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PROBLEM OF SLIP FLOW IN AERODYNAMICS
By Robert E. Street
University of Washington
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UNIVERSITY OF FLORIDA
S120 MARSTON SCIENCE LIBRARY
4:GAINESViL FL 32611-7011 USA
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NACA R1 57A50
NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS
PROBLEM OF SLIP FLOW IN AERODYNAMICS
By Robert E. Street
A survey of the present status of theory in the field of slip-flow
aerodynamics leads to the conclusion that the Navier-Stokes equations of
motion together with first-order velocity slip and temperature jump at
any boundary are sufficient insofar as experimental confirmation is avail-
able. The use of the Burnett terms in the equations of motion as well as
second-order terms in boundary conditions does not seem justified, at
least for semirarefied gas flows. Further kinetic theoretical study and
additional experimental data are urgently needed.
The purpose of the present report is to survey the present status
of theory in the field of slip-flow aerodynamics. No new problems will
be set up and solved, but a fairly complete listing of books and papers
which would seem to have some connection with this problem has been
assembled and will be commented upon. The present report is therefore
essentially preliminary, a small-scale map of the field, but with some
suggestions as to areas which would seem to be the most desirable to
explore in more detail. The realm of interest is the near-continuum flow
domain (modern compressible viscous flow) in which the effects of slip in
velocity and jump in temperature at the boundary of the flow begin to
The original survey of the field made for the aerodynamicist is the
paper of Tsien (ref. 1). Although this paper was very fertile in that it
led to the considerable activity in the investigation of flow of rarefied
gases which has taken place since its publication, it is no longer a sat-
isfactory introduction to the field. A better and more up-to-date survey
of the whole problem of rarefied gas flow is the treatment by Schaaf and
Chambrd (ref. 2). The present report was outlined and planned just before
a copy of the manuscript of this reference became available, so there is
considerable overlapping, although much material which is included in
the cited reference has been deleted from this report. A similar but
less comprehensive survey was also published by Schaaf a fev years ago
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The dimensionless parameter which is considered of paramount impor-
tance in the flow of semirarefied gases is the Knudsen number K, which
is defined as the ratio of the mean free path I of the gas to some
significant length L in the problem. The well-known connection between
the Knudsen number and the Mach and Reynolds numbers which is taken as
the governing factor determining the nature of the flow regime is
K 1.25371/2 M (1)
where 7 is the ratio of specific heats of the gas, M = U/a is the
Mach number, and Re = pLU/p is the Reynolds number. The velocity U
is some reference speed such as the mass velocity of the gas or the
Some writers suppose that for large values of Re the character-
istic length is the boundary-layer thickness 6 and define the Knudsen
number as K = M/\fRe. This seems to be the proper parameter for
boundary layers in slip flow, but it is not as yet known just how the
boundary-layer thickness depends on Reynolds number except that it is
certainly larger and hence may increase inversely as the square root of
Re. The first-order slip on flat plates does depend upon the parameter
V(~4/\ifRe at low speeds, but in the hypersonic flow over flat plates
the more significant parameter is a constant times M5/\fRe. Physically
this seems to be due to the greater importance of M since it indicates
compressibility in hypersonic flows while II occurs in conjunction with
Re as an indication of rarefaction alone in equation (1). Since
M U v v IV _
Re a LU aL 2aL ivy L
independently of the magnitude of U, it follows that at any Mach number
or Reynolds number the Knudsen number or the ratio M/Re remains as an
indication of the amount of rarefaction of the gas. For K << 1 the
laws of continuum mechanics must hold and for K >> 1 the flow is highly
rarefied or of free-molecule type. Nothing more can be said about the
relevancy of one form of parameter as compared with that of another at
the present time.
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It is usually stated that the problem of skin friction and heat
transfer for high-speed flight at extreme altitudes K > 1 must be
based upon kinetic theory rather than on classical hydrodynamics (ref. 4).
The question is, where does classical hydrodynamics as such break down
and kinetic theory take over? It seems that modern phenomenological aero-
dynamics together with properly extended boundary conditions (that is,
the compressible Navier-Stokes equations with first-order slip in the
boundary conditions) is capable of extension to a broader domain of flow
regimes than is sometimes assumed. In other words, it appears desirable,
first, to see what the limitations are before abandoning hydrodynamics.
For example, the problem of shock-wave structure appears to be contained
within the classical equations refss. 5 and 6) insofar as any experimental
check has been made. As for other problems in semirarefied flow, the
empirical theories based upon the phenomenological equations of motion
have been surprisingly successful. Therefore, it seems of first impor-
tance to determine just how rarefied a gas must be or how fast a body
must fly before this approach has to be abandoned. The free-molecular
flow regime of highly rarefied gas is not considered here at all, except
it is referred to as a limiting case in some places. The kinetic theory
would be necessary in this flow regime. Also, the more theoretical problem
of statistical mechanics of nonequilibrium and transport processes is not
a proper subject. It has already been surveyed elsewhere (ref. 4), and
the foundations of the subject are questions of extreme mathematical com-
plication. The immediate question here is to exhibit those problems in
semirarefied flow about which enough is now known to enable the design
engineer to make some attempt at determining optimum aerodynamic shapes
for aircraft and missiles of the immediate future.
This investigation was conducted at the University of Washington
under the sponsorship and with the financial assistance of the National
Advisory Committee for Aeronautics.
a speed of sound
Cf local skin friction, dimensionless
CS constant of proportionality in linear viscosity-temperature
7c instrinsic velocity, | u
components of instrinsic velocity
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Cp specific heat at constant pressure
Cy specific heat at constant volume
d distance between plates
f distribution function depending on i, J, and t
G coefficient of slip
K Knudsen number, z/L, dimensionless
L reference length
1 mean free path of gas
M Mach number, U/a, dimensionless
P probability that molecule will collide with surface before it
collides with another molecule
P.. components of complete stress
Pr Prandtl number, cpp/i/, dimensionless
p pressure in gas
Q any aerodynamic function of K such as drag coefficient or
qi components of heat flux vector
R gas constant
Re Reynolds number
r recovery factor on circular cylinders
T absolute temperature
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macroscopic flow velocity
components of flow velocity
mean molecular speed of gas
Cartesian coordinate components of position vector x
accommodation coefficient, dimensionless
ratio of specific heats of gas, dimensionless, cp/cv
8 boundary-layer thickness
K coefficient of heat conduction
1 second coefficient of viscosity
p density of gas
a reflection coefficient of Maxwell
p coefficient of viscosity of gas
v coefficient of kinematic viscosity, u/p
molecular velocity vector
i components of
T average time between molecular collisions
c value in continuum range
f value in free-molecule range
o free-stream conditions
s value in slip flow
w value at wall
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The basis for the aerodynamics of rarefied gases is necessarily the
kinetic theory of gases. Only the latter will be valid in highly rarefied
flows where the mean free path is so large that the effects of all inter-
molecular collisions may be completely neglected as compared with those of
collisions between the molecules and the surface of the body moving through
the gas. At the other extreme of continuum flow, where the mean free path
is so small that collisions between molecules themselves predominate, the
Navier-Stokes equations for viscous compressible fluids are definitely
established both theoretically and experimentally. However, the Navier-
Stokes equations are also integrals of motion of the Boltzmann equation
of kinetic theory. Thus it follows that some form of the Boltzmann equa-
tion is valid throughout the flow regime from a highly rarefied gas to
the conventional gaseous continuum of phenomenological theory. A study
of the assumptions underlying the derivation of the Boltzmann equation,
such as stated clearly and succinctly by Grad (ref. 7), indicates that
although restrictions upon the gas are necessary they are not unreason-
able. In other words, real gases differ so inconsequentially from the
gas described by the Boltzmann equation (or else the extraneous effects
can be separated out) that the assumptions underlying the Boltzmann equa-
tion will be adhered to in the following discussion. Any really serious
criticisms of the theory can be embodied within the range of this simpli-
fied gas theory.
