Modeling Expansions of Nitrogen–Xe Gas Mixtures
Alfred E. Beylich
Technische Hochschule Aachen, 52056 Aachen, Germany
Abstract. Existing experiments using orifice expansions of gas mixtures provide evidence of collisional alignment of the rotor spin vectors. Expansion flow of N2 Xe gas mixtures is modeled on the basis of kinetic equations, extending the previously developed interlaced system and the model for N2 . Numerical results of rotational
relaxation are in qualitative agreement with what is known from experiments in pure N2 , and velocity slip is in
good agreement with existing experiments. Some first results on alignment, defined as the ratio nn =n p (with n as
number density, and index n=spin vector normal to flow direction, p= parallel), indicate that this ratio first grows
with p0 D, then decays again, and that a large number of collisions is required to build the alignment.
INTRODUCTION
In gas dynamic flows at low densities where large deviations from equilibrium with large anisotropies in the velocity
distributions and slip velocities between the gas components can exist, one may also expect a deviation from an
isotropic distribution of the rotor spin vectors. One might conceive two major examples where a collisional alignment
of vectors could be expected: A rapid expansion of gas mixtures with components of different masses, causing a
velocity slip between the components; another example might be rarefied flow through channels where gas-surface
interaction plays a dominant role. The first example seems to be the more simple problem dealing with angular
dependent particle-particle interaction, and in the present study we shall concentrate on this problem.
Recent experiments [1] of expansions from orifices, using gas mixtures consisting of a noble gas and a molecular
gas with internal degrees of freedom, provide evidence of collisional alignment of the rotor spin vectors. The results
indicate that, up to a certain level of stagnation pressure, the degree of alignment seems to increase with p0 D (p0 =
stagnation pressure, D = orifice diameter). One of the physical reasons that might be responsible for this alignment is
seen in the velocity slip between the components which develops during the rapid expansion in the orifice region and
which is a typical non-equilibrium effect caused by limited collisional transfer between the particles of the components.
However, velocity slip, as a typical non-equilibrium effect, decreases with p0 D [2]. In the present work we try to model
expansion flows of a typical gas mixture, here nitrogen-xenon, on the basis of kinetic equations, by extending the
previously developed interlaced system [3] and the model for N2 [4]. We start with calculations of classical trajectories
for N2 Xe interactions, from which we deduce overall collision cross sections and transition probabilities. Then,
kinetic equations are developed for the two components, restricting to only small fractions of N2 so that N2 N2
collisions can be neglected and by simplifying the gain terms of the N2 Xe collisions. The kinetic equations are
used in two versions: As path integrals which provide the distribution functions after a time step (lower level system),
and as sets of moment equations obtained from the equations of transfer (upper level system), which impose global
constraints upon the lower level distributions. A stream tube model is used to describe the expansion though an orifice
on the axis.
CROSS SECTIONS AND TRANSITION PROBABILITIES
On the level of molecular dynamics individual trajectories of interacting particles can be calculated. On the level of
kinetic theory, however, positions of particles are not known, and it is necessary to simplify the collision integral
by introducing cross sections and transition probabilities. These quantities shall be obtained from classical particle
trajectory calculations.
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
125
10
0
Pϑ’
ϑ
10
0
10
-1
10
0
Pϑ’
ϑ
10
-1
PJ’J
10-2
-0.5
0
FIGURE 1. Xe
Right: ϑ0 = 900 .
-0.25
0
∆er/2etot
0.25
0.25
0.5
ϑ’/π
0.75
10
10-2
0.5
1
0
0
10
-1
PJ’J
-1
10-2
-0.5
10
-0.25
0
∆er/2etot
0.25
0.25
0.5
ϑ’/π
0.75
10-2
0.5
1
N2 . Transition probabilities PJJ and Pϑϑ for J = 5 and Tc = 100K. Sample: 5 105 trajectories. Left: ϑ0 = 00 .
0
0
To start, we have to choose a proper interaction potential. We shall use the model of two Lennard-Jones (6-12)
potentials with their centers separated by a distance h0 in the Nitrogen molecule (index 1) interacting with the noble
gas atom, here Xenon (index 2). Following Hirschfelder et al.[5] empirical relations for the potential parameters are
used
1
p
ε0 = ε1 ε2 and σ0 = (σ1 + σ2 ):
(1)
2
We shall use ε1 =k = 91:5K ; σ1 = 3:68Å, and ε2 =k = 229K ; σ2 = 4:055Å. These parameters, however, still have to be
adapted to a two-center Lennard-Jones system. Here we choose ε2cl j = 0:5ε0 and σ2cl j = 0:87σ0. The separation
of the atoms, r0 , in the N2 molecule is obtained from the rotational constant[6] Tr0 = Bhc=k = 2:89K; we obtain
r0 = 1:09Å, and we put h0 = r0 .
