Question

If the Prandtl number \( \text{Pr} = 0.01 \), which of the following statements is/are correct?

• A. The momentum diffusivity is much larger than the thermal diffusivity.
• B. The thickness of the momentum boundary layer is much smaller than that of the thermal boundary layer.
• C. The thickness of the momentum boundary layer is much larger than that of the thermal boundary layer.
• D. The momentum diffusivity is much smaller than the thermal diffusivity.

Hint:

For small Prandtl numbers (\( \text{Pr} \ll 1 \)), thermal diffusivity dominates over momentum diffusivity, leading to a thicker thermal boundary layer compared to the momentum boundary layer.

Answer 1

The Correct Option is B

Step 1: Understand the Prandtl number. The Prandtl number is defined as: \[ \text{Pr} = \frac{\nu}{\alpha}, \] where: - \( \nu \) is the momentum diffusivity (kinematic viscosity), - \( \alpha \) is the thermal diffusivity. For \( \text{Pr} = 0.01 \): \[ \nu \ll \alpha. \] This means that the momentum diffusivity is much smaller than the thermal diffusivity. Step 2: Boundary layer thicknesses. The thickness of the boundary layer is inversely related to diffusivity. For small Prandtl numbers (\( \text{Pr} \ll 1 \)): - The thermal boundary layer is much thicker than the momentum boundary layer because thermal diffusivity dominates. Step 3: Evaluate the statements. 1. (A): The momentum diffusivity is much larger than the thermal diffusivity. - This is incorrect because \( \nu \ll \alpha \) for \( \text{Pr} = 0.01 \). 2. (B): The thickness of the momentum boundary layer is much smaller than that of the thermal boundary layer. - This is correct because \( \text{Pr} \ll 1 \) implies the thermal boundary layer is thicker than the momentum boundary layer. 3. (C): The thickness of the momentum boundary layer is much larger than that of the thermal boundary layer. - This is incorrect because the opposite is true for \( \text{Pr} \ll 1 \). 4. (D): The momentum diffusivity is much smaller than the thermal diffusivity. - This is correct because \( \nu \ll \alpha \) for \( \text{Pr} = 0.01 \). Step 4: Conclusion. The correct statements are:
(B) The thickness of the momentum boundary layer is much smaller than that of the thermal boundary layer, (D) The momentum diffusivity is much smaller than the thermal diffusivity.