Question

Mercury at \(0^{\circ} \mathrm{C}\) is flowing over a flat plate at a velocity of \(0.1 \mathrm{~m} / \mathrm{s}\). Using appropriate software, determine the effect of the location along the plate \((x)\) on the velocity and thermal boundary layer thicknesses. By varying \(x\) for \(0

Short answer

Discuss the obtained results.

Step-by-step solution

1. Finding the velocity and thermal boundary layer thickness formulæ:

The velocity boundary layer thickness 𝛿v (in meters) can be calculated using the formula: $$ \delta_v = \frac{5x}{\sqrt{Re_x}} $$ where x is the distance from the leading edge of the plate (in meters) and Re_x is the Reynolds number at distance x along the plate. The thermal boundary layer thickness, 𝛿t (in meters), can be calculated using the formula: $$ \delta_t = \delta_v\sqrt{\frac{Pr}{2}} $$ where Pr is the Prandtl number.

2. Calculating the Reynolds number at each x:

The Reynolds number Re_x is calculated using the following formula: $$ Re_x = \frac{\rho u_\infty x}{\mu} $$ where ρ is the density of the fluid (in kg/m³), 𝑢∞ is the free-stream velocity of the fluid (0.1 m/s), x is the distance from the leading edge of the plate (in meters), and μ is the dynamic viscosity of the fluid (in kg/m.s).

3. Using appropriate software:

There are many suitable software to perform such calculations, including Excel, Matlab, and Python. Select one of these and create an algorithm to perform the above calculations for different x values in the given range 0<x≤0.5 m. Use the physical properties of Mercury at 0°𝐶 (density and dynamic viscosity) for fluid flow calculations.

4. Plotting the boundary layer thicknesses as a function of x:

Once we determine boundary layer thicknesses at the various points along the plate, we have to plot the results (x as the independent variable and 𝛿v and 𝛿t as dependent variables). Create a clear graph which represents the thicknesses of both velocity and thermal boundary layers as x varies.

5. Discussing the results:

Analyze the plot, discussing any trends observed as x increases along the plate, noting any significant relationships between the thicknesses of the two boundary layers of Mercury at 0°𝐶 flowing at 0.1 m/s.