Since in the trajectory calculations the back transformation from the body fitted coordinate system [xyz]00 to the
laboratory system [xyz] suffers from a singularity when using the Euler angles, the system is extended to quaternions[7].
This works very reliably in connection with a Runge-Kutta integrator of 6th/8th-order.
On the basis of extended studies the following model of particle interaction was developed to be integrated later in
the kinetic equations: A cross section bmax (g) being a function of relative speed g independent of the angle between g
and J is introduced, and a transition probability P(g; er ; e0r ; ϑ ; ϑ 0 ) will describe the change of internal energy from er
to e0r and the angular shift of the spin vector from J to J0 as a function of vector g. Here ϑ is the angle between J and
g. It p
turns out that bmax is practically independent of J. For a translational energy corresponding to Tc < 500K, where
g = 2kTc =mN , one can approximate the relation by
σXe N2
σ0
100
= 1:63
Tc
0:1775
:
(2)
The transition from J; ϑ to J0 ; ϑ 0 is a rather complicated process, however, basic properties can be recovered by using
an ansatz written as a product of projections
PJJϑϑ PJJ Pϑϑ :
(3)
0
0
0
0
In Fig. 1 typical distributions of PJ over e0r and Pϑ over ϑ 0 are plotted for the two cases of ϑ = 00 and 900 , respectively.
One may notice that for ϑ = 00 , where J is parallel to g, the angular scattering decay is considerably slower than in
the case where J is perpendicular to g. In sample calculations J; ϑ and g were varied. The plots suggest that the decay
of P can be approximated by some combinations of exponential functions.
126
BASIC EQUATIONS
The expansion of the gas mixture will be modeled using a stream tube. The flow along the axis z of a supersonic jet
issued from an orifice of diameter D has been experimentally determined[8], and empirical relations are available from
which a stream tube cross section A(z) can be deduced. We extend these relations also to the upstream subsonic region
and write the radius R(z) of the stream tube as R=R = a1 j x j +1 + a2fexp[ j x j a1 =a2 ] 1g; where x = z=D 0:2.
Here R is a reference radius at x = 0, a1 = 1:42, and a2 = 0:5.
Stagnation conditions (index 0) will be used to non-dimensionalize corresponding quantities: Temperature T0 ,
density ρ0 = ρ10 + ρ20 , number density n0 = n10 + n20 , and pressure p0 = n0 kT0 . Furthermore we define a ratio
X10 = n10 =n0 , an average mass m0 = ρ0 =n0 , a specific heat capacities ratio γ0 = (7n10
p + 5n20)=(5n10 + 3n20) and
p
a corresponding speed a0 = γ0 kT0 =m0 . For a reference mean free path λ0 = 1=( 2π n0σ02 ) the cross section
σ0 = 12 (σ1 + σ2 ) is used (here σs are the hard sphere values).
When considering the dimensionality of the distribution functions, we take advantage of the symmetry on the
jet axis (in the stream tube); thus, to describe J, we only have to keep ϑ , here the angle between J and axis
z. Then the distribution function for nitrogen (index 1) will be six-dimensional (not including time dependence)
f1 (cx ; cy ; cz ; J; ϑ ; z) = f1Jϑ , with f1J = ∑πϑ =0 f1Jϑ ∆ϑ , and f1 = ∑JJmax
=0 f1J . If the space of internal energy is subdivided
into portions of Bhc=(kT0 ), and if discretized at J (J + 1), then in each cell (2J + 1) portions are to be grouped together.
The distribution function for the noble gas, here Xenon (index 2), is four-dimensional, f2 (cx ; cy ; cz ; z). The general
form of the system of kinetic equation will be
D f1Jϑ
D f2 = J22 + J21:
= J11 + J12 ;
(4)
Here D = ∂∂t + c ∇ is the streaming operator, and J are collision integrals. At this point we simplify the problem by
restricting only to cases where X10 1. This permits to neglect J11 , the N2 N2 collision integrals which are of high
dimension.
