Problem 6.101 - Problem 6.101[Difficulty 3 Given Data from...

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Problem 6.101 [Difficulty: 3] Given: Data from Table 6.2 Find: Stream function and velocity potential for a source in a corner; plot; velocity along one plane Solution: From Table 6.2, for a source at the origin ψ r θ , ( ) q 2 π θ = ϕ r θ , ( ) q 2 π ln r ( ) = Expressed in Cartesian coordinates ψ x y , ( ) q 2 π atan y x = ϕ x y , ( ) q 4 π ln x 2 y 2 + ( ) = To build flow in a corner, we need image sources at three locations so that there is symmetry about both axes. We need sources at ( h , h ), ( h ,- h ), (- h , h ), and (- h ,- h ) Hence the composite stream function and velocity potential are ψ x y , ( ) q 2 π atan y h x h atan y h + x h + atan y h + x h + + atan y h x h + + = ϕ x y , ( ) q 4 π ln x h ( ) 2 y h ( ) 2 + x h ( ) 2 y h + ( ) 2 + q 4 π x h + ( ) 2 y h + ( ) 2 + x h + ( ) 2 y h ( ) 2 + = By a similar reasoning the horizontal velocity is given by u q x h ( ) 2 π
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