Past exam of the probability, statistics and options I course of the University of Oxford / 2008 / K / 1

Use the two-dimensional streamfunction convention $u={\psi }_y$, $v=-{\psi }_x$. HereThe divergence is $u_x+v_y=-\cos x\sin y+\cos x\sin y=0$, so the flow is incompressible.

The vorticity is perpendicular to the plane:It is generally nonzero, so this is a rotational flow even though it has a streamfunction.

For steady inviscid flow of constant density without a body-force potential, Bernoulli's theorem givesThe constant can differ between streamlines, because this flow has nonzero vorticity. Dotting the steady Euler equation with $\mathbf{u}$ gives $\mathbf{u}\cdot \nabla (p/\rho +|\mathbf{u}{|}^2/2)=0$, which proves precisely the along-streamline form.

Direct differentiation gives the convective accelerationThe steady Euler equations therefore require $p_x=-\rho \sin x\cos x$ and $p_y=\rho \sin y\cos y$. Integrating both derivatives consistently givesThe additive constant is arbitrary. With a corresponding redefinition of this constant, the Bernoulli function is $p/\rho +|\mathbf{u}{|}^2/2=C_0-{\psi }^2$, making its constancy along streamlines explicit.