GENERALIZATEON GF NORMAL SHOCK THEOREMS TO MAGNETOGASDYNAMICS wsm RAmATmM ' Thesis for the Degree of Ph. D. MICHIGAN STATE UNIVERSITY RAM MOHAN SRWASTAVA ' 1958* _—~»-~_‘-_-§M—. H”-.—< .. mats 0-169 L] P" ' 4 R Y lVEiCl‘u‘ _ IS..a‘£t? Univ. .sity This is to certify that the thesis entitled Generalization of Normal Shoo}: Theorems to I.Ea310toga:sd3mamic-s with Radiation presented by Ram Liohnn Srivastava has been accepted towards fulfillment of the requirements for lug—degree inMQQhanignl Engineering /[/~fi <2 V fl/jfiww/ggn/Z; " Major profefj/ I 17 A HAW ¢ (/ a» F H1 0 g ‘f i ABSTRACT GENERALIZATION OF NORMAL SHOCK THEOREMS T0 MAGNETOGASDYNAMICS WITH RADIATION by Ram Mohan Srivastava The present work is primarily concerned with the general- ization of normal shock theorems by Courant and Friedrichs valid in isentropic, inviscid, non-heat conducting fluids to radiation- magnetogasdynamics. In the process, few generalized Rankine- Hugoniot relations and generalized Prandtl relation have been derived. Also, the generalized Hugoniot function has been de- fined, and the shape of the Hugoniot curve in the (p,v)-p1ane has been determined. In the main part of this work one-dimensional, uniform, and steady state flow of an electrically conducting, fully ionized and compressible gas under a planar magnetic field perpendicular to the velocity vector has been assumed. Only first approximations for radiation parameters, for an optically thick medium, have been considered. The shape of the Hugoniot curve, in the (p,v)-p1ane, has been found to be similar to the one in classical gasdynamics. The generalized Rankine-Hugoniot relations are in implicit form, hence, successive approximation technique has to be used to find the corresponding state behind the shock front. Theorems 1 and 3 refer to change in modified entropy across a shock front. Theorem 2 compares the pressure rise across a shock front to Ram Mohan Srivastava that in reversible adiabatic change. Finally, Theorem 4 refers to the flow velocities in front and behind the shock wave. GENERALIZATION OF NORMAL SHOCK.THEOREMS TO MAGNETOGASDYNAMICS WITH RADIATION By Ram Mohan Srivastava A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mechanical Engineering 1968 ACKNOWLEDGEMENTS The author is very grateful for guidance and encour- agement during the period of research by his Guidance Committee members, Professors Maria Z. v. Krzywoblocki, Charles R. St. Clair, Jr., George E. Mase and Norman L. Hills. Special thanks are due to Professor Krzywoblocki for serving as Chairman of this committee; and for his excellent guidance and constructive criticism throughout the course of this work. Grateful acknowledgment is made to the Department of Mechanical Engineering of Michigan State University for financial support during graduate study and research. Finally, the author would like to thank his family for their understanding and encouragement throughout his graduate program. ii TABLE OF CONTENTS ACKNOWLEDGEMENTS ................................ LIST OF APPENDICES ... ........................... NOMENCLATURE .... ..... .. ..... ..... ............... INTRODUCTION ..... . .............................. Chapter I. II. III. IV. THEORY OOOOOOOOOOOOOOOOO 0.0.0... OOOOOOOOOOOOOOOOO 1.1 Inviscid Isentropic Flow-Shock and Rankine- Hugoniot Relation (Continuum) ........ ...... Viscous Flow-Shocks in Continuous Media .... I-‘H DON Kinetic Theory Treatment of the Viscous Flow-Shocks ...... .. ....................... Shocks in Magnetogasdynamics ............... Shocks in Radiation-Magnetogasdynamics ..... HHv—I oxUt-P Fundamental Properties of the Shock Transition ...... ........ ................. 1.7 Table of Shock Layer Thickness ............. FUNDAMENTAL ASPECTS OF SHOCIG IN RADIATICN- MAGNETOGASDYNAMICS ... ........ . .......... .. ...... 2.1 Fundamental Systems of Equations .. ......... 2.2 Reduction of the System of Equations ....... FUNDAMENTAL EQUATIONS FOR NORMAL SHOCK .......... 3.1 Generalized Rankine-Hugoniot Relations for an Optically Thick Medium .......... ........ .2 Generalized Prandtl Relation ............... WW .3 Shock Relation in Terms of Change in Internal Energy ............................ 4 Modified Equations ...... ........ ........... .5 Auxiliary Inequalities ..................... 6 Table of Stepwise Shock Relations .......... WWW GENERALIZED HUGONIOT FUNCTION ................... 4.1 Generalized Hugoniot Function .............. 4.2 Hugoniot Curve ... ...... ....... ............. iii Page ii vi O\U'IJ-‘ WW \1 12 12 13 17 17 20 21 21 22 3O 31 31 31 VI. VII. VIII. THEOREM 1 ....................................... 5.1 The Present Formulation of Theorem 1 ....... 5.2 Theorem 1 in Radiation-Magnetogasdynamics 6.1 The Present Formulation of Theorem 2 ....... 6.2 Theorem 2 in Radiation-Magnetogasdynamics TIIEOREMSOOOOIOOOOOOOOOOO0.00.0.00... 0000000 0000 7.1 The Present Formulation of Theorem 3 ....... 7.2 Theorem 3 in Radiation-Magnetogasdynamics THEOREM 4 .......... . ........ . ................... 8.1 The Present Formulation of Theorem 4 ....... 8.2 Theorem 4 in Radiation-Magnetogasdynamics LIST OF REFERENCES .............................. iv 36 36 36 39 39 39 42 42 42 48 48 48 51 LIST OF APPENDICES Appendix Page A Fortran Program 54 B Effective Speed of Sound in Radiation-Magneto- gasdynamics 56 NOMENCLATURE effective velocity of sound in radiation-magneto- gasdynamics Stefan-Boltzmann constant magnetic flux density velocity of light specific heat at constant pressure specific heat at constant volume Rosseland diffusion coefficient of radiation specific internal energy = CvT total internal energy per unit mass = ch + 3vpr + vph electric field total radiation energy per unit volume = arT total internal energy per unit mass 1 2 =ch+§u +Er/p electromagnetic force magnetic field Hugoniot function electric current mechanical equivalent of heat electric current density coefficient of heat conduction constant = pu Mach number critical Mach number vi hydrostatic pressure magnetic praisure 1 2 9 Wt.