Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 49, 2, pp. 501-513, Warsaw 2011 SELF-SIMILAR FLOW OF A NON-IDEAL GAS WITH INCREASING ENERGY BEHIND A MAGNETOGASDYNAMIC SHOCK WAVE UNDER A GRAVITATIONAL FIELD Kaushal K. Singh Bineeta Nath North Eastern Hill University, Department of Mathematics, Shillong, India e-mail: kaukumar@yahoo.com; binita nath@yahoo.com A self-similar solution for the propagation of a spherical shock wave in a non-ideal gas in the presence of an azimuthal magnetic field is inve- stigated. The medium is assumed to be under a gravitational field due to a heavy nucleus at the origin(RocheModel). The unsteady model of Roche consists of a gas distributed with spherical symmetry around the nucleus having a large mass. It is assumed that the gravitational effect of the medium itself can be neglected compared with the attraction of the heavy nucleus. The total energy of the flow-field behind the shock is supposed to be increasing with time. Similarity solutions are obtained, and the effects of variation of the parameter of non-idealness of the gas, the shock-Mach number and the Alfven-Mach number on the flow-field behind the shock are investigated. Key words: shock wave, non-ideal gas, self-similar flow, magnetogasdy- namics, gravitational field 1. Introduction Carrus et al. (1951) studied the propagation of shock waves in a gas under the gravitational attraction of a central body of the fixedmass (RocheModel) and obtained similarity solutions by numerical method. Rogers (1957) discus- sed a method for obtaining an analytical solution to the same problem. Ojha et al. (1998) discussed the dynamical behaviour of an unstable magnetic star by employing the concept of the Roche Model in an electrically conducting atmosphere. Singh (1982) studied the self-similar flow of a non-conducting 502 K.K. Singh, B. Nath perfect gas, moving under the gravitational attraction of a central body of the fixed mass, behind a spherical shock wave driven out by a propelling contact surface into a quite solar wind region. Singh and Srivastava (1989) studied the self-similar flow of a perfect gas, moving under the gravitatio- nal attraction of a central body of the fixed mass, behind a spherical shock wave moving into a conducting gas of spatially decreasing density and per- vaded by a spatially decreasing magnetic field. Total energy content between the inner expanding surface and the shock front is assumed to be increasing with time.Ratkiewicz et al. (1994) studied similarity solutions for synchrotron emission from a supernova blast wave. In all of the works, mentioned above, the medium is taken to be a gas satisfying the equation of state of a perfect gas. The assumption that the gas is ideal is no longer valid when the flow takes place at high temperatures. Anisimov and Spiner (1972) studied a problem of point explosion in anon-ideal gasby taking the equation of state ina simplified form,which describes the behaviour of themedium satisfactorily at low densi- ties. RangaRao andPurohit (1976), Ojha (2002) andVishwakarma andNath (2007) also studied the propagation of shock waves in gases with the above equation of state. Roberts and Wu (1996, 2003) used an equivalent equation of state to study the shock wave theory of sonoluminescence. Vishwakarma et al. (2007) studied the propagation of the magnetogasdynamic cylindrical shockwave in a rotating non-ideal gas by using the equation of state taken by Roberts andWu (1996, 2003). In the present work, we therefore investigate the self-similar flow behind a spherical shock wave propagating in a non-ideal gas in the presence of an azimuthal magnetic field. Themedium is assumed to be under a gravitational field due to a heavy nucleus at the origin (RocheModel). The unsteadymodel of Roche consits of a gas distributed with spherical symmetry around the nucleus having a large mass m. It is assumed that the