Question

Consider a laminar boundary layer flow over a flat plate. Determine the \(\delta / \delta_{\mathrm{t}}\) ratios for air (at 1 atm), liquid water, isobutane, engine oil, and mercury. Evaluate all properties at $50^{\circ} \mathrm{F}$.

Short answer

Question: Calculate the thickness ratio (\(\frac{\delta}{\delta_{\mathrm{t}}}\)) for different fluids: air, liquid water, isobutane, engine oil, and mercury at a temperature of \(50^{\circ} \mathrm{F}\) (laminar boundary layer flow over a flat plate). Use the same reference distance and free-stream velocity for all fluids. Answer: To find the thickness ratio (\(\frac{\delta}{\delta_{\mathrm{t}}}\)) for air, liquid water, isobutane, engine oil, and mercury at \(50^{\circ} \mathrm{F}\), first convert the temperature to Celsius (which is \(10^{\circ}\mathrm{C}\)). Next, determine the kinematic viscosity (\(\nu\)) and density (\(\rho\)) for each fluid at \(10^{\circ}\mathrm{C}\) using reference tables. Calculate the thickness ratio for each fluid using the formula \(\frac{\delta}{\delta_{\mathrm{t}}} = \sqrt{ \frac{2}{\pi} \cdot \frac{Ux}{\nu} }\) with the same reference distance and free-stream velocity for all fluids.

Step-by-step solution

Convert Temperature

First, we will convert the given temperature of \(50^{\circ} \mathrm{F}\) to Celsius. The conversion formula we will use is: \(T_{\mathrm{C}} = (T_{\mathrm{F}} - 32) \times (5/9)\) Now, plug in the values: \(T_{\mathrm{C}} = (50 - 32) \times (5/9) = 10^{\circ} \mathrm{C}\). From now on, we will use the temperature of \(10^{\circ}\mathrm{C}\) to evaluate all properties.

Fluid Properties

We will now find the properties for each fluid at \(10^{\circ}\mathrm{C}\), which include the kinematic viscosity (\(\nu\)) and density (\(\rho\)). We will use reference tables or databases to find these values.

Calculate Thickness Ratio

To calculate the thickness ratio, we first need to find the boundary layer thickness (\(\delta\)) and the displacement thickness (\(\delta_{\mathrm{t}}\)) for each fluid. The thickness ratio is given by: \(\frac{\delta}{\delta_{\mathrm{t}}} = \sqrt{ \frac{2}{\pi} \cdot \mathrm{Re}_{x} }\) where \(\mathrm{Re}_{x}\) is the Reynolds number at the reference distance. The Reynolds number for laminar flow is given by \(\mathrm{Re}_{x} = \frac{Ux}{\nu}\) where \(U\) is the free-stream velocity and \(x\) is the reference distance. Assuming a constant value of \(Ux\) for all fluids (since we are looking for the ratio), we can find \(\frac{\delta}{\delta_{\mathrm{t}}}\) for each fluid as: \(\frac{\delta}{\delta_{\mathrm{t}}} = \sqrt{ \frac{2}{\pi} \cdot \frac{Ux}{\nu} }\) Now, calculate the thickness ratio for each fluid using their viscosity values. For example, let's calculate the thickness ratio for air: \(\frac{\delta}{\delta_{\mathrm{t}}} = \sqrt{ \frac{2}{\pi} \cdot \frac{Ux}{\nu_{\mathrm{air}}} }\) Repeat the same process for liquid water, isobutane, engine oil, and mercury to find their respective thickness ratios.