Question

The hydrodynamic and thermal boundary layers 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:

The hydrodynamic and thermal boundary layers merge when \( \text{Pr} = 1 \), as momentum and heat diffuse at the same rate, making their thicknesses equal.

Answer 1

The Correct Option is A

Step 1: Understand hydrodynamic and thermal boundary layers. 
The hydrodynamic boundary layer is the region near a surface where the fluid velocity transitions from zero (at the surface, due to the no-slip condition) to the free stream velocity. The thermal boundary layer is the region where the temperature transitions from the surface temperature to the free stream temperature. The thickness of these layers depends on the fluid properties and flow conditions. 
Step 2: Analyze the condition for merging of boundary layers. 
The relative thickness of the hydrodynamic (\( \delta \)) and thermal (\( \delta_t \)) boundary layers is determined by the Prandtl number (Pr), defined as: \[ \text{Pr} = \frac{\nu}{\alpha} = \frac{\text{kinematic viscosity}}{\text{thermal diffusivity}}, \] where: \( \nu = \frac{\mu}{\rho} \): kinematic viscosity \( \text{m}^2\text{s} \),
\( \alpha = \frac{k}{\rho c_p} \): thermal diffusivity (\( \text{m}^2\text{s} \)),
\( \mu \): dynamic viscosity (\( \text{Pas} \)),
\( \rho \): density (\( \text{kg/m}^3 \)),
\( k \): thermal conductivity (\( \text{W/mK} \)),
\( c_p \): specific heat capacity (\( \text{J/kgK} \)).
The Prandtl number compares the rate of momentum diffusion (hydrodynamic) to thermal diffusion. The ratio of the boundary layer thicknesses is approximately: \[ \frac{\delta}{\delta_t} \approx \text{Pr}^{1/3}. \] When \( \text{Pr} = 1 \), \( \nu = \alpha \), meaning momentum and heat diffuse at the same rate, so the hydrodynamic and thermal boundary layers are of the same thickness (\( \delta \approx \delta_t \)), effectively merging.
When \( \text{Pr}>1 \) (e.g., oils), the thermal boundary layer is thinner (\( \delta_t<\delta \)).
When \( \text{Pr}<1 \) (e.g., gases like air), the thermal boundary layer is thicker (\( \delta_t>\delta \)). 
Step 3: Evaluate the options. 
(1) Prandtl number is one: Correct, as \( \text{Pr} = 1 \) means the hydrodynamic and thermal boundary layers have the same thickness, effectively merging. Correct.
(2) Schmidt number tends to infinity: Incorrect, as the Schmidt number (\( \text{Sc} = \frac{\nu}{D} \), where \( D \) is mass diffusivity) relates to mass transfer, not thermal boundary layers. Incorrect.
(3) Nusselt number tends to infinity: Incorrect, as the Nusselt number (\( \text{Nu} = \frac{hL}{k} \)) relates to the heat transfer coefficient, not directly to boundary layer thickness. A high Nu indicates enhanced heat transfer, but does not imply merging of layers. Incorrect.
(4) Archimedes number is greater than 10,000: Incorrect, as the Archimedes number (\( \text{Ar} = \frac{gL^3 \rho (\rho_s - \rho)}{\mu^2} \)) relates to buoyancy-driven flows, not boundary layer merging. Incorrect. 
Step 4: Select the correct answer. 
The hydrodynamic and thermal boundary layers will merge when the Prandtl number is one, matching option (1).