The assumptions themselves are:
(1) The gas is monatomic. This statement means that each gas molecule
behaves classically with only translational degrees of freedom, three in
number. This is not a serious restriction since a monatomic gas, helium,
is used in some wind tunnels, especially in hypersonic flows, and studies
of shock-wave structure have included data on monatomic gases. The effects
of internal degrees of freedom in more common gases, such as air, have been
satisfactorily accounted for by the use of a second coefficient of vis-
cosity in the Navier-Stokes equations. There are much more fundamental
problems to be solved before the significance of the other degrees of
freedom must be considered. The elimination of second viscosity from
the Navier-Stokes equations by means of Stokes' relation (ref. 8)
5A + 2P = 0
has in the past not seriously hampered the development of a fruitful
boundary-layer theory of viscous and heat-conducting gases in the con-
tinuum flow regime. Only one paper so far has appeared in which an
attempt has been made to develop a Jinetic theory of polyatomic gases.
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In this, Wang Chang and Uhlenbeck (ref. 9) have shown that internal
degrees of freedom in polyatomic gases can be embodied in a second vis-
cosity coefficient. Although this assumption restricts the discussion
to gases at relatively low temperatures, this again is not serious,
except for very strong shock waves and extremely high Mach number. Other
considerations such as the deviation of the gas from a classical state
would have to be taken into account in the case of high temperatures, or
extremely low temperatures, anyway. Finally, the effects of relaxation
times, which are also connected with the coefficient of second viscosity,
are eliminated from the theory through this assumption.
(2) The gas is ideal. This statement means that the equation of
state connecting pressure, temperature, and density is the simple relation
p = RpT
This is commonly assumed for gases in the continuum state and is certainly
valid unless the gas is compressed to a pressure of many atmospheres. As
the gas becomes more rarefied this equation of state is even better in
the slip-flow and free-molecule flow regimes (ref. 10). There is no great
restriction here, especially for monatomic gases.
(3) Point molecules, complete collisions (binary collisions only),
a slowly varying distribution function f and, finally, molecular chaos.
These assumptions are embodied in the derivation of the Boltzmann equation
for f and are completely discussed by Grad (ref. 7). Later, another
assumption made is that the molecules are either spherical (in that the
collisions behave like those between perfectly elastic spheres) or
Maxwellian (are repelled according to the inverse fifth power of the dis-
tance between them).
(4) The state of the gas is one of equilibrium or near equilibrium.
This is implicit in the previous assumption, but is not commonly stressed.
The strength of a good theory lies in its range of validity being so much
broader than its basic assumptions would seem to imply. Such, for example,
is true of the Prandtl-Glauert approximation in the subsonic flow of a
compressible nonviscous fluid and the Prandtl boundary-layer theory in
the flow of viscous compressible fluids. Since the Navier-Stokes equa-
tions work so well for shock-wave phenomena, it is not too much to expect
this to be true of the kinetic theory based upon Boltzmann's equation.
However, the amount of nonequilibrium allowed is neither known nor can
as yet be estimated. The most significant approximation so far developed
is that of Grad (ref. 7) and it is definitely limited to very near equi-
8 NACA RM 57A50
It must be emphasized that the difficulties with present theories
of slip flow and the transition regime do not lie in any of the above
assumptions but rather in the method of deduction of the suitable type
of distribution function which determines the moments, the ordinary
macroscopic variables: the density p, the velocity of the gas flow u,
the stress components Pij, the temperature T, and the heat flux vector
components qi. If the Boltzmann equation is multiplied by certain sum-
mational invariants and integrated over all the velocities, the collision
integral vanishes, corresponding to the conservation of mass, momentum,
and energy during collision, with the result that the following five con-
servation equations are obtained refss. 2 and 7)
L + -(pui) = 0 (2)
6ui+ uj 1+ i( = (5)
at ax. P dx.
dp + () 2 dui 2 6i
-- + --(pui) + P. -- + -- = 0 (4)
6t 6xi 5 J x 5 6xi
where i,j = 1,2,5 and the double summational convention is used.
To these must be added the equation of state
p = RpT (5)
These equations can be deduced by elementary means and form the con-
tinuum basis of fluid dynamics, for which see any standard reference such
as Lamb, Goldstein, Howarth or Kuethe, and Schetzer. As they stand there
are 6 equations, but many more unknowns (15 in all). For complete equi-
librium the solution of Boltzmann's equation is the well-known distribu-
tion function of Maxwell
f(0) P e- c2/2RT (6)
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where c2 is the square of the intrinsic velocity whose components are
the difference between the components of molecular velocity |i and mass
velocity ui. All odd moments of f(O) vanish as do the second-order
ones, leading to Pi = paij and qi = 0, and equations (2) to (5) reduce
to the completely determined set for a nonviscous non-heat-conducting
ideal fluid, the Euler equations. In macroscopic theory the same result
is obtained by assuming that the coefficients of viscosity and heat con-
duction are zero.
From the phenomenological point of view the assumption that departure
from equilibrium is slight, such that the stress and heat flux are linear
functions of the rate of deformation of the fluid and the temperature
gradient, respectively, leads to the relations
Pij = Piij + Pij
Pij = -L(uij + uji) Abijuk,k
qi = -KTi (8)
Then equations (2) to (5) reduce to the Navier-Stokes equations of a
compressible viscous fluid, which are also completely determined. The
coefficients p, ~, and < are then left to experiment for determina-
tion. A determination of higher order, nonlinear terms in the relations
between stress and heat flux with velocity and temperature gradients based
upon the theory of invariants in tensor analysis has been carried out by
Truesdell (ref. 11) but leads to an enormous number of undetermined coef-
ficients. To date no corresponding variety of solutions and experiments
has been devised which would determine these additional coefficients.
