3.4.3.3 Analytical Solutions for the Fluid Equations
In the general case, the solution of the fluid equations requires a numerical model.
There are, however, analytical expressions in the fluid approach that are limited in
applicability but useful for illustrative and verification purposes. For a Mach number
Ma < 1, there is no analytical solution and so the applicability of the solutions
requires assumptions. If one assumes that collisions are so frequent that the gas is
always in thermal equilibrium then one can imagine that the mean free path is much
smaller than any length scale of the gas flow which then allows a set of analytical
solutions.
The principal assumption is that the gas flow is at Ma ¼ 1 at the subliming
surface—in essence, one is assuming that the Knudsen layer is infinitely small. The
number density, the drift velocity, and the temperature at a distance, r, can then be
determined using (Sone and Sugimoto 1993)
r
R N
¼
1
ffiffiffiffiffiffi ffi
Ma
p
1 þ
2
γÀ1
Ma
2
þ
2
γÀ1
! À
γþ1
4 γÀ1
ð
Þ
ð3:69Þ
n g
n g
à ¼
1 þ
2
γÀ1
Ma
2
þ
2
γÀ1
!1
γÀ1
ð
Þ
ð3:70Þ
T
T
à ¼
1 þ
2
γÀ1
Ma
2
þ
2
γÀ1
ð3:71Þ
v g
v g
à ¼ Ma
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 þ
2
γÀ1
Ma
2
þ
2
γÀ1
v
u
u
t
ð3:72Þ
where the asterisk indicates that the quantity should be taken at the sonic point using
the macroscopic jump condition. The theoretical maximum gas velocity can be
computed from
v
max
g
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
γ 1 þ
2
γ À 1
kT
Ã
m
s
ð3:73Þ
and is approached asymptotically with increasing distance from the source. In
comparing with the literature, it is useful to recall that
x þ 1
x À 1
¼ 1 þ
2
x À 1
ð3:74Þ
as authors can express the basic equations in γ using either of these forms. Consequently, in the case of three internal degrees of freedom (e.g. water), γ ¼ 4/3, and,
3.4 Gas Expansion
221
In the general case, the solution of the fluid equations requires a numerical model.
There are, however, analytical expressions in the fluid approach that are limited in
applicability but useful for illustrative and verification purposes. For a Mach number
Ma < 1, there is no analytical solution and so the applicability of the solutions
requires assumptions. If one assumes that collisions are so frequent that the gas is
always in thermal equilibrium then one can imagine that the mean free path is much
smaller than any length scale of the gas flow which then allows a set of analytical
solutions.
The principal assumption is that the gas flow is at Ma ¼ 1 at the subliming
surface—in essence, one is assuming that the Knudsen layer is infinitely small. The
number density, the drift velocity, and the temperature at a distance, r, can then be
determined using (Sone and Sugimoto 1993)
r
R N
¼
1
ffiffiffiffiffiffi ffi
Ma
p
1 þ
2
γÀ1
Ma
2
þ
2
γÀ1
! À
γþ1
4 γÀ1
ð
Þ
ð3:69Þ
n g
n g
à ¼
1 þ
2
γÀ1
Ma
2
þ
2
γÀ1
!1
γÀ1
ð
Þ
ð3:70Þ
T
T
à ¼
1 þ
2
γÀ1
Ma
2
þ
2
γÀ1
ð3:71Þ
v g
v g
à ¼ Ma
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 þ
2
γÀ1
Ma
2
þ
2
γÀ1
v
u
u
t
ð3:72Þ
where the asterisk indicates that the quantity should be taken at the sonic point using
the macroscopic jump condition. The theoretical maximum gas velocity can be
computed from
v
max
g
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
γ 1 þ
2
γ À 1
kT
Ã
m
s
ð3:73Þ
and is approached asymptotically with increasing distance from the source. In
comparing with the literature, it is useful to recall that
x þ 1
x À 1
¼ 1 þ
2
x À 1
ð3:74Þ
as authors can express the basic equations in γ using either of these forms. Consequently, in the case of three internal degrees of freedom (e.g. water), γ ¼ 4/3, and,
3.4 Gas Expansion
221