Question

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

Short answer

Answer: In laminar flow over a flat plate, the friction coefficient decreases with increasing distance from the leading edge, as per the Blasius solution. The heat transfer coefficient also changes with distance from the leading edge and depends on fluid properties and flow parameters, such as the Reynolds and Prandtl numbers.

Step-by-step solution

Understanding Laminar Flow Over a Flat Plate

Laminar flow over a flat plate refers to the smooth and orderly fluid movement in parallel layers without mixing, over a flat surface. This flow regime is characterized by the smooth sliding of fluid layers and the absence of turbulence.

Define the Friction Coefficient

The friction coefficient (\(C_f\)) is a dimensionless number that represents the ratio of wall shear stress (\(\tau_w\)) to the fluid dynamic pressure (\(\frac{1}{2} \rho U^2\)), where \(\rho\) is the fluid density and \(U\) is the free-stream velocity of the fluid. It is given by the formula: \(C_f = \frac{2 \tau_w}{\rho U^2}\)

Define the Heat Transfer Coefficient

The heat transfer coefficient (\(h\)) is a parameter that evaluates the efficiency of heat transfer between a solid surface and a fluid in contact with it. It depends on the fluid properties, the flow conditions, and the surface geometry.

Friction Coefficient in Laminar Flow Over Flat Plate

For laminar flow over a flat plate, the friction coefficient changes along the length of the plate. The variation can be quantified using the Blasius solution, which gives the following relation for \(C_f\): \(C_f = \frac{0.664}{\sqrt{Re_x}}\) where \(Re_x = \frac{\rho U x}{\mu}\) is the local Reynolds number, \(x\) is the distance from the leading edge and \(\mu\) is the dynamic viscosity of the fluid. From this relation, it is evident that the friction coefficient decreases with increasing distance from the leading edge.

Heat Transfer Coefficient in Laminar Flow Over Flat Plate

Similar to the friction coefficient, the heat transfer coefficient also changes along the length of the flat plate in laminar flow. It is related to the Nusselt number (\(Nu_x\)), which is the ratio of convective to conductive heat transfer and is given as: \(Nu_x = \frac{hx}{k}\) where \(k\) is the thermal conductivity of the fluid. For laminar flow over a flat plate, the Nusselt number can be expressed as a function of Reynolds and Prandtl numbers (\(Pr\)): \(Nu_x = 0.332Re_x^{1/2}Pr^{1/3}\) From this relation, we can see that the heat transfer coefficient, \(h\), changes with the distance from the leading edge as well.

Conclusion

In conclusion, for laminar flow over a flat plate, both the friction coefficient and the heat transfer coefficient change with distance from the leading edge. The friction coefficient decreases with increasing distance, while the heat transfer coefficient depends on fluid properties and flow parameters, such as the Reynolds and Prandtl numbers.