As mentioned earlier, the fact that the Navier-Stokes equations lead
to correct solutions in the continuum range substantiates the correctness
of equations (2) to (5) together with relations (7) and (8). Any kinetic
theory must therefore begin with the first approximation (6) and as second
approximation give equations (7) and (8). The Chapman-Enskog-Burnett
method does essentially do so by taking the complete distribution function
f = f(C')(l + f(l) + f(2) + (9)
Substitution of this expression into Boltzmann's integral equation and
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the determination of f(0), f(l), f(2), by a complicated scheme
of iteration leads in succession to the Euler, Navier-Stokes, and Burnett
equations of motion.
The Burnett expressions are not written down here for several reasons.
First, they are long and can be found in many places, for example, refer-
ences 1 to 5 and 11. Secondly, they can be deduced from Grad's distri-
qi ici( c2)
f= f(0) 1 + cic -- 1 (10)
2pRT pRT 5RT
as was first shown by Schaaf and Chambrd refss. 2 and 5). Thirdly, no
solution of the Burnett equations has yet been found which gives any
better results than the Navier-Stokes equations, if as good. Finally,
it has been pointed out by Grad, Schaaf, Truesdell, and others that only
a very limited number of possible solutions of the Boltzmann equation can
be found by an iteration such as equation (9). Since the iteration is
made upon the collision integral and only collisions between molecules
are contained therein, it is strange that the Burnett terms are considered
valid in the transition regime between slip flow and free-molecule flow
where collisions between the surface and the molecules are as important
as those between the molecules themselves.
For this last reason, it is to be expected that the Burnett terms
would be more significant in the continuum regime, where collisions
between molecules predominate and large deviations from equilibrium take
place, such as in shock-wave structure or in hypersonic flow. Yet in
these two cases in particular the Burnett terms, when applied, have not
yet led to any significant results (refs. 5 and 12) different from the
Navier-Stokes terms. In hypersonic flow Von Kryzwoblocki has deduced
equations to boundary-layer order of magnitude (ref. 15) and has indicated
means of solving these but has not worked out a solution. Yang (ref. 14)
has found a solution to Rayleigh's problem of the impulsive motion of a
flat plate in low-speed flow which agrees with both limiting cases of
free-molecule and continuum flow. This solution was obtained from Grad's
equations and could not be deduced from the Burnett equations.
The Couette flow between parallel plates at low Mach number and the
heat transport between the plates with small temperature difference have
been calculated by Wang Chang and Uhlenbeck refss. 15 and 16) for arbitrary
Knudsen numbers starting with the Boltzmann equation. As expansions in
powers of the reciprocal of the Knudsen number K = l/d, where d is the
distance between the plates, they obtain solutions in the far transition
regime approaching free-molecule flow. In the terminology used by Wang
Chang and Uhlenbeck, K = d/l, or tie reciprocal of the Knudsen number
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defined in this report, and the continuum and free-molecule flow regimes
are called a Clausius gas and Knudsen gas, respectively. Since their
expansions in powers of 1/d do not converge while those in powers of
d/l do converge, except for a minor difficulty due to the special geometry
of parallel plates, the results imply, although they do not prove, that
an expansion such as equation (9) is not valid at all. Thus, while a
solution of the Boltzmann equation in a power series may be quite possible
in highly rarefied flows, there is more doubt of such a solution in the
continuum or slip-flow regimes. This is the solution which Yang showed
to be useless in the Rayleigh problem.
It is worthwhile to take a closer look at Yang's thesis, which is
the only attempt so far to solve the flat-plate problem by purely kinetic
theory. Yang uses Grad's 15-moment approximation to the Boltzmann equa-
tion which assumes the distribution function (10) above and deduces there-
from a system of nine additional equations for the tensors Pij and 'qi,
which together with equations (2) to (5) above give a determined system
with unknowns p, p, T, uij, qi, and Pij. Thus Grad's equations con-
tain Euler's, Navier-Stokes', and Burnett's equations as successive approxi-
mations, for which Schaaf and Chamribr4 (refs. 2 and 5) and Yang give the
method of iteration. In his thesis Yang summarizes Grad's theory and
gives a clearer and more physical derivation of Grad's boundary conditions,
about which more will be stated later in the present report (cf. section
entitled "Boundary Conditions"). Applying the equations of Grad to a
flat plate starting suddenly from rest with velocity U in its own plane,
Yang finds it possible to linearize these and the boundary conditions by
assuming that U is small and the temperature difference between the
plate and undisturbed gas is also small. Solving the linearized equa-
tions and boundary conditions for the adiabatic plate by an approximate
method Yang obtains series expansions for small and large times t. Con-
sidering only the dominant terms in these two limiting expressions he
deduces a closed expression which is approximately valid for all time.
This is done for shear stress, tangential heat flux, and slip velocity
on a plate where the interaction between plate surface and molecules is
completely diffuse or a. = 0. For example, the result for local skin
friction Cf is
MoCf 0.683 exp 3.52 Re erfc (0. 342 Ie1 (11)
Mo Mo /
Here Mo = U/ao and Re = Po U2t, the subscript o denoting free-stream
conditions. The ratio of the speed of sound to the rate of diffusion of
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vorticity is \fRe/Mo = a.o t = a-t or, alternatively, Re/Mo2 = ao2t/vo
y ^o y ^O
is proportional to the ratio of elapsed time to the average time between
molecular collisions T, that is, Re/Mo2 4- -t.
Schaaf and Sherman (ref. 17) obtained a similar expression for C.
as did Mirels (ref. 18) by considering Rayleigh's problem using Navier-
Stokes equations and first-order slip at the surface. Their result for
MoCf is the same as equation (11) in functional form but differs in the
magnitude of the constants. Because of this, both formulas give twice
the correct value of Cf for free-molecule flow and Schaaf's equation
fails to give the correct value for Cf in continuum flow. Many papers
have been published lately in which Rayleigh's problem has been extended
to compressible flow, but in most cases only with the no-slip boundary
conditions. Besides the two mentioned papers by Schaaf and Mirels, a
report by Martino (ref. 19) has just appeared which repeats the calcula-
tions of M4irels but considers in addition the case of heat transfer.
Although Martino surveys the kinetic-theory approach and includes
another method of solving Boltzmann's equation, attributed to Chapman,
which is to appear in a forthcoming book by Dr. G. N. Patterson entitled
"The Molecular Flow of Gases," he deduces the usual boundary-layer equa-
tions of motion and first-order slip from the Chapman-Patterson equations.
The parameter used is = 1.50 where Cs = (- 12 To + S is the
1MoK \T0/ Tw + S
constant of proportionality in the linear viscosity-temperature formula.
