On the structure of MHD shock waves in a viscous gas

Transcription

1 On the stuctue f MHD shck waves in a viscus gas On the stuctue f MHD shck waves in a viscus gas R. K. Anand and Haish C. Yadav Depatment f Physics, Univesity f Allahabad, Allahabad-, India anand.ajkuma@ediffmail.cm Abstact The exact slutins f MHD shck waves in an ideal gas ae btained taking int cnsideatin nly the viscsity f the gas. In view f an axial magnetic field, the analytical expessins f the paticle velcity, tempeatue, pessue and change-in-entpy within the shck tansitin egin ae btained. The flw vaiables ae numeical analysed t exple the influence f static magnetic field, shck stength, specific heat ati, initial pessue, initial density and cefficient f viscsity n the flw vaiables. The findings cnfim that thickness f MHD shck fnt inceases with incease in the viscsity f the gas and the change in thickness is me nticeable f lage values f the stength f magnetic field. The esults pvided a clea pictue f whethe and hw the viscsity f gas and the magnetic field affect the thickness f shck fnt. PACS numbes: 5.5.T c, 47.4-X, 47.Hj, 47..ad Keywds: Magnethyddynamics (MHD), viscus shck waves, axial magnetic field, entpy-pductin. Intductin Magnethyddynamic (MHD) shck waves ae fequently encunteed in vaius astphysical and gephysical phenmena. A bette undestanding f the mechanisms gvening emegence and evlutin f distubances in a viscus MHD is essential f the develpment f efficient methds f cntlling the tubulent tansitin in the flw aund the hypesnic flying bjects (at lwe Mach numbe). Amng the industial applicatins invlving applied extenal magnetic fields ae dag eductin in duct flws, design f efficient clant blankets in tkmak fusin eacts, cntl f tubulence f immesed jets in the steel casting pcess and advanced ppulsin and flw cntl schemes f hypesnic vehicles and missiles. Studies f the stuctue f shck waves in an ideal gas have been extensively epted in the liteatue. Ealy theetical investigatins wee esticted t the pefect gas equatin f state and the Navie-Stkes elatins with cnstant cefficients f heat cnductin and viscsity. Rankine [] published his dissetatin n the stuctue f shck waves in 87 in which he gave an explicit slutin f the case f heat cnductin nly. Hugnit s wk [] was published in 889. Thei equatins f shck jumps in paticle velcity, stess, and specific intenal enegy have becme knwn as the Rankine-Hugnit cnditins. Late, an explicit slutin f viscsity alne was given by Tayl [], wh als gave an explicit fmula valid f weak shck waves with bth viscsity and heat cndu ctin pesent. The slutins wee deived fm the cnsevatin laws f mass, mmentum and enegy unde the assumptin that the shck is single steady wave f finite thickness. A cnclusin was that, f a weak shck wave, the thickness becmes vey lage, vaying as the ecipcal f the shck stength. Hffmann and Telle [4] extended the Rankine-Hugnit cnditins f classical hyddynamics t shck waves in an infinitely cnducting fluid with supepsed magnetic field. The mathematical discntinuity in the physical vaiables given by the Rankine-Hugnit cnditins at a shck fnt is, hweve, nt physically pssible, and it is well knwn that cnsideatins f /5