“ ‘27 radiation pressure = P + P r total pressure = + p + 91. ph Prandtl number total heat flux Gas Constant Reynolds number radiation pressure number = pr/p magnetic pressure number = ph/p modified entropy time absolute temperature flow velocity specific volume space coordinate ratio of specific heats =c/c p v shock wave thickness dielectric constant joule heat #fi = E.J vii cons tant ==/E;yH mean free path Rosseland mean free path of radiation coefficient of viscosity magnetic permeability magnetic diffusivity density (mass per unit volume) excess electric charge electrical conductivity subscript 1 signifies the state in front of the shock wave subscript 2 signifies the state behind the shock wave viii INTRODUCTION In the magnetogasdynamics, the generalized Rankine-Hugoniot relations referring to a normal shock have been developed by sev- eral authors [4, 11, 13, 14]. In the radiation-magnetogasdynamics, the generalized Rankine-Hugoniot relation, as the ratio of velocities U (i.e. 52) only, has been developed by [15, 16]. The normal shock theoremi, referring to some characteristic properties of normal shock in classical gasdynamics, have been developed by [2, 6, 28] and condensed by [3]. In the present work, we derive generalized Rankine-Hugoniot and Prandtl relations, not covered in [15, 16], in radiation- magnetogasdynamics. Further, we have generalized the normal shock theorems of [3] with their proofs, to radiation-magnetogasdynamics. In Chapter I, some aspects of normal shocks have been dis- cussed based upon the existing literature and a table of shock layer thickness have been compiled. In Chapter II the fundamental equations of radiation-magnetogasdynamics have been collected and reduced for the case of one-dimensional steady state flow. Chapter III contains the derivation of the generalized Rankine-Hugoniot and Prandtl relations. Further, modified first law of thermo- dynamics and modified Rankine-Hugoniot relations have been in- troduced in Section (3.4). In addition, several auxiliary in- equalities have been derived in Section (3.5). In Chapter IV the generalized Hugoniot function has been defined, and the shape of the Hugoniot curve has been determined. The crucial part of the work, i.e., the generalizing the four normal shock theorems with their proofs, occupies Chapters V to VIII. I. BRIEF REVIEW OF THE PRESENT STATUS OF THE SHOCK THEORY 1.1 Inviscid Isentropic Flow-Shock and Rankine-Hugoniot Relation (Continuum): All the approaches in this chapter are phenomenological. The considerations referring to the shock phenomena should emphasize two kinds of assumptions which form the basis of any approach to the shock theory. The first group refers to the fundamental laws govern- ing the flow, i.e., three conservation laws and equation of state. We use a unique nomenclature for these laws, namely fundamental dynamic laws (f.d.1.). The second group refers to the fundamental assumptions governing the structure of the shock (f.s.1.). The Rankine-Hugoniot relation [3, 10, 20, 27] is derived with f.d.l. as standard laws of ideal, perfect gas. The f.s.l. assume that the shock is a step-wise transition, in all the variables, of zero thickness. 1.2 Viscous Flow Shocks in Continuous Media: Shock in real gases exhibit very steep but continuous transition from the state 1 to the state 2. As the shock wave becomes very steep, viscous stresses and heat conduction effects become appreciable, no matter how small be the coefficient of viscosity and the coefficient of thermal conductivity, and so a particle of fluid is subject to diabatic effects. The effects of viscosity and heat conduction tend to wipe out discontinuities in velocity and temperature. Therefore these effects control the thickness of a shock wave. Below, we discuss briefly the funda- mental assumptions in a few representative works on the subject. 3 In [21] a perfect gas, satisfying f.d.1. with two different uniform U, P,and T as initial and end boundary conditions, has been assumed. The f.s.1. assume special functions describing the velocity distribution inside the shock. The thickness of the shock is obtained from the entire formalism. In [23] the "Shock-Thickness Reynolds Number" is derived for air. The f.d.l. and f.s.1. assumptions are the same as in [21]. The viscosity H has been assumed to be proportional to T“. In [26] the f.d.l. assumptions are the same as in [21], the f.s.1. assume that the quotient u/k remains approximately constant with temperature variation. No other constraints are introduced. This allows the author to calculate only the upper and lower bounds of the thickness but not the actual thickness of the shock itself. In [9] the f.d.l. assumptions are the same as in [21], except the viscosity terms are retained in the momentum equation and ne- glected in the energy equation. The f.s.1. assume the inflection point (i.e. dZu/dx2 = O, at x = 0) inside the shock wave and this is sufficient for the existence of the transition region. The re- sults obtained are in close agreement with the exact solutions of [12, 19] for the structure of the shock wave and its thickness. 