<DOC>
<DOCNO>
149
</DOCNO>
<TITLE>
expansions at small reynolds number for the flow past
a sphere and a circular cylinder .
</TITLE>
<AUTHOR>
proudman,i. and pearson,j.r.a.
</AUTHOR>
<BIBLIO>
j.fluid mech. 2, 1957, 237.
</BIBLIO>
<TEXT>
  this paper is concerned with the problem of obtaining higher
approximations to the flow past a sphere and a circular cylinder
than those represented by the well-known solutions of stokes
and oseen .  since the perturbation theory arising from the
consideration of small non-zero reynolds numbers is a singular
one, the problem is largely that of devising suitable techniques for
taking this singularity into account when expanding the solution
for small reynolds numbers .
  the technique adopted is as follows .  separate, locally valid
the regions close to, and far from, the obstacle .  reasons are
presented for believing that these 'stokes' and 'oseen' expansions
are, respectively, of the forms
where are spherical or cylindrical polar coordinates made
dimensionless with the radius of the obstacle, r is the reynolds
number, and and vanish with r .  substitution
of these expansions in the navier-stokes equation then yields a
set of differential equations for the coefficients and, but
only one set of physical boundary conditions is applicable to each
expansion (the no-slip conditions for the stokes expansion, and
the uniform-stream condition for the oseen expansion) so that
unique solutions cannot be derived immediately .  however, the
fact that the two expansions are (in principle) both derived from
the same exact solution leads to a 'matching' procedure which
yields further boundary conditions for each expansion .  it is thus
possible to determine alternately successive terms in each
expansion .
  the leading terms of the expansions are shown to be closely
related to the original solutions of stokes and oseen, and detailed
results for some further terms are obtained .
</TEXT>
</DOC>