2 On the stuctue f MHD shck waves in a viscus gas dissipatin f enegy by viscsity and heat cnductivity enable the physical quantities t vay cntinuusly and esult in a finite width f the shck fnt. Applying simila cnsideatins Sen [5] descibed the stuctue f a magnethyddynamic shck wave in infinitely cnducting plasma (a macscpically neutal, inized gas). Richtmye and Vn Neumann [6] suggested a methd which avids such difficulties, paticulaly f numeical calculatin. They bseved that the additin f a paticula viscsity like tem int gas dynamics equatins culd lead t the cntinuus shck flw in which the finite thickness f discntinuities at the shck wave was emved and eplaced by a egin in which physical paametes changed apidly and smthly. The thickness f a thin tansitin laye epesenting the shck acss which the gas undeges tansitin fm the initial t the final state, and this laye is geneally knwn as a shck fnt. In this laye the density, the pessue and the velcity f fluid change as entpy inceases. The incease in entpy indicates that thee is a dissipatin f mechanical enegy and thus an ievesible cnvesin f mechanical enegy int heat enegy takes place in the tansitin laye. The dissipative pcesses f viscsity (intenal fictin) and heat cnductin ae attibutable t the mlecula stuctue f a fluid. Such pcesses ceate an additinal, nn-hyddynamic t ansfe f m mentu m and enegy, and esult in nnadiabatic flw and in the themdynamically ievesible tansfmatin f mechanical enegy int heat. Viscsity and heat cnductin appea nly when thee ae lage gadients in the flw vaiables within a shck fnt. Zel dvich and Raize [7] studied the entpy pductin within a plana viscus shck wave and gave an analytical mdel f the shck pcess with effects f viscsity and heat cnductin based n Huguent cuves. Landau and Lifshitz [8] investigated the weak viscus shck waves with espect t the small changes in the flw vaiables. Painte [9] studied a viscus shck wave in an elastic tube. Maslv [] studied the wave pcesses in a viscus shck laye and cntl f fluctuatins. The auths [] investigated the ppagatin f plana, cylindically and spheically viscus shck waves in an ideal gas. Recently, Anand [] fmulated the shck jump elatins f MHD shck waves in nn-ideal gas and discussed the change-in-entpy acss the shck fnt. The main pupse f this pape is t pesent an exact slutin f ne dimensinal MHD shck wave in an ideal gas, by assuming that the viscsity is pesent, and that the heat cnductiv ity is neglected, t find the effect f viscsity n the shck wave pfile. F this pupse, a mdel was develped t pvide a simplified, cmplete teatment f the stuctue f plana MHD shck waves in ideal gas. The geneal nn-dimensinal fms f the analytical expessins f the distibutin f paticle velcity, tempeatue, pessue and change-in-entpy within the shck tansitin egin ae deived, assuming the medium t be viscid, nn-heat cnducting, electically infinitely cnducting, initially unifm and at est. The magnetic field is assumed having nly cnstant axial cmpnent which is pependicula t the shck fnt. The numeical estimatins f flw vaiables ae caied ut using MATLAB cde. The effects f viscsity ae investigated n the MHD shck tansitin egin. This mdel apppiately makes bvius the effects due t an incease in (i) the ppagatin distance fm the cente f fnt, (ii) the stength f magnetic field, (iii) the stength f shck wave i.e., Mach numbe, ( iv) cefficient f viscsity and (v) the adiabatic index, n the paticle velcity, tempeatue, pessue and change-in-entpy within the shck tansitin egin. The est f the pape is ganized as fllws. Sectin descibes the geneal assumptins and ntatins, the equatins f mtin and bunday cnditin. In sectin the analytical expessins f flw vaiables ae btained. /5

3 On the stuctue f MHD shck waves in a viscus gas A bief discussin with esults is pesented in sectin 4. The findings ae pesented in sectin 5 with details n which effects wee accunted f and which wee nt.. Basic equatins and bunday cnditin The cnsevatin equatins gvening the flw f a ne-dimensinal, viscus, ideal gas unde an equilibium cnditin in pesence f a magnetic field can be expessed cnveniently in Eulaian cdinates as fllws: ρ ρ u u ρ t, () ( ρu) ( ρu P q) H t, () [ ρ E ρu t / ] [ u( E u / ) pu qu], () H t u H u H, (4) whee ρ (, t), u (, t), q (, t), H (, t), p (, t), t and ae density, paticle velcity, viscus stess tens, axial magnetic field, pessue, time cdinate and psitin cdinate with espect t the igin in the diectin nmal t the shck fnt, espectively, and being the cnstant magnetic pemeability f the gas taken t be unity thughut the pblem. It is t be nted that the diffusin tem is mitted in the Equatin () by vitue f the assumed pefect electical cnductively. The viscus stess tens q is given by q ( 4 / ) μ( du/d) (5) whee, μ is the cefficient f viscsity. F simplicity, it is assumed that μ is independent f tempeatue. It is emakable that with cnditins H H and H z H, Equatin 4 can be witten as H / t ( H u) and futhe( H ) / t. Thus, the Maxwell equatin H is included in Equatin 4. In a cdinate system with statinay shck fnt, the shck stength emains pactically unchanged duing the small time inteval t equied t tavel a distance f the de f the shck fnt thickness, as a esult the tem cntaining the patial deivative with espect t time ( patial deivative ( /t ) is dpped in the Equatins () t (4) and futhe the / ) is eplaced by the ttal deivative (d/d ). Thus, the Equatins () t (4) gvening the ne dimensinal plane symmetical flw f a viscus gas unde the influence f an axial magnetic field ae witten as dρ du u ρ, (6) d d /5

