<DOC>
<DOCNO>
962
</DOCNO>
<TITLE>
contributions to the theory of heat transfer through
a laminar boundary layer .
</TITLE>
<AUTHOR>
m. j. lighthill
</AUTHOR>
<BIBLIO>
department of mathematics, university of manchester
communicated by s. goldstein, f.r.s.
</BIBLIO>
<TEXT>
an approximation to the heat transfer rate across a laminar
incompressible boundary layer, for arbitrary distribution of
main stream velocity and of wall temperature, is obtained by
using the energy equation in von mises's form, and approximating the
coefficients in a manner which is most closely correct near the
surface .  the heat transfer rate to a portion of surface of length l
breadth is given as
where k is the thermal conductivity of the fluid, o its prandtl number,
p its density, u its viscosity, r(x) is the skin friction,
and t(x) the excess of wall temperature over main stream temperature .
a critical appraisement of the formula indicates that it should be very
accurate for large, but that for of order 0.7 (for most gases) the
constant should be replaced by 0.73, when the error should not exceed
this yields a formula for nusselt number in terms of the reynolds number
r and the mean square root of the skin friction coefficient c, in the
case of uniform wall temperature .
however, for the boundary layer with uniform main stream, the
original formula is accurate to within 3 percent even for .  by
known transformations an expression is deducted for heat
transfer to a surface, with arbitrary temperature distribution
along it, and with a uniform stream outside it at arbitrary
mach number (equation (42)) .  from this the temperature distribution
along such a surface is deduced in the case (of importance at high mach
numbers) when heat transfer to it is balanced entirely by radiation
from it .  this calculation, which includes the solution of a non-linear
integral equation, gives higher temperatures near the nose,
and lower ones farther back (figure 2), than are found from a theory
which assumes the wall temperature uniform and averages the heat
transfer balance .  this effect will be considerably mitigated
for bodies of high thermal conductivity., the author is not
in a position to say whether or not it will be appreciable for
metal projectiles .  but for stony meteorites at a certain stage
of their flight through the atmosphere it indicates that
melting at the nose and re-solidification farther back may occur, for
which the shape and constitution of a few of them affords evidence .
an appendix shows how the method for approximating and solving
von mises's equation could be used to determine the skin friction
as well as heat transfer rate, but this line seems to have no advantage
over established approximate methods .
</TEXT>
</DOC>