We now describe the terms in detail and begin with the most simple one
ν22 f2 = ν22 ( fg2
J22 = G22
f2 );
(5)
where the gain function fg2 = G22 =ν22 is written as the ratio of the gain rate G22 and the loss frequency ν22 . In the
present problem we approximate ν22 by the corresponding Maxwellian function and fg2 by the local Maxwellian, thus,
p
J22 = νM22 ( fM2 f2 ), where νM22 (c2 ) = n2 2π kT2 =m2 σ02 S22 , with
S22 = 4 exp[ Ĉ2 ]
Z
∞
0
σ22
σ0
2
sinh 2Ĉĝ
exp( ĝ2 )ĝ3 d ĝ;
2Ĉĝ
(6)
Ĉ2 = (c2 v2 )2 =(2kT2 =m2 ), and ĝ2 = g2 =(2kT2 =m2 ). Here T2 is related to the center of gravity of fM2 , i.e. to v2 . The
speed dependent cross section σ22 (g) for Lennard-Jones potential with a cutoff angle at χ = 0:05 has been obtained
by solving the corresponding integrals[9].
The collision term J21 deals with collisions of f2 with those out of f1 integrated over all ϑ ; J and c1
J21 = ν21 ( fg21
f2 ):
(7)
This again will be approximated by J21 = νM21 ( fM2 f2 ).
There remains the very complicated term J12 that covers elastic and inelastic collisions between f1Jϑ and f2 for all
c2 . For equilibrium we request a detailed balance, and PJJϑϑ can be replaced by the forward reaction term PJJϑϑ
Z
Z
2J + 1 J ϑ
PJϑ gbdbd ε dc2 f1Jϑ f2 ∑ PJJϑϑ gbdbd ε dc2:
J12 = ∑ f1J ϑ f20 0
(8)
2J + 1
Jϑ
Jϑ
0
0
0
0
0
0
0
0
0
0
0
0
0
0
We shall use a probability averaged over b; then the sum in the loss term reduces to one. In the gain term we shall
approximate the distribution functions by Maxwellians at v; T of the mixture f1J ϑ ζJ ϑ fM1 ; f20 fM2 . Here
ζJ ϑ is the number density nJ ϑ of level J 0 ; ϑ 0 related to the total number density n1 . This allows one to replace the
distributions after collision by those before collision, and we obtain
0
0
0
0
0
0
0
0
+ (c ; J ; ϑ )
J12 = fM1Jϑ ν12
1
127
f1Jϑ ν12 (c1 );
(9)
where
+=
ν12
Z
fM2
ζ
∑ ζJeqϑ
Jϑ
0
0
0
0
Jϑ
0
0
2
PJJϑϑ (g)gπσ12
(g)dc2 ;
0
ν12 =
0
Z
2
f2 gπσ12
(g)dc2 :
(10)
+ will
Here ζJeqϑ is the equilibrium value of ζJ ϑ for which isotropy with respect to ϑ should exist. For equilibrium ν12
+ cannot be further reduced, because now one has to determine, for a given [c ; J; ϑ ], for
reduce to ν12 . The gain rate ν12
1
each c2 the angle θ between g and J and sum over all possible positions of J0 , with θ 0 as the angle between J0 and g.
+ one has to perform a summation over 11 dimensions. Again we shall approximate the loss frequency by
To obtain ν12
the Maxwellian term. With νM2 = νM22 + νM21 , finally, the set of kinetic equations can be written in the form
0
0
0
0
D f1Jϑ
= νM12
+
ν12
f
νM12 M1Jϑ
f1Jϑ
;
D f2 = νM2 ( fM2
f2 ):
(11)
Moments
The distributions and their moments will be defined at equidistant points on the axis z, at zm = zmin + ∆z m,
m = 0; 1; 2; : : : ; M. In c-space the distributions will be defined on an equidistant cartesian grid with a grid constant
hi (m) dependent on the species and on z. This allows to use the allotted space in an optimal way. Offsets are chosen
such that the center of gravity of each distribution is near the center of the c-space cube. With the indices [i jk] for
[cx ; cy ; cz ] one can write the moments as sums over all grid points in c-space; thus, for the number density
nJ ϑ
3
= h1
∑ f1mi jkJϑ
i jk
;
nJ = ∑ nJ ϑ ;
ϑ
n1 = ∑ nJ ;
J
also ζJϑ
=
nJ ϑ
;
n1
and ζJ
=
nJ
;
n1
(12)
and for the internal energy er = Tr0 =T0 ∑J J (J + 1)ζJ . The velocity v1 , the temperature T1 , related to the center of
gravity, v1 , the stress tensor σ1i j , and the heat flux vector q1z are obtained similarly. The moments for the second
species are obtained in the same manner.
