Question

The force (F) required to maintain the flow of layers of a liquid is equal to (A = area of contact of layers, dz = distance between the layers, du = change in velocity, \(\eta\) = coefficient of viscosity)

• A. \( \eta \frac{du}{dz} \frac{1}{A} \)
• B. \( \eta \frac{dz}{du} A \)
• C. \( \eta A \frac{du}{dz} \)
• D. \( \eta \frac{dz}{A} \frac{1}{du} \)

Hint:

Remember Newton's law of viscosity as "Stress is proportional to Strain Rate". Stress = Force/Area. Strain Rate = Velocity Gradient (\(du/dz\)). The proportionality constant is the viscosity (\(\eta\)). This gives \(F/A = \eta (du/dz)\), which is the fundamental equation for viscous force.

Answer 1

The Correct Option is C

This question relates to Newton's law of viscosity.
Newton's law of viscosity states that the shear stress (\(\tau\)) in a fluid is directly proportional to the rate of shear strain, which is also called the velocity gradient.
Shear stress is defined as the force per unit area: \( \tau = \frac{F}{A} \).
The velocity gradient is the rate of change of velocity with respect to the distance perpendicular to the flow: \( \frac{du}{dz} \).
The proportionality is expressed as: \( \tau \propto \frac{du}{dz} \).
The constant of proportionality is the coefficient of viscosity, \(\eta\).
So, the equation is \( \tau = \eta \frac{du}{dz} \).
Substituting the definition of shear stress, we get:
\( \frac{F}{A} = \eta \frac{du}{dz} \).
To find the force (F), we multiply both sides by the area (A).
\( F = \eta A \frac{du}{dz} \).
This matches option (C).