Understanding Laminar Boundary Layer Thickness

The concept of a boundary layer is crucial in fluid dynamics when a fluid flows over a surface, like a flat plate. The boundary layer is the thin layer of fluid near the surface where viscous effects are significant and the fluid velocity changes from zero at the surface (due to the no-slip condition) to the free-stream velocity away from the surface.

Laminar Boundary Layer on a Flat Plate

When the flow is smooth and orderly, it is called laminar flow. For a flat plate with zero pressure gradient, the thickness of the laminar boundary layer ($\delta$) can be determined using the Blasius solution.

The local thickness of the laminar boundary layer at a distance $x$ from the leading edge of the flat plate is given by the formula:

\(\delta \approx 5 \frac{x}{\sqrt{Re_x}}\)

where:

• \(\delta\) is the boundary layer thickness at distance \(x\).
• \(x\) is the distance from the leading edge of the flat plate.
• \(Re_x\) is the local Reynolds number at distance \(x\), defined as \(Re_x = \frac{\rho U_\infty x}{\mu}\).
• \(\rho\) is the fluid density.
• \(U_\infty\) is the free-stream velocity.
• \(\mu\) is the dynamic viscosity of the fluid.

Substituting the expression for \(Re_x\) into the formula for \(\delta\), we get:

\(\delta \approx 5 \frac{x}{\sqrt{\frac{\rho U_\infty x}{\mu}}} = 5 \sqrt{\frac{\mu x^2}{\rho U_\infty x}} = 5 \sqrt{\frac{\mu x}{\rho U_\infty}}\)

This shows that the boundary layer thickness \(\delta\) is proportional to \(\sqrt{x}\).

\(\delta \propto \sqrt{x}\)

Analyzing the Relationship with Distance and Reynolds Number

Based on the formula \(\delta \approx 5 \frac{x}{\sqrt{Re_x}}\) or \(\delta \propto \sqrt{x}\):

• Relationship with distance from leading edge (\(x\)): As the distance \(x\) from the leading edge increases, the value of \(\sqrt{x}\) also increases. Therefore, the thickness of the laminar boundary layer ($\delta$) increases with increasing distance from the leading edge.
• Relationship with Reynolds number (\(Re_x\)): As the local Reynolds number \(Re_x\) increases, the value of \(\sqrt{Re_x}\) increases. Since \(Re_x\) is in the denominator of the formula \(\delta \approx 5 \frac{x}{\sqrt{Re_x}}\), an increase in \(Re_x\) at a given location \(x\) would lead to a decrease in the boundary layer thickness \(\delta\). Note that \(Re_x\) itself depends on \(x\), so when considering a fixed location \(x\), changing \(Re_x\) implies changing fluid properties (\(\rho, \mu\)) or free-stream velocity \(U_\infty\).

Evaluating the Options

Let's examine the given options based on our understanding of laminar boundary layer thickness on a flat plate:

• Option 1: Increases with increase in distance from leading edge. This aligns with our finding that \(\delta \propto \sqrt{x}\). As \(x\) increases, \(\delta\) increases.
• Option 2: Decreases with increase in distance from leading edge. This contradicts our finding.
• Option 3: Constant with increase in distance from leading edge. This contradicts our finding.
• Option 4: Increases with increase in Reynold number at given location. This contradicts our finding that \(\delta\) decreases as \(Re_x\) increases (for a fixed \(x\)).

Therefore, the thickness of the laminar boundary layer on a flat plate increases with an increase in distance from the leading edge.