Question

The hydrodynamic and thermal boundary layer will merge, when______

• A. Prandtl number is one
• B. Schmidt number tends to infinity
• C. Nusselt number tends to infinity
• D. Archimedes number is greater than 10,000

Hint:

For mass transfer, the analogous relationship between the hydrodynamic and concentration boundary layers is:
\[ \frac{\delta}{\delta_c} \approx Sc^{1/3} \]
where \( Sc \) is the Schmidt number. When \( Sc = 1 \), the hydrodynamic and concentration boundary layers merge.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The question asks for the condition under which the thickness of the hydrodynamic (velocity) boundary layer (\( \delta \)) is equal to the thickness of the thermal boundary layer (\( \delta_t \)).
This is a key concept in boundary layer theory and convective transport.

Step 2: Key Formula or Approach:
For laminar flow over a flat plate, the relationship between the hydrodynamic boundary layer thickness and the thermal boundary layer thickness is given by:
\[ \frac{\delta}{\delta_t} \approx Pr^{1/3} \]
where \( Pr \) is the dimensionless Prandtl number.

Step 3: Detailed Explanation:

• Prandtl Number Definition: The Prandtl number is defined as the ratio of momentum diffusivity (kinematic viscosity) to thermal diffusivity:
\[ Pr = \frac{\nu}{\alpha} \]

• Boundary Layer Growth: The momentum diffusivity governs the rate of growth of the velocity boundary layer, while the thermal diffusivity governs the growth of the thermal boundary layer.

• Condition for Merging: If the rate of momentum diffusion equals the rate of thermal diffusion:
\[ \nu = \alpha \quad \implies \quad Pr = 1 \]
Under this condition:
\[ \frac{\delta}{\delta_t} = (1)^{1/3} = 1 \quad \implies \quad \delta = \delta_t \]
Thus, the hydrodynamic and thermal boundary layers will grow at the same rate and merge.

• This behavior is commonly observed in gases (e.g., air has \( Pr \approx 0.7 \), which is close to 1).


Step 4: Final Answer:
The hydrodynamic and thermal boundary layers will merge when the Prandtl number is equal to one.