p1126 - y , the velocity potential is ( x, y ) = U y + e x/...

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30 CHAPTER 11. POTENTIAL FLOW 11.26 Chapter 11, Problem 26 The solution strategy is to first compute the velocity components from the velocity poten- tial. Then, introducing the streamfunction, we obtain two equations relating the derivatives of the streamfunction and the velocity components. Integration of these equations yields the streamfunction. Hence, we begin by writing u = ∂φ x = U ± 1 e x/ λ sin( y/ λ ) = v = ∂φ y = Ue x/ λ cos( y/ λ ) By definition, the velocity components are related to the streamfunction as follows. u = ∂ψ y and v = ∂ψ x So, we have ∂ψ y = U ± 1 e x/ λ sin( y/ λ ) = Integrating over
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Unformatted text preview: y , the velocity potential is ( x, y ) = U y + e x/ cos( y/ ) = + f ( x ) where f ( x ) is a function of integration. Now, we differentiate with respect to x to evaluate v , wherefore, v = Ue x/ cos( y/ ) df dx Comparing this equation to the value obtained above from v = / y , clearly df dx = 0 = f ( x ) = constant With no loss of generality, we can set the constant equal to zero. Therefore, the stream-function for this flow is ( x, y ) = U y + e x/ cos( y/ ) =...
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