Introduction - Atomic Nanoscale Technology in the Nuclear Industry - page 4

The.continuity.equation.is

1 ∂

∂

1 ∂

∂

u

+ ∂
∂

u
z

θ
θ

z

. ( ru ) + = 0 . (1.5)

r

r x r

For. spherical. coordinates. the. momentum. equation. is,. with. r , θ, ϕ (ϕ =
colatitudes):

2 u
r

2

u
r

u +

∂
∂

u
t

∂
∂

u
r

u ∂

∂

u ∂

∂

u

r r θ r r θ

ρ + u + + −

r

r sin( ) θ

2

= − ∂
∂

p
r

1 ∂

∂

∂
∂

u
r +

1 ∂

∂

u

g r

2 r r

+ ρ + µ

r

2 2 2 2

r r r sin( ) θ

+ ∂
∂

u

u + u cot( )

r

−

1 ∂

∂

∂
∂

u 2 ∂ u

r θ

+ sin( ) 2 −

r

2

sin( ) r

2

r

2

sin( ) ∂ θ

∂
∂

u
t

∂
∂

u
r

u ∂

∂

u u

r

∂ u u u + u u

r

cot( )

r θ θ

θ

u

θ θ θ θ

−

ρ + + +

r

r sin( ) θ ∂

1 ∂

∂

p 1 ∂

∂

∂
∂

u
r

g r

2 θ

= − + ρ + µ (1.6)

θ

2

r sin( ) θ r r

∂
∂

u r ∂

∂

u

2 2 cos( )

+ + u

1

2

u 1 u

+

∂ ∂

∂

∂
∂

θ

θ θ

θ θ

+ + sin( )

2 2 2 2 2 2

r sin( ) ∂ θ r sin( ) r sin( )

2

∂
∂

u
t

∂
∂

u u ∂

∂

u u

r

∂
∂

u u u − u

r

cot( )

r

θ

ρ + u + + −

θ

r

r r sin( ) θ

2

1 p 1 ∂

∂

u
r +

1 ∂

∂

u

∂
∂

∂
∂

2

= − + ρ g + µ r

2 2 2 2

r r r r sin( ) θ

∂
∂

u

θ

u + 2 cos( )

u

1 ∂

∂

∂
∂ +

2 ∂

∂

u

θ

r

+ sin( ) −

2 2 2 2

r sin( ) r r sin( )

.

and.the.continuity.equation.is

1 ∂

∂

1 ∂

∂

u 1 ∂

∂

2 θ

( r u ) + + ( sin r ( ) u ) = 0 . (1.7)

r

2

r r r sin( ) θ r sin( )

.