Homework Help: Geophysical Fluid Dynamics

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

1. The problem statement, all variables and given/known data

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;[tex]\phi[/tex] ; z) with the corresponding unit vectors ([tex]\hat{s}[/tex]; [tex]\hat{\phi}[/tex]; [tex]\hat{z}[/tex]), with [tex]\hat{z}[/tex] 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

[tex]\frac{\delta \textbf{u}}{\delta t}[/tex] + 2[tex]\hat{ \textbf{z}}[/tex] x [tex]\textbf{u}[/tex] = -[tex]\nabla[/tex]p; (1)
AND

[tex]\nabla[/tex] . [tex]\textbf{u}[/tex] = 0; (2)

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

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

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


2. Relevant equations

They are above

3. The attempt at a solution

So my attempt at this is ,

 

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

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

itu_{[tex]\phi[/tex]} + 2[tex]\hat{\textbf{z}}[/tex] x [tex]u_{\phi}[/tex] = - [tex]\frac{\delta p}{\delta \phi}[/tex]


itu_z + 2[tex]\hat{\textbf{z}}[/tex] x [tex]u_{z}[/tex] = - [tex]\frac{\delta p}{\delta z}[/tex]

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

[tex]\frac{\delta u_{s}}{\delta s}[/tex] + [tex]\frac{\delta u_{\phi}}{\delta \phi}[/tex] + [tex]\frac{\delta u_{z}}{\delta z}[/tex] = 0

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

 

ah nm this is wrong , seen my mistake