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1 THE CONDITIONS NECESSARY FOR DISCONTINUOUS MOTION IN GASES G I Taylor Proceedings of the Royal Society A vol LXXXIV (90) pp The possibility of the propagation of a srface of discontinity in a gas was first considered by Stokes in his paper On a difficlty in the theory of sond This paper begins with a physical interpretation of Poisson s integral of the eqation of motion of a gas in one dimension The integral in qestion is [ z ( a+ w t] m = f ) ; and it represents a distrbance of finite amplitde moving in a gas for which the velocity of propagation of an infinitesimal distrbance is a; w is the velocity of the gas in the direction of the axis z It is shown that the parts of the waves in which the velocity of the gas is w travel forward with a velocity a + w and that there is in conseqence a tendency for the crests to catch p the troghs After a certain time and at a certain point in space the vale of w z will become negatively infinite; a discontinity will then occr and Poisson s integral will cease to apply Stokes then leaves the sbject of oscillatory waves and proceeds to consider whether it is possible to maintain a sharp discontinity in a gas which obeys Boyle s law ( p= a ρ ) His argment slightly modified by Lord Rayleigh is as follows: Sppose that a travelling discontinity can exist Give the whole gas sch a motion that the discontinity is broght to rest Consider then a gas which is moving with niform velocity p to a discontinity At this point the velocity sddenly changes to ; and the gas moves on niformly at this speed Let ρ and ρ be the corresponding densities p and p the corresponding pressres The eqation of continity of mass is ρ = () ρ The eqation of conservation of momentm is p p = a ρ a ρ = ρ ( ) () If and ρ be given these two eqations determine and ρ Against this theory however Lord Rayleigh eqation of energy raised the objection that the Phil Mag XXXIII (848); Collected Papers vol I Theory of Sond II p 4

2 = a logρ a log cannot in general be satisfied Simltaneosly with () and () In a recent note 3 he adds a remark that it is possible that energy might be lost at the discontinity bt it cannot be spposed that energy is gained Lord Rayleigh frther points ot that the energy lost mst be converted into heat and that this complication mst be taken in accont This has been done by C V Brton 4 and by Hgoniot 5 bt their eqations have the same disadvantage as those of Stokes in that they contain no indication that the motion represented by them is irreversible In the case considered by Stokes it is evident that the motion is irreversible; in fact it is only the front of a compression that can possibly travel nchanged For if for an instant the sharp discontinity were to disappear leaving a small transition layer in which the velocity might vary continosly from to then the back part of the layer wold travel forward relatively to the front part with a velocity Hence if exceeds any sch transition layer will become obliterated owing to the greater velocity behind and the discontinity will ths be maintained This is the case of the front of a wave of condensation If however the wave is a wave of rarefaction that is if is less than then the layer of transition will get wider and the sharp discontinity will not be re-established The object of this paper is to discss in detail what actally does occr at a discontinity and to determine in the general case of a gas whose characteristics are known whether a discontinity obtained by the method of Stokes is a physically possible featre It is evident that a plane of absoltely sharp or mathematical discontinity cannot occr in any real gas When owing to change of type there is a sdden compression or rarefaction of the material in crossing any bondary modified physical laws mst come into operation whose effect is to prevent abrpt discontinity from being formed Some cle to the natre of the processes involved in this case is afforded by the kinetic theory of gases; for when the change in velocity is very sdden the molecles which are moving faster will penetrate among those which are moving more slowly and an irreversible redistribtion of velocities will ense This sggests that heat condction and viscosity are in the case of a real gas the cases of the prodction of dissipative heat; it will be shown that nder certain conditions they are also sfficient to ρ 3 Proc Roy Soc A LXXXI (908) Phil Mag xxxv (893) 37 5 See Lamb s Hydrodynamics note on p 466 3rd edition

3 prodce permanence of type in the layer of transition Consider a continos distrbance of permanent type in a gas whose characteristic eqations are known Give the whole system sch a velocity that the distrbance is broght to rest; the motion is then steady Let A and B be two planes which move with the gas and let p ρ ' E and p ρ E be the pressre density velocity and internal energy of nit mass of the gas at A and B respectively Since θ the temperatre is a fnction of p and ρ and E is a known fnction of p ρ and θ therefore E may be regarded as a fnction of independent variables p and ρ The eqation of continity of matter is ρ ' ' = ρ= ω (3) The rate of gain of momentm between A and B is ω ( ' ) The eqation of momentm for the gas between A and B is therefore ( p+ X ) + ω = ( p' + X ') + ω' (4) where X and X are the viscos normal forces which act over the planes B and A respectively The work done on the gas between A and B in a small interval of time dt is ( p ' + X ') ' dt ( p+ X ) dt The increase of its kinetic energy in time dt is ω ( ' ) dt The increase in internal energy in the same time is ( E E' ) ωdt The amont of heat condcted away from the mass of gas between A and B in time dt is where ωξ and ω(' ξ ξ ')dt ωξ ' are the rates at which heat measred in mechanical nits is condcted across the planes B and A The eqation of energy for the gas between A and B is therefore ( p + X ) + ω + ω( E+ ξ) = ( p' + X ') ' + ω' + ω( E' + ξ ') (5) Since E is a fnction of p and ρ and ω = the state of the gas at any ρ time may be completely represented by a point in a plane diagram sch as fig ; pressre is represented by the ordinates and density by the abscissae If C and D are the points which represent the state of the gas at the two ends of the transition layer in which the velocity changes from to then the state of the gas along that layer are represented by the points on some crved line L joining C and D It will be possible by means of eqations (4) and (5) to