But using equation (1) MoK = 1.25o61/2 and since C. = 1 for the
adiabatic no-heat-transfer case wRe/M, or t is essentially the
same parameter used by the others, including Yang.
The results of Martino's analysis are essentially the same as Mirel's
insofar as the skin friction is concerned, although no explicit form of
the relation is given. Again the limiting value of skin friction in free-
molecule flow is twice its correct value. Martino is primarily interested
in heat transfer expressed by means of the Stanton number and finds the
ratio of the wall temperature to the free-stream gas temperature a signif-
icant parameter. This agrees with the result of Maslen (ref. 20), who
expanded the boundary-layer equations and slip-boundary conditions in
powers of the parameter e = y7 M/Rel/2, retaining only terms to the first
power in E.
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Maslen's paper is the only one to treat the flat-plate boundary-layer
problem with first-order slip by the usual boundary-layer analysis instead
of by Rayleigh's method. His expressions for skin friction and heat trans-
fer show a different functional form. However, the recent measurements by
Kendall (ref. 21) tend to confirm Maslen, and Martino's independent deter-
mination of the Nusselt number also confirms Maslen's heat-transfer rela-
tion. On the other hand, Martino's heat-transfer characteristics expressed
as the Stanton number differ from the value for free-molecule flow by the
factor of two, as does the skin-friction coefficient. Also, the recovery
factor, deduced by Martino, exhibits exactly the opposite characteristics
observed with increasing rarefaction. Instead of increasing to a value
greater than 1 as observed by Stalder, Goodwin, and Creager (ref. 22),
the calculated recovery factor actually decreases with decreasing values
of the Chapman-Patterson parameter t and becomes zero in free-molecule
flow. On the other hand, the value of Nusselt number obtained from the
Rayleigh solution by Martino does check experimental data of reference 22
according to the authors, although no graph of the curve is presented.
Thus, on the whole, the attempt to solve the flat-plate boundary
layer with slip boundary conditions is hardly to be considered a complete
success. The significant parameter does seem to be (MoK)1/2 or M/f
for relatively low-speed flow and for conditions in the near slip flow
regime where the flow is not too far from continuum conditions. A study
of experimental results, discussed below, tends to lead to the belief
that effects of interaction between thickening boundary layer and external
flow are more significant than slip in the observed decrease in heat trans-
fer and, for supersonic flow, increase in skin friction with increasing
Knudsen number. A particularly simple analysis by Schaaf and Chambrd
(ref. 2) shows why there is no first-order effect of slip upon the skin
friction of a flat plate. If the stream function is expanded in a series
of terms multiplied by increasing powers of the mean free path 1, it
turns out that the term in Cf containing I to the first power also
contains the pressure gradient as a factor. Hence, slip is effective
only when an induced pressure gradient is present or else flow over sur-
faces other than flat plates is considered..
Schaaf and Lin suggest the flow over a wedge as a case for which the
pressure gradient is not zero, or the stagnation point flow (ref. 25),
both of which have been analyzed only in incompressible flow. However,
the effect of slip is to reduce skin friction in both cases. The experi-
mental data of Schaaf and Sherman (ref. 17) show the effects of both slip
and boundary-layer interaction, the first tending to lower the drag coef-
ficient while the latter tends to increase it. This interaction effect is
expected to be larger at higher Mach numbers and near to the leading edge
of a flat plate as borne out by the experiments of Kendall (ref. 21) per-
formed at a Mach number of 5.8.
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SYang 's attempt to solve the flat-plate problem by applying Grad's
equations does not seem sufficiently more fruitful than the attempts of
others who have used the Navier-Stokes equations with only first-order
slip and like them is definitely limited to extremely low Mach numbers.
He exposes one of the characteristics of Grad's equations in that his
solution for the shear stress away from the surface contains a term
showing the presence of waves in the flow which disappear only after an
infinite time lapse, the limit of continuum flow. This is not at all
surprising since Grad's equations are hyperbolic, not parabolic like the
Havier-Stokes equations, and, as Yang points out, the shear stress sat-
isfies the wave equation at the start of the motion rather than the heat-
The most serious objection to the application of Rayleigh's technique
to the flat-plate problem is its neglect of the leading-edge conditions,
however. In high-speed flow the leading-edge region is where the effects
of mean free path are most important; rarefaction is greatest just at the
leading edge and becomes less as the flow goes farther downstream. The
Knudsen number, if based upon the distance from the leading edge, the
same as Rex, decreases rapidly from infinity to any finite value as x
increases. The rarefaction parameter M/j/Re exhibits the same behavior
for any fixed value of M. At high Mach numbers approaching the hyper-
sonic range, the interaction between the shock wave and boundary layer
becomes more serious as well. Attempts to develop a second-order boundary-
layer theory valid for small values of Re have been made by various
investigators, but so far without much success. The Low Pressures Research
Group at the University of California, Berkeley, are supposedly working on
this problem as is Sidney Goldstein at Harvard. Until their studies are
complete little can be said about the best theoretical approach to the
slip flow over flat plates.
The same statement probably holds true for blunt bodies or bodies
of revolution. For the time being all that theory can contribute to the
problem is to apply the usual boundary-layer equations or the Navier-
Stokes equations themselves, but with first-order slip and temperature
jump in the boundary conditions. Thus far the results so obtained do
not seem to be too far from agreement with experiment where available,
especially when interaction between the boundary layer and external flow
is included in the analysis. Otherwise, definitely empirical relations
such as equation (ll) or some of those suggested in experimental literature
should be used for engineering design and analysis. A more serious criti-
cism of the 15-moment equation as well as of the Burnett equations has
been made by a purely theoretical but rigorous analysis of kinetic theory
by Ikenberry and Truesdell in two papers which appeared in print just as
this was written (ref. 24). A brief summary of this new theory is pre-
sented in the appendix. A study of this theory, either applying it to
the shock-wave problem or determining some compatible set of boundary
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conditions which can lead to a simple problem such as Couette.flow might
be carried out with some hope of success, at least to obtaining working
A thorough analysis of the proper macroscopic boundary conditions
would have to be based upon an analysis of the molecular forces between
molecules in the gas and the atoms of the solid wall which forms the
boundary of the flow. It is clear that such an interaction would be
statistical in nature and that the conditions sought would be some sort
of integrated average over an interval of time or of velocity and space.
The problem is essentially the same as that known as the physical adsorp-
tion of gases on solid surfaces. Various models have been proposed for
the latter phenomenon, but so far the intuitive picture is the best avail-
able. That is, the molecules approaching the wall are in an equilibrium
state which is Maxwellian. At the wall this state is disturbed to a con-
siderable extent and probably destroyed as the molecules become attached
temporarily. The time of adsorption is really so very short from the
macroscopic point of view (about 10-10 second) that incidence and reemis-
sion can be considered almost instantaneous. The reemission is called
diffuse since the speed and direction of the reemitted molecules is random
and has no connection with the same values they had at incidence.
