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Problem 5.87 - r θ-z plane z V θ 1 r θ V z z-r plane z V...

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(a) Rates of linear and angjular deformation for this flow (b) Expression for the vorticity vector Problem 5.87 [Difficulty: 2] Given: Velocity field for fully-developed flow in a circular tube Find: Solution: We will apply the definition of vorticity to the given velocity field. V Governing Equation: (Definition of vorticity) Assumptions: (1) Steady flow 0 1 1 z V V r r rV r V z r Rates of linear deformation in all three directions is zero. The volume dilation rate of the flow is: The angular deformations are: r- θ plane: 1 r r V θ r θ
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Unformatted text preview: r θ-z plane: z V θ 1 r θ V z z-r plane: z V r r V z V max 2 r R 2 angdef V max 2 r R 2 k V r r rV r e r V z V e z V V r V r z r r z ˆ 1 1 ˆ ˆ 1 The vorticity in cylindrical coordinates is: e R r V ˆ 2 2 max...
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