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## Homework Statement

We consider a homogeneous fluid in a sphere of radius r

_{0}= 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.

## Homework Equations

They are above

## The Attempt at a Solution

So my attempt at this is ,