From the newer point of view the molecules have wave properties and
can be diffracted by the potential field of the surface. Thus, diffuse
reflection can occur without any adsorption actually taking place. The
crystals of the solid surface are randomly orientated with the result
that diffracted molecules are reflected in all directions. Some may
reflect in the right direction and with no loss of tangential momentum,
whence the term specular reflection is appropriately applied to them.
Upon reemission, whether diffuse or specular, the molecules rapidly
approach another state of equilibrium which is also Maxwellian. Actually,
collisions between incident and reemitted molecules tend to smooth out
the temperature difference between them, but this effect is a second-
order one and not considered likely to take place right at the wall.
Also, some molecules may be bounced back and forth between the gas and
wall many times except when the flow velocity is extremely high.
The original theoretical approach to the boundary conditions was
made by Maxwell (ref. 25) who assumed that a fraction a of the incident
molecules is temporarily adsorbed by the surface and then reemitted in
arbitrary directions, but with an energy corresponding to the tempera-
ture of the wall, while a fraction 1 a is reflected perfectly like
light rays at a plane mirror. The fraction a thus reflects diffusely
while the fraction 1 a reflects specularly. If x is the coordinate
NACA RM 57A30
normal to the surface and v is the velocity of the gas parallel to the
surface, the original calculations of Maxwell led to the velocity of slip
v(O) = G -
2 pT KaYx =Oy
where G = 2a
- 1)1 and
I is the mean free path of the molecules.
If there is no temperature gradient then
v(O) = CL ) X
The coefficient of slip G is defined by Maxwell to be
G = p(2)1/2(p)-1/2 1
with p = 4cpl(2itRT)-1/2 and c = 0.499 (ref. 26, pp. 292 to 297)
G = 2c 2 1>
Modern derivations omit the temperature gradient normal to the wall
(ref. 26) and obtain as the first-order slip velocity at the wall
2 a ,/y \
v(O) = 2-a I V
a Yx LO
+ ( R6) (12)
The last term containing the temperature gradient in the direction of
flow is called "thermal creep;" it is a flow along the surface induced
by the temperature gradient.
NACA RM 57A50
Both Kennard and Knudsen refss. 26 and 27) found experimentally a
temperature jump at any surface in contact with a rarefied gas as was
originally predicted by Poisson
Tk T = o(t)
where Ty is the wall temperature and Tk is the temperature which the
gas would have if the temperature curve near the wall is linearly extra-
polated up to the wall. Knudsen introduced an accommodation coefficient a
to account for the fact that adsorbed molecules are not reemitted from the
wall with energies corresponding to the wall temperature Tw but cor-
responding to some temperature Tr which is intermediate between Twy
and the temperature Ti of the incident molecules. The definition is
given by Ei Er = a(Ei Ew) in terms of the energies, or in terms of
the temperatures it becomes
Ti Tr = a(Ti Tw)
It is seen that for a = 1 the accommodation is complete so that the
emitted molecules do leave at the temperature of the wall. Although
time of attachment of molecules is very short, there can be considerable
accommodation and a is only slightly less than 1 for aerodynamical sur-
faces. For a complete discussion see the book by Kennard or Loeb (ref. 26
or 28) or the monograph of Schaaf and Chambrd (ref. 2).
The elementary kinetic-theory derivation of the temperature jump
(ref. 26) gives g in terms of a, whence the temperature jump at a wall
is to the first order
T(0) T, = 2 _K 2- at (13)
7 + 1 Pc a \Bx/o
or, in terms of the Prandtl number of the gas Pr = pCp/K,
T(O) Tw 2=-a 2-7 ( = (14)RT 1
L 7 + 1 Pr\x p V rx
18 NACA RM 57AO50
For polyatomic gases
cl 2 a 2
2 a 7 + 1 Pr
and for monatomic gases
cL 12- (14a)
6 a f2
because K- (97 5) and y = 2.
pcv 4 5
This expression for the temperature jump in a polyatomic gas is
carefully derived by Weber (ref. 29) but by intuitive means which are
ascribed to Knudsen. He arrives at the above expression for a monatomic
gas as long as the accommodation coefficient is the same for translational
and inner energies of a molecule. Then Weber concludes from the experi-
mental evidence that equation (15) together with equation (14) gives the
correct temperature jump for all gases, polyatomic as well as monatomic.
The Weber formula (identical with Smoluchowski's) is then
15 P a (T
A' = T(0) -T,= T 2----a-(- (15)
which is independent of 7 and Pr.
Welander (ref. 50) has recently redetermined the first-order tempera-
ture jump for Maxwellian molecules by finding a solution of the linearized
Boltzmann equation (i.e., small disturbances from a Maxwellian distribu-
tion) under the assumption that the reflected molecules have a Maxwellian
distribution. His method is similar to the Chapman-Enskog expansion with
one additional term and his result simply replaces the factor (2 a)/a
by (2 ka)/a where k = 0.827. Welander also gives an excellent survey
of the literature up to 1954. His results imply that except for the
lightest gases, hydrogen and helium, accommodation is perfect. In that
case equation (1i) reduces for air to
LT = 2.21 T
NACA RM 57A50
The assumption that the distribution function for the gas molecules
is Maxwellian in the neighborhood of the wall is the one most often
criticized. Payne (ref. 31) attempted to remove the assumption that the
molecules approaching the wall have the same distribution in velocity and
temperature over the last short distance of the order of magnitude of
1 mean free path but had to make additional assumptions as to the order of
magnitude of velocities and temperatures. His result is that the lower
limit on the velocity of slip is the first term in equation (12) and the
upper limit is 1.27 times the lower limit for Maxwellian molecules. The
temperature jump which Payne derives has the same minimum value as equa-
tion (14), but with the factor 27/(7 + 1) in cl replaced by 7/2(7 1).
For monatomnic molecules 7 = 5/5 and both of these factors are 5/4, so
Payne's result is equation (14) with c1 given by equation (14a). He
obtains 1.52 times the minimum value as the upper bound to LU. The
result obtained by Welander would therefore be about the average of Payhe's
maximum and minimum values.
It thus appears that all attempts to derive the first-order boundary
conditions lead to the same result, equation (12) for the slip velocity
and equation (15) or (14) for the temperature jump, but with different
values for the coefficients cl and a, = 2- Since the param-
V 2 0r
eters a and a are not predicted by theory anyway, but must be deter-
mined from experiment, it would seem that the boundary conditions (12)
and (14) are perfectly adequate to handle any problem in the slip-flow
regime. Parameters a and a are apparently a macroscopic expression
of unknown but intuitively understood adjustment of state and equilibrium
of gas molecules near solid boundaries.
