RM 57A30 NACA RESEARCH MEMORANDUM PROBLEM OF SLIP FLOW IN AERODYNAMICS By Robert E. Street University of Washington UNIVERSITY OF FLORIDA DOCUMENTS DEPARTMENT 120 MARSTON SCIENCE LIBRARY RO. BOX 11 70 11 GAINESVILLE. FL 32611-7011 USA NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS WASHINGTON March 4, 1957 K^O'7'^^*'' ^^5-? 2/^ NACA RM 57A5O NATIONAL ADVISORY C0I4MITTEE FOR AERONAUTICS RESEARCH MEMORANDUM PROBLEM OF SLIP FLOW DJ AERODYTTAMICS By Robert E. Street SUMMARY 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 lirgently needed. INTRODUCTION 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 j\mp in temperature at the boundary of the flow begin to exhibit themselves. 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 Chambr^ (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 few years ago (ref. 5). NACA EM 57A50 The dlmensionless parameter which is considered of paramount impor- tance in the flow of semirarefied gases is the Knudsen niimber 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 = i . X..„W^ ^ (1) where 7 is the ratio of specific heats of the gas, M = u/a is the Mach number, and Re = pLU/^i is the Reynolds number. The velocity U is some reference speed such as the mass velocity of the gas or the flight speed. Some writers suppose that for large values of Re the character- istic length is the boimdary- layer thickness S and define the Knudsen number as K = — ~ M/\/Re. This seems to be the proper parameter for 5 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 Sfj^/ \fRe at low speeds, but in the hypersonic flow over flat plates the more significant parameter is a constant times M^/\fRe. Physically this seems to be due to the greater importance of M since it indicates compressibility in hypersonic flows while M occurs in conjunction with Re as an indication of rarefaction alone in equation (l). Since Re a LU aL 2aL \j -ny 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. NACA m. 57A50 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. k) . 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 ass-amed. 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 (refs. 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 m\ist 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. k) , 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 knovm to enable the design engineer to make some attempt at determining optimimi aerodynamic shapes for aircraft and missiles of the immediate future. This investigation was conducted at the Iftiiversity of Washington under the sponsorship and with the financial assistance of the National Advisory Committee for Aeronautics. SYMBOLS a speed of sound Cf local skin friction, dimensionless Cg constant of proportionality in linear viscosity- temperature formula ■^ instrinsic velocity, I - u Cj_,C4 components of instrinsic velocity k NACA RM 57A50 Cp specific heat at constant press\:ire Cy specific heat at constant voltmie d distance between plates f distribution function depending on x, |*, and t G coefficient of slip g function E energy K Knudsen number, ^/^j dimensionless L reference length I mean free path of gas M Mach n\:miber, 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, c-q\i/k, dimensionless p pressure in gas Q any aerodynamic f\mction of K such as drag coefficient or recovery factor qj^ components of heat flux vector R gas constant Re Reynolds number r recovery factor on circular cylinders T absolute temperature t time * NACA RM 57A50 U reference velocity u macroscopic flow velocity Uj_,U; components of flow velocity V mean molecular speed of gas x.,x^ Cartesian coordinate components of position vector x a accommodation coefficient, dimensionless 7 ratio of specific heats of gas, dimensionless, Cp/cy 5 botmdary-layer thickness K coefficient of heat conduction A second coefficient of viscosity p density of gas a reflection coefficient of Maxwell |j, coefficient of viscosity of gas V coefficient of kinematic viscosity, u/p I molecular velocity vector Ij^ components of | T average time between molecular collisions Subscripts: c value in continuum range f value in free-molecule range o free-stream conditions s value in slip flow w value at wall NACA RM 57A30 THEORETICAL CONSIDERATIONS 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 laxge 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 ass\miptions underlying the derivation of the Boltzmann equation, such as stated clearly and succinctly by Grad (ref. j) , 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: (l) 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 struct\are 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 + 2|i = 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 kinetic theory of polyatomic gases. NACA RM 57A30 7 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. Althoiogh 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) . (h) The state of the gas is one of equilibriim or near equilibrium. This is implicit in the previous assxamption, 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- librium conditions. 