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130 3. Further applications

and we must choose or simply and then seek a solution

with

The matching condition, from (3.98), gives so

The simplest solution available for involves using the method of separation of variables
and so, noting the boundary value on y = 0, i.e. Y = 0, given in (3.48), we write

Thus we obtain

where C and D are arbitrary constants; a bounded solution valid away from the bound-
ary layer requires C = 0 and then as The condition on
Y = 0 is then satisfied if we select D = 1, and therefore the solution in the boundary
layer near y = 0 is

Near y = 1, the boundary layer is clearly the same thickness, so here we write
and set to obtain the equation

cf. equation (3.50) with The matching condition this time, again from (3.98),
is

we seek a solution where

with

The solution is obtained altogether routinely by writing so that

with