gravitational effect of themedium itself can be neglected compared with the attraction of the heavy nucleus. The total energy of the flow-field behind the shock is supposed to be increasing with time (Freeman, 1968; Director and Dabora, 1977). This increase can be obtained by the pressure exerted on themedium by the inner expanding surface (Rogers, 1958). In order to obtain the similarity solutions of theproblem, thedensity of theundisturbedmediumis assumed tobeconstant. Effects of variation of the parameter of the non-idealness of the gas b, the shock-Mach number M and the Alfven-Mach number MA on the flow field behind the shock are investigated. Self-similar flow of a non-ideal gas... 503 2. Fundamental equations and boundary conditions The fundamental equations for one-dimensional adiabatic unsteady spherically symmetric flow of a perfectly conducting non-ideal gas in which an azimuthal magnetic field is permeated, in the generalizedRocheModel are (Rogers, 1957; Singh and Srivastava, 1989; Vishwakarma, 2000) ∂ρ ∂t +u ∂ρ ∂r +ρ ∂u ∂r + 2ρu r =0 ∂u ∂t +u ∂u ∂r + 1 ρ ∂p ∂r + µh ρ ∂h ∂r + µh2 ρr + G∗m r2 =0 (2.1) ∂h ∂t +u ∂h ∂r +h ∂u ∂r + hu r =0 ∂e ∂t +u ∂e ∂r − p ρ2 (∂ρ ∂t +u ∂ρ ∂r ) =0 where u, p, ρ and h are the velocity, pressure, density and azimuthal ma- gnetic field, respectively, at a radial distance r from the center of the core at time t, µ is themagnetic permeability, e is the internal energy per unitmass, m denotes the constantmass of the core and G∗ is the gravitational constant. Here, it is assumed that the gravitating effect of themedium itself is negligible in comparison with the attraction of the heavy nucleus. Theabove systemof equations shouldbe supplementedwithan equation of state.Todiscover howdeviations fromthe ideal gas can affect the solutions,we adopta simplemodel.Weassumethat thegas obeysa simplifiedvanderWaals equation of state of the form (Roberts and Wu, 1996, 2003; Vishwakarma et al., 2007) p= R∗ρT 1− bρ e=CvT = p(1− bρ) ρ(γ−1) (2.2) where R∗ is the gas constant, Cv =R ∗/(γ−1) is the specific heat at constant volume and γ is the ratio of specific heats. The constant b is the ”van der Waals excluded volume”; it places a limit, ρmax =1/b, on the density of the gas. We assume that the spherical shock wave is propagating outwards from the center of symmetry in a perfectly conducting non-ideal gas with constant density and a variable azimuthal magnetic field, which is at rest. The flow variables immediately ahead of the shock front are u1 =0 ρ1 = const h1 = cR −k (2.3) 504 K.K. Singh, B. Nath where R is the shock radius, cand k are constants and the subscript1denotes the condition immediately ahead of the shock. At the equilibrium state, the pressure ahead of the shock is p1 =G ∗mρ1R −1+µc2(1−k) R−2k 2k (2.4) where 2k=1. The jump conditions across the magnetogasdynamic shock are ρ2(Ṙ−u2)= ρ1Ṙ h2(Ṙ−u2)=h1Ṙ p2+ 1 2 µh2 2+ρ2(Ṙ−u2)2 = p1+ 1 2 µh21+ρ1Ṙ 2 (2.5) e2+ p2 ρ2 + 1 2 (Ṙ−u2)2+ µh22 ρ2 = e1+ p1 ρ1 + 1 2 Ṙ2+ µh21 ρ1 where subscript 2 denotes conditions immediately behind the shock and Ṙ(= dR/dt) denotes the velocity of the shock front. From eqautions (2.5), we obtain u2 =(1−β)Ṙ ρ2 = ρ1 β h2 = h1 β (2.6) p2 = [ 1 γM2 + 1 2M2A ( 1− 1 β2 ) +(1−β) ] ρ1Ṙ 2 where M = √ ρ1Ṙ2/(γp1) is the shock-Mach number referred to the frozen speed of sound √ γp1/ρ1, and MA = √ ρ1Ṙ2/(µh 2 1) is the Alfven-Mach num- ber. The quantity β(0<β< 1) is given by the relation β3−β2 [ 2 (γ+1)M2 + γ ( 1 M2 A +1 ) +2b−1 γ+1 ] +β [γ−2+ b γ+1 ] 1 M2A + (2.7) + b (γ+1)M2A =0 where, b= bρ1 is the parameter of non-idealness of the gas. The shock-Mach number Me referred to the speed of sound in non-ideal gas √ γp1/[ρ1(1−b)] is given by Me =M √ 1− b (2.8) Self-similar flow of a non-ideal gas... 505 The total energy E of the flow-field behind the shock is not constant, but assumed to be time dependent and varying as (Rogers, 1958; Freeman, 1968; Director and Dabora, 1977) E=Ect s (2.9) where s is a non-negative number and Ec is a constant. The positive values of s correspond to the class in which the total energy increases with time. This increase can be achieved by the pressure exerted on the gas by the expanding surface (a contact surface or a piston). Thus the flow is headed by a shock front and has the expanding surface as the inner boundary. 