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Problem 6.113 - Problem 6.113[Difficulty 4 Given Complex...

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Problem 6.113 [Difficulty: 4] Given: Complex function Find: Show it leads to velocity potential and stream function of irrotational incompressible flow; Show that df/dz leads to u and v Solution: Basic equations: u y ψ = v x ψ = u x φ = v y φ = First consider x f x z z f d d = 1 z f d d = z f d d = (1) and also y f y z z f d d = i z f d d = i z f d d = (2) Hence 2 x f 2 x x f = z z f d d d d = 2 z f d d 2 = and 2 y f 2 y y f = i z i z f d d d d = 2 z f d d 2 = Combining 2 x f 2 2 y f 2 + 2 z f d d 2 2 z f d d 2 = 0 = Any differentiable function f(z) automatically satisfies the Laplace
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