1.3 Kinetic Theory Treatment of the Viscous Flow Shocks: In [1] a perfect gas, whose specific heat is independent of temperature, satisfies f.d.l. assumptions as that of [21]. The f.s.1. assumptions are the same as in [9]. The author concludes. that the thickness of a moderately strong shock is of the order of mean free path and must be treated directly from the relevant Boltzmann equation. Whereas very strong shocks have thickness less than the mean free path and even the Boltzmann equation cannot be used. Hence, the actual reference in [l] to the kinetic theory is a recommendation that the kinetic theory equations should be used in the shock theory. In [25] it has been pointed out that if the increase of the coefficients of thermal conductivityand of viscosity with increasing temperature and pressure is taken into account, then the shock wave thickness for a perfect gas will never be less than the mean free path and hence the Boltzmann equation can be applied even for very strong shocks. The author takes the third approximation to the Boltzmann equation. The f.d.1. and f.s.1. assumptions are the same as in [1]. In [12] the conclusions of [25] has been modified for any gas whose u and k has been assumed to be proportional to Tn, where n is a positive constant depending only on the gas in question and for Pr = 3/4. In this paper f.d.1. assumptions are the same as that in [25] and f.s.1. assumptions are the same as in [1]. In [8] the author tries to improve the results of [26], by taking the third approximation to the Boltzmann equation for f.d.1. and keeping f.s.1. assumptions the same as in [26]. But the author finds that the bounds of the shock wave thickness are not affected by the higher order Burnett terms. 1.4 Shocks in Magnetogasdynamics: The governing relations are derived from the magnetogasdynamic equations describing the steady, one-dimensional flow of a viscous, heat conducting, electrically conducting, and compressible gas under a planar magnetic field perpendicular to the velocity vector (i.e. velocity and magnetic field vectors are in the same plane but perpen- dicular one to each other). If the magnetic field vector is parallel to the velocity vector, then it will not affect the gasdynamic equations. In [13] a perfect gas, satisfying f.d.1. and Maxwell's equations with two different uniform u, p, T, and H as initial and boundary conditions, has been assumed. The f.s.1. assume that there is a point of inflection, in the transition region, for all the variables. As a special case the generalized Rankine-Hugoniot relations have been derived. Moreover, the structures of the shock wave of a finite thickness have been considered for a few special cases. In [11] the f.d.1. and the f.s.1. assumptions are the same as in [13] along with Maxwell's equations. The author determines the shock profile and width of the transition region for a few Special cases. 1.5 Shocks in Radiation-Magnetogasdynamics: The governing relations are derived exactly in the same manner as that for shocks in Magnetogasdynamics, except that the radiation pressure is added to the gas pressure in the momentum and energy equations and the radiation field energy is added to the energy equation. In [16] the generalized Rankine-Hugoniot, as well as sev- eral limiting cases of Rankine-Hugoniot relations have been derived. The f.d.1. assumptions are the same as that of [13]. The f.s.1. assume that the shock is a step-wise transition, in all the variables, of zero thickness. 1.6 Fundamental Properties of the Shock Transition: The literature reviewed in Sections (1.1) to (1.5) refers to a some sort of the phenomenological theory of shocks. Namely, it is assumed that the shock exists and the mathematical formalism helps to answer on such questions of how small or large is the thickness of the shock, etc. But there are many other important questions concerning the shock wave thoery which are of a more fundamental nature. Let us only quote from [28] two such questions: (a) What are the conditions for the equation of state of a fluid under which shocks with their distinctive qualitative features may be produced; (b) the second question refers to the physical struc- ture of the shock layer whose "infinitesimal" width is of the order of magnitude 6 provided that heat conductivity and viscosity are small of the same order. Below, we present a brief review of the literature referring to this part of the shock theory. In [2] the author derives several important theorems con- cerning the behavior of shock waves based on the three assumptions for the equation of state. In [28] the author derives some of the conclusions of [2] using more rigorous mathematical methods but made seven assumptions concerning the equation of state. In [3] the authors condensed the physical assumptions of [28] and re- arranged those of [2]. The authors follow the method of [28] and prove four basic properties of the shock transition. In [6] the author proves the existence and uniqueness of the shock layer for the general class of fluids considered in [28], for arbitrary end states satisfying the shock relations, with k and u being arbitrary functions of the state. In [22] the transient and steady state behaviors of normal shock waves are examined. 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FUNDAMENTAL ASPECTS OF SHOCKS 1N RADIATION-MAGNETOHYDRODYNAMICS 2.1 Fundamental Systems of Equations: The systems below refer to a one-dimensional non-steady flow [7, 14, 15]. (a) Hydrodynamic System: Equation of State (perfect gas): p = DRT. (2. Equation of Continuity: g3 at Equation of Motion: 8P an EU I: a 4 an —— .