(a) The flow of a viscous fluid over a flat plate surface results in the development of a region of reduced velocity adjacent to the surface. This region of reduced velocity is known as a boundary layer. Figure 1 shows a boundary layer developing on the top surface of a sharp-edged flat plate. Within the boundary layer the x-component of velocity increases from u = 0 at the plate surface (y = 0) to u = U0 at a distance 8 above the surface.
If the boundary layer thickness increases with x, show that the vertical velocity v(x, y) is everywhere positive within the boundary layer.
Hint: Use continuity equation for 2 dimensional flow.
(b) Consider the flat plate boundary layer shown in Figure 1 below. The edge of the boundary layer is shown. At the leading edge (1) of the plate the velocity profile is uniformly distributed with a value Uo. Outside the boundary layer the velocity is not affected by the plate, and so is parallel to the plate with a value Uo (same as upstream). The velocity profile u(y) at location 2 is given by the equation:
u/Uo = (y/δ)1/7
Find the volume flow rate Q through the top edge A-B of the boundary layer, between locations 1 and 2. An appropriate control volume A-B-C is given in Figure 1.
