How Does Fluid Dynamics Apply in a Rotating Sphere?

[dargar]

Heya , sorry but could someone check what I've done is right ? The remainder of the question relies on these answers :S

Homework Statement

We consider a homogeneous fluid in a sphere of radius r₀ = 1 in the limit of kinematic viscosity v = 0 which is rotating uniformly about its axis with a constant angular velocity.

We shall use cylindrical coordinates (s;\phi ; z) with the corresponding unit vectors (\hat{s}; \hat{\phi}; \hat{z}), with \hat{z} being parallel to the axis of rotation.

The small-amplitude fluid motion in a rotating reference of frame is governed by the linear dimensionless vector equations

\frac{\delta \textbf{u}}{\delta t} + 2\hat{ \textbf{z}} x \textbf{u} = -\nablap; (1)
AND

\nabla . \textbf{u} = 0; (2)

subject to the condition of vanishing normal flow
\hat{\textbf{r}} . \textbf{u} = 0 at r = 1: (3)

Let \textbf{u}(\textbf{x}, t) = \textbf{u}(s, z)e^{i(\phi + t )}. (4)

Write down the four equations by projecting the equations (1) and (2) onto cylindrical coordinates.

Homework Equations

They are above

The Attempt at a Solution

So my attempt at this is ,

 

[dargar]

Using (1) , I find out \frac{\delta \textbf{u}}{\delta t} which I get to be it\textbf{u} from (4) so plugging this into (1) gives us three equations of the 4

itu_s + 2\hat{\textbf{z}} x u_{s} = - \frac{\delta p}{\delta s}

itu_\phi + 2\hat{\textbf{z}} x u_{\phi} = - \frac{\delta p}{\delta \phi}itu_z + 2\hat{\textbf{z}} x u_{z} = - \frac{\delta p}{\delta z}

and then using equation (2) the last equation of the 4 is

\frac{\delta u_{s}}{\delta s} + \frac{\delta u_{\phi}}{\delta \phi} + \frac{\delta u_{z}}{\delta z} = 0

I think I probably have gone wrong somewhere tho , would someone please check to see what I'm doing is correct please ?

 

[dargar]

ah nm this is wrong , seen my mistake