Past exam of the options I course of the University of Oxford / 2008 / K / K1

Use the stream function convention $u={\psi }_y$, $v=-{\psi }_x$. It gives $u=-\sin x\sin y$, $v=-\cos x\cos y$. Thenso the flow is incompressible.

The vorticity points normally to the plane and has scalar componentIt is nonzero in general, so the flow is rotational.

For this steady, inviscid, constant-density flow without a body force, Bernoulli's theorem says thatIts constant may vary from one streamline to another because the flow has vorticity.

The convective acceleration isSteady Euler equations therefore give $p_x=-\rho \sin x\cos x$ and $p_y=\rho \sin y\cos y$. Integrating these compatible equations yieldswith arbitrary additive constant $p_0$. As a check, $p/\rho +(u^2+v^2)/2=p_0/\rho +1/2-{\psi }^2$, indeed constant on each streamline. This is a Steady Euler flow with vorticity proportional to its stream function.