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

Short answer

Statement (A) is true as resistive force is inversely proportional to height h. Statement (C) is true as shear stress increases with velocity u_0. Statement (D) is true as shear stress varies linearly with viscosity eta. Statement (B) is false.

Step-by-step solution

Understand the Physical Scenario

Identify and understand the physical principles involved. A square plate is floating on a viscous liquid and is being pulled with a constant velocity. The problem involves fluid dynamics and the principles of viscosity.

Analyze Statement (A)

Recall that the resistive force due to viscosity is given by Newton's law of viscosity, which states the force is proportional to the area of the plate, the velocity gradient, and the viscosity and inversely proportional to the distance over which the velocity changes (in this case, the height h). Thus, as h decreases, the resistive force increases.

Analyze Statement (B)

Consider that the resistive force formula includes the area of the plate. Therefore, the resistive force is not independent of the area of the plate. This statement is false.

Analyze Statement (C)

The tangential or shear stress on the floor is related to the velocity gradient from the plate to the floor. Because there is no explicit dependence on the height h, and the shear stress is caused by the motion of the plate, an increase in velocity u_0 leads to an increase in the velocity gradient and thus an increase in shear stress.

Analyze Statement (D)

The tangential or shear stress is directly proportional to the viscosity of the liquid, according to Newton's law of viscosity. As the viscosity of the liquid increases, the shear stress also increases linearly.

Key concepts

Newton's Law of Viscosity

When we talk about viscosity, we're referring to the property of a fluid that quantifies its resistance to flowing freely. Imagine a sweet, sticky syrup pouring slowly from a bottle—that's high viscosity in action. Conversely, water, which flows readily, has low viscosity. Now, Newton's law of viscosity brings clarity to how this resistance works: it tells us that the internal friction within a fluid layer is directly proportional to the rate at which the fluid layers move past each other.

In mathematical terms, the law is typically expressed as \( \tau = \eta \frac{du}{dy} \) where \( \tau \) is the shear stress, \( \eta \) is the dynamic viscosity, \( du \) is the change in the fluid's velocity, and \( dy \) is the distance over which this change occurs. In the context of the floating plate in our exercise, as the plate moves, it drags the fluid beneath it, creating a velocity gradient. The greater the viscosity or velocity gradient, the greater the resistive force—or shear stress—imposed by the fluid.

Resistive Force

A resistive force is exactly what its name suggests: a force that resists the motion of an object. In fluid dynamics, this resistance comes from the viscosity of the fluid. When an object, like our thin square plate, moves through a viscous fluid, it experiences a drag force. This force opposes the motion and depends on several factors, including the object's speed, surface area, and the fluid's viscosity.

The thicker the fluid or the faster the object moves, the greater the drag. Our exercise also hints at an interesting relationship between the resistive force and the height \( h \) of the liquid—a decrease in height correlates to an increase in resistive force, proving the resistive force is not just a simple hindrance but a variable player in fluid dynamics.

Shear Stress

Shear stress, denoted by \( \tau \) in formulas, is a form of stress that causes layers of fluid to move parallel to each other at different speeds. This is what happens with our thin square plate; as it glides over the liquid, it exerts a force parallel to its surface.

Shear stress is not exclusive to fluids; solids can also experience it—think of scissors cutting through paper. But in fluids, it's pivotal for understanding flow behaviors under different conditions. The direct proportionality of shear stress to viscosity, as mentioned in statement (D), is a key aspect of Newton's law of viscosity. As we increase the plate's speed (thus increasing the velocity gradient), we see a direct effect on the shear stress exerted on the floor of the tank. The exercise reminds us that shear stress is a phenomenon deeply rooted in both speed and the intrinsic properties of the fluid in question.

Velocity Gradient

Imagine sliding your finger across a book's page—the friction you feel is somewhat analogous to the concept of a velocity gradient in fluid dynamics. It is the rate of change of the fluid's velocity relative to a fixed position. The velocity gradient is at the heart of the relationship between the motion of the plate and the fluid in our exercise.

Because the floating plate is moving at constant velocity \( u_{0} \) and the fluid at the bottom is stationary (due to no-slip condition at the solid boundary), there exists a gradual change in velocity from the top surface of the fluid (moving) to the bottom surface (not moving). This gradient is a measure of how quickly the velocities vary, playing a crucial role in determining how the fluid will behave—in this case, how it resists the plate's movement. The velocity gradient is a linchpin in calculating shear stress and understanding the fluid's response to forces applied to it.