Equations of Transfer
For stream tube flow, there are only four moments for which the rhs of the equation of transfer vanishes,
i.e.
and (ρ e) = ρ1 (u1 + v21 =2) + ρ2 (u2 + v22 =2), where ρ1 u1 = n1 32 kT1 + er ,
sufficient to describe the different velocities vs and temperatures Ts . One
therefore has to add two more equations, now having nonzero rhs integrals which have to be obtained from the
distribution functions. In the present problem it seemed practical to use the moments ρ1 v1 and ρ1 e1 , here with
e1 = 32 n1 kT1 + ρ1v21 =2.
For a stream tube A(z) the general form of the equations for the moments Q = [ρ1 ; ρ2 ; ρ v; (ρ e); ρ1 v1 ; ρ1 e1 ] will be
[ρ1 ; ρ2 ; ρ v; (ρ e)], with ρ v = ρ1 v1 + ρ2 v2
and ρ2 u2 = 32 n2 kT2 . However, this is not
∂
1 ∂
Q+
(AF) = S:
∂t
A ∂z
(13)
The Path Integrals
The general form of the basic kinetic equations D f = ν ( fg f ), with fg = G=ν , will be transformed into an integral
along a path s for velocity c. At a point r1 = r0 + ct we have
Zt
Zt
Zt
0
ν dt ] + exp[
ν dt 00 ] fg ν dt 0 :
(14)
f (r1 ; c;t ) = f (r0 ; c;t = 0) exp[
0
0
t0
For a small time step we shall approximate the integral, assuming piece-wise constant integrands, and solve for a
two-point integral. The distribution at m and time level n + 1 will be constructed [9] from the functions at time level n
by
fmn+1 = fon exp[
Gn0
n
n
(ν0 + νm )∆t =2] +
ν0n
exp[ νmn ∆t =2] (1
128
exp[ ν0n ∆t =2]) +
Gnm
(1
νmn
exp[ νmn ∆t =2]) :
(15)
Here G0 ; ν0 ; f0 are to be interpolated at level n; Gm ; νm formally would have to be taken at n + 1; this results from the
fact that Eq. 14 is an integral equation; however, there are two arguments to take Gm ; νm at level n: One can regard Eq.
15 as a first step in an integral iteration, and for a steady state problem it does not matter how realistically the steady
state is approached.
In the evaluation of the fmn+1 two aspects need special attention: The variation of the grid constants hs (m) and the
rotation of vectors cs for a time step due to the curvilinear stream tube system.
Global Constraints
+ , and the gain functions fg result in a
The approximations introduced for the loss frequencies ν , the gain term ν12
lack of orthogonality of the rhs integrals of the equations of transfer for the summational invariants and, as a result, in
small deviations of the moments obtained from the distribution functions after a time step. In addition, the lower level
system cannot handle by itself the change along the stream tube as it is performed at the upper level. As a consequence,
we expect for each time step for the path integral a corresponding small deviation of the fs . We therefore impose global
constraints upon the distributions using the moments of the upper level. This corrector step will be performed for each
gas component separately. In particular for N2 we get a corrected distribution from the ansatz
f1corr
5
= f1
∑ α1 j Ψ1 j
j =1
(16)
;
where Ψ1 = m1 [1; cx ; cy ; cz ; c2 =2]. The corrected distribution must produce the upper level moments
Z
Ψ1i f1 ∑ α1 j Ψ1 j dc = Q1i ;
(17)
with Q1 = [ρ1 ; 0; 0; ρ1 v1 ; ρ1 e1 ]. One solves the set of equations ai j α1 j = Q1i for the α1 j . The correction will be imposed
on all f1Jϑ . In a similar way this step is performed for Xe, the second species, where now Ψ2 = m2 [1; cx ; cy ; cz ; c2 =2]
and Q2 = [ρ2 ; 0; 0; (ρ v ρ1v1 ); ((ρ e) n1er ρ1 e1 )].
