Viscosity

For a small sphere falling through a viscous fluid, terminal velocity is given by Stokes’ law: where:

• is the radius of the sphere,
• is density of the sphere,
• is density of the fluid,
• is coefficient of viscosity.

From this formula:
Hence, the correct answer is the option corresponding to:

Concept:
• Capillary rise:

Step 1: Given radii ratio


Step 2: Inverse relation
Final Conclusion:
Option (E)

Concept:
• Relative viscosity remains nearly constant over moderate temperature range

Step 1: Physical observation
Blood viscosity ratio with water is approximately constant from to Final Conclusion:
Option (C)

Step 1: Understanding the Concept:
When a spherical body falls through a viscous fluid, it initially accelerates due to gravity. As its velocity increases, the viscous drag force opposing its motion also increases (according to Stokes' Law). Eventually, a state is reached where the upward forces (buoyant force + viscous drag) perfectly balance the downward gravitational force. At this point, net force is zero, and the body falls with a constant maximum velocity known as the terminal velocity.
Step 2: Key Formula or Approach:
At terminal velocity ( ):


Where:
= radius of the spherical ball
= density of the ball material (iron)
= density of the fluid
= coefficient of viscosity of the fluid
= acceleration due to gravity
Step 3: Detailed Explanation:
Rearranging the balance equation to solve for :



Looking at the final expression, for a given fluid (constant , constant ) and a given material (constant ), all terms are constant except the radius .
Therefore, the relationship is:

The terminal velocity is directly proportional to the square of the radius of the ball.
Step 4: Final Answer:
Terminal velocity is proportional to .