Question

For the same initial conditions, one can expect the laminar thermal and momentum boundary layers on a flat plate to have the same thickness when the Prandtl number of the flowing fluid is (a) Close to zero (b) Small (c) Approximately one (d) Large (e) Very large

Short answer

Answer: (c) Approximately one

Step-by-step solution

Identify the formulas for the boundary layer thicknesses

The formulas for the thermal and momentum boundary layer thicknesses are given by: Momentum boundary layer thickness (δ): δ = 5 * sqrt(νx / U) Thermal boundary layer thickness (δt): δt = 5 * sqrt(κx / U) where ν is the kinematic viscosity, κ is the thermal diffusivity, x is the distance along the flat plate, and U is the free stream velocity.

Introduce the Prandtl number

The Prandtl number (Pr) is the ratio of kinematic viscosity (ν) to the thermal diffusivity (κ), i.e., Pr = ν / κ. Considering the formulas for the boundary layer thicknesses, we can derive the relationship between the momentum and thermal boundary layer thicknesses as: δ/δt = sqrt(κ/ν). Now we can implement the Prandtl number into this relationship: δ/δt = sqrt(1/Pr).

Analyze the relationship between the boundary layer thicknesses and the Prandtl number

From the relationship δ/δt = sqrt(1/Pr), we can infer that the thicknesses will be equal when the value inside the square root is equal to 1. In other words: 1/Pr = 1, which gives us Pr = 1. So we can conclude that the laminar thermal and momentum boundary layers on a flat plate have the same thickness when the Prandtl number of the flowing fluid is approximately one. Thus, the correct answer is (c) Approximately one.