Past exam of the mathematics course of the University of Cambridge / 2008 / ib / Paper 1 / 5B / Solution

The divergence is ${\partial }_x(e^y\sinh x)+{\partial }_y(-e^y\cosh x)=e^y\cosh x-e^y\cosh x=0$, proving incompressibility. Use the stream function convention $u_x={\psi }_y$, $u_y=-{\psi }_x$. Integration of either component then givesThe streamlines are its level curves. Setting the irrelevant constant to zero, a nonzero level $K$ haswith $x>0$ for $K>0$ and $x<0$ for $K<0$. Each curve rises to $y=+\infty$ as $x\to 0$ from its side and approaches $y=\ln (2|K|)-|x|$ at large $|x|$. The zero streamline is $x=0$. Flow is downward everywhere, outward to the right when $x>0$ and outward to the left when $x<0$; on $x=0$ it is vertically downward.