Animated Solution for Physics - Properties of Solids and Liquids: 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?

Imagine a thin square plate of area $A$ floating on a viscous liquid in a large tank. The depth of the liquid is $h$, which is extremely small compared to the lateral dimensions (width) of the tank. This geometric constraint ($h\ll \text{width}$) is a crucial hint: it allows us to ignore edge effects and assume a steady, fully developed laminar flow beneath the plate.

When the plate is pulled horizontally with a constant velocity $u_0$, the fluid layer in direct contact with the plate moves at the same velocity $u_0$ due to the no-slip condition. Conversely, the fluid layer in contact with the bottom floor of the tank remains stationary ($v=0$). This difference in velocities across the small height $h$ establishes a velocity gradient within the fluid.

To analyze the forces at play, we turn to Newton's Law of Viscosity. This fundamental law states that the tangential viscous force $F_v$ between adjacent layers of fluid is directly proportional to the contact area $A$ and the velocity gradient $\frac{dv}{dy}$:

Here, $\eta$ is the dynamic viscosity of the liquid. The negative sign indicates that the viscous force opposes the relative motion of the plate.

Because the liquid layer is very thin ($h\ll \text{width}$), the velocity profile $v(y)$ can be approximated as linear. Thus, the velocity gradient is constant throughout the depth:

Substituting this back into Newton's law gives the magnitude of the resistive force acting on the plate:

Let's evaluate each statement systematically using our derived formula:

Resistive force is inversely proportional to $h$

Looking at our force equation:

Since $h$ is in the denominator, the resistive force is indeed inversely proportional to the height of the liquid. As the liquid layer becomes thinner, the velocity gradient becomes steeper, requiring a larger force to maintain the same speed $u_0$.

Therefore, Statement (a) is TRUE.

Resistive force is independent of the area of the plate

From the same equation, we see that:

The resistive force is directly proportional to the area of the plate. A larger surface area means more fluid molecules are in contact with the plate, leading to greater total viscous drag.

Therefore, Statement (b) is FALSE.

Tangential (shear) stress on the floor of the tank increases with $u_0$

Tangential (shear) stress $\tau$ is defined as the force per unit area:

By Newton's third law, the shear stress exerted by the fluid on the floor of the tank is equal in magnitude to the shear stress on the plate. Since $\tau \propto u_0$, pulling the plate faster (increasing $u_0$) directly increases the velocity gradient, which in turn increases the shear stress on the floor.

Therefore, Statement (c) is TRUE.

Tangential (shear) stress on the plate varies linearly with viscosity $\eta$

The shear stress on the plate is given by:

This shows a direct, first-power linear relationship between the shear stress and the coefficient of viscosity $\eta$.

Therefore, Statement (d) is TRUE.

This elegant problem beautifully illustrates how a simple linear approximation of a velocity gradient can simplify complex fluid dynamics. By mastering Newton's Law of Viscosity and understanding the physical meaning of shear stress, you can confidently tackle any conceptual fluid mechanics question on the JEE!