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208 5. Some worked examples arising from physical problems

Figure 10. Sketch of a laser drilling through a block of material; the bottom of the drill hole moves
according to

the vaporisation condition at the bottom of the drill hole at see figure 10.
The speed of the drilling process is controlled by

which is based on Fourier’s law applied at the bottom of the hole. This set, (5.23)–(5.26),
is an example of a Stefan problem; see Crank (1984) for more details (and important
additional references) about this problem and other moving-boundary, heat-transfer
examples. (A discussion of this particular problem can also be found in Andrews &
McLone, 1976, and, in outline, in Fulford & Broadbridge, 2002.)
Our intention is to seek a solution of this set of equations for For many
common metals, is fairly small (about 0.2); small signifies that rather more (latent)
heat than heat content is required to vaporise the material, once it has reached the
vaporisation temperature. We shall approach the solution by seeking a straightforward
expansion in powers of but we will need to take care over the evaluation on the
moving boundary. We will find that the resulting solution is not uniformly valid as
and the way forward requires a careful examination of what is happening in
the early stages of the heating process.
We write

so that equations (5.25) and (5.26) yield

and