All the hypersonic viscous flow theory has so far been based upon
the well-known Prandtl boundary-layer equations which originally were
derived for low-speed flow from the Navier-Stokes equations. No kinetic
theory has been proposed which has special qualities that would make it
more correct for high Mach numbers; in fact, both Grad's theory and the
Burnett equations have been shown to break down at a relatively low Mach
number. The study of shock-wave structure is excellent for the determina-
tion of the range of validity of a theory since it is independent of bound-
ary conditions, as also is the absorption and dispersion of high-frequency
sound waves. The Navier-Stokes equations have been completely confirmed
by the known data in both of these fields refss. 5, 6, and 52) so at the
present time there seems to be no theory more adequate than the Navier-
Stokes equations together with the first-order slip boundary conditions.
This is not to imply that the Navier-Stokes equations are proven to be
valid for all densities and all Mach numbers but only that present experi-
mental evidence agrees best with them for Mach numbers less than 2 and
mean-free-path lengths of 0.005 to 0.05 inch (ref. 6). Theoretical work
does show refss. 5 and 12) that nonlinear terms in the stress-deformation
NACA RM 57A50
relation do lead to thicker shock waves at Mach numbers above 2 but
experimental data are as yet too meager to determine the significance of
these nonlinear terms. The one measurement of Sherman at M = 5.70 has
not apparently been compared with either of these two theories.
Second-order-slip boundary conditions are implicitly contained in
the formalization of Epstein (ref. 55) but were not explicitly written
out until Schamberg used Burnett's theory to obtain terms to order
(p/p)2 in the nonlinear pressure and temperature gradients. Since Grad
(ref. 7) obtains more rigorously and simply a set of boundary conditions
by iteration in the sane manner in which an iteration on the 15-moment
equations give the Burnett terms (ref. 2), there is no reason to consider
Schamberg's results at all. Yang (ref. 14) has rederived Grad's boundary
conditions from a more physical point of view while Kryzwoblocki (ref. 54)
has reconsidered and generalized Epstein's derivation. Since Von
Kryzwoblocki does not work out the second-order terms, showing only that
the boundary conditions are consistent with those for the Navier-Stokes
equations with no slip, there is need for an analysis of his integrals,
which look like those of Grad, to determine whether or not they are the
same as Grad's. A superficial examination indicates that they are, but
both Yang's and Von Kryzwoblocki's papers arrived too late to determine
if they are the same for the present report.
Experimental checks of theory or disagreement therewith have been
touched upon in the earlier section on theory. While there is not so
much experimental data available as would be desired in order definitely
to prefer one theory above all others, there is a definite tendency for
the data to agree best in the slip flow regime with the predictions of
the Navier-Stokes order of approximation. Furthermore, there is a smooth
transition of data all the way from the continuum flow domain through slip
flow and the transition regime into free-molecule flow. No theory yet
proposed is capable of predicting when the Navier-Stokes equations break
down and kinetic theory considerations take over. Experiment can help
in this respect but is not a complete answer, since the conditions which
distinguish one theory from another are too fine to be caught in the
coarse-grained experiments so far performed. For example, measurements
of drag in the rotating-cylinder apparatus do not determine the pressure
and temperature distribution between the cylinders which is necessary to
distinguish one theory from another. The Navier-Stokes equations with
first-order-slip boundary conditions are sufficient to predict the meas-
Since Schaaf and Chambrd refss. 2 and 5) have presented excellent
surveys of all the experimental data, no attempt will be made here to
NIACA RM 57A50
refer to all the significant papers. All data so far obtained have come
from the Ames Laboratory of the NACA or the Low Pressures Research Project
of the University of California at Berkeley.
Some of the most interesting experimental results are the definite
increase in drag coefficient with decreasing Reynolds numbers and increasing
Mach number, the same for the recovery factor, and the decrease in heat
transfer as expressed in the form of a Nusselt number for both spheres
and cylinders. The changes apparently begin in the slip flow regime and
the experimental curves fair into the free-molecule flow values as pre-
dicted by the theory for the latter. Hence, it is possible to obtain a
rough indication of the Knudsen number at which slip begins, which it is
not yet possible to do theoretically. However, no precise value of K
can be said to be that at which slip flow starts, only that it seems to
be of the order of 0.1.
Experimental data on flat plates and cones have so far shown negli-
gible effects of slip, the effects of interaction between the thick bound-
ary layer and the external flow (as well as the transverse curvature cor-
rection for cones) tending to overshadow any slip which might be present.
For cones there is also the decrease in heat transfer and increase in the
recovery factor with increasing rarefaction.
The considerable increase in the temperature, recovery factor from
the value Pr 0.9 for continuum flow to values of 1.5 or so in free-
molecule flow is a significant result. Martino (ref. 19) has proposed a
simple method of expressing the recovery factor for circular cylinders,
as measured by Stalder, Goodwin, and Creager (ref. 22), which is definitely
empirical but is suggested here as the answer to those who need simple
relations for immediate design problems.
If P is the probability that a molecule will collide with the sur-
face before colliding with another molecule then
(Slip system = P(Free-molecule flow system)
S(1 P)(Continuum system)
Evidently P is a function of the Knudsen number K such that
lim P = 0
lim P = 1
22 NACA RM 57A30
since Knudsen interpreted what is now called the Knudsen number as the
ratio of the number of collisions of molecules with the wall to the num-
ber of collisions between molecules themselves. Martino then takes P
as equal to the fraction of molecules colliding with the surface and
1 + K
1 P= 1
1 + K
If Q is any aerodynamic function of K such as the drag coefficient
or recovery factor, Q% is its value in the continuum range, and Qf
is its value in the free-molecule flow, the value Qs in slip flow is
Qs Qc + -K- Qf
For the recovery factor on circular cylinders
r = c Krf
and with ro = 0.95 and rf = 1.55 this formula agrees very closely
with the data of figure 6 in reference 22. On the other hand, an attempt
to use this method for the skin-friction coefficient and Nusselt number
in heat transfer did not work. The reason lay in the dependence of both
of these quantities upon the Reynolds number and through the latter on
the Knudsen number. Thus, Martino's suggestion would appear to be valid
for cases in which the quantity Q is independent of Knudsen number in
both continuum and free-molecule flow. Yang's empirical approach appears
better for quantities like skin friction, heat transfer, and velocity of
IJACA EM 57A30
Most of the experimental papers contain empirical or semiempirical
formulas which correlate the data given. In general, some of the best
fitting curves are derived from low-speed incompressible analysis, but
with first-order slip in the boundary conditions. This is the real
justification for using the Navier-Stokes equations with first-order slip
at the present time. No theoretical justification has been found, but
the agreement with experiment is all the designer has to go on.
Besides reference 22, reference 55 also gives data on heat transfer
from circular cylinders and correlates the data by an analytical expres-
sion. Data on skin friction on flat plates are given in reference 17.
