Answer Happy

2. (a) [4 marks] Define the streamfunction 4, in cylindrical polar coordinates (r, 0, z), for the steady two-dimensional flow of an incompressible fluid when the motion is inde- pendent of z and there is no component of velocity in the z-direction. (b) [6 marks] Show that the streamfunction is related to the vorticity (0,0,8) for such motions by V2y = -Ě, where V2 denotes the two-dimensional Laplace operator. (c) (20 marks] Incompressible inviscid fluid of density p is contained between two coaxial circular cylinders of radius a and b respectively, where a <b, and between two rigid planes perpendicular to the axis at a distance L apart. The cylinders and planes are at rest and the fluid is circulating in irrotational motion. (i) If the speed of the fluid at the surface on the inner cylinder is V, show that the streamfunction takes the form y (r) = -Va logr. (ii) Show that the total kinetic energy possessed by the fluid volume is pra²V²L log (b/a).