<DOC>
<DOCNO>
417
</DOCNO>
<TITLE>
on the stability of two dimensional parallel flows .
</TITLE>
<AUTHOR>
lin,c.c.
</AUTHOR>
<BIBLIO>
pt.iii - stability in a viscous fluid. q. app. math. 3, 1945, 273.
</BIBLIO>
<TEXT>
  this is the last part of the author's theory of the stability
of plane laminar motion .  (for parts 1 and 2, cf. the same
quart. 3, 117-142, 218-234 (1945),. these rev. 7,225,226 .)
the stability character of a viscous fluid is considered in
detail .  the author proceeds first to give a proof of a criterion
of stability due to heisenberg ..  if a velocity profile has an
number and phase velocity, the disturbance with the same
wave number is unstable in the real fluid when the reynolds
number is sufficiently large .  this destabilizing effect of viscosity
is one of the most interesting phenomena in the
general stability theory,. its physical and mathematical significance
is carefully discussed .
  the author then discusses the behavior of the so-called
neutral curve for the two characteristic types of
velocity distribution, the boundary layer type profile and
the symmetrical profile .  the asymptotic behavior of the
neutral curve is discussed first .  the main difference between
profiles with and without a point of inflection is that the two
branches of the neutral curve approach and for
profiles with a flex, but both converge to for the profile
without a flex .  the most important results are as follows .
for sufficiently large reynolds number r .  (2) there always
exists a minimum r below which the motion is stable .  a
similar result was obtained by synge from energy considerations
.  synge found a limiting curve below which the
motion is necessarily stable .  the author's discussion of the
asymptotic behavior of the curves shows further
that there always exists a maximum value of a beyond
which the motion is stable for all reynolds numbers .  hence
the qualitative shape of the curve is determined .
  the author proceeds to show that simple approximate
expressions for the stability limit can be obtained from his
general analysis for a given velocity profile .  these approximate
stability limits for plane poiseuille flow and blasius
flow are found to be r=5906 and r=502 .  the reynolds
numbers are based on the width of the channel and the
displacement thickness, respectively .  finally, the method
for computing the complete instability curve is presented
and the plane poiseuille case and the blasius problem worked
out in detail .  the stability limit for blasius flow had been
given before by tollmien and schlichting .  the present more
exact computations agree well with tollmien's result as far
as the minimum critical reynolds number is concerned .
the value found here is r=420 .  the neutral curve for
poiseuille motion had not been obtained before .  the minimum
critical number here is found to be r=5314 .  the
agreement with the estimate from the simple criterion mentioned
above is thus very good .
  a discussion of the physical significance of the viscous
effects and of future developments concludes the paper .
</TEXT>
</DOC>