For spheres, drag data are to be found in reference 56, while the measured
heat transfer from spheres is presented and analyzed in references 57
As a result of this study of the problem of slip flow in aerodynamics,
it is possible to make several significant conclusions:
1. Two extreme cases in the flow regime are well known and established
both theoretically and experimentally: (a) Continuum flow (Knudsen number
much less than 1). Here the Navier-Stokes equations with no slip and
no temperature jump at the boundaries are perfectly valid and seem to
hold for laminar flow to extremely large Mach number well into the hyper-
sonic range (turbulent flow included); (b) Free-molecule flow (Knudsen
number much greater than 1). The gas is in equilibrium with a Maxwellian
distribution of velocities and collisions between molecules can be neg-
lected; the collisions between the body and the molecules can be accounted
for by means of the accommodation coefficient and reflection coefficient.
2. The slip-flow regime near to continuum flow can best be described
by the Navier-Stokes equations together with first-order slip and tempera-
ture jump at any bounding surfaces. In this region, rarefaction is slight,
and collisions between molecules predominate, but some effect of collisions
between surfaces and the molecules must be taken into account.
5. The transition regime between first-order slip flow and the free-
molecule regime has no valid theoretical basis. The collisions between
molecules themselves and between the molecules and surfaces are apparently
of equal importance so any correct theory must embody both. Since the
new theory of Ikenberry and Truesdell contains no treatment of boundary
conditions, it is not yet considered suitable to describe flow about
bodies or between surfaces. For problems not involving solid boundaries,
such as the propagation of sound or the structure of shock waves, the
Ikenberry-Truesdell theory holds the greatest promise.
24 NACA RM 57A50
4. Experimental data to some extent exist for both subsonic and
supersonic flows which span the region of rarefaction or the complete
Knudsen number range. Empirical or semiempirical relations can be
devised to agree with these data and these should be used in engineering
analyses at the present time.
5. Finally, some problems of interest which could be attacked with
some hope of success, at least for obtaining working results, are: (1) A
first-order slip investigation of a flow involving pressure gradients such
as the flow over a wedge or biconvex profile, and (2) a study of the
Ikenberry-Truesdell kinetic theory, either applying it to the shock-wave
problem or determining some compatible set of boundary conditions which
can lead to a solution of a simple problem such as Couette flow.
University of Washington,
Seattle, Wash., March 20, 1)6b.
NACA RM 57TA50
THEORY OF TRUESDELL AND IKENBERRY
Thanks to Professor Truesdell, copies of the manuscripts of the
two papers of reference 24 were obtained before the present paper was
completed. A hurried reading indicated that severe criticism of present
kinetic theory was contained therein. On the other hand, the work of
Truesdell and Ikenoerry included a new approach, figuratively speaking,
to the kinetic-theory problem. However, their approach is not exactly
new, since the authors go back to a method which was used by Maxwell
himself and which they call Maxwellian iteration. All developments in
kinetic theory since Maxwell are criticized as wrong in principle, if
not in results.
The new iteration method does lead directly from a few basic assump-
tions to nine exact equations of transfer. No knowledge of any distribu-
tion function is needed, and it is precisely in this respect that the new
scheme seems to be the most powerful approach so far advanced. It is
restricted to Maxwellian molecules, but, as pointed out earlier in this
report, that is not considered serious when the question considered is
the one of the foundations of kinetic theory.
While reference 24 is critical of some aspects of Grad's theory there
remains an impression of a great debt to some of the ideas and methods
which Grad proposed in his examination of kinetic theory. The new theory
is still based upon Boltzmann's integro-differential equation, although
it nowhere explicitly appears in the papers. The foundations of kinetic
theory are still the same; only the manner of constructing the edifice
thereupon is new. By using spherical harmonics as originally suggested
by Maxwell, the technique is straightforward, although just as tedious as
Maxwell's in other approaches. The great advantage is in the exactness
due to the fact that any iteration depends only upon iterates of lower
The treatment is thorough. Other methods of iteration are touched
upon and comparisons are made with other approaches, those of Hilbert,
Enskog, Chapman, and Grad. One exact solution is found, that of shearing
flow, which is the first ever to be obtained from kinetic theory for
On the other hand, there are new difficulties as well as successes.
The solutions are not general but special and may or may not turn out to
be correct representations of physical phenomena. The complete validity
of the method is shown only for a contrived mathematical model, not for
the kinetic theory itself. Despite the author's claim for simplicity,
NACA RM 57A50
the techniques developed are no more elementary than in the methods of
Grad or Enskog. Similar integrals have to De evaluated and systems of
equations have to be solved. However, this treatise in 120 pages of text
certainly contains a more thorough and less forbidding introduction for
aerodynamicists to the foundations of advanced kinetic theory than the
thicker treatise by Chapman and Cowling. Actually, there is no real com-
parison, since the latter book was really concerned with the derivation
of the physical coefficients of diffusion, viscosity, and heat conduction
in transport processes. Given the latter, this new treatment is more
suitable for the setting up of the equations to be solved in a physical
There are two questions which arise from this work and they are:
(1) Are there any more exact solutions than the one found or even any
valid approximate solutions? (2) What happens if boundary conditions are
added? There is no mention at all of boundary conditions, particularly
of slip, and hence no indication as to whether the method is compatible
with the addition of conditions at the walls. In the concluding remarks
some indications of incompatibility with continuum mechanics is mentioned
with the remark that a new theory of the latter nature is urgently needed.
It is not mentioned that, no matter how elaborate a correct continuum
theory must be, the EJavier-Stokes equations will be contained therein and
will reduce the the latter when classical conditions are valid.
NACA RM 57A50
1. Tsien, Hsue-Shen: Superaerodynamlcs, Mechanics of Rarefied Gases.
Jour. Aero. Sci., vol. 15, no. 12, Dec. 1946, pp. 655-664.
2. Schaaf, S. A., and Chambre, P. L.: Flow of Rarefied Gases. Vol. IV,
Pt. G, High Speed Aerodynamics and Jet Propulsion Laminar Flows
and Transition to Turbulence. Princeton Univ. Press (Princeton)
(to be publ.).
5. Schaaf, S. A.: Theoretical Considerations in Rarefied-Gas Dynamics-.
Ch. 9. Experimental Methods and Results of Rarefied-Gas Dynamics.
Ch. 10. Heat Transfer Symposium (1952), Eng. Res. Inst., Univ. of
Mich., 105, pp. 257-282.
4. Montroll, E. W., and Green, M. S.: Statistical Mechanics of Transport
and Non-Equilibrium Processes. Annual Rev. of Phys. Chem., vol. 5,
1954, pp. )44-476.
5. Gilbarg, D., and Paolucci, D.: The Structure of Shock Waves in the
Continuum Theory of Fluids. Jour. Rational Mech. and Anal., vol. 2,
no. 4, Oct. 1955, pp. 617-642.