—— = — ——— —— — —— . 2. p By 4 o and {3(v 1)Rp2 + 1} o, sz 2 0. (3.5.2) The case 3y - 4 = 0 when T approaches infinity. Therefore from (3.5.1)2 and (3.5.2), we get the following inequality: 4 4 ‘e 3 0’ or 3 'e, Q.E.D. (30503) 1. (ii) Ye S Y, everywhere. From (3.5.1)1 we can write: (Y- 1) (TY-4) sz Y - Ye = 3(Y-1)R + 1 (3.5.4) p2 Since (v-l) > O everywhere, hence, from (3.5.2) and (3.5.4) we get: Y - Ye 2 0, or 'Ye S Y, Q.E.D. (3.5.5) Now combining the two inequalities, (3.5.3)2 and (3.5.5)2, we get: 4 1— s s . 3 Ye y, Q.E.D. (3.5.6) Y-1 1 e X'1 . —S——S Lemma 2. 7 Ye+1 Y+1’ everywhere. Proof: The proof is divided into two parts. 1 Ye-1 2. (1) -7 s Ye+1, everywhere. From (3.5.3)2, we have: 4 4 4 4 4 7 _g _ _ .. _ g... 3 Ye’ or 3 Ye + 3 S 3 Ye + Ye’ or 3 (Ye+1) 3 Ye’ I I‘ll ' lllll Ill $11. 24 fl 1 s Ye-1 7 4 4 ._ + g _. - S - - ____ .——_— y -1 l .3 1 e or 7+(7-——Y+1)S—---—Y+l. (3.5.7) e e From (3.5.3)1, we have: y-EZO or (Y+1)-(£+1)20 e 3 ’ e 3 ’ 7 3 1 -—2 —- 2 o 00 or (Ye+l) 3 0, or 7 Ye+1 0 (3 5 8) Therefore, from (3.5.7) and (3.5.8)2, we get: 1 Ye-l — s 7 Ye+1 , Q.E.D. (3.5.9) y -1 ,. e y-l ———— S 2. (11) Ye+1 Y+1 , everywhere. From (3.5.5)2 we have: S S Ye Y, or Ye + 1 Y + 1, 1 1 l 1 ——-s—— s -— or Y+1 Ye+1’ or 0 [§e+1 Y+{]. (3.5.10) Again from (3.5.5)2, we have: s ' s Ye Y, or Ye +’YeY Y + YeY, or Y (Y+1) S Y(Y +1) or Ye S _X_ e e ’ ye+l y+l’ Y Y or e -J—Si-i,or ———e—__1_.+1 _ 1 SE, ye+l y+1 y+1 y+l ye+1 ye+1 ye+l y+1 y+l Y -1 e 1 1 X'1 — - or ell + l-z—T _—1| 5 1. (3.5.11) 25 Therefore, from (3.5.10)2 and (3.5.11) we get: Y -1 e S ve+1 x-_1 Y“, Q.E.D. (3.5.12) Now combining the two inequalities (3.5.9) and (3.5.12), we get: Y -1 1 y-l _ S —— 7 ye+1 Sy+1’ Q°E°D' V LemmaIQ: 7 everywhere. Proof: From (3.1.5) we get: u 2 > U 1 since all other terms in (3.1.5) are greater than zero. u from continuity equation, :2 1 3.212;}, +’ v1 Ye 1 we.1 1 From (3.5.13), Ye+1 7 r 2 Y-1 .11.. ye+1 v2 = -—3 and (3.5.14) we get: v 1 2__1>Ye‘1_1 v 7 Y +1 7' H m - —.2 0, hence from (3.5.15) we get: (3.5.13) The ratio of the specific volumes, Si, is less than (3.5.14) Then (3.5.15) 26 :2. V \HP‘ > O, or 7 --- > O. Q.E.D. (3.5.16) H d N * Lemma 4: dzp 1 > O at the point of state 1. * Proof: By definition, p = p + pr + ph, * a 4 2 or p = p +—r£(pv) +97, (3.5.17) 3R 2v * * Thus, p = g(v,p). where p = p(V.S )- * * Therefore, p = g(v,p(v,S )), * * or p = G(v,S ), (3.5.18) * where, v and S are assumed to be independent agruments, so that: * * * ‘k * dp = G vdv + G d8 5 p vdv + p S*dS , (3.5.19) 3* d d2 * * d + * d * a d * * 3 5 an p = (p vv V p,S*V S )dv + *( p )dS . ( . .20) 3 as At the point of state 1, we have from (5.2.4), (see below), * dS1 = 0, thus: dzp: = {pfvv(dv)2}|1. (3.5.21) Differentiating (3.5.17) twice with respect to v, we obtain: 27 2 4a ,._. 39 r 2 2 I1 {v4 + R4 (PV) (P + VP,V) }|1 * ,W a +{§-§3.<2p +vp >}l1+{p 1| R ,v ,vv ,vv 1. (3.5.22) Let p*W|1 - {1} + {II} + {111}. (3.5.23) (1) {I} > 0 everywhere, since each term of {I} is positive everywhere. (ii) From (4.2.5) and (4.2.10), we get: respectively, p1 ,v 1 2 V1(p,v|1) T] 2 3 P y 123l3%1.pr1 + 16.1%3fiiy. ‘1 3 + 12288x—El (Y-l) 1’1 (Y-l) p1 Y(Y+1)P1 p 12 p v2[ 1 + r 133 (3.5.24) 1 y-l p1 Therefore, 28 YP 12 p ...l _1_ r1 2 -2{(Y_1 + 16 pr1)-(.Y_1 + p1 ) 1 p2 Y(Y+1)P p3 + y.——Yl52'242.pr1 + 16.1%333Y.pr1 + 31 + 12288.-—;—1 (211,, + vp,W)I1 0, since all other functions on the right hand side of (3.5.25) are greater than zero everywhere. Therefore, {II} > 0 at the point of state 1, since all other quantities in {II} is positive everywhere. (iii) From (3.5.24), {III} > 0 at the point of state 1 for 1 < y < 2. 29 * Hence, from (3.5.23), p W|1 > o and then from (3.5.21), we get: * dzpl > o, Q.E.D. (3.5.26) 3O Ame + Hav . AN> - H>V m . + H u a .H $522332: 39. HHee+Heem+Nee+HeecH> Ne - Ha Ne - He 3 . ||I||||||.+ NHIIIIIII.+ H u a NSQIHSQ QIHHQ w HH +> e «HHWIH-+>H_AHI NZvMIAIHINvIWI+Hu\HJ1 MJ+MM+HQ leJflmB Iw - Hv z» + gm + H m + H a «a. N 1 H+%12>N H: Her: eHH+HzHIIHfln|~ HHIHVMHHIH +H.HH+He +HHHuNe w: H+> HHH H+ or H+m> H+o> H+m> +m> H .HTH...H-: :HIHH NH... + iii... .21... .33.. HNHH+ $5 H-> NH: .55 H-» . N NH A~>.H>V me u He-~m e+ a N H .A av dBflNDHD Na-Hm. r We Hate He mmlw+>ud AH- mic N411: non-Inn. H .AH+> - Mz>~v www_u mm B a H H H: . e e. N H+> HHH wluwl A|~+HL$JIumI 5 a 6 HH acoHumHmm xoosm mmHsmoum mo mHnme o.m IV. GENERALIZED HUGONIOT FUNCTION 4.1 Generalized Hugoniot Function: Transferring all quantities from the right hand side to the left hand side in the generalized Rankine-Hugoniot relation (3.3.1) and abandoning the subscript 2 in this equation, we obtain a function defined as the generalized Hugoniot function. This function is, with e = E!- - Y-l ' ”=PV _p1v1+_]_-_( - ) { + +:£.[( )4+( v)4] v-1 Y-l 2 V V1 ' P p1 3R4 P" P1 1 2 a 2 e 1 1 r 4 5 4 5 e _1_ __ 1 + 2 [v2 + 2]} + R4 {p v - p1v1}+ 2 {V VI}. (4.1.1) v1 The right side of (4.1.1) is a function of p and v only, the variables at the state 1 being fixed. Therefore the Hugoniot function, 1’ = N(v,p). We get the Hugoniot relation across a normal shock by substituting NKv5p) = O in (4.1.1), and p = p2, and VEVZ. 4.2 Hugoniot Curve: The graph of the Hugoniot relation, NYp,v) = 0, in the (p,v)-p1ane is called the Hugoniot curve. The general shape of this curve can be determined if the signs of first and second derivatives of p with respect to v are known. With dV'= 0 along this curve, we have: W=wvdv+3rpdp=o, (4.2.1) 31 d 3! thus, 35- '7’1' (4.2.2) From (4.1.1), we have: 1 2 1 1 11. 