Question

Under what conditions will the normalized velocity, thermal, and concentration boundary layers coincide during flow over a flat plate?

Short answer

**Answer:** The normalized velocity, thermal, and concentration boundary layers coincide during flow over a flat plate when the Prandtl number is equal to the Schmidt number (Pr = Sc). This ensures that the momentum, heat, and mass transfer processes are occurring at the same rate, leading to similar boundary layer behavior.

Step-by-step solution

Determine the dimensionless parameters for each boundary layer

For the velocity boundary layer, we'll use the Reynolds number (Re), which is the ratio of inertial forces to viscous forces. For the thermal boundary layer, we'll use the Prandtl number (Pr), which is the ratio of momentum diffusivity to thermal diffusivity. For the concentration boundary layer, we'll use the Schmidt number (Sc), which is the ratio of momentum diffusivity to mass diffusivity. Re = \frac{ρuL}{μ}, Pr = \frac{ν}{α}, Sc = \frac{ν}{D}

Analyze the boundary layer equations

For a steady laminar flow over a flat plate, the boundary layer equations are governed by the following partial differential equations for each parameter: - Velocity: \frac{∂u}{∂x} + \frac{∂v}{∂y} = 0 - Temperature: u\frac{∂T}{∂x} + v\frac{∂T}{∂y} = α\frac{∂^2 T}{∂y^2} - Concentration: u\frac{∂C}{∂x} + v\frac{∂C}{∂y} = D\frac{∂^2 C}{∂y^2}

Set up boundary conditions

For each boundary layer, the boundary conditions at the flat plate (y = 0) and far away from the plate (y → ∞) are as follows: - Velocity: u(0) = 0, u(∞) = U - Temperature: T(0) = T_w, T(∞) = T_∞ - Concentration: C(0) = C_w, C(∞) = C_∞

Find the relationship between dimensionless parameters

For the boundary layers to coincide, the dimensionless parameters must satisfy the following relationship: RePr = ReSc This equation implies that the velocity and thermal boundary layers will coincide when the Prandtl number is equal to the Schmidt number.

Determine the conditions for boundary layers to coincide

By analyzing the relationship between dimensionless parameters, we find that the normalized velocity, thermal, and concentration boundary layers will coincide during flow over a flat plate when the Prandtl number is equal to the Schmidt number (Pr = Sc). This condition ensures that the momentum, heat, and mass transfer processes are occurring at the same rate, leading to similar boundary layer behavior.