3. Similarity solutions Following the general similarity analysis of Sedov (1959), we define two cha- racteristic parameters a and dwith independent dimensions as [a] = [mG∗ρ1] [d] = [mG ∗] = [ Ec ρ1 ] 3 5 The single dimensionless independent variable in this case will be η=(αmG∗) −δ1 2 rt−δ1 (3.1) where δ1 = 2 3 = 2+s 5 (3.2) and α is a constant to be determined by the condition that η assumes the value 1 at the shock front. Second of equations (3.2) shows that the similarity solution of the present problem exists only when the total energy of the flow-field behind the shock increases as t4/3, that is only when s=4/3. From (3.1), we find that Ṙ2 = 4αmG∗ 9R dṘ dt =− Ṙ2 2R (3.3) From equations (2.4) and (3.3)1, we obtain the following expression for α in terms of the shock-Mach number M and Alfven-Mach number MA mG∗ RṘ2 = 9 4α = 1 γM2 − 1 2M2A (3.4) 506 K.K. Singh, B. Nath The quantity 9/(4α) (= δ, say) may be taken as a parameter of gravitation. To obtain similarity solutions, we write the unknown variables in the fol- lowing form (Vishwakarma and Yadav, 2003) u= ṘU(η) ρ= ρ1g(η) p= ρ1Ṙ 2P(η) √ µh= √ ρ1ṘH(η) (3.5) where U, g,P and H are functions of the non-dimansional variable (similarity variable) η only. The condition to be satisfied at the inner expanding surface is that the velocity of the fluid is equal to the velocity of the surface itself. This kinematic condition, from equations (3.1) and (3.5), can be written as U(ηp)= ηp (3.6) where ηp is the value of η at the inner expanding surface. Using similarity transformations (3.5), the equations of motion are trans- formed into −(η−U) dg dη +g (dU dη + 2U η ) =0 −(η−U) dU dη − U 2 + 1 g dP dη + H g dH dη + H2 gη + 9 4αη2 =0 (3.7) −(η−U) dH dη − H 2 +H dU dη + HU η =0 −(η−U) dP dη + γ(η−U) 1− bg P g dg dη −P =0 From equations (3.7), we have dU dη = [2γPU 1− bg + (η−U)g η2 ( 1 γM2 − 1 2M2A ) +(η−U) H2 η +H2 (U η − 1 2 ) + −(η−U) Ug 2 −P ][ (η−U)2g−H2− γP 1− bg ] −1 dg dη = g η−U (dU dη + 2U η ) dH dη = H η−U (dU dη − 1 2 + U η ) (3.8) dP dη =(η−U)g dU dη + Ug 2 − H2 η−U ( H dU dη − H 2 + HU η ) − H2 gη + − g η2 ( 1 γM2 − 1 2M2A ) Self-similar flow of a non-ideal gas... 507 The transformed shock conditions are U(1)= 1−β g(1)= 1 β P(1)= 1 γM2 + 1 2M2A ( 1− 1 β2 ) +1−β H(1)= 1 βMA (3.9) where β is given by equation (2.7). For exhibiting the numerical solutions, it is convenient to write the field variables in the non-dimensional form as u u2 = U(η) U(1) ρ ρ2 = g(η) g(1) p p2 = P(η) P(1) h h2 = H(η) H(1) (3.10) Ordinary differential equations (3.8) with boundary conditions (3.9) can now be numerically integrated to obtain the solution for the flow behind the shock surface. 4. Results and discussion Distribution of the flow variables in the flow-field behind the shock front are obtained by numerical integration of equations (3.8) with boundary condi- tions (3.9) by the Runge-Kutta method of the fourth order. For the purpose of numerical integration, the values of constant parameters are taken to be (Roberts and Wu, 1996, 2003; Rosenau and Frankenthal, 1976; Vishwakarma et al., 2007) γ =5/3; b=0, 0.025, 0.05; M =5, 10; M−2A =0, 0.02, 0.1. For a fully ionized gas γ = 5/3, and therefore it is applicable to stellar medium. Rosenau and Frankenthal (1976) have shown that the effects of magnetic field on the flow-field behind the shock are significant when M−2A ­ 0.01; therefore the above values of M−2A are taken for calculations in the present problem. The value b=0 corresponds to the perfect gas case. Figures 1-4 show the variation of the flow variables u/u2, ρ/ρ2, h/h2 and p/p2 with η at various values of the parameters b,M −2 and M−2A . It is shown that, as wemove inwards from the shock front towards the inner contact sur- face, the reduced fluid velocity u/u2 and the reduced azimuthalmagnetic field h/h2 increase, and the reduced density ρ/ρ2 decreases whereas the reduced pressure p/p2 increases when M −2 A 6= 0 and decreases when M −2 A = 0 (non- magnetic case). 