Define the terms irrotational flow and incompressible flow. The two-dimensional flow of an incompressible fluid is given in terms of a streamfunction as
in Cartesian coordinates . Show that the line integral
along any path joining the points and , where is the unit normal to the path. Describe how this result is related to the concept of mass conservation.
Inviscid, incompressible fluid is contained in the semi-infinite channel , , which has rigid walls at and at , apart from a small opening at the origin through which the fluid is withdrawn with volume flux per unit distance in the third dimension. Show that the streamfunction for irrotational flow in the channel can be chosen (up to an additive constant) to satisfy the equation
and boundary conditions
if it is assumed that the flow at infinity is uniform. Solve the boundary-value problem above using separation of variables to obtain