Past exam of the mathematics course of the University of Cambridge / 2007 / ib / Paper 1 / 5D (Fluid Dynamics)

The vorticity is the curl of the velocity, perpendicular to the flow plane:The planar rotation rate is half this scalar vorticity, not the vorticity itself.

The velocity divergence is ${\partial }_x{u}_x+{\partial }_y{u}_y=\alpha -\alpha =0$, satisfying the constant-density incompressible flow condition. Choose the stream function convention $(u_x,u_y)=({\psi }_y,-{\psi }_x)$. Integration then givesIndeed, ${\psi }_y=\alpha x-\beta y$ and $-{\psi }_x=\beta x-\alpha y$. Streamlines are contours of this function. The vector field is also compatible with steady inviscid momentum balance without body forces: $(\mathbf{u}\cdot \nabla )\mathbf{u}=({\alpha }^2-{\beta }^2)(x,y)$ is balanced by pressure $p=p_0-\rho ({\alpha }^2-{\beta }^2)(x^2+y^2)/2$.

Rotate coordinates to $s=(x+y)/\sqrt{2}$ and $t=(x-y)/\sqrt{2}$. The stream function becomesFor $\alpha <\beta$, both coefficients are negative, so every nonzero contour is a bounded ellipse surrounding the stationary origin. For $\alpha >\beta$, the form is indefinite: nonzero contours are unbounded hyperbolas, and the zero contour consists of straight separatrices. Consequently all nonstationary streamlines are bounded exactly when $\alpha <\beta$. The origin is itself a stationary bounded trajectory in every case; the assertion about boundedness concerns the nonstationary streamlines. The equality case is handled separately below. These are the linear planar strain and rotation streamlines.

If $\alpha =\beta$, $\psi =-\alpha (x-y{)}^2/2$. Its contours are the parallel lines $x-y=d$. On each such line,For $d\ne 0$ this is an unbounded straight streamline. Every point on $x=y$ has zero velocity, so that diagonal is a line of equilibria, rather than one moving streamline.