8 NACA RM 57A30 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 Uj_, the stress components Pij> the temperature T, and the heat flux vector components q^. If the Boltzmama 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 (refs. 2 and 7) |P + -^(pui) = (2) ot ax^ -^ + U. ^ + i ii = (3) ht J bx. p ax. SP ^ / ^ 2 ^ ^^i 2 ^^i ^ ,,, ht bx. 3 ^^ bx. 3 Sx^ where i,j = 1,2,3 • • • aJid- 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 ^(0) = e__ e-<^2/2RT (6) 5/2 (2jtRT) NACA RM 57A50 9 where c^ is the square of the intrinsic velocity whose components are the difference between the components of molecular velocity |j_ and mass velocity Uj^. All odd moments of f^^) vanish as do the second-order ones, leading to P^j = p6ij and qj^ = 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 fliox are linear functions of the rate of deformation of the fluid and the temperature gradient, respectively, leads to the relations Pij = P5ij + Pij Pij = -^^(ui,j + ^j,i) - ^5ij^k,k (7) (li = -t«Ti (8) Then equations (2) to (5) reduce to the Navier-Stokes equations of a compressible viscous fluid, which are also completely determined. The coefficients |i. A, and k 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 continuiom 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 to be f = f(0)(l + f(l) + f(2) + . \ (9) Substitution of this expression into Boltzmann's integral equation and 10 NACA RM 57A30 the determination of f^O), f\^), f\^), ... 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 3 and 11. Secondly, they can be deduced from Grad's distri- bution function f = f(0) 1 + ^ C-iC. 2pRT ^ (10) as was first shown by Schaaf and Chambrd (refs. 2 and J)- 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 n\jmber 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 axe 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 continuimi 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 sudi 12) different from the Navier-Stokes terms. In hypersonic flow Von Kryzwoblocki has deduced equations to boundary-layer order of magnitude (ref. 13) ajid has indicated means of solving these but has not worked out a solution. Yang (ref. Ik) has found a solution to Rayleigh's problem of the impiilsive 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 Biornett 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 (refs. I5 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 \ised by Wang Chang and Uhlenbeck, K = d/l, or the reciprocal of the Knudsen number NACA RM 5TA30 11 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 l/d do not converge while those in powers of d/Z 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 Boltzmaiin 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 IJ-moment approximation to the Boltzmann equa- tion which assumes the distribution function (lO) above and deduces there- from a system of nine additional equations for the tensors P^j and 'q^, which together with equations (2) to (5) above give a determined system with unknowns p, p, T, Uj_, q-i_, and P^- ^' Thus Grad's equations con- tain Euler's, Navier-Stokes ' , and Burnett's equations as successive approxi- mations, for which Schaaf and Chambrd (refs. 2 and 3) 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 fl\ax, 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 « 0.685 exp (0.3^+2 ^ ) erfc(0.3i+2 ^ ] (ll) Here Mq = u/ao and Re = -^ U^t, the subscript o denoting free-stream conditions. The ratio of the speed of sound to the rate of diffusion of 12 NACA RM 5TA30 vorticity is \/Re/Mo = aQ\ -2— = \ -2 — or, alternatively, Rs/Mq = slq^/vq y M-o V ^o is proportional to the ratio of elapsed time to the average time between molecular collisions t, that is, Re/Mo = — — . Schaaf and Sherman (ref • 17 ) obtained a similar expression for Cj as did Mirels (ref. l8) 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 (ll) 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 Mirels 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. jn /m \l/2 m + g The parameter used is ^ = 1.30\/— ^ where Cg = — — is the VMoK \To/ Tv + S constant of proportionality in the linear viscosity- temperature formula. But using equation (l) MqK = 1.2567^/ — and since Cg = 1 for the Re adiabatic no-heat-transfer case ^ « \jRe/M, or ^ 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. Meirtino 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 temperatirre 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 = 1/7" M /Re ' , retaining only terms to the first power in e . NACA RM 57A50 15 Maslen's paper is the only one to treat the flat-plate boiondary- layer problem with first-order slip by the usiial boundary- layer analysis instead of by Rayleigh's method. His expressions for skin friction and heat trans- fer show a different fionctional form. However, the recent meas\rrements by Kendall (ref . 