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Figure 15. Coordinates and velocity components for the uniform flow past a sphere.

(see figure 15) are given by

Note that equation (5.114) does not exhibit the conditions for a boundary-layer struc-
ture, as because the highest derivatives are retained in this limit—indeed, this
term dominates. It is therefore unclear what difficulties we may encounter.
Let us seek a solution

then from (5.114) we simply have that

and all the boundary conditions appear to be available. Indeed, there is an exact solution
(Stokes, 1851) which satisfies all the given conditions:

and for a number of years this was thought to be acceptable, and that higher-order
terms would simply provide small corrections in the case However, difficulties
were encountered when a more careful analysis was undertaken, and a little thought
suggests why this should be so. At infinity the motion (convective terms) dominate, i.e.
the left-hand side of the equation, but near the sphere the viscous terms dominate (the
right-hand side); thus an approximation which uses only the right-hand side (as
above does) cannot be uniformly valid—it must break down as
We introduce where as and then from (5.117)
we see that we must also scale Equation (5.114) yields immediately