508 K.K. Singh, B. Nath Fig. 1. Variation of reduced velocity u/u2 in the flow-field behind the shock front for γ=5/3 Fig. 2. Variation of reduced density ρ/ρ2 in the flow-field behind the shock front for γ=5/3 Self-similar flow of a non-ideal gas... 509 Fig. 3. Variation of reduced azimuthal magnetic field h/h2 in the flow-field behind the shock front for γ=5/3 Fig. 4. Variation of reduced pressure p/p2 in the flow-field behind the shock front for γ=5/3 510 K.K. Singh, B. Nath Table 1.Density ratio β across the shock front and the position of the inner expanding surface ηp for different values of b,M −2 and M−2A with γ=5/3 b M−2 M−2A β ηp 0 0 0.400000 0.7741250 0.04 0.02 0.418474 0.7519095 0.1 0.488109 0.6844721 0 0.325000 0.8236720 0.01 0.02 0.344752 0.8000735 0.1 0.417444 0.7327128 0.025 0 0.418750 0.7645350 0.04 0.02 0.435571 0.7432718 0.1 0.500606 0.6779128 0 0.343750 0.8150435 0.01 0.02 0.361215 0.7927513 0.1 0.428142 0.7280513 0.05 0 0.437500 0.7546676 0.04 0.02 0.452864 0.7342914 0.1 0.513580 0.6709548 0 0.362500 0.8061769 0.01 0.02 0.377997 0.7850806 0.1 0.439469 0.7230134 The density ratio β across the shock front and the position of the inner expanding surface ηp are tabulated in table 1 for γ = 5 3 and various values of b, M−2 and MA −2. It is found that the effects of an increase in the value of the parameter of non-idealness b of the gas are: (i) to increase the value of β, i.e. to decrease the shock strength (see Ta- ble 1); (ii) to increase the reduced velocity u/u2 and the reduced density ρ/ρ2 at any point in the flow field behind the shock (see Figs.1 and 2); (iii) to slightly increase the reduced pressure p/p2 and the reducedmagnetic field h/h2 at any point in the flow field near the shock front and to decrease these quantities at any point in the flow field near the inner expanding surface in the magnetic case whereas to increase the reduced pressure p/p2 at any point in the flow field behind the shock in the non-magnetic case (see Figs.3 and 4); Self-similar flow of a non-ideal gas... 511 (iv) to increase the distance of the inner expanding surface from the shock front (see Table 1). Physically it means that the gas behind the shock is less compressed, i.e. the shock strength is reduced, which is the same as given in (i) above. The effects of an increase in the value of M (i.e. a decrease in the value of M−2) are: (i) to decrease the value of β, i.e. to increase the shock strength (see Ta- ble 1); (ii) to decrease the reduced velocity u/u2 and the reduced density ρ/ρ2 at any point in the flow field behind the shock (see Figs.1 and 2); (iii) to increase the reduced pressure p/p2 and the reduced azimuthalmagne- tic field h/h2 at any point in the flow field behind the shock, in general; in the magnetic case, whereas to decrease the reduced pressure p/p2 at any point in the flow field behind the shock in the non-magnetic case (see Figs.3 and 4); (iv) to decrease the distance of the inner expanding surface from the shock front (see Table 1). The effects of an increase in the value of M−2A (i.e. the effects of an increase in the strength of ambient magnetic field) are: (i) to increase the value of β, i.e. to decrease the shock strength (see Ta- ble 1); (ii) to decrease the reduced velocity u/u2 and the reduced magnetic field h/h2, and to increase the reduced density ρ/ρ2 and the reduced pressure p/p2 at any point in the flow field behind the shock (see Figs.1 to 