21) tend to confirm Maslen, and Martino's independent deter- mination of the Nusselt nijimber 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 (, 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 (MqK) ' or MlURe 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 bo\andary 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 n\:miber. A particularly simple analysis by Schaaf and Chambr^ (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 I, it turns out that the term in Cf containing I to the first power also contains the pressinre 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 pressvire gradient is not zero, or the stagnation point flow (ref. 23), 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. I7) 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- Ik NACA RM 57A30 Yang'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 n\ambers. 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 Navier-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- conduction equation. 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 Rejr^ decreases rapidly from infinity to any finite value as x increases. The rarefaction parameter M/wRe 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 bo\;indary- 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 liiiversity 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 ji:mip 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 literatiore should be used for engineering design and analysis. A more serious criti- cism of the IJ-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. 2^4-). 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 NACA EM 57A30 15 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 results. BOUNDARY CONDITIONS A thoroiigh analysis of the proper macroscopic boundary conditions would have to be based upon an analysis of the molecular forces between molec\iles 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 occixr without any adsorption actually taking place. The crystals of the solid surface axe 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 tang;ential momentimi, 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 16 NACA RM 57A30 normal to the siirface and v is the velocity of the gas parallel to the surface, the original calculations of Maxwell led to the velocity of slip v(0) = J^.lAL^ \dx 2 pT dx a, + f 1 iL ^' By/ ^ \k PT hy, ^ •^/x=0 V •^/x=0 2/2 where G = — f— - 1]^ and. I is the mean free path of the molecules. If there is no temperature gradient then v(0) = ofe x=0 The coefficient of slip G is defined by Maxwell to be G = 1 ,(2„)^/2(pp)-l/2| - l) With 11 = i+Cpl(2jtRT)"^/2 and c = O.U99 (ref. 26, pp. 292 to 297) G= 2c|-lh-(|-lU 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 Jbv\ 5 |i /St\ |i § ^ ^^i bv\ 3 ^(bT + ^ R ^ ^x/o k \ay/o (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 I7 Both Kennard and Knudsen (refs. 26 and 27) found experimentally a temperature jump at any siirface in contact with a rarefied gas as was originally predicted by Poisson ^ ' o where Tv is the wall temperature and Tj^ 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 accoimt for the fact that adsorbed molecules are not reemitted from the wall with energies corresponding to the wall temperat\u:e T^ but cor- responding to some temperature T-^. which is intermediate between T-^ and the temperature Ti of the incident molecules. The definition is given by Ei - E^ = a(Ej_ - E^) 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. Althoiigh 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 temperatiore jump at a wall is to the first order T(0) - T^ = -2— -J5_ L^cc Jar (13) 7 + 1 HC^ a \Sx/o or, in terms of the Prandtl number of the gas Pr = ^Cp/n, ^(0) -- ^ ^ htl = ^ =^^11 (li^) 18 NACA M 57A50 For polyatomic gases ■^ 1/2 a 7+lPr and for monatomic gases 8 a \ 2 because -^ = —(97 - 5) and. 7 = X This expression for the temperature jump in a polyatomic gas is carefully derived by Weber (ref . 29) but by intuitive meajis 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 (I5) together with equation (lU) gives the correct temperature jump for all gases, polyatomic as well as monatomic. The Weber formula (identical with Smoluchowski's) is then Ar = T(o) -T^^f^ilm (15) which is independent of 7 and Pr. Welander (ref. 30) has recently redetermined the first-order tempera- ture jump for Maxwellian molecules by finding a solution of the linearized Boltzmann equation (i.e., small distiirbances from a Maxwellian distribu- tion) under the assumption that the reflected moleciiLes 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 = O.827. Welander also gives an excellent survey of the literatiire up to 195^. His results imply that except for the lightest gases, hydrogen and heliiom, accommodation is perfect. In that case equation (15) reduces for air to AT = 2.2z(^ \OX/q NACA EM 57A30 19 The assumption that the distribution function for the gab molecules is Maxwellian in the neighborhood of the wall is the one most often criticized. Payne (ref . 