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244 5. Some worked examples arising from physical problems

finite plate.) By virtue of the existence of a stream function, we see that we must also
scale we write

and then we must choose to give

This set is to be solved, subject to the boundary conditions (5.110c,d), written in
boundary-layer variables, and matching conditions for The leading-order
problem (zero subscripts) satisfies

with

and the matching conditions

To solve equations (5.111), we note, first, that and then the matching
condition requires throughout this region. Next we use (5.111c) to allow the
introduction of a stream function and then (5.111a) can be
written

with

and

The relevant solution (Blasius, 1908) takes a similarity form:
direct substitution then yields the ordinary differential equation
for

with

This equation must be solved numerically; the properties of the solution agree well
with experimental data for laminar flows. It is left as an exercise to show that there are