The boundary layer analysis performed in Section 12.6.1 assumed that the fluid was flowing over a stationary plate. However, there is no reason why the fluid cannot be quiescent while the plate moves within it. Consider the situation where the plate is moving from right to left at a speed of \(v_{\infty}\).

a. What are the boundary conditions for the velocities associated with this situation?

b. Using the procedure in Section 12.6.1, derive a formula for the boundary layer thickness.

c. Evaluate the velocity profile using the boundary conditions of part

(a) and derive the local boundary layer thickness as a function of Reynolds number.

d. For air at \(40^{\circ} \mathrm{C}\) and a plate that is \(100 \mathrm{~cm}\) long with a plate velocity of \(1 \mathrm{~m} / \mathrm{s}\), how does the boundary layer thickness for this situation compare with the boundary layer thickness for a similar situation but where the plate is stationary?

Transcribed Image Text:

12.6.1 FORCED CONVECTION, LAMINAR FLOW Our first target will be the hydrodynamic boundary layer since hydrodynamics will influence all the other transport operations. We consider the flat plate of Figure 12.6. Fluid flows in the free stream at a constant velocity, v., implying that we have no pressure force acting to accelerate the fluid across the surface of the plate (dP/dx - 0). The surface of the plate remains stationary, v = 0, and the plate will be insoluble in the liquid. The hydrodynamic boundary layer equations in their original dimensional form are: