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248 5. Some worked examples arising from physical problems

that the appropriate choice is (Oseen, 1910), but unfortunately this scaling
recovers the full equation—the small parameter is removed identically! However, the
good news is that this scaling (obviously) is associated with the region far away from
the sphere (the ‘far field’), where the uniform flow exists and, presumably, this should
be the first term in an asymptotic solution valid here; we expect, therefore, that

Of course, (5.118) and (5.117) match directly, and we should now regard (5.117) as
valid only for r = O (1) (the ‘near field’) and then (5.118) is valid for
When we express (5.117) in far-field variables, we obtain

and so we require a term in the far-field expansion; let us write

The equation for from (5.114), is

where

with

and

the former condition being given by the matching, and the latter ensuring that the
flow at infinity is unchanged.
The relevant solution of equation (5.120a) is

where A – E are arbitrary constants, and then written in near-field
variables, gives

The term in is unmatchable, and so it must be removed, and otherwise this