Problem 6.112 - Problem 6.112 [Difficulty: 4] Given:...

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Problem 6.112 [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: Irrotationality because φ exists u y ψ = v x ψ = u x φ = v y φ = Incompressibility x u y v + 0 = Irrotationality x v y u 0 = fz () z 6 = xi y + () 6 = Expanding () x 6 15 x 4 y 2 15 x 2 y 4 + y 6 i6xy 5 6x 5 y + 20 x 3 y 3 + = We are thus to check the following φ xy , ()x 6 15 x 4 y 2 15 x 2 y 4 + y 6 = ψ , 6 x y 5 5 y + 20 x 3 y 3 = uxy , x φ , = so , ( ) 60 x 3 y 2 5 30 x y 4 = vxy , y φ , = so , ( ) 30 x 4 y 60 x 2 y 3 6y 5 + = An alternative derivation of u and v is , y ψ , = , ( ) 60 x 3 y 2 5 30 x y 4 = , x ψ , = , ( ) 30 x 4 y 60 x 2 y 3 5 + = Hence x ,
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