4 On the stuctue f MHD shck waves in a viscus gas d ( ρu P q) d dh d, (7) d [ u( E u / ) pu qu], (8) d dh d u u H. (9) d d whee γ is the adiabatic index, i.e., γ C p /C v. The bunday cnditin n the slutin f these diffeential Equatins (6) t (9) equies that the gadient f flw vaiables must vanish ahead f the shck fnt (at as well as behind the shck fnt (at subscipt ae p, ρ, u, ) ). With these limits, the initial flw vaiables designated by the and the final flw vaiables with n subscipt ae p, ρ, u, H. If the shck fnt is mving with velcity U, then in the cdinate system fixed with the shck fnt, the initial paticle velcity will be u u U (). Exact Slutins f the flw vaiables In de t btain the exact slutins f the flw vaiables, we need t slve the flw Equatins (6) t (9) using the bunday cnditin given by Equatin () in the equilibium state. F this, we integate the Equatins (6) t (9) which yields, ρ ρ U/u () p p q ρ U ρu H / H / () p u γ / ( γ ) ρu / qu p U γ / ( γ ) ρ U / () H H U/u (4) Nw using Equatins () t () and (4), the Equatin () becmes p u qu U u H u / Uu H U / u ( ) Uu p U ( ) U / / ( ) qu. (5) Let us intduce tw new dimensinless quantities called paticle velcity η and shck stength M as η u/u and M U/a (6) whee a is the speed f sund in the unpetubed state. Using Equatins (5) and (6) in Equatin (5), we get [( ) / ] ( M H / p M ) ( M ( ) / ) H [(4 / ) / M ( p ) / / p M ] d / d (7) 4/5

5 On the stuctue f MHD shck waves in a viscus gas Thus, the Equatin (7) can be witten as a b c d e d / d (8) Whee a ( γ ) /, b ( M H / p M )/, c ( M ( γ ) / ) /, d H / p M and e [(4/) / M( ) ] Since utside the tansitin egin, thee is n gadient in the flw vaiables in the equilibium state. Theefe, in the equilibium state, we can wite d / d with eq With this equilibium cnditin, the Equatin (8) becmes a cubic equatin with espect t the paticle velcity in p / equilibium state eq. F seaching eal slutins, we put cubic equatin in the fm as Z FZ G (9) whee Z a b, F a c b and G a eq d a b c b Nw defining, tan K / G, whee G 4F K. The Equatin 9 is slved using the Cdm s methd and thus, the algebaic slutins ae / / eq ( F) cs( / ),, ( F) cs[( ) / ] () 4 The thee ts, given by Equatin () will be eal if the cnditin G F is satisfied. Using this cnditin and Equatin (), we can numeically cmpute the paticle velcity cespnding t the equilibium state in which thee ae n gadients in the flw vaiables. Let us chse the igin at the pint f inflectin f the velcity pfile. The pint f inflectin is btained by using, the cnditin d / d int the Equatin (8) which again yields a cubic equatin given as in G () F in whee F c / a, G d / a Defining, tan K/ G, whee G 4F K. Using the Cdn s methd, the algebaic slutins f the Equatin () ae / / in ( F) cs( /),, ( F) cs[( )/] () 5/5

