An Analytical Study of Strong Non Planer Shock. Waves in Magnetogasdynamics

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1 Adv Theo Appl Mech Vol no An Analyical Sdy of Song Non Plane Shock Waves in Magneogasdynaics L P Singh Depaen of Applied Maheaics Insie of Technology Banaas Hind Univesiy Vaanasi-5 India Akal Hsain Depaen of Applied Maheaics Insie of Technology and DST-CIMS Banaas Hind Univesiy Vaanasi-5 India ahsainsap@ibhacin M Singh Depaen of Applied Maheaics Insie of Technology Banaas Hind Univesiy Vaanasi-5 India Absac An analyical appoach is sed o deive a new exac solion of a poble of onediensional nseady adiabaic flow of a plane and cylindical song shock wave popagaing in a plasa whose densiy ahead of he shock fon is assed o vay as a powe of he disance fo he soce of explosion The plasa is assed o be an ideal gas wih infinie elecical condciviy peeaed by a ansvese agneic field A coplee invesigaion is ade fo he cases of plane and cylindical flows in he pesence of agneic field An analyical solion of he poble is obained in es of flow vaiables velociy densiy and he pesse in he pesence of he agneic field which exhibis space ie dependence Also he analyical expession fo he oal enegy nde he inflence of ansvese agneic field is deeined

2 9 L P Singh A Hsain and M Singh Maheaics Sbjec Classificaion: 76L5 76W5 Keywods: Shock waves Magneogasdynaics Rankine-Hgonio condiion Analyical solion Inodcion The occence of shock waves in a gaseos edi has dawn he aenion of seveal invesigaos ding pas decades The popagaion of shock waves nde he inflence of song agneic field consies a poble of gea inees o eseaches in a vaiey of fields sch as asophysics nclea science geophysics and plasa physics Koobeinikov [] Geifinge and Cole [] and Hne [] sdied he poble of blas wave popagaion in a hoogeneos and inhoogeneos edi The pioneeing sdies of his phenoenon wee caied o by Taylo and Sedov [4] and hei neical solion based on self-siilaiy consideaion wee fond in good ageeen wih expeienal esls A nbe of analyical solions fo he blas wave popagaion have been obained by Roges [5] Bach and Lee [6] Labach and Pobsein [7] Sachdev [8] Poslavskii [9] Chisnell [] and Maa [] Labach and Pobsein [7] and Sachdev [8] sed an appoach based on he shock popagaion heoy of Binkley and Kikwood [] which peis a siple analyical solion o be obained diecly fo he govening eqaions Chisnell [] povided an analyical solion of he poble of conveging shock waves by he sdy of singla poins of he diffeenial eqaions A fhe conibion owads he deeinaion of exac solion of gasdynaic eqaions involving disconiniies via Lie gop ansfoaion has been caied o by any ahos eg Olivei and Speciale [-4] Radha and Shaa [5] Pandey e al [6] Singh e al [7] Olivei and Speciale [-4] sed sbsiion pinciple o obain an exac solion fo nseady eqaion of pefec gas and ideal agneogasdynaic eqaion Roges [5] and Maa [] obained he closed fo solion fo spheical blas wave poble when he densiy of he gas ahead of he shock fon vaies as a powe of he disance fo he oigin Singh e al [7] sed he ehod of Lie gop ansfoaion o obain an appoxiae analyical solion o he syse of fis ode qasi-linea paial diffeenial eqaions ha govens a one diensional nseady plane cylindically syeic and spheically syeic oion in a non-ideal gas involving song shock waves In he pesen pape we have consideed he poble of popagaion of a onediensional nseady non-plane flow of an inviscid ideal gas peeaed by a ansvese agneic field wih infinie elecical condciviy I is assed ha ass densiy disibion in he edi follows a powe law of he adial disance fo he poin of explosion An analyical solion of he poble is obained in es of flow vaiables velociy densiy and he pesse in he pesence of he agneic field Also he analyical expession fo he oal enegy nde he

