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

A velocity boundary layer is defined as the region in the fluid flow adjacent to a solid surface where the flow velocity is less than 99% of the free-stream velocity. A thermal boundary layer is defined as the region where the temperature difference between the solid surface and the fluid is significant.

The velocity boundary layer thickness (\(\delta\)) can be expressed using the following formula: $$ \delta = 5 \sqrt{\frac{\nu x}{U_0}} $$ where \(\nu\) is the kinematic viscosity of the fluid, \(x\) is the location along the plate, and \(U_0\) is the free-stream velocity. The thermal boundary layer thickness (\(\delta_T\)) can be expressed using the following formula: $$ \delta_T = \sqrt{\frac{\alpha x}{U_0}} $$ where \(\alpha\) is the thermal diffusivity of the fluid.

For Mercury at \(0^{\circ} \mathrm{C}\), the kinematic viscosity \(\nu\) is approximately \(1.14 \times 10^{-7} \mathrm{~m}^2 / \mathrm{s}\), and the thermal diffusivity \(\alpha\) is approximately \(1.175 \times 10^{-7} \mathrm{~m}^2 / \mathrm{s}\). Now, for \(x\) ranging from \(0\) to \(0.5 \mathrm{~m}\), we can calculate the velocity and thermal boundary layer thicknesses using the formulae from Step 2.

Using the calculated values of velocity and thermal boundary layer thicknesses in Step 3, we plot these values against \(x\) to create a graph showing the effect of location along the plate on the respective boundary layers.

From the plot, we can observe that both the velocity and thermal boundary layer thicknesses increase with increasing \(x\). This indicates that, as we move along the plate, the regions in which the flow velocity and the temperature difference are significant, increase in thickness. At the beginning of the plate, the boundary layers are thin, and the velocity and temperature gradients are steep. As we move further along the plate, the gradients become less steep, although the velocity and thermal boundary layers continue to grow. In conclusion, the location along the plate significantly impacts the velocity and thermal boundary layer thicknesses, as they both increase with increasing \(x\). This information is important when designing systems where control of fluid flow or heat transfer is critical. The plot effectively demonstrates the relationship between the location along a flat plate and the thicknesses of the velocity and thermal boundary layers for Mercury at \(0^{\circ} \mathrm{C}\).