The Propagation of Strong Shock Waves in Magnetohydrodynamics

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1 IOS Journal of Alied Physics (IOS-JAP) e-issn: Volume 6, Issue 6 Ver. III (Nov.-Dec. 4), PP 7-75 The Proagation of Strong Shock Waves in Magnetohydrodynamics T.N.Singh, A. K. ay Deartment of Alied Sciences, B.B.S. College of Engineering and Technology Phahamau, Allahabad-4 (U.P.) India Deartment of Mathematics, M. G. P. G College Gorakhur-73 (U.P.) India Abstract: In this aer, we have studied non-self-similar gas motions in resence of magnetic field which result from the roagation of lane, cylindrical and sherical shock waves through the gas requires comlicated and cumbersome calculations. An aroximate method of calculation of such motions is taken from [,3,4]. Key Words: shock waves, roagation of lane, cylindrical and sherical shock waves Mathematics subject classification : 76L5, 76W5 I. Introduction: The investigation of non-self-similar gas motions which result from the roagation of shock waves through the gas requires comlicated and cumbersome calculations. In a few cases these have actually been carried out, e. g. on the roblem of oint exlosion []. In [,3,4] an aroximate method of calculation of such motions is given, valid for a high gas density jum across the shock wave, i.e. for the roagation of lane, cylindrical & sherical shock waves of large intensity in a gas. This method is based on the reresentation of gas dynamical quantities in the form of series in a secial form for the owers of arameter,which characterizes the ratio of the gas density in front of the wave to the gas density behind the wave. The successive terms of the series are found from the equations by means of quadratures. When only the two first terms of the series are taken into account, the gas arameters in a disturbed region behind the shock wave are exressed in terms of the function t in [3], which treats of the law of roagation of a shock wave. For the determination of this function in the roblems of motion resulting from the exlosion in a gas and from the exansion the exansion of a movable boundary (iston) in a gas, a law of conservation of energy in integral form may be used, ertaining to the whole of the region of disturbed gas motion [4]. Shock waves are characterized by an abrut, nearly discontinuous change in the characteristics of the medium Anderson []. Analytical solutions to the wave equations for steady vertical comression waves in a isothermal hydrostatics atmoshere with uniform horizontal magnetic field have been resented by Musielak et.al. [6]. Arora [], Arora et.al. [4], Bhardwaj [9] and Blyth [3] have studied shock waves characteristics in a magnetic field. Cheng-Yue et.al. [] resented one dimensional relativistic shock model for the light curve of gamma-ray bursts. Ghose et.al. [5] investigated relativistic effects on the modulation instability of electron lasma waves in quantum lasma. II. Flow Governing Equation The total energy of a moving gas ( the sum of its internal and kinetic energies) at each instant must equal the sum of the energy E which was generated by the exository the initial energy of the gas affected by the motion, magnetic field and the work done by the iston. In the resence of magnetic field the ressure term H H become change which is given by (Where is fluid ressure and is magnetic ressure), then we have used effective ressure for. Taking the exression to the internal energy er unit mass of a gas (where is the effective ressure, is the density, is the ratio of secific heats), then we have 7 Page

2 v v v The Proagation Of Strong Shock Waves In Magnetohydrodynamics t dv E dv dv t. t () Where v v is the volume occuied by the moving gas, is the initial effective gas ressure articles and t is the time. is the effective ressure on the iston, v is the volume dislaced by the iston, t is the velocity of gas To use this integral relationshi along with the reresentation of the desired quantities, and in the form of a series in owers of, we shall also reresent the function (t) which gives the law of roagation of a shock wave in the form of a series for the function which determines the form of a bow shock wave for the steady flow ast a body. We shall follow the method of Liubimo used for the case of non- stationary one dimensional motions, t t t Substituting series for, and in equation () and equating the terms on the right with the terms on the left for the same owers of, after aroriate transformations, we obtain a sequences of differential equations for the determination of functions, etc. As will be shown below, by roer choice of the main terms in the exansions of the quantities t and in owers of, we can obtain a satisfactorily accurate first aroximation for the determination of the law of roagation of a shock wave (and evidently all arameters of the stream immediately behind it) and the effective ressure acting on the iston. In accordance with the results of [] let a O ( a t ) dm r dr a m O Where is the initial gas density, m is the Lagrange coordinate which is introduced by the relation, where r is the initial coordinate of a article,,,3 corresond resectively to the flow with lane, cylindrical and sherical waves. The main terms are chosen such that in the case where the law of roagation of a shock wave, they will yield exact values of the corresonding quantities immediately behind the shock wave, i.e. For m. r After substitution of the exressions for and into the integral relationshi () for the t determination of functions t we obtain the following equation (index o is subsequently omitted): t a E dt () Where a,, 3 And t is 7 Page