3 Analyical sdy of song non plane shock wave 9 inflence of ansvese agneic field is deeined To o knowledge sch an analyic solion inflenced by he ansvese agneic field which exhibis space ie dependence has no been discssed in he pas Folaion of he Poble Assing he elecical condciviy o be infinie and he diecion of he agneic field ohogonal o he ajecoies of he gas paicles he govening eqaions fo a one-diensional nseady non-plane oion can be wien as [8] ρ ρ ρ ) ρ / ) ρ p h ) ) p p a ρ ρ ) ) h h h h ) / 4) whee is he gas velociy; ρ is he densiy; p is he pesse; is he consan specific hea aio; is he ie; is he single spaial co-odinae being eihe axial in flows wih plane geoey o adial in cylindically syeic flows; a p / ρ is he eqilibi speed of sond ; h is he agneic pesse defined by h μh / wih μ as agneic peeabiliy and H is he ansvese agneic field; and coespond especively o plane and cylindical syey A coa followed by a sbscip o denoes paial diffeeniaion nless saed ohewise The syse of eqaion )-4) is sppleened wih an eqaion of sae p ρrt whee R is he gas consan and T is he epeae I is well known ha a shock wave ay be iniiaed in he flow egion and once i is foed i will popagae by sepaaing he poions of coninos egion A shock he coec genealized solion saisfies he Rankine- Hgonio jp condiions Le χ) be he song shock wih he shock speed W dχ d popagaing ino he edi chaaceized by ρ ρ ) p p ) h h ) 5) Theefoe he bonday condiions a he shock fon can be wien as [] a ρ ρ 6) W W a W p ρ W - a ) - C ρ W W ) a W 7) 8)

4 94 L P Singh A Hsain and M Singh ) a h Cρ W 9) ) W whee a is he sond speed of he ndisbed edi C o h / ρw is he shock Cowling nbe and he sffix denoe evalaion of he flow paaees js ahead of he shock especively Since he iniial enegy inp E of explosion is vey lage he shocks speed W >> a so ha a W in he song shock lii Theefoe he Rankine-Hgonio jp condiions in he case of song shock waves can be wien as ρ ρ W ) ) ) p ρw CρW h C ρw ) ) ) I is assed ha a ie an explosion akes place ove a plane o along a line accopanied by elease of a finie aon of enegy E A plane o cylindical song shock is insananeosly foed which begins o popagae owad ino a pefecly condcing gas a es The densiy ρ is assed o vay as he invese powe of he adial disance fo he soce of explosion δ ρ ρ c χ ) whee δ and ρ c ae consans The oal enegy E inside a blas wave is eqal o he enegy spplied by he explosive and hs consan The oal enegy is given by he expession W p ) E 4π ρ h) d ) which epesens he s of he kineic and inenal enegies of he gas Analyical Solion wih Shocks Wih he help of eqaion ) eqaion ) can be wien as C ) ) p ρ ρ h Cρ 8 ) 8 ) 4) Afe sing eqaion 4) he govening eqaions ) ) and 4) can be ansfoed o ρ ) ρ 5) ) ) 6)

5 Analyical sdy of song non plane shock wave 95 ) 7) Using eqaions 5) and 6) and hen inegaing wih espec o we ge ) ) ρ 8) whee ) is an abiay fncion of inegaion Using he solion 8) eqaion ) edces of he fo ) d d 9) On solving eqaions 6) and 9) we have d d η whee ) ) ) ) η ) Plgging in eqaion ) in eqaion 9) and hen inegaing we obained whee ) ) ) ) ) ) ) and is abiay consan Wih he help of bonday condiion ) we can obain he analyical expession of he disance χ as χ whee ) ) ) ) Using he Rankine-Hgonio jp condiion ) and powe law of he densiy ) gives a vale of he consan δ as ) ) ) ) δ ) Conseqenly wih he help of eqaion ) he analyical solions of he flow vaiables and oal enegy in he pesence of he agneic field is given by ) ) ) ) ) 8 8 ) ) ρ C h C p 4) In view of above solion 4) he analyical expession fo he oal enegy is given by ) ) ) 8 π C E 5) I ay be noed hee ha in he absence of he agneic field he analyical solions 4) and 5) obained in his anne which exhibis space ie

