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National Advisory Committee for Aeronautics, Technical Notes - Estimate of Slip Effects on Compressible Laminar Boundary Layer Skin Friction

Rayleigh's problem for a compressible viscous gas, subject to slip
at the wall, is considered. Expressions for slip velocity and skin
friction are derived with the gas temperature at the wall and the product
of viscosity times density assumed constant, or nearly so. The solution
is related to that for the flow past a flat plate by a transformation
which, in the continuum regime, results in exact agreement with the
expression for laminar skin friction obtained by Chapman and Rdbesin. The
expressions for local skin friction are considered to'define the extent of
the continuum regime and to estimate the effect of slip in the border
regime between continuum and slip flow. An estimated upper limit for the
over—all drag of a flat plate in the slip flow regime is also obtained.

For the entire range from free—molecule flow to continuum flow, the
parameter governing skin friction (expressed in the form 1/? M3Uf*) was
found to be w/ReF/q/y M*’ (where Cf*, r, Re*, and M* are the local
skin-friction coefficient, the ratio of specific heats, Reynolds number,
and Mach number, respectively, all based On gas properties at the wall).
Continuum theory for local skin friction was found to agree with the
expressions for skin friction developed herein to within 1 percent for
1/Refiyq/— M* greater than 5.2 and to within 5 percent for 1/Reg/1/_fM*
greater than 2. 4.

Fluid mechanics can be classified, as indicated in reference 1, into
continuum, slip, intermediate, and free-molecule flow regimes. In the
continuum regime the mean free path of the fluid molecules is negligible
compared with body dimensions, and the conventional equations of motion
apply. When the mean free path is small, but not negligible, compared
with body dimensions or boundary-layer thickness, the fluid in contact
with the body surface has a nonzero tangential velocity relative to the
surface; this flow regime is consequently termed a slip flow. A free-
molecule flow is one wherein the mean free path is very large compared
with body dimensions so that the chances for collision of molecules
among themselves are much smaller than the chances for collision of
molecules with the body surface. Intermediate between the free-molecule
and slip flew regimes, collisions between molecules and collisions of
the molecules with the wall have the same order of probability.