Problem 5.37 - Problem*5.37 Given[Difficulty 3 Rigid body...

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Problem *5.37 [Difficulty: 3] Given: Rigid body motion in Example Problem 5.6 e r C V ˆ C 0.3 m 2 s Find: (a) Stream function for the flow field (b) Volume flow rate per unit depth between r = 0.20 m and 0.24 m (c) Sketch velocity profiles along a line of constant θ (d) Check the volume flow rate calculated from the stream function by integrating the velocity profile along this line. Solution: We will generate the stream function from the velocity field. r V r V r 1 Governing Equations: (Definition of stream function) Assumptions: Incompressible flow Flow is in the r- θ plane only Integrating the θ -component of velocity yields the stream function: ψ r C r d f θ ( ) C ln r ( ) f θ ( ) Now take the derivative of the stream function: V r 1 r df d θ 0 Therefore, f θ ( ) C 1 ψ C ln r ( ) C 1 The volume flow rate per unit depth is: Q ψ r 2
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