Consider steady, laminar, two-dimensional flow over an isothermal plate. Does the wall shear stress increase, decrease, or remain constant with distance from the leading edge?

Given that the flow is steady, laminar, and two-dimensional, we know that the flow velocity changes smoothly within the boundary layer without causing significant turbulence. The isothermal plate ensures that temperature variations do not affect the flow properties.

To analyze the flow over the plate, we can use the boundary layer theory. In this theory, the boundary layer is the region near the plate where the flow velocity varies from zero at the plate surface (no-slip condition) to the free-stream velocity far from the plate. This flow behavior follows the Navier-Stokes equations, which describe the conservation of mass, momentum, and energy for a fluid flow. For this particular case, the equations can be further simplified by assuming the boundary layer is thin, the pressure gradient along the plate is zero, and the viscosity is constant. With these assumptions, we can derive the Blasius equation for the laminar boundary layer: $$\frac{d}{dy}\left(v \frac{d^2 v}{dy^2}\right) = 0$$ where v is the streamwise velocity and y is the distance normal to the plate.

By solving the Blasius equation, we can obtain the velocity profile in the boundary layer, which is given by: $$v(y) = u_\infty \left[1 - \frac{y}{\delta} - \frac{1}{3} \left(\frac{y}{\delta}\right)^3\right]$$ where \(u_\infty\) is the free-stream velocity and \(\delta\) is the boundary layer thickness.

Now that we have the velocity profile, we can calculate the wall shear stress as follows: $$\tau_w = \mu \left. \frac{dv(y)}{dy}\right|_{y = 0}$$ Differentiating the velocity profile with respect to y, and evaluating it at y = 0 to find the wall shear stress: $$\tau_w = \mu \left[u_\infty \left(-\frac{1}{\delta} + \frac{1}{\delta^3} \right) \right]$$

As we move along the plate, the boundary layer thickness \(\delta\) increases due to the accumulation of fluid particles near the wall. Since the wall shear stress is inversely proportional to the boundary layer thickness and its cube (\(\delta\) and \(\delta^3\)), the wall shear stress will decrease as we move away from the leading edge. Therefore, the wall shear stress decreases with distance from the leading edge.