Fluid Dynamics | Part IB, 2001

Starting from the Euler equation, derive the vorticity equation for the motion of an inviscid incompressible fluid under a conservative body force, and give a physical interpretation of each term in the equation. Deduce that in a flow field of the form u=(u(x,y,t),v(x,y,t),0)\mathbf{u}=(u(x, y, t), v(x, y, t), 0) the vorticity of a material particle is conserved.

Find the vorticity for such a flow in terms of the stream function ψ\psi. Deduce that if the flow is steady, there must be a function ff such that

2ψ=f(ψ)\nabla^{2} \psi=f(\psi)

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