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Assignment 8 - Problem 8.23

# Assignment 8 - Problem 8.23 - Problem8.23...

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Problem 8.23 Shell balance – see Special Problem 1 of this assignment – the details of developing the EOM with a shell balance are given there. The current problem does not have a pressure driving force, so we obtain only the stress term. Also, a shell balance in each fluid will produce the same form of the equation. The stress equations for the two fluids are: Provide the following: Governing equation from shell balance, the stress field, the velocity profile, the maximum and average velocities, and the total drag of walls on the fluid 1 1 1 1 Top fluid: 0 or ˆ ˆ ˆ Bottom fluid: 0 or ˆ BC 1: at 0 (the centerline of the 2h channel) ˆ Hence, Use Newton's law of v yx yx yx yx yx yx d c dy d c dy y c c τ τ τ τ τ τ = = = = = = = 1 1 1 1 2 2 ˆ iscosity for and : ˆ ˆ and Integrating the these equations: ˆ ˆ and ˆ ˆ BC 2: yx yx x x x x x x dv dv c c dy dy c c v y c v y c v v τ τ μ μ μ μ = = = + = + = 1 2 2 2 1 2 1 1 ˆ ˆ at 0 and ˆ ˆ BC 3: at and ˆ ˆ BC 4: 0 at x x

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