Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 331 / 3 / d

Table of contents
- Solution d

Solution

Differentiate the

y

-momentum equation with respect to

x

, subtract the

y

derivative of the

x

-momentum equation, and use incompressibility. With vertical perturbation vorticity

ζ^{'} = v_{x}^{'} - u_{y}^{'},

one obtains

Ro D ζ^{'} - Ro w_{z}^{'} - Ro^{2} w_{y}^{'} = 0.

Therefore

(\frac{\partial}{\partial t} + z \frac{\partial}{\partial x}) (\frac{\partial v ^{'}}{\partial x} - \frac{\partial u ^{'}}{\partial y}) = \frac{\partial w ^{'}}{\partial z} + Ro \frac{\partial w ^{'}}{\partial y} .

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