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 (Pr) is equal to the Schmidt number (Sc).

Step-by-step solution

Identify dimensionless parameters governing boundary layers

The three main dimensionless parameters governing boundary layers are Reynolds number (Re), Prandtl number (Pr), and Schmidt number (Sc). The Reynolds number represents the ratio of inertial forces to viscous forces, the Prandtl number represents the ratio of momentum diffusivity to thermal diffusivity, and the Schmidt number represents the ratio of momentum diffusivity to mass diffusivity. Re = \(\frac{ρUL}{μ}\) Pr = \(\frac{μ}{α}\) Sc = \(\frac{μ}{D}\) where: ρ = fluid density U = free stream velocity L = length of the flat plate μ = dynamic viscosity α = thermal diffusivity D = mass diffusivity

Determine the conditions for coinciding boundary layers

For the boundary layers to coincide, their thicknesses should be related by the dimensionless parameters Re, Pr, and Sc. The thickness of the boundary layers can be related to the dimensionless parameters as follows: δ = \(\frac{L}{\sqrt{Re}}\) δ_t = \(\frac{δ}{\sqrt{Pr}}\) δ_c = \(\frac{δ}{\sqrt{Sc}}\) where: δ = thickness of the velocity boundary layer δ_t = thickness of the thermal boundary layer δ_c = thickness of the concentration boundary layer For the boundary layers to coincide, we need to have: δ = δ_t = δ_c Which means: \(\frac{L}{\sqrt{Re}} = \frac{δ}{\sqrt{Pr}} = \frac{δ}{\sqrt{Sc}}\)

Find the relation between Pr and Sc for coinciding boundary layers

Now, we can derive the relation between Pr and Sc by setting the expressions for δ_t and δ_c equal: \(\frac{δ}{\sqrt{Pr}} = \frac{δ}{\sqrt{Sc}}\) Then, solving for Pr: Pr = \(\frac{δ^2}{δ^2}\) Sc Thus, Pr = Sc

State the conditions for coinciding boundary layers

The conditions under which the normalized velocity, thermal, and concentration boundary layers coincide during flow over a flat plate are when the Prandtl number (Pr) is equal to the Schmidt number (Sc).

Key concepts

Reynolds Number

The Reynolds Number is a crucial concept in fluid dynamics that helps predict flow patterns in different fluid flow situations. It is dimensionless, meaning it has no units, and is symbolized as Re. This number evaluates the ratio of inertial forces to viscous forces in a fluid. The formula for calculating the Reynolds Number is:
\[ Re = \frac{ρUL}{μ} \]
Where:

• \( ρ \) is the fluid density, representing how much mass the fluid has in a given volume.
• \( U \) is the free stream velocity, indicating the speed at which the fluid flows across a surface.
• \( L \) is the length of the flat plate, used as a characteristic dimension in flow conditions.
• \( μ \) is the dynamic viscosity, a measure of fluid's resistance to flow.

Smaller Reynolds Numbers indicate laminar flow, where fluid particles move in layers without much mixing. Higher Reynolds Numbers suggest turbulent flow, with fluid particles moving in chaotic patterns and greater mixing. By understanding the Reynolds Number, engineers and scientists can design more efficient systems, from aircraft to chemical reactors.

Prandtl Number

The Prandtl Number is another dimensionless parameter important in the study of fluid flow, heat transfer, and boundary layers. It is represented as Pr and quantifies the ratio of momentum diffusivity (kinematic viscosity) to thermal diffusivity. The equation to find the Prandtl Number is:
\[ Pr = \frac{μ}{α} \]
Where:

• \( μ \) stands for dynamic viscosity, helping to calculate kinematic viscosity when divided by fluid density.
• \( \alpha \) is the thermal diffusivity, describing how quickly heat spreads through a material.

The Prandtl Number informs us about how fast momentum is transferred relative to heat within a fluid. A high Prandtl Number indicates that momentum diffuses quicker than heat, typical for oils and high-viscosity fluids. Low Prandtl Numbers, like those found in liquid metals, mean heat moves faster than momentum. In practical terms, this helps in designing systems with optimal cooling or heating, such as heat exchangers.

Schmidt Number

The Schmidt Number, symbolized as Sc, is yet another dimensionless number used in fluid dynamics. It describes the ratio of momentum diffusivity (kinematic viscosity) to mass diffusivity, providing insight into mass transfer within a fluid. The formula for the Schmidt Number is:
\[ Sc = \frac{μ}{D} \]
Where:

• \( μ \) signifies dynamic viscosity, essential for determining how forcefully the fluid resists flow.
• \( D \) represents mass diffusivity, which indicates how fast a substance spreads out in another.

The Schmidt Number is particularly useful when studying mass transfer in gases and liquids, such as in chemical processing and environmental engineering scenarios. If the Schmidt Number is large, momentum diffusivity dominates, meaning the fluid resists shear stress more than mass diffusion. Conversely, a small Schmidt Number suggests mass diffusivity prevails, vital in understanding how gases mix in various applications, from pollution control to fermentation.