1 11 ”w {2 pr +2(p1+pr1)} +{2 pr(35 - 4v)} +{2(ph1-ph)-ph(1- V H, (4.2.3) fl' = l'v {r2 + 28 R - :1 (1 + 4 R )}. (4-2-4) .p 2 p v 13 Therefore: {r2+( + )}+{ (35-43)}+{< - >-2 (131)} 'dp = _ p pl pr1 pr v phl ph ph v dv 2 v1 vr +28 RP-7(1+4RP)} (4.2.5) Equation (4.2.5) gives the slope at a point of the Hugoniot curve in (p,v)-p1ane. Theorem 4.2: The derivative g5 is everywhere less than zero along the entire Hugoniot curve. Proof: The proof is divided into two parts. 1. (i) fl'v_> 0 along the entire Hugoniot curve. From (4.2.3), let: Arv={1} +{11}+{111}. (4.2.6) ’ (i.a) {I} > O everywhere, since r2 > O for Y > 1 and all the functions in {I} are positive everywhere. 1 4V1 1 V1 (1.1:) {11} = 3 pr[35 - -v—] = -2- pr[(35 - 28) + 4(7 - 7)]. 1 V1 or {11} = -2- pr[7 + 40 - 7)]. Therefore, {II} > 0 everywhere from (3.5.16). (1. c) {111} = {——2- - —2 - ——2- (1 - 7)}. 33 92(v1 - v)2(2v1 + v) or {III} = 4 V VI Therefore, {III} > O everywhere, since v and v1 are always positive. Thus, from (4.2.6), we get: 9.x >0, ,v along the entire Hugoniot curve. Q.E.D. 2. (ii) fl'p > 0 along the entire Hugoniot curve. From (4.2.4), let: Y +1 V Y +1 = l ‘ e - ..l 11.1 _ e 12p 2 v[ {(1+411p)(Ye_1 v )} +{v-1 + 28 RP Ye'l (1+4Rp)}], let fi’p a é-vEfI} +-{II}]. (4.2.7) (ii.a) {I} > 0 everywhere from (3.5.15)1. Ye+1 7(Y-1)Rp + Y+1 Ye”1 (Y'1)(Rp+1) (ii.b) From (3.1.7), (4.2.8) From (4.2.8), we have: 6Rp(3y-4) ‘ (Y-1)(Rp+1) {11} Therefore, {II} 2 O everywhere, since Y 21% from (3.5.6). Thus, from (4.2.7) we get: .3} > 0 along the entire Hugoniot curve. Q.E.D. Since .&:v > O and .fl; > O everywhere along the Hugoniot curve, hence, from (4.2.2) %$‘< 0 along the entire Hugoniot curve. Q.E.D. With dév 8 O and dp = 35 dv along the Hugoniot curve, we can write from (4.2.1) after differentiation: 34 2 = fig 2 d 2 <1sz (awe. pvj< —5-+) Nppq‘) +%,p<—Edv2>1 Since dv # 0, therefore: 2 92 2 $12 = ”NH pv—jfl >+ ”ppqv) +:’P(dv2> o, 2 v 15‘?” gm or La .. at”? :W , 2 ‘ fi'v dv ,p . QR _ Substituting the value of dv from (4.2.2), we get. 2 21! A! M -A/ - L12 . JV .1: .v .vv(311a)2fi(,1>1>wv)2 dv2 0V p)3 Differentiating (4.2.3) with respect to v, we get: 2pr 3ph vav = :2- (35v - 3v1) +-;§- (v - v1). Differentiating (4.2.4) with respect to v and p, we get: r2 2pr fihpv = E— +-E;— (35v - 4V1). 691, fl' = ——— 7v - v .pp p2 (1) (4.2.9) (4.2. (4.2. (4.2. (4.2. (4.2. Equation (4.2.11) gives the second derivative of p with respect to v at a point of the Hugoniot curve in (p,v)-plane. second derivative is positive everywhere along the Hugoniot curve. To verify this statement the positive real roots, of (4.1.1) with fiKp,v) = 0, are determined by programming this Hugoniot relation on CDC 3600 computer [see Appendix A].Then corresponding to these roots the values of ‘g-g, are calculated by programming (4.2.11) dv 10) 11) 12) 13) 14) 35 2 on CDC 3600 computer which gives 2—5- always positive for each dv and every root of the Hugoniot relation. Hence, it can be inferred that 2 IS p031tive everywhere along the Hugoniot curve. dv 2 Since g5~< 0 and ‘9—5 > 0 along the entire Hugoniot dv curve, hence, the shape of the Hugoniot curve is convex downwards in (p,v)-plane. I‘.il llllll'llI-lll llr‘llll" I V. THEOREM 1 5.1 The Present Formulation of Theorem 1: Courant and Friedrichs [3] have proven a series of theorems referring to the mathematical formulation of the description of a normal shock in an isentropic flow. We quote below the first of these theorems (denoted by letters C.F.): Theorem 1 [C.F.|: "The increase of entropy across a [normal] shock front is of the third order in the shock strength." Here the shock strength refers to any of the differences p2 - p1, I. .. V'V 2 1. 2 1| 5.2 Theorem 1 in Radiation-Magnetogasdynamics: We generalized the theorem and proof by [C.F.] to the case of radiation-magnetogasdynamics. Theorem 1: "The increase of modified entropy across a normal shock front in radiation-magnetogasdynamics is of the third order in the Shock strength." ,Egggf: The proof is a straightforward one. Along the Hugoniot curve 6V = 0, (4.2.1), hence from (3.4.3), we get by differentiation: * * * * 25V = 2de + (p + p1)dv + (v - v1)dp = 0. (5.2.1) From the modified first law of thermodynamics (3.4.1), we get: * * * de = TdS - p dv. (5.2.2) * Substituting the value of de from (5.2.2) into (5.2.1), we get: 36 {liliIIIIIIIIII 'lllll 37 * * * * 2TdS + (p1 - p )dv + (v - v1)dp = 0. (5.2.3) At the point of state 1, we have: * * p =p1,V=V1, and thus from (5.2.3) with T1# 0, we get: * dS1 = 0. (5.2.4) Differentiating (5.2.3) again along the Hugoniot curve and con- sidering v as the independent variable, we get: * * 2d(TdS ) + (v - v1)d2p = 0. (5.2.5) Therefore at the point of state 1, we have from (5.2.5): * 2 * * d(TdS )I1 = 0, or (Td s + de3 )I1 = 0, * But dsl = 0 from (5.2.4), hence: 2* d 31 = 0. (5.2.6) Differentiating (5.2.5) again along the Hugoniot curve and con- sidering v as the independent variable, we get: * * * 2d2(TdS ) + dv dzp + (v - v1)d3p = 0. (5.2.7) Therefore at the point of state 1, we have from (5.2.7): 2 * 2 * 2d (TdS )I1 = -(dv d p )l1’ 2 * * * 2 * or 2(Td38 + d TdS + 2dT dZS )I1 = -(dv d p )|1’ li’l‘l’ll'i'l‘fll‘l‘ull" l .lll'li'l‘l 38 * * But dS1 = 0 and dZS1 = 0 from (5.2.4) and (5.2.6) respectively, hence: 3 * 2 * d(Td s )|1 = -(dv d p )|1. (5.2.8) Therefore from (5.2.8), we get for T1 > 0: 'k 833 > 0, (5.2.9) 1 * when dv1 < 0, since de1 > 0 from (3.5.26). Hence from (5.2.9) the increase of the modified entropy is exactly of the third order in the shock strength. Q.E.D. I!‘[lll‘l"llll llll“: ill VI. THEOREM 2 6.1 The Present Formulation of Theorem 2: [C.F.] have proven the following theorem 2 in an isentropic flow: Theorem 2 [C.F.