∂e u
∂τ
¼ À
λ
2
e υ λ 0 , 0
ð
Þ
γ
∂
∂χ
e p þ e q
ð
Þ,
ð6:12Þ
where the artificial viscosity is also in units of the ambient pressure p 0 . Clearly, the
particle velocity is given by
u ¼
∂r
∂t
ð6:13Þ
and with the variables defined as above we have the following equation for the
normalized particle velocity,
e u ¼
∂λ
∂τ
:
ð6:14Þ
6.2.2 Continuity Equation
Mass conservation gives the following equation,
4πr
2
ρ r 0 , t
ð
Þdr ¼ 4πr
2
0 ρ r 0 , 0
ð
Þdr 0
ð6:15Þ
where
dr ¼ r r 0 þ dr 0 , t
ð
ÞÀr r 0 , t
ð
Þ,
and it follows that
∂r
∂r 0
¼
ρ λ 0 , 0
ð
Þ
ρ
λ
2
0
λ
2
:
ð6:16Þ
Writing this in the form;
∂r
∂χ
∂χ
∂r 0
¼
ρ λ 0 , 0
ð
Þ
ρ
λ
2
0
λ
2
;
noting that dχ ¼ r
2
0 =ε
3
À
Á
dr 0 , then the latter equation becomes
∂r
∂χ
λ
2
0
ε
¼
ρ λ 0 , 0
ð
Þ
ρ
λ
2
0
λ
2
and using λ ¼ r/ε we have
6.2 Lagrangian Equations in Spherical Geometry
285