4); (iii) to increase the distance of the inner expanding surface from the shock front (see Table 1). References 1. Anisimov S.I., Spiner O.M., 1972, Motion of an almost ideal gas in the presence of a strong point explosion, J. Appl. Math. Mech., 36, 883-887 2. Carrus P.A., Fox P.A., Hass F., Kopal Z., 1951, Propagation of shock waves in the generalized Rochemodel,Astrophys. J., 113, 193-209 3. Director M.N., Dabora E.K., 1977, An experimental investigation of va- riable energy blast waves,Acta Astron., 4, 391-407 512 K.K. Singh, B. Nath 4. Freeman R.A., 1968, Variable energy blast wave, J. Phys. D, 1, 1697-1710 5. Ojha S.N., 2002, Shockwaves in non-ideal fluids, Int. J. Appl. Mech. Eng., 7, 2, 445-464 6. Ojha S.N., Nath Onkar, Takhar H.S., 1998, Dynamical behaviour of an unstable magnetic star, J. MHD Plasma Res., 8, 1-14 7. Ranga Rao M.P., Purohit N.K., 1976, Self-similar piston problem in non- ideal gas, Int. J. Eng. Sci., 14, 91-97 8. Ratkiewicz R., Axford W.I., McKenzie J.F., 1994, Similarity solutions for synchrotron emission froma supernovablastwave,Astron.Astrophys.,291, 3, 935-942 9. Roberts P.H., Wu C.C., 1996, Structure and stability of a spherical implo- sion,Phys. Lett. A, 213, 59-64 10. Roberts P.H., Wu C.C., 2003,The shock wave theory of sonoluminescence. In shock focussing effect in medical science and sonoluminescence, edited by R.C. Srivastava, D. Leutloff, K. Takayama andH. Gronig, Springer-Verlag 11. Rogers M.H., 1957, Analytic solutions for the blast waves problem with an atmosphere of varying density,Astrophys. J., 125, 2, 478-493 12. Rogers M.H., 1958, Similarity flows behind strong shock waves, Quart. J. Mech. Appl. Math., 11, 4, 411-422 13. Rosenau P., Frankenthal S., 1976, Shock disturbances in a thermally con- ducting solar wind,Astrophys. J., 208, 633-637 14. Sedov L.I., 1959, Similarity and Dimensional Methods in Mechanics, Acade- mic Press, London 15. Singh J.B., 1982, A self-similar flow in generalized Roche model with incre- asing energy,Astrophys. Space Sci., 88, 269-275 16. SinghV.K., SrivastavaG.K., 1989,Propagationof exponential shockwaves in magnetogasdynamics,Astrophys. Space Sci., 155, 215-224 17. Vishwakarma J.P., 2000, Propagation of shock waves in a dusty gas with exponentially varying density,Eur. Phys. J. B, 16, 369-372 18. Vishwakarma J.P., Maurya Anil Kumar, Singh K.K., 2007, Self-similar adiabatic flow headed by a magnetogasdynamic cylindrical shock wave in a rotating non-ideal gas,Geophysical and Astrophysical Fluid Dynamics, 101, 2, 155-168 19. Vishwakarma J.P., Nath G., 2007, Similarity solutions for the flow behind an exponential shock in a non-ideal gas,Meccanica, 42, 331-339 20. Vishwakarma J.P., Yadav A.K., 2003, Self-similar analytical solutions for blast waves in inhomogeneous atmosphere with frozen-in-magnetic field, Eur. Phys. J. B, 34, 247-253 Self-similar flow of a non-ideal gas... 513 Automorficzny przepływ gazu niedoskonałego o wzrastającej energii za magnetogazodynamiczną falą uderzeniową w obecności pola grawitacyjnego Streszczenie W pracy przedstawiono automorficzne rozwiązanie dla problemu propagacji sfe- rycznej fali uderzeniowej w gazie niedoskonałymw obecności azymutalnego polama- gnetycznego.Wrozważaniachprzyjęto, żeośrodekpodlegawpływowipolagrawitacyj- nego pochodzącego od jądra (model Roche’a) Nieustalonymodel Roche’a opisuje gaz o sferycznej symetrii dookoła jądra o dużejmasie m. Założono, że efekt grawitacyjny od samego ośrodka jest pomijalny w porównaniu do przyciągania od ciężkiego jądra. Przyjęto również, że całkowitaenergiapola przepływugazuza faląuderzeniową rośnie z upływem czasu. Otrzymano automorficzne rozwiązania dla tego zagadnienia oraz zbadanowpływ zmienności parametru określającegoniedoskonałość gazu b, zmienno- ści liczbyMacha M oraz Alfven-Macha MA na pole przepływu za falą uderzeniową. Manuscript received June 30, 2010; accepted for print December 1, 2010