Past exam of the mathematics course of the University of Cambridge / 2008 / ib / Paper 3 / 18B / iv / Solution

Substitute the rotating velocity field into the Euler equations for an inviscid fluid. Its acceleration is $(-\dot{\Omega }y-{\Omega }^{2}x,\dot{\Omega }x-{\Omega }^{2}y,0)$, soThe preceding angular acceleration makes both mixed coefficients $(\beta +\gamma )/2$, ensuring integrability. ThusPressure is determined up to the spatially uniform function $p_0(t)$. The specified force has no vertical component; if gravity were additionally included in the body force, the usual hydrostatic term $-\rho gz$ would also appear. It is not implied by the displayed force here.