Page 104 -

87

and some and write Then we have

and so equation (2.80) becomes

where We will further assume, whatever the choice of (and in almost
all problems that we encounter, that the terms in and dominate as
Thus the ‘classical’ choice for the balance of terms, applies here for
general g(x) (which we have yet to determine).
The differential equation valid in the boundary layer can now be written

which has the first term in an asymptotic expansion, satisfying

At this stage, has yet to be determined; let us choose (we may not
choose then we are left with the simple, generic problem for

which also solves the difficulty over the mixing of the x and X notations in (2.83).
Thus all boundary-layer problems in this class have the same general solution, from
(2.84),

However, we are no nearer finding the position of the boundary layer itself; this we
now do by examining the available solutions for g(x).
The general form for g(x) is

where C is an arbitrary constant (and this proves to be the most convenient way of
including the constant of integration). First, we suppose that a(x) > 0, and examine