6 96 L P Singh A Hsain and M Singh dependence is a well known solion o he blas wave poble nde he consideaion ha he enegy eleased in he blas wave is conseved is caied o by vaios appoaches {[][]} 4 Conclding Reaks In he pesen invesigaion a siple analyic ehod is sed o obain he solion of he poble of he popagaion of a one-diensional nseady adiabaic non-plane shock wave hogh a pefecly condcing inviscid gas peeaed by a ansvese agneic field The densiy ahead of he shock fon is assed o vay accoding o powe of he disance fo he soce of explosion Hee i is assed ha he aospheic pesse and agneic pesse saisfy he elaed Rankine-Hgonio condiions aoaically Then he govening eqaions ae inegaed o povide he shock fon as a fncion of ie An exac solion of he poble in fo of powe in he disance and ie is obained I ay be eaked ha in he absence of he agneic field sch an analyical solions ae in close ageeen wih ealie esls obained {[] []} Acknowledgeens Ahos acknowledge he financial sppo fo DST- CIMS Banaas Hind Univesiy Vaanasi India Refeences [] V P Koobeinikov Pobles in he Theoy of Poin Explosions in Gases Povidence RI: Aeican Maheaical Sociey 976 [] C Geifinge and J D Cole Siilaiy solions fo cylindical agneohydodynaic blas waves Phys Flids 5 96) [] C Hne Siilaiy solions fo he flow ino a caviy J Flid Mech 5 96) 89-5 [4] L I Sedov Siilaiy and Diensional Mehods in Mechanics Acadeic Pess New Yok 959 [5] M H Roges Analyical solion fo he blas wave poble wih an aosphee of vaying densiy Asophysical J 5 957) [6] GG Bach and J H S Lee An analyical solion fo blas waves AIAA 8 97) 7-75 [7] DD Labach and R F Pobeisein A poin explosion in a cold aosphee Pa II Radiaing flow J Flid Mech 4 969) [8] P L Sachdev Popagaion of a blas wave in nifo o non-nifo edia: a nifoly valid analyic solion J Flid Mech 5 97) [9] S A Poslavskii A new class of exac solions wih shock waves in gas dynaics PMM )

7 Analyical sdy of song non plane shock wave 97 [] R F Chisnell An analyic descipion of conveging shock waves J Flid Mech ) [] S Maa New exac solion of he blas waves poble in gas dynaics Chos Solions & Facals 8 6) 7- \ [] S R Binkley and J G Kikwood Theoy of he popagaion of he shock wave Phys Rev Le 7 947) 66-6 [] F Olivei and M P Speciale Exac solions o he nseady eqaions of pefec gases hogh Lie gop analysis and sbsiion pinciples In J Non-linea Mech 7 ) [4] F Olivei and M P Speciale Exac solions o he ideal agneogasdynaic eqaions of pefec gases hogh Lie gop analysis and sbsiion pinciples J Phys A 8 5) [5] V D Shaa and R Radha Exac solions of Ele eqaions of ideal gasdynaics via Lie gop analysis ZAMP 59 8) 9-8 [6] M Pandey V D Shaa and R Radha Syey analysis and exac solion of agneogasdynaic eqaions Q J Mech Appl Mah 6 8) 9- [7] L P Singh Akal Hsain and M Singh An appoxiae analyical solion of iploding song shocks in a non-ideal gas hogh Lie gop analysis Chinese Phys Le 7 ) 47-)-47-4) [8] G B Whiha Linea and Non-linea Waves John Wiley & Sons New Yok 974 Received: Ocobe 9

The sudden release of a large amount of energy E into a background fluid of density

The sudden release of a large amount of energy E into a background fluid of density 10 Poin explosion The sudden elease of a lage amoun of enegy E ino a backgound fluid of densiy ceaes a song explosion, chaaceized by a song shock wave (a blas wave ) emanaing fom he poin whee he enegy