= 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 gives
$$
\boxed{\psi(x,y)=e^y\sinh x+C.}
$$
The <streamlines> are its level curves. Setting the irrelevant constant to zero, a nonzero level $K$ has
$$
y=\ln|K|-\ln|\sinh x|,
$$
with $x>0$ for $K>0$ and $x<0$ for $K<0$. Each curve rises to $y=+\infty$ as $x\to0$ 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.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2008/ib/paper-1-streamlines.png]
{title=Streamlines and flow direction for the exponential hyperbolic flow}
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