫ ViV j Fcol dV , I 3 =
−∞
∞
m
ViV 2 Fcol dV
2 −∫∞
After some analytical manipulations, for all of three specific molecular models mentioned above, we obtain
I2 = −
p
µ
pij , I 3 = −
2 p
qi ,
3µ
(28)
The above relations are also numerically ensured for several power potential of positive s, within the accuracy of
tolerance. Thus, the system of moment equations is identified with the one primarily proposed by Grad. As the
results, the consistency of the presumption of Eq. (21) is also confirmed.
302
CONCLUDING REMARKS
We proposed the kinetic model equation, in which the collision integral is given by a closed analytical formula in
terms of the thirteen moments of the velocity distribution function. In view of the underlying approximation, this
model may be applicable to analyses of the weakly rarefied gases flows of the monatomic molecules obeying the
inverse power-law potentials. Actual computational analyses on rarefied gas flows will be made based upon the
kinetic model together with the recent evolution of computational techniques (for example, references8,9). Moreover,
further extension of the present model may be expected for the molecules obeying such interaction potentials that the
viscosity cross-section σμ is explicitly specified as a function of the relative speed g (=2r) in the form of Eq. (13).
ACKNOWLEDGMENTS
The author would like to express his sincere thanks to Professor A. Sakurai of Tokyo Denki University, and
Professors H. Honma and K. Maeno of Chiba University for their helpful advices and stimulating discussions
throughout this study.
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APPENDIX
According to Chapman-Enskog procedure, we can derive the well-known relations for the viscosity as well as
heat conduction of gases from Eqs (26) and (27) along with Eq. (28) as follows. Namely, we have the lower–order
relations by use of Eqs. (26), (27) and (28) as follows:
µ(
∂u i ∂u j 2 ∂u r
) = − pij ,
+
− δ ij
∂x j ∂xi 3 ∂x r
15 ∂RT
µ
= −qi
4
∂xi
From the second relation, the coefficient of heat conduction λis related to the coefficient of viscosity μ, by
λ = (15/4) Rμ
and thus we have the correct Prandtl number Pr =2/3.
303