Let V = [−y, x, pi] be the velocity vector of a steady fluid flow. Is the flow irrotational? Incompressible?
Since δV/δt = 0, then the flow is steady
The flow is irrotational if the curl of V is zero.
Curl V = (δV3/δy – δV2/δz, δV1/δz – δV3/δx, δV2/δx – δV1/δy),
where V1 = -y, V2 = x and V3 = pi
curl V = ((0 – 0), (0 – 0), (1 + 1)) = (0, 0, 2) not equal to zero!
Since curl V is not equal to zero, the flow is not irrotational.
If the fluid is incompressible, then div V is zero.
Div (V) = δV1/δx + δV2/δy + δV3/δz
Dic(V) = 0 + 0 + 0 = 0
Since div V = 0, then the fluid is incompressible.