6 On the stuctue f MHD shck waves in a viscus gas 4 The thee ts, given by the Equatin (), will be eal if the cnditin G F is satisfied. Using this cnditin and Equatin (), we can detemine the pint f inflectin at the velcity pfile. On integatin the Equatin (8) yields an analytic slutin f as ( e/ a)[ Alg{( )/( in )} Blg{( )/( in )} Clg{( )/( in )}] () whee A /( )( ), B /( )( ) and C /( )( ) Equatin () gives a elatin between the paticle velcity and the distance. Thus, the paticle velcity depends n the distance within the shck tansitin egin. Using Equatins () and (8), we can wite the tempeatue acss the shck fnt as T T + ( γ-) M [ γ η / - ( M / p M ) ( M / ) / p M ] (4) The Equatin (4) gives a elatin between the tempeatue and the paticle velcity and it is bvius fm Equatin () that the paticle velcity depends n the distance. Hence using the Equatin (4) we can study the vaiatins f the tempeatue with espect t the distance within the shck tansitin egin. Using the Equatins (5), (), (4) and (8), we can wite the expessin f the pessue acss the shck fnt as p / p ( ) M ( ) H / p M ( a b c d)/ (5) This Equatin (5) gives a elatin between the pessue and the paticle velcity, and fm Equatin it is bvius that the paticle velcity depends n the distance. Thus, using Equatin (5) we can study the vaiatins f the pessue with distance within the shck tansitin egin. Futhe, the change-in-entpy acss the shck fnt is given by ( ΔS/R) η ( γ ) lg( T/T ) lg( p / p ) (6) Thus, the entpy pductin acss the MHD shck fnt is easily btained by substituting Equatins (4) and (5) int Equatin (6). 4. Results and Discussins In the pesent pape, an exact slutin was btained f MHD shck waves in a viscus gas. The geneal nn-dimensinal fms f the analytical expessins f the distibutin f the flw vaiables (the paticle velcity, the tempeatue T / T, the pessue p / p, and the change-in-entpy S / R ) within the shck tansitin egin ae given by Equatins () t (6), espectively. These analytical expessins wee deived by assuming that the distubances due t the eflectins, wave inteactins in the wake, etc., d nt vetake the shck waves. The geneal nn-dimensinal analytical expessins f the paticle velcity, tempeatue, pessue and change-inentpy ae the functins f the distance, cefficient f viscsity, stength f magnetic field H, stength f shck M and adiabatic index f the gas. Theefe, the values f the cnstant paametes ae taken t 6/5

7 On the stuctue f MHD shck waves in a viscus gas be μ 5, 7., pascal. sec,,.,,,. 8, M.,.5,.,.,.4,.66, initial pessue p,.,. ba and initial density geneal pupse f numeical cmputatins. ρ.,.9,.4 Kg/m f the T/ T p/ p S/ R Fig. Nn-dimensinal paticle velcity, tempeatue, pessue and change-in-entpy distibutin with distance f vaius values f and cnstant values fm,., p = ba, ρ =.Kg/m and pascal.sec. μ 5 Fig. shws the vaiatins f the paticle velcity, the tempeatue, the pessue and the change-in-entpy with espect t the ppagatin distance f the cnstant values f M,., p.9ba ρ. Kg/m μ 5 pascal.sec and diffeent values f axial magnetic field H,.,,,. 8. It is ntable that f small values f magnetic field the speading f the flw vaiables is lesse than that f lage values f magnetic field. Hweve, the effect f incease in the stength f magnetic field n the speading f flw vaiables is appeciable ahead f the pint f inflectin. Theefe, the pesence f magnetic field inceases the thickness f shck fnt and it is bseved that the thickness is maximum cespnding t the highest value f the stength f magnetic field. Thus, the thickness f MHD shck fnt inceases with inceasing stength f the magnetic field. It is als bseved that the ange f vaiatins f the flw vaiables deceases with inceasing magnetic field. 7/5

8 On the stuctue f MHD shck waves in a viscus gas.7.4 p.9 p =. p =. T/ T p.9 p =. p = p/ p p.9 p =..5 p = S/ R H.4. p..9 p =.. p = Fig. Nn-dimensinal paticle velcity, tempeatue, pessue and change-in-entpy distibutin with distance f vaius values f p and and cnstant values f M,., and μ 5 pascal.sec. ρ =. Kg/m Fig. shws the vaiatins f the paticle velcity η, the tempeatue T / T, the pessue p / p and the change-in-entpy S / R with distance f the diffeent values f initial pessue p,,. ba and axial magnetic field H,,. 8 Tesla and cnstant values f M,., ρ =. kg/m, μ 5 pascal.sec. The speading f the flw vaiables deceases with incease in the value f initial pessue. Hweve, the decease in the thickness f shck fnt with incease in the initial pessue is me nticeable f lage values f the stength f magnetic field than that f small values f the stength f magnetic field. Thus, the thickness f MHD shck fnt inceases with decease in the initial pessue. 8/5

