Question

A flow has streamfunction ψ(x,y)=Axy where A is a real constant with units s−1. a) Does the flow satisfy 2D conservation of mass? How do you know? b) Is the flow ir

Answer 1

The flow satisfies 2D conservation of mass but is not irrotational.

Step-by-step explanation:

The student has provided a streamfunction, ψ(x,y) = Axy, where A is a real constant with units s−1. To determine if the flow satisfies 2D conservation of mass, we need to check if the divergence of the velocity field is zero. In this case, the velocity field can be written as v = (∂ψ/∂y, -∂ψ/∂x).

Taking the divergence of v, ∇⋅v, we get (∂^2ψ/∂x∂y) - (∂^2ψ/∂y∂x), which is equal to 0 when mixed partial derivatives are equal. Therefore, the flow does satisfy 2D conservation of mass.

Regarding the question of whether the flow is irrotational, we can find the vorticity by taking the curl of the velocity field. Since v = (∂ψ/∂y, -∂ψ/∂x), the curl of v is ∇×v = (∂^2ψ/∂x^2, ∂^2ψ/∂y^2), which is not equal to zero. Hence, the flow is not irrotational.