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You have a liquid sitting between 2 plates. A shear stress

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You have a liquid sitting between 2 plates. A shear stress is applied to the top plate causing it to move. (the bottom plate is rigid) This fluid has a yield stress (ty) bellow which it will not move. Assume that the liquid, where in contact with the plate, moves at the same velocity as the plate. Also assume that the fluid has a constant viscosity above ty.

I'm trying to derive a relationship between force applied and velocity of the top plate when the stress is above the yield stress.

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The attachment contains the diagram and all calculations.

Let u be the velocity. ux is the velocity in the x-direction

Momentum equation

At steady state, the time derivative of velocity u is zero. As long as there is no body force or pressure gradient in the ...

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The solution provides detailed explanations and instructions, including diagram, for the problem on finding the relationship between force applied and velocity.