Consider laminar flow over a flat plate. Will the friction coefficient change with distance from the leading edge? How about the heat transfer coefficient?

Friction coefficient (\(C_f\)) is defined as the ratio of wall shear stress (\(\tau_w\)) to the dynamic pressure (\(\frac{1}{2}\rho U_\infty^2\)) in incompressible boundary layer flows over a flat plate: \(C_f=\frac{\tau_w}{\frac{1}{2}\rho U_\infty^2}\) Here \(\tau_w\) represents the wall shear stress, \(\rho\) denotes the fluid density, and \(U_\infty\) is the free-stream velocity of the fluid approaching the flat plate. For laminar boundary layer flow, we can derive the friction coefficient as a function of Reynolds number (\(Re_x\)), where \(x\) is the distance from the leading edge.

For laminar flow over a flat plate, the Reynolds number at any distance x along the surface of the plate can be defined as: \(Re_x=\frac{\rho U_\infty x}{\mu}\) Here, \(\mu\) is the fluid's dynamic viscosity. By using the Blasius solution for laminar boundary layer flow over a flat plate (which is applicable for \(Re_x < 5\times10^5\)), we can express the friction coefficient as a function of \(Re_x\): \(C_f=\frac{1.328}{\sqrt{Re_x}}\) Since \(Re_x\) depends on \(x\): \(Re_x \propto x\), we conclude that the friction coefficient changes with the distance from the leading edge.

The heat transfer coefficient (\(h\)) is defined as the proportionality constant between the local heat flux (\(q_w\)) and the temperature difference between the free-stream fluid temperature (\(T_\infty\)) and the wall temperature (\(T_w\)): \(h=\frac{q_w}{T_w-T_\infty}\) We will now evaluate if \(h\) changes with the distance from the leading edge for laminar boundary layer flow over a flat plate.

For laminar flow over a flat plate, we can apply the analogy between momentum transfer (wall shear stress) and heat transfer through Reynolds Analogy, which states that the Nusselt number (\(Nu_x\)) is related to \(Re_x\) as: \(Nu_x=\frac{h x}{k}=\frac{C_{fx} Re_x Pr}{2}\) Here, \(k\) represents the thermal conductivity of the fluid, and \(Pr\) is its Prandtl number. Since both \(Nu_x\) and \(Re_x\) depend on distance \(x\): \(Nu_x \propto x\) and \(Re_x \propto x\), we conclude that the heat transfer coefficient changes with the distance from the leading edge for laminar boundary layer flow over a flat plate. In summary, both the friction coefficient and the heat transfer coefficient change with the distance from the leading edge in laminar flow over a flat plate.