5I) attempted to remove the assumption that the molecules approaching the wall have the same distribution in velocity and temperatiore over the last short distance of the order of magnitude of 1 mean free path but had to make additional assvimptions as to the order of magnitude of velocities and temperatxares. His result is that the lower limit on the velocity of slip is the first term in equation (l2) and the upper limit is 1.27 times the lower limit for Maxwellian moleciiles. The temperature jump which Payne derives has the same minimiMi value as equa- tion (1^), but with the factor 27/(7 + 1) in c-^ replaced by -y/^iy - l) . For monatomic molecules 7 = 5/5 and- both of these factors are 5/^, so Payne's result is equation {ik) with cj^ given by equation (l4a). He obtains 1.52 times the minimum value as the upper boimd to /ff. The result obtained by Welander would therefore be about the average of Payne's maximum and minimum values . It thus appears that all attempts to derive the first-order boundary conditions lead to the same result, equation (l2) for the slip velocity and equation (15) or (ik) for the temperature jump, but with different values for the coefficients c^ and a-|_ = \ I . Since the param- eters a and a are not predicted by theory anyway, but must be deter- mined from experiment, it wotild seem that the boundary conditions (l2) and {ih) 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 linderstood adjustment of state and equilibrium of gas molecules near solid boundaries. All the hypersonic viscous flow theory has so far been based iipon 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 Bixrnett 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 (refs. 5, 6, and 52) so at the present time there seems to be no theory more adquate 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 O.OO5 to 0.05 inch (ref. 6). Theoretical work does show (refs. 5 and 12) that nonlinear terms in the stress -deformation 20 NACA RM 5TA30 relation do lead to thicker shock waves at Mach niombers 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 = 3- 70 has not apparently been compared with either of these two theories. Second-order-slip boxindary conditions are implicitly contained in the formalization of Epstein (ref. 35) but were not explicitly written out until Schamberg used Burnett's theory to obtain terms to order (m-/p) i^ "the nonlinear pressure and temperature gradients. Since Grad (ref. 7) obtains more rigorously and simply a set of boundary conditions by iteration in the same manner in which an iteration on the 13-moment equations give the Burnett terms (ref. 2), there is no reason to consider Schamberg's results at all. Yang (ref. 1^4-) has rederived Grad's boundary conditions from a more physical point of view while Kryzwoblocki (ref. 3^) 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 DATA 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 wo\ild 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 pressixre 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- ured drag. Since Schaaf and Chambrd (refs. 2 and 3) have presented excellent surveys of all the experimental data, no attempt will be made here to NACA HM 57A30 21 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 boimd- ary layer and the external flow (as well as the transverse ciorvature 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 temperatixre recovery factor from the value \fPr ^^ 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) 4 {l - P) (Continuum system) Evidently P is a f ■unction of the Knudsen niomber K such that and lim P = K-^0 lim P = 1 K-^oo 22 NACA RM 5TA30 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 obtains P = K 1 + K 1 + K If Q is any aerodynamic fimction of K such as the drag coefficient or recovery factor, Qq is its value in the continuum range, and Qf is its value in the free-molecule flow, the value Qg in slip flow is Qs = -^— Qc + -^- Qf 1 + K 1 + K For the recovery factor on circular cylinders Kr^ 1 + K 1 + K and with rQ = 0.95 sn^ I'f = 1'55 this formiila agrees very closely with the data of figijre 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 continuimi and free-molecule flow. Yang's empirical approach appears better for quantities like skin friction, heat transfer, and velocity of slip. NACA EM 5TA30 25 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 foimd, 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 I7. 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 and 58. CONCLIBIONS 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 nimiber 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 boionding 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 s\irfaces. 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. 2k NACA EM 57A50 k. 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: (l) 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, I956. NACA RM 57A30 25 APPENDIX THEORY OF TRUESDELL AND IKENBERRY Thanks to Professor Truesdell, copies of the manuscripts of the two papers of reference 2k 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 Ikenberry 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 order. 