If the velocity potential phi = x^2 - y^2, what is the stream function psi = psi(x,y)? Draw a few equipotential lines and streamlines for this flow. What kind of a flow can you imagine this to be?
Work:
The fluid is incompressible, flow is steady and two-dimensional.
d(phi)/dx = d(psi)/dy = 2x
psi = (integral)2x = 2xy + f1(x)
d(phi)/dy = -d(psi)/dx = -2y
d(psi)/dx = 2y
psi = (integral)2y = 2xy + f2(y)
2xy + f1(x) = 2xy + f2(y)
psi = 2xy
psi = 0: 2xy = 0, so x = y = 0 (undefined; no flow)
psi = 1: 2xy = 1, so y = 1/2x
psi = 3: 2xy = 3, so y = 3/2x
p>Solutionpsi = 2xy
streamlines drawn in top right and bottom left quadrants are curves; see submitted drawing