B$8.($a$)$ A two$-$dimensional line vortex of strength $K,$ located at the origin, is defined to be such

that the flow is axisymmetric and takes the form of concentric circles about the origin with constant

circulation $K($measured in the anti$-$clockwise direction$).$ Derive the complex potential for this line

vortex from first principles.

$[7$ marks$]$

$($b$)$ Two line vortices, each of strength $K,$ are situated at $(+-a,0)$; another, of strength $-4K,$ is

situated at the origin. Show that the fluid velocity tends to zero at infinity, but that there are also

stagnation points at $(+-a\sqrt[\phantom{2}]{2},0).$

Construct an equation for the streamlines for this combination of vortices, and show that the

streamline which passes through the $($finite$)$ stagnation points meets the $y-$axis at $(0,+-ab)$ where

$b^8=16(b^2+1{)}^2.$

$[$Hint: You are given that, for a complex quantity $A,ln(A)=ln(|A|)+$iarg$(A).]$

$[11$ marks$]$