NUMERICAL EXPERIMENTS
Numerical calculations are performed in a finite region of the stream tube, typically between zmin = 1D and
zmax = 4D, and on an equidistant grid with z = zmin + m∆z, with m = 0; 1; 2; : : : ; M. The upstream boundary conditions
at zmin are obtained from assuming an isentropic change from stagnation condition to the cross section A(zmin ). At zmax
we have a supersonic flow, and the variables are linearly extrapolated. The calculation is started from an isentropic
flow as initial condition, with the distribution functions being local Maxwellians with Tm , vm , Tr = Tm , and ζJeqϑ equally
distributed over ϑ . The relaxation towards a steady state is controlled by monitoring the residuum for each time
step and by monitoring the fluxes along z. It is convenient to describe the population of the rotational levels, ζJ , by
a rotational temperature, Tr . This can be used to define a rotational temperature Tr = er =k. In particular for small
temperatures we have, for er given, to determine Tr from the corresponding sum
er = ∑ ζJ eJ
J
=
eJ
1
eJ (2J + 1) exp[
]:
Zr (Tr ) ∑
kT
r
J
(18)
In the following we shall start with calculations where Nϑ 0 = 0, i.e. ignoring the ϑ -dependence of the vector J. In
+ is considerably simplified, some pre-calculations can be made, and in a transition probability PJ (g) a
this case ν12
J
fixed angle at ϑ = π =4 is used. If there were no interaction
p between the gas species, each component would expand,
in case of isentropic changes, towards a finite speed v∞ ! 2H0 , with H0 being the stagnation enthalpy. In the present
p
p
case, we obtain v1∞ = 7m0 =(γ0 m1 ), and v2∞ = 5m0 =(γ0 m2 ), where the speeds are non-dimensionalized by a0 . In
particular for the example of X10 = 0:02, as used in the following examples, we have v1∞ = 4:409 and v2∞ = 1:723.
During the rapid expansion in the region near the orifice each component will expand towards this limiting speed; as
soon as a difference exists between v1 and v2 , collisions will also tend to reduce this difference, this effect will increase
with increasing p0 D.
0
129
3
2
10
10
1.75
T, ρ
Knc
101
10-1
u
0.5
1
Tr
T
0.75
ρ
Knc
-1
0
1
2
3
4
0
10-1
0.75
ρ
Knc
0
10
0.5
0.25
10-2
10
1.25
u
T, ρ
1
T
0
1.5
10
1.25
u
10
1.75
2
1.5
10-1
1
2
0
Knc
Tr
0
u
10
-2
10
0.25
10-1
10-2
-1
0
1
z/D
2
3
4
0
10-3
z/D
FIGURE 2. Expansion of N2 Xe gas mixture. Temperatures Ts , rotational temperature Tr , velocities vs , total density ρ , and
cell Knudsen number Knc over z=D. For s = 1: N2 , filled symbols, s = 2: Xe, open symbols. T0 = 300K, X10 = 0:02, Jmax = 16,
N (c; J ) = [193 ; 16]. Left: p0 D = 1 mbar mm, Right: p0 D = 100 mbar mm.
-3
-3
-4
-4
ln ζj/(2j+1)
ln ζj/(2j+1)
We shall now compare two examples with p0 D = 1mbar mm and p0 D = 100mbar mm. In Fig. 2 the temperatures Ts ,
the velocities vs , the rotational temperature Tr , the density ρ , and the cell-Knudsen number Knc are plotted as function
of z=D. The position of the orifice is at z = 0. The main acceleration takes place at the orifice within one diameter D.
For small p0 D a velocity slip ∆v =j v1 v2 j is observed, and it reaches about 17% at z = 4D, whereas for the large
p0 D the velocity slip becomes negligible. For small p0 D at z = 0:25D a Knc = 1 is reached, and beyond this point
the drifting terms in the kinetic equations become more and more important; this point shifts to z = 4:8D for the large
p0 D . We note a large deviation of Tr from T for small p0 D and a small deviation for large p0 D: This is in qualitative
agreement with experiments of N2 expansions [10]; direct comparison, however, is not possible, since in the present
case the dominating noble gas component expands more rapidly than the pure nitrogen, and the collision cross sections
are different.
-5
m=5
-6
-7
-8
-5
m=5
-6
-7
0
50
100
150
200
-8
250
j(j+1)
0
50
100
150
200
250
j(j+1)
FIGURE 3. Boltzmann plot for rotational levels of N2 at z=D = 1 + m=M, with m = 5; 10; : : : ; M, and M = 50. Parameters same
as in Fig. 2
In Fig. 3 Bolzmann plots for the rotational levels ζJ are presented for positions m = 5; 10; : : : ; 50 along the jet
axis. For small p0 D the typical concave shape of the lines in the downstream region is observed that is known from
experiments. For the large p0 D the state is close to equilibrium, and only far downstream the lines are slightly concave.
