Problem 5.33 - Problem*5.33[Difficulty 3 Approximate...

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Problem *5.33 [Difficulty: 3] Given: Approximate profile for a laminar boundary layer: u U sin π y 2 δ δ cx (c is constant) Find: (a) Stream function for the flow field (b) Location of streamlines at one-quarter and one-half the total flow rate in the boundary layer. Solution: We will generate the stream function from the velocity field. x v y u Governing Equations: (Definition of stream function) Integrating the x-component of velocity yields the stream function: ψ y U sin π y 2 δ df x () 2U δ π cos π y 2 δ fx If we set ψ 0 at y0 the stream function would be: ψ δ π cos π y 2 δ The total flow rate per unit depth within the boundary layer is: Q ψδ ψ 0 δ π cos π 2 cos 0 δ π At one-quarter of the flow rate in the boundary layer: Q 1 4 δ π U δ 2
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