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 11 Top fluid: 0 or ˆ ˆˆ 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 cc τ ττ == = 22 ˆ iscosity for and : ˆ ˆ and Integrating the these equations: and ˆ ˆ BC 2: yx yx xx dv dv dy dy vy c c vv μμ ⎛⎞ =+ ⎜⎟ ⎝⎠ = 1 2 1 21 1 at 0 and ˆ ˆ BC 3: at and ˆ ˆ BC 4: 0 at x x x x c yc c v y c ch y h vV y h cV c V h c μ =→ = = + = = + = 1 1 2 ˆ ˆ (1 ) ˆ ˆ () 1 ˆ Since / 3, ˆ 4 ) 3 4 V hh h V h h V = + −− = + =− =
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Problem 8.23 The stress fields are then: 1 ˆ 4 The velocity fields: 13 3 ˆ and
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Assignment 8 - Problem 8.23 - Problem8.23...

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