ABSTRACT
The two-dimensional time dependent Navier-Stokes equations may be derived from equations 1.4 and 1.7, setting all terms in z to zero.
∂Ux ∂x
+ ∂Uy ∂y
= 0 . (3.1)
∂Ux ∂t
+ Ux
( ∂Ux ∂x
) + Uy
( ∂Ux ∂y
)
= − 1 ρo
∂P
∂x + gx +
µ
ρo
( ∂2Ux ∂x2
+ ∂2Ux ∂y2
) . (3.2)
∂Uy ∂t
+ Ux
( ∂Uy ∂x
) + Uy
( ∂Uy ∂y
)
= − 1 ρo
∂P
∂y + gy +
µ
ρo
( ∂2Uy ∂x2
+ ∂2Uy ∂y2
) . (3.3)
If we wish to replace the x direction with a radius r, we introduce a geometry term α, where α = 0 for plane (Cartesian) coordinates and α = 1.0 for cylindrical coordinates. Equations 3.1-3.3 become, after some manipulation and substitution,
1 rα ∂rαUr ∂r
+ ∂Uy ∂y
= 0 . (3.4)
∂Ur ∂t
+ 1 rα ∂rαUr
∂r + ∂UrUy ∂y
= − 1 ρo
∂P
∂r + gr +
µ
ρo
∂
∂y
( ∂Ur ∂y
− ∂Uy ∂r
) .
