Question

Three layers of liquid are flowing over a fixed solid surface as shown below. The correct order of velocity of liquid in these layers is:

• A. \( V_1 > V_2 > V_3 \)
• B. \( V_1 = V_2 = V_3 \)
• C. \( V_3 > V_2 > V_1 \)
• D. \( V_3 > V_1 > V_2 \)

Hint:

In fluid dynamics, the velocity of a liquid flowing over a solid surface follows \( V_{\text{top}} > V_{\text{middle}} > V_{\text{bottom}} \) due to the no-slip condition.

Answer 1

The Correct Option is C

Step 1: Apply the No-Slip Condition - The layer of liquid in direct contact with the solid surface has zero velocity. - As we move away from the surface, velocity increases.

Step 2: Identify Velocity Profile From the diagram:
- Layer 1 (bottom-most, near the solid surface) → Lowest velocity (\( V_1 \)).
- Layer 2 (middle layer) → Higher velocity (\( V_2 \)).
- Layer 3 (top-most, farthest from the surface) → Highest velocity (\( V_3 \)).

Step 3: Correct Order of Velocities \[ V_3 > V_2 > V_1 \] Thus, the correct answer is \( V_3 > V_2 > V_1 \).

Answer 2

Step 1: Understanding the No-Slip Condition
- When a fluid flows over a solid surface, the fluid layer that is in immediate contact with the surface does not slip or move relative to it. This is known as the no-slip condition.
- According to this condition, the velocity of the fluid right at the solid boundary is exactly zero.
- As we move away from the solid surface into the fluid, the velocity gradually increases due to reduced frictional effects.

Step 2: Analyzing the Velocity Profile from the Diagram
- The diagram shows three distinct layers of fluid flowing over the surface:
- Layer 1: This is the fluid layer closest to the solid surface. Because of the no-slip condition, this layer has the lowest velocity, denoted as \( V_1 \).
- Layer 2: The middle fluid layer experiences less friction than Layer 1 and therefore has a higher velocity, represented as \( V_2 \).
- Layer 3: The topmost fluid layer is farthest from the solid boundary and faces minimal frictional resistance. Consequently, this layer has the highest velocity, labeled as \( V_3 \).

Step 3: Establishing the Correct Order of Velocities
- Based on the no-slip condition and the velocity profile described, the velocities of the layers increase as the distance from the solid surface increases.
- Therefore, the velocity order is: \[ V_3 > V_2 > V_1 \]
- This indicates that the topmost fluid layer moves fastest, the middle layer moves at an intermediate speed, and the bottom layer in direct contact with the solid surface has zero or the lowest velocity.

Final Conclusion:
- The correct velocity relationship among the fluid layers over a solid surface is \( V_3 > V_2 > V_1 \), perfectly illustrating the effect of the no-slip condition on fluid flow near a boundary.