6. Sherman, F. S.: A Low-Density Wind-Tunnel Study of Shock Wave Struc-
ture and Relaxation Phenomena in Gases. NACA TN 3298, 1955.
7. Grad, H.: On the Kinetic Theory of Rarefied Gases. Comm. on Pure
and Appl. NhMath., vol. 2, 1049, pp. 33551-407.
8. Truesdell, C.: On the Viscosity of Fluids According to the Kinetic
Theory. Zs. Physik, Bd. 151, 1952, pp. 275-280.
9. Wang Chang, C. S., and Uhlenbeck, G. E.: On the Transport Phenomena
in Polyatomic Gases. Rep. No. CM-681, Univ. of Mich., Eng. Res.
Inst., Project M604-6, Eng. Res. Inst., Univ. of Mich., July 1951.
10. Zemansky, Mark W.: Heat and Thermodynamics. Second ed., McGraw-Hill
Book Co., Inc., 1945, ch. 6.
11. Truesdell, C. A.: The Mechanical Foundations of Elasticity and Fluid
Dynamics. Jour. Rational Mech. and Analysis, vol. 1, nos. 1 and 2,
1952, pp. 125-500.
12. Zoller, K.: Zur Struktur des Verdichtungs-stosses. Zs. Phys.,
Bd. 150, 1951, pp. 1-58.
NACA RM 57A50
15. Von Kryzwoblocki, M. Z.: On Two Dimensional Laminar Boundary Layer
Equations for Hypersonic Flow in Continuum and in Rarefied Gases.
Jour. Aero. Soc. India, vol. 5, nos. 1 and 2, Feb. 19553, pp. 1-15,
14. Yang, H. Sun-Tiao: Rayleigh's Problem at Low Mach Number According
to the Kinetic Theory of Gases. Thesis, C.I.T., 1955.
15. Wang Chang, C. S., and Uhlenbeck, G. E.: The Heat Transport Between
Two Parallel Plates as Functions of the Knudsen Number. Proj-
ect M999, Eng. Res. Inst., Univ. of Mich., Sept. l115.
16. Wang Chang, C. S., and Uhlenbeck, C. E.: The Couette Flow Between
Two Parallel Plates as a Function of the Knudsen Number.
Rep. lo90-1-T, Eng. Res. Inst., Univ. of Mich., June 1954.
17. Schaaf, S. A., and Sherman, F. S.: Skin Friction in Slip Flow.
Jour. Aero. Sci., vol. 21, no. 2, Feb. 1954, pp. 85-90, 144.
18. Mirels, Harold: Estimate of Slip Effect on Compressible Laminar-
Boundary-Layer Skin Effect. NACA TIN 2609, 1952.
19. Martino, R. L.: Heat Transfer in Slip Flow. UTIA Rep. No. 55, Inst.
Aero., Univ. of Toronto, Oct. 155.
20. Maslen, Stephen H.: Second Approximation to Laminar Compressible
Boundary Layer on Flat Plate in Slip Flow. IIACA Tl 2818, 1952.
21. Kendall, J. M., Jr.: Experimental Investigation of Leading Edge
Shock Wave-Boundary Layer Interaction at Hypersonic Speeds.
Preprint No. 611, 24th Annual Meeting, Inst. Aero. Sci., Jan. 25-26,
22. Stalder, Jackson R., Goodwin, Glen, and Creager, Marcus 0.: Heat
Transfer to Bodies in a High-Speed Rarefied-Gas Stream. NACA
Rep. 1095, 1952.
25. Lin, T. C., and Schaaf, S. A.: Effect of Slip on Flow Near a Stagna-
tion Point and In a Boundary Layer. NACA TN 2568, 1951.
2k. Ikenberry, E., and Truesdell, C.: On the Pressures and the Flux of
Energy in a Gas According to Maxwell's Kinetic Theory. Pts. I and
II. Jour. Rational Mech. and Analysis, vol. 5, no. 1, Jan. 1956,
25. Maxwell, J. C.: On Stresses in Rarefied Gases Arising From Inequal-
ities of Temperature. Phil. Trans. Roy. Soc., Pt. 1, vol. 2, 1879,
NACA RM 57A30
26. Kennard, Earle H.: Kinetic Theory of Gases With an Introduction to
Statistical Mechanics. First ed., McGraw-Hill Book Co., Inc., 1958.
27. Knudsen, Martin: The Kinetic Theory of Gases. Second ed., Methuen &
Co., Ltd. (London), 1946.
28. Loeb, Leonard B.: The Kinetic Theory of Gases. Second ed., McGraw-
Hill Book Co., Inc., 19534.
29. Weber, S.: Dependence of the Temperature Jump on the Accommodation
Coefficient for a Heat Conduction Problem in Gases. (Title in
German.) Det. Kgl. Danske Videns. Selsk., Math.-Fysik Medd.,
vol. 16, no. 9, 1959.
30. Welander, P.: On the Temperature Hump in a Rarefied Gas. Arkiv
fUr Fysik, Bd. 7, nr. 44, 1954, pp. 507-555.
31. Payne, H.: Temperature Jump and Velocity of Slip at the Boundary of
a Gas. Jour. Chem. Phys., vol. 21, 1955, pp. 2127-2151.
32. Truesdell, C.: Precise Theory of the Absorption and Dispersion of
Forced Plane Infinitesimal Waves According to the Navier-Stokes
Equations. Jour. Rational Mech. and Analysis, vol. 2, no. 4,
Oct. 1955, pp. 645-741.
33. Epstein, Paul S.: On the Resistance Experienced by Spheres in Their
Motion Through Gases. Phys. Rev., vol. 25, no. 6, June 1924,
34. Von Krzywoblocki, M. Z.: On Some Problems in Free Molecule-Slip Flow
Regimes. Acta Phys. Austriaca, vol. 9, 1955, pp. 216-257.
35. Sauer, F. M., and Drake, R. M., Jr.: Forced Convection Heat Transfer
From Horizontal Cylinders in a Rarefied Gas. Jour. Aero. Sci.,
vol. 20, no. 5, Mar. 1955, pp. 175-180, 209.
56. Kane, E. D.: Sphere Drag Data at Supersonic Speeds and Low Reynolds
Numbers. Jour. Aero. Sci., vol. 18, no. 4, Apr. 1951, pp. 259-270.
37. Drake, R. M., Jr., and Backer, 0. H.: Heat Transfer From Spheres to
a Rarefied Gas in Supersonic Flow. Trans. A.S.M.E., vol. 74, no. 7,
Oct. 1952, pp. 1241-1249.
58. Kavanau, L. L.: Heat Transfer From Spheres to a Rarefied Gas in
Subsonic Flow. Trans. A.S.M.E., vol. 77, 1955, pp. 617-625.
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