@q
@t
þ r Á qv
ð Þ ¼ 0
ð10:3Þ
where
q Gas density;
t Time;
v Velocity vector;
∇ Hamiltonian operator.
(2) Momentum Conservation Equation
The change rate of fluid momentum to time in a microelement is equal to the sum of
forces acting on the microelement by the outside, that is
@ qv
ð Þ
@t
þ r Á qvv
ð Þ ¼ ÀrÁp þ r Á s þ qg þ F
ð10:4Þ
where
p
Static pressure on fluid microelement;
g and F Gravitational volume force acting on the microelement and other external
volume forces (such as magnetic force);
s
Viscous stress tensor acting on the surface of microelement due to
molecular viscous action.
(3) Energy Conservation Equation
The increment rate of the energy of the microelement is equal to the net heat flux
into the microelement plus the work done by the volume force and the surface force
on the microelement, that is
@ qE
ð Þ
@t
þ r Á v qE þ p
ð
Þ
½
Š¼rÁ k eff rT À
X
j
h j J j þ s eff v
!
þ S h
ð10:5Þ
where
E
Unit control volume total energy, that is, the sum of internal energy and
kinetic energy;
k eff Effective thermal conductivity;
h j The enthalpy of component j;
J j
Diffusion flux of component j;
S h Exothermic and endothermic caused by chemical reactions.
10.2 Laval Nozzle for Attitude Control of Aircraft
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