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246 5. Some worked examples arising from physical problems

We will not proceed further with this analysis here (but far more detail is available in
many other texts e.g. van Dyke, 1975), although we should add one word of warning.
The details that we have presented suggest that we may continue, fairly routinely,
to find the next term in the boundary-layer expansion, and then the next in the
outer, and so on, and that these will develop according to the asymptotic sequence
However, this is not the case: a term ln appears and this considerably
complicates the procedure (again, see e.g. van Dyke, 1975).

The two essential types of problem that are usually of most interest in fluid mechanics
are associated with (a) (the previous example) and (b) (the next
example). Problems for small Reynolds number (sometimes referred to as Stokes flow
or slow flow) have become of increasing interest because this limit relates to important
problems in, for example, a biological context. Thus the movement of platelets in
the blood, and the propulsion of bacteria using ciliary hairs, are examples of these
small-Reynolds number flows. We will describe a simple, classical problem of this
type.

E5.20 Very viscous flow past a sphere
In this example, we take (and since where U is a typical speed of
the flow, d a typical dimension of the object in the flow and v the kinematic viscosity,
this limit can be interpreted as ‘highly viscous’ or ‘slow flow’ or flow past a ‘small
object’). We consider the axisymmetric flow, produced by a uniform flow at infinity
parallel to the chosen axis, past a solid sphere; see figure 15. (This could be used as a
simple model for flow past a raindrop.) It is convenient to introduce a stream function
(usually called a Stokes stream function, in this context), eliminate pressure from the
Navier-Stokes equation and hence work with the (non-dimensional) equation

for and where

with

and

this latter condition ensuring that there is an axisymmetric flow of speed one at infinity.
(The subscripts here denote partial derivatives; we have mixed the notation because,
we submit, this is the neatest way to express this equation.) The velocity components