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A thin square plate floats on a viscous liquid in a large tank. The liquid depth is much smaller than the tank width. The plate is dragged horizontally at a constant speed u0. Considering the linear (Couette) velocity profile in the liquid film beneath the plate, which of the following statements are correct?

1. The resistive force of the liquid on the plate is inversely proportional to the liquid depth h
2. The resistive force of the liquid on the plate is independent of the plate area
3. The tangential (shear) stress on the floor of the tank increases with u0
4. The tangential (shear) stress on the plate varies linearly with the viscosity eta of the liquid

Correct answer: The resistive force of the liquid on the plate is inversely proportional to the liquid depth h

Solution

With a linear velocity profile, shear stress tau = eta*u0/h. Force F = tau*A = eta*A*u0/h. (A) F proportional to 1/h - TRUE. (B) F depends on A (proportional to A) - FALSE. (C) Floor shear stress = eta*u0/h, which increases with u0 - TRUE. (D) Stress = eta*u0/h is linear in eta - TRUE. So correct statements are A, C, and D.

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