9 On the stuctue f MHD shck waves in a viscus gas.7.4 =. =.9 = T/ T =. =.9 = p/ p =. =.9 = S/ R... =. =.9 = Fig. Nn-dimensinal paticle velcity, tempeatue, pessue and change-in-entpy distibutin with distance f vaius values f and H and cnstant values f M,., p = ba and μ 5 pascal.sec. Fig. shws the vaiatins f the paticle velcity η, the tempeatue T / T, the pessue p / p and the change-in-entpy S / R with distance f the diffeent values f the magnetic field H,,. 8 Tesla and the initial density.,.9,. 4 Kg/m and cnstant values f M,., p.9ba and 5 pascal.sec. It is ntable that the speading f the flw vaiables deceases with incease in the value f initial density. Thus, the thickness f MHD shck fnt inceases with deceasing initial density. Hweve, the decease in the thickness f shck fnt with incease in the initial density is independent f the stength f magnetic field. μ 9/5

10 On the stuctue f MHD shck waves in a viscus gas..7 =5* -6 =7.* -6 * -6.7 =5* -6 =7.* -6 * -6 T/ T =5* -6 =7.* -6.4 * T/ T =5* -6 =7.* -6. * p/ p =5* -6 =7.* -6 * -6 p/ p..8 =5* -6 =7.* -6 * S/ R =5* -6 =7.* -6. * S/ R =5* -6 =7.* -6 * Fig. 4 Nn-dimensinal paticle velcity, tempeatue, pessue, and change-in-entpy distibutin with distance f vaius values f and H and cnstant values fm,., ρ. Kg/m and p = ba. /5

11 On the stuctue f MHD shck waves in a viscus gas Fig. 4 shws the vaiatins f the paticle velcity η, the tempeatue T / T, the pessue p / p and the change-in-entpy S / R with distance f diffeent values f the magnetic field H,. 8 Tesla and the cefficient f viscsity 5,7., pascal.sec. It is ntewthy that the speading f the flw vaiables inceases with incease in the value f the cefficient f viscsity. Thus, the thickness f MHD shck fnt inceases with incease in the value f the cefficient f viscsity. Hweve, the incease in the thickness f shck fnt with incease in the viscsity cefficient is me nticeable f lage values f the stength f magnetic field. T/ T.7 M=. M=.5 M= M=. M=.5 M= T/ T....7 M=. M=.5 M= M=. M=.5 M= /5

12 On the stuctue f MHD shck waves in a viscus gas p / p S/ R M=. M=.5 M= M=. M=.5 M= p / p S/ R M=. M=.5 M= M=. M=.5 M= Fig. 5 Nn-dimensinal paticle velcity, tempeatue, pessue and the change-in-entpy distibutin with distance f vaius values f M and and cnstant values f., ρ =. Kg/m, μ 5 pascal.sec and p = ba. Fig. 5 shws the vaiatins f the paticle velcity η, the tempeatue T / T, the pessue p / p and the change-in-entpy S / R with distance f diffeent values f the magnetic field H,. 8 Tesla and the shck stength M,5, and cnstant values f., ρ =. Kg/m, μ 5 pascal.sec and p = ba. It is emakable that f small values f shck stength the speading f the flw vaiables is me than that f lage values f the shck stength. Thus, the thickness f MHD shck fnt deceases with inceasing stength f the shck wave. Hweve, the change in the speading f the flw vaiable due t the change in shck stength is me f lage values f the stengths f magnetic field. It is t be nted that the effect f magnetic field is appeciable f small values f shck stength, while f lage values f shck stength it is vey small. /5

13 On the stuctue f MHD shck waves in a viscus gas S/ R p/ p T/ T.7 =. =.4 = =. =.4.8 = =. 4 =.4 =.66.5 H =.. =.4 = T/ T p/ p S/ R =. =.4 = =. =.4 = =. =.4 = =. =.4 = Fig. 6 Nn-dimensinal paticle velcity, tempeatue, pessue and change-in-entpy distibutin with distance f vaius values f and and cnstant values f M, ρ =. Kg/m, μ 5 pascal.sec and p. 9 ba. /5