3 The Proagation Of Strong Shock Waves In Magnetohydrodynamics For simlicity it is assumed that at the start the gas occuies all sace. We shall evaluate the accuracy of determination of functions t and from equation () by comaring the solutions of this equation with the know exact solutions of roblems on self- similar gas motions. Let III. Imulsive Motion Of Piston: ct n. For n the motion is self-similar only under the condition that a, i.e. only n as long as the shock wave may be considered to be strong. Assuming equation () we find and,, n ( ) 4 n E and taking from Where is the ratio of the volume dislaced by the iston to the volume bounded by the shock wave is the effective gas ressure immediately after the shock wave; 4 and It is interesting that in the aroximation under consideration the values and do not deend uon each of the arameters n and searately, but only uon their combination. Grahs of theses function for.4, i.e. for are reresented in figure. In this figure the values of and are reresented, obtained at the results of numerical integration of corresonding exact solutions for (hollow squares 3) and for 3 (hallow circles ) for and n the aroximate values, redicated uon the choice of the main terms in the - exansions, coincide with the exact values (hollow triangles 6); for and n the results of exact calculations are not available. Half- shaded symbols, 5, 7 for corresond to the exact solution of the roblem of a strong exlosion [5]. Finally, the black squares 4 corresond to the values obtained for the exact solution of the roblem with a cylindrical iston ( ), exanding according to the indicated law. This case may be considered as the limiting case of imulsive iston exansion for n. Figure show that in all the cases enumerated aroximate solutions for 6 satisfactory accuracy. for If, have a quite IV. Exansion Of Piston With Constant Velocity UT, then the motion will be rogressive also for a. Substitution of this exression into Equation () for E leads to the relations Dt U a, D D Where For these relations are exact; their curves for.4 are reresented in Figure. by solid lines. For and 3 these relations are only of aroximate validity; relations obtained for 3 and.45 by numerical integration of exact equations, are reresented in this figure by the dashed line. For 7 Page

4 The Proagation Of Strong Shock Waves In Magnetohydrodynamics a.4 3 aroximate exressions retain satisfactory accuracy u to the values, which corresonds to D and u to the effective ressure ratios in the shock wave of the order [,5,6]. V. Strong Exlosion Assuming in equation (),, E Where t 4 6 E and resuming, 3 4 Figure 3 shows the curves of the aroximate functions obtained for the quantities Z t E and functions, and the exact values of these quantities [6] for,,3., we find From figure 3 it follows that in the case of the solution of the roblem of the strong exlosion, the aroximate exressions for and satisfactorily agrees with the exact exressions u to the values.6.8,, i.e. u to the values (Note that the relative error in the determination of is times smaller that the difference corresonding to the quantity Z between the exact and the aroximate values Z in figure. 3). Thus, the examles resented of comarison of the aroximate and exact solutions suort the conclusion that the functions (t) and t, determined by equation, retain a satisfactory accuracy u to the values.5.3. Equation () allows the comutation of any non-self- similar motions resulting from an exlosion and from the exansion of a iston, (the equation is easily modified for the cases when the initial volume of a iston is different from zero), rovided the intensity of the resulting shock waves is sufficiently large, so that does not exceed.-.3. In articular, using the law of lane cross-sections, in solving this equation one may determine the form of a shock wave, which is created by the flow ast a rofile ( ) or a body of revolution ( ) of a gas with large suersonic velocity. The effective ressure distribution on the surfaces of these bodies may likewise be determined, even in the cases when the front art is some what blunt [7]. VI. esult In resent aer, we have studied the roagation of strong shock waves in magneto hydrodynamics. We did not get any significant change in nature of shock wave. Only we get small change in ressure. Hence the surfaces of shock wave become smooth in resence of magnetic field. The results are shown in figures. eferences: []. G.G. Chernii; Odnomornye niustanovivshiesin dvizheniin severshenogo gazas silnymy udarnyml volnnmi (Onedimensional non-steady motions of an ideal gas with strong shock waves). Dokl. Akad Nauk SSS, Vol. 7, No. 5 (956). []. E.E.Okhotimskii; I.L.Kondrasheva; Z.P.Vlasova and.k.kasakkova, aschet sil nogo vzryva suchetom nachal nogo davleniia (calculation of a strong exlosion with consideration of the initial ressure). Tr. Matem in-ta ANSSS ix B.I. Steklova, Vol.5, (957). [3]. G.G.Chernii; Adiabaticheskie divzheniia sovershennogo Gaza & udarnymi bolnami bol shoi intensivnosti (Adia batic motions of an ideal gas with shock waves of great intensity). Izv ANSSS, OTN No 3, (957). [4]. G.G. Chernii; zadacha o tochechnox vzryve (The roblem of a oint exlosion).dokl. Akad. Nauk SSS Vol., No. 4 (957). [5]. G.I. Liublmov; Method of solution of roblems in gas dynamics and magneto hydrodynamics of the flows with strong shock waves, MGU. (958). [6]. H.Mark; The interaction of a reflected-shock wave with the boundary layer in a shock tube, NACA TM 48 (958). and 73 Page

5 The Proagation Of Strong Shock Waves In Magnetohydrodynamics [7]. G.G.Chernii; Priminenie intergral nykh soothoshenii v zadachakh o rasrostranenii sil nykh udarnykh voln, (Alication of integral relationshis in roblems of roagations of strong shock waves), PMM Vol. 4, No., -5 (96). [8]. Z.E Musielak,; C.H An.,.L. Moore and S.T. Suess, :Proagating and Non Proagating Comression Waves in a Isothermal Atmoshere with Uniform Horizontal Magnetic Field. The Astrohysical Journal, Vol.344 (989); [9]. D. Bhardwaj :Formation of Shock Waves in eactive Magnetogasdynamics flow. Int. J. Engg. Sc. Vol. 38(); []. John D. Jr Anderson: Fundamental of Aerodynamics (3 rd Ed.) Mc Graw-Hill Science/Engineering/Math(), IBBN []. Cheng-Yue, Su; Yi-Ping Qin;Jun-Hui Fan and Zhang-Yu Han Chin.J.Astrohys; Vol.6(6); 59. [].. Arora Non-Planar Shock Waves in a Magnetic Field. Com. Math. Al., Vol. 56 (8); [3]. P.A Blythe,.(8) Wave Proagation at High Dissociation Temerature. J.Engg. Math., Vol. 6, [4]..Arora; A Tomar,and V.P. Singh. (9): Shock Waves in eactive hydrodynamics.shock Waves; Vol. 9, [5]. B Ghose; S. Chandra, and S.N. Paul, () Pharma J.Phys., Vol. 78, Page

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