- Fluid flows between two non-parallel plane walls, towards the intersection of the planes, in such a manner that
if is measured along a wall from the intersection of the planes then , where is
a positive constant. Verify that a solution of the boundary layer equation (704) can be found
such that is a function of only. Demonstrate that this solution yields

where , and

subject to the boundary conditions and . Verify that

is a suitable solution of the above differential equation, where . - A jet of water issues from a straight narrow slit in a wall, and mixes with the surrounding water, which is at rest.
On the assumption that the motion is non-turbulent and two-dimensional, and that the approximations of boundary
layer theory apply, the stream function satisfies the boundary layer equation

Here, the symmetry axis of the jet is assumed to run along the -direction, whereas the -direction is perpendicular to this axis. The velocity of the jet parallel to the symmetry axis is

where , and as . We expect the momentum flux of the jet parallel to its symmetry axis,

to be independent of .Consider a self-similar stream function of the form

Demonstrate that the boundary layer equation requires that , and that is only independent of when . Hence, deduce that and .Suppose that

Demonstrate that satisfies

subject to the constraints that , and as . Show that

is a suitable solution, and that

- The growth of a boundary layer can be inhibited by sucking some of the fluid through a porous wall.
Consider conventional boundary layer theory. As a consequence
of suction, the boundary condition on the normal velocity at the wall is modified to , where
is the (constant) suction velocity. Demonstrate that, in the presence of suction, the von Kármán velocity
integral becomes

Suppose that

where . Demonstrate that the displacement and momentum widths of the boundary layer are

respectively. Hence, deduce that

Consider a boundary layer on a flat plate, for which . Show that, in the absence of suction,

but that in the presence of suction

Hence, deduce that, for a plate of length , suction is capable of significantly reducing the thickness of the boundary layer when

where .