|: "The pressure rise across a {normal] shock front agrees with the pressure rise in the adiabatic [reversible] change up to terms of the second order in the shock strength." Also, [C.F.] had shown that geometrically the Hugoniot curve and the adiabatic [reversible] curve, passing through the point of state 1, have a contact of second order at this point. It is assumed here that the initial state and one quantity (say specific volume) in the final state, i.e. at the end of the shock process, are the same for both, the isentropic flow and Hugoniot curve (process). This assumption implies that the both curves, isentropic and Hugoniot, pass through the point 1, but do not meet at the final state . 6.2 Theorem 2 in Radiation-Magnetogasdynamics: We generalize the theorem and proof by [C.F.] to the case of radiation-magnetogasdynamics. Theorem 2: "The pressure rise across a normal shock front in radiation-magnetogasdynamics is not equal to the pressure rise in the reversible adiabatic change." Geometrically, the Hugoniot curve and the reversible adiabatic curve, passing through the point of state 1, intersect 39 40 each other at this point. Additionally, the slope of the adiabatic curve at l is steeper than the slope of the Hugoniot curve at the same point. Egggf: The proof is divided into two parts. 1. The Hugoniot curve intersects the adiabatic curve at the point of state 1. We obtain the slope of the Hugoniot curve at the point of state 1, from (4.2.5) after substituting p = p1, v = v1, pr = Prl’ 2 p1(r +1) + 32 pr1 .%2| 1 = 2 H , 9 v v1{(r -1) + 24 Rpl} sz + 16(Y 1)p p or 92 1 1 ‘1 (6.2.1) dv H1 vlfp1 + 12(Y-1)pr1} ' The equation of the reversible adiabatic curve, passing through the point of state 1, is: PVY = PIVIY. (6.2.2) We obtain the slope of the adiabatic curve at the point of state 1 by differentiating (6.2.2) with respect to v and then sub- stituting p = p1 and v = v1. The result is: YP £2 H-__1 dv|A1 v1 . (6.2.3) Since the slope of the Hugoniot curve (6.2.1) is not equal to the slope of the adiabatic curve (6.2.3) and the point of state 1 is 41 common in both curves, hence the Hugoniot curve intersects the adiabatic curve at the point of state 1. Thus the pressure rise across a normal shock front, in radiation-magnetogasdynamics, is not equal to the pressure rise in the reversible adiabatic change at all. Q.E.D. 2. The slope of the reversible adiabatic curve is steeper than the slope of the Hugoniot curve at the point of state 1. i.e., £2, -.22 > dv|Hl dv A1 0' (6'2'4) From (6.2.1) and (6.2.3), we get: 2 dv H1 dv Al vllp1 + 12(Y-1)pr1} v1 or ' 3 = dv H1 dv Al vlip1 + 12(Y-1)pr1} N 3'. (6.2.5) 2. (i) D > 0 for »Y > 1. . . . 4 2. (11) N > 0, Since .Y >'§ from (3.5.2). Thus, from (6.2.5), we get: 92 ..92 > .2. dvlH1 dv A1 0. Q.E.D. (6 6) VII. THEOREM 3 7.1 The Present Formulation of Theorem 3: [C.F.] have proven the following theorem 3 in an isentropic flow: Theorem 2 [C.F.|: "Along the whole Hugoniot curve the entropy increases with decreasing specific volume." 7.2 Theorem 3 in RadiationrMagnetogasdynamics: We generalize the theorem and proof by [C.F.] to the case of radiation-magnetogasdynamics. * Theorem 3: "Along the whole hugoniot curve the entropy, S , increases with decreasing specific volume in radiation-magnetogasdynamics." Proof: The generalized equation of state can be written as: * * * P = P (S .V). (7.2.1) * * * Equation (7.2.1) implies that, S = S (p ,v). Therefore: * * * * S = S (p (S ,v),v). (7.2.2) * Assuming S and v as independent arguments and differentiating (7.2.1) and (7.2.2), we obtain: * dp * 'k * d + dS .2.3 p’v v p’s* , (7 ) * * * * d8 = S dp + S dv. (7.2.4) P* ,v 3 * Substituting the value of dp from (7.2.3) into (7.2.4), we get: 42 'llul.ll‘lll.fl I! I‘ll-‘1‘. 43 'k ‘k * d5 = S,p*p,S* * * * dS + (S ’p*p * ‘v'+ S,v)dvu (7.2.5) * Since S and v are assumed to be independent arguments, there- fore from (7.2.5) we get: * * S.p*p.s* =1, * * * + S,p*p.v ,v From (3.4.1) and (3.4.3) with fl'2 0, we obtain: S = 0. NS" = fi- d {(vl-v).(p"{+p*>} + p*dv. vl-v * p * ordS=2po+ (7.2.6) (7.2.7) (7.2.8) * * Since v1 > v and p > p1, which can be obtained from (3.1.10), everywhere along the Hugoniot curve; thus along this curve, we get by comparing (7.2.4) and (7.2.8): 5* * p - -'—L!-H< C 3v 8* .P* 2 4a 2 2 p p = ——— +-—Z£(pv) [(p + vp ) + 3v(2p +1vp R ,v ,v 3 (7.2.9) (7.2.10) (7.2.11) W)] + p,vv. (7.2.12) F‘I’lvful“lllll|( Ill-Ill!“ 1[.llllll 1| Ill...lll“‘l '11 44 Equation (7.2.12) is put on the CDC 3600 computer [see Appendix A] * and the values of p vv are evaluated at the number of points 3 * along the Hugoniot curve. The result gives p vv always positive 3 for each and every calculated points of the Hugoniot curve, thus: * pW>o, (7.2.13) Differentiating (7.2.11) with respect to v, we get: 5* 8* 3* 3* 8* 8* 3* 3* * W P*- v p*v R'kv 2*- v p*p* * z _ 3 3 3 3 3 3 4 3 dp,v E * 2 + { * 2 }p,v]. (7.2.14) S.p* .p* * Substituting the value of p v from (7.2.11) into (7.2.14), we obtain: * * 2 * * * * *2 * [S1vvs1p*'zs.p*vs,p*s,v+s,p*p*s.v 2 5 p‘,\’8 - *3 J. (70 .1) ’ s .P* * * But p vv > 0 and ~S 3 3 p* > 0 from (7-2-13) and (7.2.9)1 respectively, thus from (7.2.15), we get: 3* 3*2 23* 3* 3* +s* 8*2<0 (72 16) ,vv .9* ,p*v .p* .v .p*p* .v ' ' ° The increasing character of the entropy, 8*, along the Hugoniot curve, is proven below by showing that dS* > 0, along this curve except at the point of state 1 where dS*|1 = 0. The Hugoniot curve in (PTV)-plane,.flprv) = 0, is also convex downwards from (7.2.11) and (7.2.13). 