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 $$\textbf{u}$$ = -$$\nabla$$p; (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