The anisotropy of the translational distribution can be best depicted by plotting the temperature components Tp = Tzz
and Tn = (Txx + Tyy )=2, see Fig. 4. Whereas for the large p0 D drifting is still unimportant and therefore Tp Tn , for the
small p0 D the separation between Tp and Tn grows continuously moving downstream; it reaches, for both components,
130
100
100
Tp
T
T
Tp
10
-1
10
-1
Tn
Tn
10
-2
1
2
3
10
4 5
-2
1
z/D
FIGURE 4.
2
3
4 5
z/D
Parallel temperatures, Tp , and perpendicular temperatures, Tp . Diamonds: N2 ; squares: Xe.
a factor of about four at z = 4D.
The variation of the slip velocity ∆v related to the isentropic limiting speed vis∞ with p0 D for T0 = 300K and
X10 = 0:02 is plotted in Fig. 5. Limiting curves for z = 4D and z = 5D are presented. Comparison with experiments[11]
in Helium-Argon mixtures, taken at z = 6D, can be made using a slip parameter developed by Miller and Andres[2]
Ss M12 p0 D C61=3 ;
p
(19)
with M12 = µ12 m0 = j m1 m2 j, and C6 4ε0 σ06 . Here C6 is the constant of the attractive part in the interaction
potential originating from the induced dipole moments in the Lennard-Jones potential, i.e. φ = 4ε0 σ06 =r6 . The
relation here holds for a constant stagnation temperature in both the calculation and the experiment. For transforming
the experimental data we have used the following values: For Argon ε1 =k = 124K ; σ1 = 3:42Å, Helium ε2 =k =
10:2K ; σ2 = 2:576Å. The values for ε0 ; σ0 are obtained from Eq. 1. The data for Ar He were transformed using
Eq. 19. As one may note, the calculated results compare well with the experiments.
10
0
T0 = 300 K
-1
is
∆v/v∝
10
10
-2
10-3 -1
10
10
0
10
1
10
2
p0D/mbar mm
FIGURE 5. Slip velocity ∆v related to the isentropic limiting speed v∞ as function of p0 D for T0 = 300K and X10 = 0:02.
Diamonds: Calculated results for N2 Xe at z = 4D (filled symbols), and at z = 5D (open symbols). Squares: Experimental
results[11] for Ar He at z = 6D, here transformed to this scale using Eq. 19.
For the development of an anisotropy of the spin vectors two effects seem to play a major role: first, the anisotropy of
the distributions created during the rapid acceleration through the orifice; furthermore, since the alignment apparently
131
grows in small steps, additionally, a large number of collisions is required. Thus, within certain limits, one might
notice little change for the product of slip velocity (as a representative of anisotropy of the distributions) and collision
frequency being constant, which again is proportional to p0 D. We expect a lower limit when the number of necessary
collisions becomes too small, and there will be an upper limit when, downstream of the rapid acceleration in the
orifice, the gas mixture will practically return to local equilibrium destroying again the necessary anisotropy in the
distributions.
1.3
1.16
1.25
1.14
p0 D/mbar mm
1
3
5
0.5
1.12
1.1
nn/np
ζJn/ζJp
1.2
4
1.15
3
1.1
1.05
-1
0
1
2
3
4
1.04
1
1.02
5
10
1.06
2
J=0
1
1.08
1
-1
30
0
1
2
3
4
5
z/D
z/D
FIGURE 6. Alignment for T0 = 300K and X10 = 0:1. Left: Alignment of rotational levels J on the jet axis for p0 D = 1 mbar mm.
Right: Alignment for all rotational levels, nn =n p on the jet axis for different p0 D.
In Fig. 6, left, the ratio of the density ζJn , with J perpendicular to z, to ζJ p , with J parallel to z, is plotted over z=D
for different levels J. There exists a growth of the alignment for increasing J and a continuous growth outside the
orifice.
In Fig. 6, right, the ratio for all levels, nn =n p is plotted for different p0 D. One may note that with growing p0 D the
alignment goes through a maximum. The present calculations extend only to z = 5D, and one expects a further limited
growth of the alignment further downstream where the collision frequency is small but, for the small relative velocities
g, the cross sections will be very large and the ellipsoidal shape of the distributions is another cause for preference of
collisions parallel to z.
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