4 determine vales of X and ξ so that any given line joining C and D may represent the state of the gas in the transition layer; bt the motion so represented will not be thermodynamically possible nless the coefficients of condction and viscosity are both positive If x represents distance in the direction in which the gas is travelling these conditions become X and x mst have opposite signs ξ and θ mst have opposite signs x Hence at the front of a condensation X is positive and ξ is negative while for a rarefaction X is negative and ξ is positive Constrct crves M and N to represent the relations obtaining between p and ρ when X = 0 and when ξ = 0 respectively Let p m p and p n represent the pressres at points on M L and N corresponding to a particlar vale ρ of the density The eqation to M is obtained by dropping X from (4) and is p m + ω = p + ω where the sffix refers to distant points on one side of the discontinity Since (4) gives therefore ( p+ X ) + ω = p + ω p m = p+ X Similarly it can be shown that E n = E+ξ In a condensation therefore p m > p and E n < E and in a rarefaction p m < p and Also volme; and E n > E E ( θ θ ) n E= n Cv where v C is the specific heat at constant p n p mst be of the same sign as θ n θ ; hence in a

5 condensation p > m > p pn and in a rarefaction p m p< pn < Now the eqations to the lines M and N depend only on the relations which exist between pressre density temperatre and internal energy that is on the characteristic eqations of the gas and not at all on its viscosity or its condctivity; for if either X or ξ is small the other can be eliminated Hence if a discontinity is specified by the eqations ρ ρ = ω = p + ω + ω = p p + ω + ωe = p + ω + ωe connecting the two niform states between which it lies and it is desired to find ot whether it is thermodynamically possible draw the lines M and N joining C and D which are the points representing the states of the gas on the two sides of the discontinity If the line M lies above the line N (see fig ) so that greater than p m is p n then a condensation is possible If the line M lies below the line N a rarefaction is possible; If the line M cts the line N at any point between C and D neither is possible The only special case of any importance is that of a perfect gas whose characteristic eqations are p R = Rθ E E = ( θ θ ) ρ γ where γ is the ratio of the specific heats The general criterion is as above; bt if we also assme constant condctivity κ and viscosity µ the circmstances can be followed ot in detail In this case it may be shown that 6 and 4ω d X = and 3µ dθ ωξ = Jκ Sbstitting these vales in eqations (4) and (5) 4µ d p + ω= p + ω 3ω 4µ d dθ p + ω + ωe Jκ = p + ω + ωe 3ω From these two eqations together with the eqations p = Rθ ρ R ρ = ω E E ( ) = θ γ θ the qantities p ρ E and θ may be eliminated 6 See Rayleigh s Theory of Sond II p 35

6 The reslting eqation is ω 4µ d κj 4µ d d d ω B ω + + B + ( B ) = ωa γ ω R 3 3ω where ω B= p + ω ω A= p + ω + ωe It may be written in the simplified form where d y= y P dy d Q+ S + M 4µκJ P= 3ω R N+ A= 0 (6) κj 4µ Q = + Rω 3ω ( γ ) κbj S = Rω γ + M = ( γ ) Bγ N = γ If either κ = 0 or µ = 0 (6) may be solved in the form where and are the roots of x= C log( + D log( ) M N+ A= 0 If however neither µ nor κ vanish (6) cannot be solved in finite terms bt if be small compared with an approximate soltion can be obtained If be the greater of the two roots of M N+ A= 0 the soltion is Q S x= log (7) M ( ) By sbstitting for Q and S their vales and remembering that is small compared with it may be shown that κj 4µ Q S = + Rω γ 3ω ( γ ) which is positive Hence (7) represents a condensation; for when approaches x approaches and when approaches x approaches + From (7) it is possible to calclate approximately the thickness of the transition layer It is evident that the distance between the planes where the velocities are and is infinite; bt to obtain some idea of the extent of the transition layer consider the thickness T of the layer in which the velocity changes from to 9 + Sbstitting these vales for in (7) 0 0 Q S T = log 9 ( ) e M

7 and inserting the vales of γ ρ µ and κ for air 7 γ =4 3 ρ = µ = 9 0 6µ R κ = J ( γ ) there is obtained approximately Q S = 0 M ) T = ( In the case of waves of percssion it is known that the velocity differs appreciably from that of sond In that case wold be considerable and its reciprocal wold be small so that the motion wold closely approximate to an abrpt discontinos one In the case of ordinary sonds however the relative velocities of air in different parts of a wave are small so that T wold be large compared with a wavelength and nothing in the natre of a sharp discontinity wold ever be established 7 O E Meyer Kinetic Theory of Gases English edition p 9

4 Exact laminar boundary layer solutions

4 Exact laminar boundary layer solutions 4 Eact laminar bondary layer soltions 4.1 Bondary layer on a flat plate (Blasis 1908 In Sec. 3, we derived the bondary layer eqations for 2D incompressible flow of constant viscosity past a weakly crved