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134 3. Further applications

Figure 5. Schematic of the boundary-layer region, which grows from the pipe wall, and the ‘fully
developed’ region for and larger.

suitable integration by parts:

Thus we will need to match to

as for x = O(l) and 1 – r = O(l). That is, the asymptotic expansion valid
away from the boundary layer must include the exponentially small term of the form

for suitable functions and g(r, x). (It turns out that g satisfies a nonlinear, first
order partial differential equation, and that is governed by a linear, first order partial
differential equation, with coefficients dependent on g.) This result has particularly
dramatic consequences in the case for then the solution away from
the pipe wall is apparently exactly However, the requirement to match
to the boundary-layer solution introduces an exponentially small correction to the
outer solution—and this is the sole effect of the presence of the different temperature
at the pipe wall, at least for x = O(1).
Finally, we address the problem-and there is one-associated with the length of the
pipe. As we have just commented (and we will relate all this only to the simple case of
the similarity solution with if the total length of the pipe is