Write down the streamfunction for an isotropic line source of strength $2m$ in a uniform stream of speed $U.$ Take the origin to be at the source and the $x-$axis in the direction of the flow at large distance.

$($a$)$ Show that there is a stagnation point upstream of the source and find its position.

$($b$)$ The flow approaching the stagnation from upstream splits to pass either side of a streamline through the stagnation point. Find an equation for the dividing streamline.

$($c$)$ Find the points of intersection of the dividing streamline with the $y$ axis.

$($d$)$ Show that far downstream all streamlines are parallel to the $x-$axis.

$($e$)$ Calculate the far$-$downstream separation of the two branches of the dividing streamline and give a physical explanation of the result.

$($f$)$ What is the velocity potential for this flow?

$($g$)$ Sketch the motion distinguishing between fluid emitted from the source and fluid incident from upstream.