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 rr + Δ Fluid shell rz τ rz rz p p p LMB: No acceleration, 0 z F = Shell volume: 2 rrz π ΔΔ ( ) ( ) Force from shear stress on surface: 2 = 2 z rz rz r z r τπ Δ Δ ( ) ( ) Force from shear stress on surface: 2 = 2 rz rz z r z πτ Δ− Δ ( ) ( ) Force from pressure on z surface: 2 = 2 zz pr r p r r ππ Δ Δ ( ) Force from pressure on surface: 2 2 p r r p r r Δ =− Δ Sum these forces, divide by the shell volume and take limit: () ( ) 00 lim lim 0 1 or 0 rz rz r z rz rz pp z p r z ττ 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: () 0 1 1 0 00 1 1 or integrating the pressure equation: Then integrating the stress equation:
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This note was uploaded on 10/09/2011 for the course ENGINEERIN 3133 taught by Professor Sarkozi during the Spring '11 term at Texas A&M.

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Assignment 8 - Special Problem 1 - SpecialProblem1

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