45 * Let a ray E3 in the (p,v)-plane, is represented in the parametric form as: '0 ll * p1 + at; v = v1 + * * where: a = p2 - p1, b = v2 - v1. Hence, * dp = adt; dv = bdt. Therefore, from (7.2.8), we get: * * * * 2T dS + (p1 - p )dv + (v - v1)dp bt, v Fig. 1. (7.2.17) (7.2.18) ZCW. (7.2.19) * * Substituting the value of p,v, dp , and dv, from (7.2.17) and (7.2.18), into (7.2.19), we get: * T d8 = CW, along the ray P. (7.2.20) * Considering both S and .Q' as functions of t along E2 * therefore if either S (t) or .HKt) is stationary (i.e. their particular characteristic properties, like the maximum value of * S (t) or .flYt) do not change their location, are invariant) then V 46 other is also stationary which is seen from (7.2.20). Since the Hugoniot curve, fiYp*,v) = 0, is convex at 1, hence ray E’ cannot coincide with this Hugoniot curve. Since flXt)1 = NYt)2 = 0 (both are lying on the Hugoniot curve defined as fl'= 0), implies that NYt) possesses at least one extremum in between. There- fore at the point of extremum the entropy, 8*(t), is likewise stationary, or: * * * * d8 = S *dp + S vdv. (7.2.21) 3 3 * Now substituting the value of dp and dv from (7.2.18) into (7.2.21), and after dividing the resulting equation by dt we get: * dS * * -- = = . 7.2.22 dt Iextr. [S,p*a + S,vb]|extr. 0 ( ) * * Therefore, S /S = - a/b. Next: .v .p* * * * d * d(fi— g .9... (ES—H)“, +1. (Ii—)dv, dt * dt v 81» a 2 * * 2 * * 2 or d S = S * *a + 2S * ab + S b , dtZ ,p p .p v ,vv b2 (8* 8*2 28* * 8* + 8* 3* 2 - 3* 2 :P*P* .v - .p*vs.p* ,v .vv .p*)' * ’9 (7.2.23) But the expression, inside the parenthesis, on the right hand side 2 b of (7.2.23) is always less than zero from (7.2.16) and *2 > 0, 8.9* therefore, 2* d S 1< 0. (7.2.24) 47 Thus 8* and consequently' N’ possess one and only one single stationary point on E’ between the point of states 1 and 2. From the fact that 8* has just one maxima between the states 1 and 2, we infer the inequalities: * dS > dt 1 0, (7.2.25) * 1s. <. .. dt 2 0 (7 2 26) The inequality (7.2.26) excludes the possibility of the magnitude * of S being stationary along the Hugoniot curve at the point of state 2, otherwise ray E’ would be tangent to the Hugoniot curve at such a point. Thus, dV = 0 at this point would therefore imply = 0, in contradiction to (7.2.26). Thus, it has been * proven that the entropy, S , increases along the Hugoniot curve with decreasing specific volume in radiation-magnetogasdynamics. Q.E.D. VIII. THEOREM 4 8.1 The Present Formulation of Theorem 4: [C.F.] have proven the following theorem 4 in an isentropic flow: Theorem 4 [C.F.|: "The flow velocity relative to the shock front is supersonic at the front side, subsonic at the back side." 8.2 Theorem 4 in RadiationrMagnetogasdynamics: We generalize the theorem and proof by [C.F.] to the case of radiation-magnetogasdynamics. Theorem 4: "The flow velocity relative to the shock front is greater than the effective speed of sound at the front side, and is less than the effective Speed of sound at the back side." Proof: It can be demonstrated [see Appendix B] that infinitesimal pressure disturbance propagates with the effective speed of sound, ae, in radiation-magnetogasdynamics: -a2 h * p’p - e3 t US p,V ' 9 ae and from (7.2.11), we get: * S * _:_*V H - p = 92,2, (8.2.1) S ,v e .9* From (7.2.18), we get: (12* * * d = _ —-v-= - .232 dt p2 p1’ dt v2 v1’ (8 ) and from (7.2.4), we obtain after dividing this equation by dt: 48 Ill'll‘f} [Ills-[11" 49 * * d8 = * dp * d! dt S,p* dt + S,v dt ' (8'2 Hence at the point of state 1, we get from (8.2.3) using (7.2.25) and (7.2.9)1: d * s d ..2. +._L!_ ..X.) . _2 dt 8* 1 dt 0 (8 .p* d * d After eliminating 35—1 and 3% from (8.2.4) using (8.2.2), we get: * * * S - +--¢!— (v - v ) > o (8 2 (P2 P1) * 1 2 1 ' ' ' * .p* S v Substituting the value of -fi-— from (8.2.1) into (8.2.5), we S p* get: * * 2 2 _ - > . . (p2 p1) + plae1 (v2 v1) 0. (8 2 Similarly we get, at the point of state 2 from (7.2.26): * * 2 2 (p2 ’ p1) + pzaez (V2 Since v21< v1 in a normal shock, then v1 - v2 > 0; therefore from (8.2.6), we get: * * P ' P ____2 1 > 6:321, (8.2 v1 v2 e and from (8.2.7), we get: * * P ' P 2 1 2 2 . 8.2. v - v < p ae2 ( - v1) <0. (8.2. .3) .4) 5) 6) 7) .8) 9) 50 Using (3.1.l)2, we get from (3.2.1): **** zp-pp-p m = v1 _ v2 - v2 _ v1 , (8.2.10) 2112 where: m = plu1 a pzuz. Thus, from (8.2.10) and (8.2.8), we get: 2 2 > Also from (8.2.10) and (8.2.9), we get: 2 2 u2 < ae2° (8.2.12) Hence, the flow velocity relative to the shock front is greater than the effective speed of sound at the point of state 1 from (8.2.11), and is less than the effective Speed of sound at the point of state 2 from (8.2.12). Q.E.D. 10. 11. 12. LIST OF REFERENCES Becker, R. Stosswelle und Detonation. Z. Physik, Vol. 8, 1922, pp. 321-362 [Translation: NACA Tech. Mem., No. 505, 1929.]. Bethe, H.A. The Theory of Shock Waves for an Arbitrary Equation of State. Div. B, NDRC, OSRD Report No. 545, 1942. Courant, R., and Friedrichs, K.0. Supersonic Flow and Shock Waves. Interscience Publishers, Inc., New York, 1956. de Hoffman, F., and Teller, E. Magneto-Hydrodynamic Shocks. Phys. Rev., Vol. 80, No. 4, 1950, pp. 692-703. Durand, W.F., Editor-in-Chief. Aerodynamic Theory, Vol. 3; Div. H, The Mechanics of Compressible Fluids, by 0.1. Taylor and J.W. Maccoll, pp. 209-250. Dover Publications, Inc., New York, 1963. Gilberg, D. The Existence and Limit Behavior of the One- Dimensional Shock Layer. American Jour. of Math., Vol. 73, 1951, pp. 256-274. Irvine, T.F., Jr., and Hartnett, J.P., Editors. Advances in Heat Transfer, Vol. 1, 1964; Vol. 3, 1966. Academic Press Inc., New York. - Krzywoblocki, M.Z.v. On the Bounds of the Thickness of a Steady Shock Wave. App. Sci. Res., Vol. 6, Sec. A, 1956, pp. 1-14. Lieber, P., Romano, F., and Lew, H. Approximate Solutions for Shock Waves in a Steady, One-Dimensional, Viscous and Compressible Gas. Jour. Aero. Sci., Vol. 18, No. 1, 1951, pp. 55-60. Liepmann, H.W., and Roshko, A. Elements of Gasdynamics. John Wiley & Sons, Inc., New York, 1957. Marshal, W. The Structure of Magneto-Hydrodynamic Shock Waves. Pro. Roy. Soc. London, Series A, Vol. 233, 1956, pp. 367-376. Morduchow, M., and Libby, P.A. On a Complete Solution of the One-Dimensional Flow Equations of a Viscous, Heat- Conducting, Compressible Gas. Jour. Aero. Sci., Vol. 16, No. 11, 1949, pp. 674-684. 