Two-Dimensional Fluid Flow

Question

A stream function in a two-dimensional flow is A = 2xy.

Show that the flow is irrotaional(potential kind).

Let q=(u,v) be the velocity

of the two-dimensional flow.

Then w know, u = delta(A)/delta(y) and v =-[delta(A)/delta(x)] , where delta denote partial derivative.

We've been thought that the necessary and sufficient condition for a potential kind of flow is curl q= 0.

So here, I get curl q=0, which implies tgisflowis of potential kind(irrotaional)

So for this question, my professor has used an alternative appoach. Can anyone verify whether my method is an acceptable answer for this problem?

Thank you!

Answer 1

I think the easiest is to show the velocity field's curl is 0 as you suggest. The other way would be to show that you can write the velocity field as the negative gradient of some velocity potential function:

q = (-2x, 2y): Assert q = -grad(phi)

phi = x^2 + f(y), f(y) = - y^2 + C =>

phi = x^2 - y^2 + C,

Since there is a velocity potential we can assert that the curl of the velocity field is 0.

See http://farside.ph.utexas.edu/teaching/336L/Fluidhtml/node69.html for some common properties between stream functions, velocity potentials, and incompressible or irrotational fluids. Based on your formulation of the velocity potential, I'm guessing this is assumed to be an incompressible fluid?