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Assignment 8 - Special Problem 1

Assignment 8 - Special Problem 1 - SpecialProblem1...

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Special Problem 1 For laminar flow in a tube (see Section 8.3 in text), use a shell balance to develop the governing equation of motion and check this result with the tables in the Summary handout; then show all steps in solving for the stress field, the velocity profile, the maximum and average velocities, and the drag force of the tube on the fluid. z r Flow in cylindrical tube r r r + Δ Fluid shell rz τ rz τ rz τ p p p LMB: No acceleration, 0 z F = Shell volume: 2 r r z π Δ Δ ( ) ( ) Force from shear stress on surface: 2 = 2 z rz rz r r r r r r r z r τ π τ π + Δ Δ Δ ( ) ( ) Force from shear stress on surface: 2 = 2 rz rz r r r r z r z τ π τ π Δ Δ ( ) ( ) Force from pressure on z surface: 2 = 2 z z p r r p r r π π Δ Δ ( ) Force from pressure on surface: 2 2 z z z z z z p r r p r r π π + Δ Δ = − Δ Sum these forces, divide by the shell volume and take limit: ( ) ( ) ( ) 0 0 lim lim 0 1 or 0 rz rz r r r z z z r z rz r r p p r r z p r r r z τ τ τ Δ → Δ → = Δ Δ =
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Special Problem 1 (cont) Note that the first term of previous equation is a function of r and the second term is a function of z. Then: (
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