In this technical note we consider the influence of a transverse magnetic
field on the formation of a shocks wave in an electrically conducting field. We conclude
that the presence of a transverse magnetic field is Conducive to the growth of compression
waves and the decay of the expansion waves

Chopra, K P

Singh, I J

English

IACS Institutional Repository

23GROWTH OF HYDROMAGNETIC SHOCK WAVESI. J. SINQHOil. ANI> Natural Gas CommisB on, DEHRAmm, India ANBK. P. CHOjPRAE n GTNKRRING C e NTRR, U N IV E B SIT y OF SoUTHETm  C a TJFORKTALor Anofxes, CALiFOUtNiA, IT.S.A.*(Received^  Septemh  ^ 21, 1960)JABSTRACiT. In this technical note wo consider the influence of a transverse magnetic field on the formation of a shocks wave in an oleotrically conducting field. We conclude that the presonoo of a transverse magnetic field is oonducivo to tlio growth of compwjssion waves and tho dticay of the expansion waves.It ia well known that the ordinary hydrodynamic compression shock wave involves an increase in entropy and that the rarefaction shock wave decays immediately into a continuous expansion wave. This is so liccaiise in a compres­sion wave, the waves nearer tlio source tend to overtake those further from it with the result that the wave profile becomes more and more steep until the pressure gradients liecomo infinite. In this way a compression shock is formed whudi grows in strength as the process continues. In a rarefaction shock, on the other hand, the waves nearer tlie shock lag more and more behind those in front of it, tho wave profile flattens till the pressure gradients vanish and ultimately no dis continuity effects are observed. In fad; if a rarefludion sluHjk is established, even momentarily, it would decay immediately into a coniinuous expansion wave. We will consider, in this note, the influence of a transverse magnetic field on the formation of a shock wave in an electrically (iondiicting fluid. It will be seen that the presence of a transverse magnetic field is condiiciv(^ to the formation of a shock wave.For the sake of simplicity we will consider a plaixe one-dimensional shock wave propagating in a fluid of infinite electrical conductivity and specific volume r with an external magnetic fleld II oriented in a direction normal to the diretjtion in which the shock propagates. Wo define the quantities y  and r/ according to Hoffmann et aL, 1950.P and ALF) (1)♦Presont Addross: Aerodynamics Laboratory, Polytechnic Institute of Brooklyn Fr^port, N. Y*, XT. S. A.199200 1. J. Singh and K . P. ChopraThese quantities take account of the contributions o f the hydromagnetic interac­tion to the pressure p  and the velocity o f sound c through the magnetic pressure (H I^Sn) and the Alfven speed =  H{rl4n) ,^ We will call p"" and o’* the total pressure and the modified velocity of sound respectively.Let us now suppose that the properties o f the fluid at two adjacent points differ in magnitude by dr, dH, dv, dp'" and do'" where v denotes the gas speed. We also assume that the respective parts of the wave passing through these points differ in speed of propagation by For the sake of simplicity, let us further assume that the gradients o f temperature and velocity are small so that the dis­sipative effects of viscosity and heat conduction are negligible. Therefore each elementary part of the wave travels with the local speed of sound with respect to the fluid. The velocity of propagation of this part of the wave wdth reference to a fixed coordinate system is=  V f  r/ ... (2)and the velocity o f propagation o f an adjacent part o f the wave isdc"" ... (3)so that_  dv , dc^  "dp^  dp  ^ dp^ ... (4)Let us assume that the entire fluid was initally at rest with uniform pressure and temperature, and that each particle of the fluid undergoes isentropic changes. Therefore, the increments in pressure and density between adjacent particles obey the relationJr... (5)which yields on differentiation20“ d^dp= -  iL  i  / t* i?- \dp^  dr \ dr I ... (6)Again it can be shown from the equations o f constant ma»ss flux and constant momentum flux thatdvdp’'rc" ... (7)Growth of Hydromagnetic Shock Waves 201On carrying out the substitutions from (6) and (7) in (4) we finally obtaindp"" 2c" {djJ '^IdT) ... (S)N ow if Idp"") is positive, the liigh prcsstire parts of the wave overtake the low pressure parts and a wave of compression steepens as it progresses. Similarly a wave of rarefaction becomes loss steep. On the other hand if {dV^ j Idjf) is negative, a wave of compression becomes ^ss ste(^ j) and a wave of rarefaction steepens into a compression shock.A fluid is said to be thermodynamically jfttable if it does not cf)lla])sc or expand catastrophically. For a fluid to bo thermo(|yiiamit‘ally st.able, dV^ i jdp*' must be positive. It follows from (S) that the sign o f dV  ^ Idp^  deiMUids on the sign ofdY  d^ p 3//2d f ^  d r ^  ^TTT^... 00From the cumsidorations outlined above, we immediately arrive at the following conclusions :(i) Compression waves steepen «and rarefaction waves flat! cm whenf i ;  +  >  „  ... (10)dr- 47tThis happens when either (a)or (b)d-jr)/dT^  is positive d*pldr  ^ is negative, andd^ pd,T^ <37/247rr2Therefore, a flattening wave of compression will begin to steepen as soon as a magnetic field o f suitable strength is switched on.(ii) Compression waves flatten and rarefaction waves steepen when• « ’ < odf^ 4:7T ( 11)This happens when [d^pldr )^ is essentially notgative and202 I. J. Singh and K. P. Choprady_dj2 ... (12)Hence it may he eoneludcd that the magnetic field enhances the steepening of a eotnpressiou wave and flattening of a rarefaction wave. Hence the presence of a transverse magnetic! field is conducive to the growth of compression waves and flic decay of the expansion waves.R E P E R E N C E f?Hoffmann, F. do and Toiler, E., 1050. Phyn. Rev., 80, 692.

Growth of Hydromagnetic Shock Waves

http://arxiv.iacs.res.in:8080/jspui/bitstream/10821/2424/1/Growth%20of%20hydromagnetic%20shock%20waves_I%20J%20%20Singh.pdf

Growth of Hydromagnetic Shock Waves

Abstract

Similar works

Full text

Available Versions

IACS Institutional Repository