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132 3. Further applications

is transmitted to (or possibly lost by) the fluid as it flows along the pipe. (Note that,
in order to avoid a discontinuity in temperature at the start of the pipe—which is
not an essential requirement in the formulation of the problem—then we must have
Finally, observe that multiplies the highest derivative terms, so we
must expect a boundary-layer structure.
We will choose the velocity profile to be that associated with a laminar, viscous flow
i.e. and then we seek a solution

where we have been careful not to commit ourselves to the second term in this
asymptotic expansion. Thus we have, from (3.53),

when we invoke the boundary condition (3.54a); this solution is, apparently, valid for
all but (in general) it cannot possibly accommodate the boundary condition on
r = 1 in x > 0, (3.54b). This observation, together with the form of the governing
equation, (3.53), suggests that we need a boundary layer near r = 1; let us set
with as and write
Thus equation (3.53) becomes

and, as we must use the balance (using the ‘old’ term/‘new’ term concept):
which is satisfied by the choice and so we have

We seek a solution of this equation in the form

so that

with and a matching condition for
Although it is possible to find the appropriate solution of (3.57), satisfying the given
boundary conditions, it is somewhat involved and we are likely to lose much of the
transparency of the results. Thus we will complete the solution in the special case:
constant wall temperature x > 0, and we will seek a solution
in x > 0, thereby ignoring the discontinuity that is evident as