Page 221 -

204 5. Some worked examples arising from physical problems

Figure 9. Circular tube (centre r = 0) penetrating the surface of a liquid whose undisturbed level is z = 0.

Interior problem
It is a familiar observation, at least for that the narrower the tube then
the higher the liquid rises in the tube; this suggests that the height of the liquid will
increase as When this is coupled with the property (the confirmation of which
is left as an exercise) that no relevant solution of equation (5.13) exists if we ignore the
term we are led to write the solution in the form

where h(0) = O(1) (and h(0) > 0). Now we seek an asymptotic solution

and so the leading-order problem becomes

with

(The prime denotes the derivative with respect to r.) One integration of (5.14) gives
directly that