14 On the stuctue f MHD shck waves in a viscus gas Fig. 6 shws the vaiatins f the paticle velcity η, the tempeatue T / T, the pessue p / p and the change-inentpy S / R with distance f diffeent values f the adiabatic index.,.4,. 66 and magnetic field H,. 8Tesla and f cnstant values fm, ρ =. Kg/m, μ 5 pascal.sec and p. 9 ba. It is ntable that the speading f the flw vaiables inceases ahead f inflectin pint and deceases behind the inflectin pint with incease in the value f adiabatic index. Thus, the thickness f MHD shck fnt inceases with incease in the value f adiabatic index. Hweve, the change in the speading f the flw vaiables with the adiabatic index is independent f the stength f magnetic field. 5. Cnclusins The investigatins made in the pesent pape ae intended t cntibute t the undestanding f the stuctue f MHD shck waves in a viscus gas, by giving, f the fist time, the full exact slutins f the flw field within the shck tansitin egin. The analysis pesented in the pape shws the fundamental le played by viscsity in detemining MHD shck stuctue. The fllwing cnclusins may be dawn fm the findings f the cuent analysis:. The thickness f MHD shck fnt inceases with inceasing stength f the magnetic field. The influence f magnetic field n the speading f the flw vaiables is appeciable ahead f the pint f inflectin.. The thickness f MHD shck fnt inceases with deceasing initial pessue and the change in thickness is me nticeable f lage values f the stength f magnetic field than that f small values f the stength f magnetic field.. The thickness f MHD shck fnt inceases with deceasing initial density and the change in thickness with initial density is independent f the stength f magnetic field. 4. The thickness f MHD shck fnt inceases with incease in the viscsity and the change in thickness is me nticeable f lage values f the stength f magnetic field. 5. The thickness f MHD shck fnt deceases with inceasing stength f the shck wave and the change in thickness is me f lage values f the stengths f magnetic field. It is t be nted that the effect f magnetic field is appeciable f small values f shck stength, while f lage values f shck stength it is vey small. 6. The thickness f MHD shck fnt inceases with incease in the value f adiabatic index and the change in thickness with the adiabatic index is independent f the stength f magnetic field. Thus, the esults btained hee ae imptant f the cases whee the viscsity f fluid plays an imptant le in additin t the vaius industial and mechanical engineeing pcesses whee fe fluids ae used. The findings may als be useful in the study f the effects f magnetic fields n bld ciculatin, cadivascula events, cude il tansptatin, etc. 4/5

15 On the stuctue f MHD shck waves in a viscus gas Refeences. Rankine, W.J.M.: On the themdynamic they f waves f finite lngitudinal distubances. Phil. Tans. Ry. Sc. Lndn. 6, (87).. Hugnit, P.H.: Memie su la ppagatin du mvement dans les cps et plus specialement dans les gaz pafaits. J. Ecle Plytec. Pais. 58, -5 (889).. Tayl, G.I.:The cnditins necessay f discntinuus mtin in gases. Pc. R. Sc. Lnd. A 84, 7-77 (9). 4. de. Hffmann, F., Telle, E.:Magnet-hyddynamic shcks, Phys. Rev. 8, 69-7 (95). 5. Sen, H.K.: Stuctue f magnet-hyddynamic shck wave in plas ma f infinite cnductivity. Phys. Rev., 5- (956). 6. vn Neumann, J., Richtmye, R.D.: A Methd f the Numeical Calculatin f Hyddynamic Shcks. J. Appl. Phys., -7 (95). 7. Zeldvich, Ya.B., Raize, Yu.P.: Physics f Shck Waves and High Tempeatue Hyddynamics Phenmena. Dve Publicatin, New Yk (). 8. Landau, L.G., Lifshitz, E.M.: Fluid Mechanics. Pegaman, England (959). 9. Painte, P.R.: The velcity f the ateial pulse wave: a viscus-fluid shck wave in an elastic tube. Theetical Bilgy and Medical Mdelling 9, 5-5 (8).. Maslv, A.A., Minv, S.G., Kudyavtsev, A.N., Pplavskaya, T.V., Tsyyulnikv, I.S.: Wave pcesses in a viscus shck laye and cntl f fluctuatins, Fluid Mech ().. Yadav, H.C., Anand, R.K.: Ppagatin f shck waves in a viscus medium. Phys. Sc , ().. Anand, R.K.: Jump elatins f magnethyddynamic shck waves in nn-ideal gas flw. Astphys Space Sci 4, 7 7 (). 5/5