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 dlssipative flows. 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. 26 NACA RM 57A30 the techniq.ues developed are no more elementary than in the methods of Grad or Enskog. Similar integrals have to be 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 aerodynamlcists 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 problem. There are two questions which arise from this work and they are : (l) 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 Navier-Stokes equations will be contained therein and will reduce the the latter when classical conditions are valid. NACA RM 57A50 27 REFERENCES 1. Tslen^ Hsue-Shen: Superaerodynamlcs, Mechanics of Rarefied Gases. Jour. Aero. Sci., vol. 13, no. 12, Dec. 19k6, pp. 655-66U. 2. Schaaf, S. A., and Chambre', P. L. : Flow of Rarefied Gases. Vol. IV, Ft. G, High Speed Aerodynamics and Jet Propulsion - Laminar Flows and Transition to Turbulence. Princeton Univ. Press (Princeton) (to be publ. ) . 3. 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., 1955, pp. 257-282. h. Montroll, E. W., and Green, M. S. : Statistical Mechanics of Transport and Non-Equilibrium Processes. Annual Rev. of Phys. Chem., vol. 5> I95U, pp. 1+1+9-1+76. 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. 1+, Oct. 1955, pp. 6I7-6I+2. 6. Sherman, F. S.: A Low-Density Wind-Tunnel Study of Shock Wave Struc- ture and Relaxation Phenomena in Gases. NACA TN 5298, 1955. 7. Grad, H. : On the Kinetic Theory of Rarefied Gases. Comm. on Pure and Appl. Math., vol. 2, 19^9, pp. 53I-U07. 8. Truesdell, C: On the Viscosity of Fluids According to the Kinetic Theory. Zs . Physik, Bd. I5I, 1952, pp. 273-289. 9. Wang Chang, C. S., and Uhlenbeck, G. E. : On the Transport Phenomena in Polyatomic Gases. Rep. No. CM-68I, Univ. of Mich., Eng. Res. Inst., Project M60I+-6, Eng. Res. Inst., Univ. of Mich., July 1951. 10. Zemansky, Mark W. : Heat and Thermodynamics. Second ed., McGraw-Hill Book Co., Inc., 19^3, 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. 125-300. 12. Zoller, K. : Zur Struktur des Verdichtimgs-stosses . Zs. Phys., Bd. 130, 1951, pp. 1-58. 28 NACA RM 57A30 13. 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. 1953j PP- 1-13) 25-55. Ik. 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 T^vTO Parallel Plates as Functions of the Knudsen Number. Proj- ect M999, Eng. Res. Inst., Univ. of Mich., Sept. 1955. 16. Wang Chang, C. S., and Uhlenbeck, G. E. : The Couette Flow Between Two Parallel Plates as a Function of the Knudsen Number. Rep. 1999-l-T, Eng. Res. Inst., Univ. of Mich., June 195^- 17. Schaaf, S. A., and Sherman, F. S.: Skin Friction in Slip Flow. Jour. Aero. Sci., vol. 21, no. 2, Feb. 195^+, pp. 85-9O, lUU. 18. Mirels, Harold: Estimate of Slip Effect on Compressible Laminar- Bo\indary- Layer Skin Effect. NACA TI^ 2609, 1952. 19. Martino, R. L. : Heat Transfer in Slip Flow. UTIA Rep. No. 55, Inst. Aero., Univ. of Toronto, Oct. 1955- 20. Maslen, Stephen H. : Second Approximation to Laminar Compressible Boundary Layer on Flat Plate in Slip Flow. NACA TN 2818, 1952. 21. Kendall, J. M. , Jr.: Experimental Investigation of Leading Edge Shock Wave-Boundary Layer Interaction at Hypersonic Speeds. Preprint No. 6II, 2i+th Annual Meeting, Inst. Aero. Sci., Jan. 25-26, 1956. 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 rfexwell's Kinetic Theory. Pts. I and II. Jour. Rational Mech. and Analysis, vol. 5, no. 1, Jan. I956, pp. 1-128. 25. Maxwell, J. C: On Stresses in Rarefied Gases Arising From Inequal- ities of Temperature. Phil. Trans. Roy. Soc, Pt. 1, vol. 2, I879, pp. 681-712. NACA RM 57A50 29 26. Kennard, Earle H. : Kinetic Theory of Gases With an Introduction to Statistical Mechanics. First ed., McGrav-Hill Book Co., Inc., 1958- 27. Knudsen, Martin: The Kinetic Theory of Gases. Second ed., Methuen & Co., Ltd. (London), 19k6. 28. Loeb, Leonard B. : The Kinetic Theory of Gases. Second ed., McGraw- Hill Book Co., Inc., I95U. 29. Weter, 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. hk, 195^, pp. 507-555- 31. Payne, H. : Temperature Jump and Velocity of Slip at the Boundary of a Gas. Jour. Chem. Phys., vol. 21, 1953, PP- 2127-2131. 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. h, Oct. 1953, PP- 6I4-3-7UI. 33- Epstein, Paul S.: On the Resistance Experienced by Spheres in Their Motion Through Gases. Phys. Rev., vol. 23, no. 6, June 192^, pp. 710-733- 3^. 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. 3, Mar. 1953, PP- 175-l8o, 209- 36. Kane, E. D. : Sphere Drag Data at Supersonic Speeds and Low Reynolds Numbers. Jour. Aero. Sci., vol. I8, no. h, Apr. 1951, pp. 259-270. 37. Drake, R. M. , Jr., and Backer, G. H. : Heat Transfer From Spheres to a Rarefied Gas in Supersonic Flow. Trans. A.S.M.E., vol. jk, no. 7, Oct. 1952, pp. 12Ul-12i+9. 38. Kavanau, L. L. : Heat Transfer From Spheres to a Rarefied Gas in Subsonic Flow. Trans. A.S.M.E., vol. 77, 1955, PP- 617-623. NACA - La.igley Field, Va. 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