Question

Consider a thin square plate floating on a viscous liquid in a large tank. The height h of the liquid in the tank is much less than the width of the tank. The floating plate is pulled horizontally with a constant velocity $u_0$. Which of the following statements is (are) true?

• A. The resistive force of liquid on the plate is inversely proportional to h.
• B. The resistive force of liquid on the plate is independent of the area of the plate.
• C. The tangential (shear) stress on the floor of the tank increases with $u_0$.
• D. The tangential (shear) stress on the plate varies linearly with the viscosity $\eta$ of the liquid.

Answer 1

Answer: (A), (C), (D)

The tangential (shear) stress on the plate varies linearly with the viscosity $\eta$ of the liquid.

Viscous force is given by $F=-\eta A\frac{dv}{dy}$ since h is very small, so magnitude of viscous force is given by
$F=\eta A\frac{△v}{△y}$
$\therefore F=\frac{\eta A{u}_0}{h}$ (Because $v=u_o$ and y = height of fluid (h))
$\Rightarrow F\propto \eta ,F\propto u_0$
$F\propto \frac{1}{h},F\propto A$
Since plate is moving with constant velocity, same force must be acting on the floor.

So Option (A), (C) and (D) are correct