51 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 52 Pai, 8.1. On Exact Solutions of One Dimensional Flow Equations of Magneto-Gasdynamics. Proc. IX Inter- national Congress of App. Mech., 1956, pp. 17-25. Pai, S.I. Magnetogasdynamics and Plasma Dynamics. Prentice-Hall, Inc., Englewood Cliffs, N.J., 1963. Pai, S.I. Radiation Gas Dynamics. Springer-Verlag New York Inc., 1966. Pai, 8.1., and Speth, A.I. Shock Waves in Radiation-Magneto- Gas Dynamics. Phys. of Fluids, Vol. 4, No. 10, 1961, pp. 1232-1237. Patterson, G.N. Molecular Flow of Gases. John Wiley & Sons, Inc., New York, 1956. Pucket, A.E., and Stewart, H.J. The Thickness of a Shock Wave in Air. Quart. App. Math., Vol. 7, No. 4, 1950, pp. 457-463. Reissner, H.J., and Meyerhoff, L. A Contribution to the Exact Solutions of the Problem of a One-Dimensional Shock Wave in a Viscous, Heat Conducting Fluid. PIBAL Report No. 138, 1948. Shapiro, A.H. The Dynamics and Thermodynamics of Compressible Fluid Flow, Vol. 1. The Ronald Press Co., New York, 1953. Shapiro, A.H., and Kline, S.J. 0n the Thickness of Normal Shock Waves in a Perfect Gas. Trans. ASME, Vol. 76, 1954, pp. 185-192. Shapiro, A.H., and Kline, S.J. 0n the Normal Shock Wave in any Single Phase Fluid Substance. Heat Transfer and Fluid Mech. Inst. 1953, Stanford Univ. Press, pp. 193-210. Shapiro, A.H., and Kline, S.J. 0n the Thickness of Normal Shock Waves in Air. VIII Intern. Cong. App. Mech., Istanbul, Turkey, 1952. Shercliff, J.A. A Textbook of Magnetohydrodynamics. Pergamon Press, New York, 1965. Thomas, L.H. Note on Becker's Theory of the Shock Front. The Jour. of Chemical Phys., Vol. 12, No. 11, 1944, pp. 449-453. 26. 27. 28. 53 von Mises, R. 0n the Thickness of a Steady Shock Wave. Jour. Aero. Sci., Vol. 17, No. 9, 1950, pp. 551-554. von Mises, R. Mathematical Theory of Compressible Fluid Flow. Academic Press Inc., New York, 1958. Weyl, H. Shock Waves in Arbitrary Fluids. Communs. on Pure and App. Math., Vol. 2, 1949, pp. 103-122. APPENDIX A Fortran Program The following fortran program for H-CURVE, help us to . (122 * verify the statements that 2 and p vv are both greater dv ’ than zero along the entire Hugoniot curve. In this program SUBROUTINE POLYRT, supplied by the MSU Computer Center, has been used to find the roots of (4.1.1), with .RXp,v) = 0. The variables used in the program are: AR = at, GMA = Y, R = Gas Constant, VI = v1, V = v, P1 = p1, THETA = 9, PR1 = prl’ PHl - phl’ P = p, PRnpr,PH-ph,.HV-N’V,HP-H’p,HVV=Y , ,vv HPV 3 N pv’ I-IPP -= V pp’ DPV = dp/dv, DSPV = dzp/dvz, ’ 3 * PSTVV = p vv' 9 54 55 sz Am.ommq.mav Hezmoa oom ANH.HuH.Aav>>emm.AHv>mmn.Ava.AHV>.HV.oom Hszm A\\s>>ammsxaaa>mmnsxea«msxo~*>axqasa.xs\\v Hazmom 0mm 0mm Hszm mDZHHZOU on Aav>mma+AAHv>mmoefiHv>+>ma1Nv*m«sAAHv>aAHVmv*q**m\m¢*M\e+N«*A>aoH aAHv>+AHVmv¥¢x%M\N«¥AAHv>sAHVmvsm\Nss>Hmm mm\ANsx>maemmm+>mna>mma~+>>mv-uAHV>mmn mm\>m-u>mn AH>uAHv>xmvx~kkAva\mmsoummm Aa>se-AHv>smmvaAAHv>sfiHvmv\mma~+m\ommu>mm AH>-AHV>V«N«*AHV>\mm*m+AH>*m-AHV>emmvswaxAHv>\ma«Nu>>z AAAHVm\mmae+HV*AHV>\H>-Ava\mmewN+ommvem\AHv>umm AAHV>H \H>-Hv«mm-~\Amm-ammv+m\AAHV>\H>*e-mmV«mm+N\Aamm+amv+m\ommsAHVau>m AmaaAHv>*Nv\N«s*AHVmV*maxm.NHs u stmv Hazmom Aq.anw .Amva.Anvmmv .AHV> .H .mmm HZHMm 35.1.3: Awaeama.Hm.mm.mmnmoz.Hm.mmv Hawaom aaau ouAmvaquvauAmvHaquvauAHva eeefiav>aAAHv>+o\AH>-AHV>Vv*e*«m\m¢uflmvmm oufievmmuAmem * ~\AH>-AHV>*ommVuANme AAH>\H-AHV>\HV+~\AH>-Aav>vafim«sa>\aa +NesfiHv>\va*~\~***a-AHV>Vsamm+~\AH>*omm-AHV>V«HmuAvam .eH\AH-mHV*H>uAHV> NH.HuH on on Amaea>emv\ma«*Hmvsm.m.§o.m< .02 2mm AooHv>>emmH .AooHv>mma.AooHva.Aooavmm.AooHva.Aoovam.AooHv>.Aoova onmzman m>mao m zamoomm mNN OOH APPENDIX B Effective Speed of Sound in Radiation-Magnetogasdynamics Considering the speed of sound, ae, in radiation-magneto- gasdynamics, it is derived by means of the small perturbation theory [27], in which the second and higher order terms of small quantities are negligible in comparison to the first-order terms. For the simplicity sake we restrict our presentation to one dimension and time. Consider a small perturbation of the state of rest, caused by an initial disturbance: to each (x,t) there will correspond small values of u, p - p0, etc. Hence, the equation of continuity (2.1.2) is reduced to: 32., ..52 - PO ax ac’ (B 1) and the equation of motion (2.1.3), using (2.2.4), is reduced to: 32- p0 at ap iloax ’ (3‘2) Now differentiating (B-l) with respect to t and (B-2)2 with respect to x, we get: respectively, 2 2 a u Q 9 p = ' a (3'3) 0 atax at2 §___.= _ 22. .:_E. _ 90 atax IO (B 4) Eliminating the value of po gzg; , using (B-3), from (B-4), we get: 56 57 2 * 2 §2=62 82 _5 2 an o 2 ° (B ) at 5x The one dimensional wave equation for p, with the small dis- turbance propagation velocity, ae, can be written as: 2 2 fi-% = a2 h_% . (B-6) at e ax Now comparing (B-5) and (B-6), we obtain the effective speed of sound, ae, in radiation-magnetogasdynamics as: * 2 ae - BAP—‘0' (B-7) p 970 mmmm RHII mlll L“ Y” T“ 8" RI" EH Vl N“ U" H H I 3 1293 03177 I'IHIIWIIH