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Problem 6.111 - Problem 6.111 Given Velocity potential...

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Problem 6.111 [Difficulty: 2] Given: Velocity potential Find: Show flow is incompressible; Stream function 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 = φ x y , ( ) x 6 15 x 4 y 2 15 x 2 y 4 + y 6 = u x y , ( ) x φ x y , ( ) = u x y , ( ) 60 x 3 y 2 6 x 5 30 x y 4 = v x y , ( ) y φ x y , ( ) = v x y , ( ) 30 x 4 y 60 x 2 y 3 6 y 5 + = Hence x u x y , ( ) y v x y , ( ) + 0 = Hence flow is INCOMPRESSIBLE Hence u y ψ = so ψ x y , ( ) y u x y ,
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