IONIZING SHOCK WAVE IN A ROTATING AXISYMMETRIC NON- IDEAL GAS WITH HEAT CONDUCTION AND RADIATION HEAT FLUX

Transcription

1 International Journal of Physics and Mathematical Sciences ISSN: 77- (Online) 7 Vol. 7 () January-March, pp. 5-6/Nath and Talukdar esearch rticle IONIZING SHOCK WVE IN OTTING XISYMMETIC NON- IDEL GS WITH HET CONDUCTION ND DITION HET FLUX *B. Nath and S. Talukdar Department of Mathematics, Girijananda Chowdhury Institute of Management and Technology (GIMT)- Tezpur, Dist.: Sonitpur, ssam, India, PIN: 7845 *uthor for Correspondence BSTCT The propagation of a cylindrical ionizing shock wave in a rotational axisymmetric non-ideal gas with heat conduction and radiation heat flux, in presence of an azimuthal magnetic field is investigated. The electrical conductivity of the medium behind the shock is assumed to be negligible, which becomes infinitely large due to the passage of the shock. The ambient medium is assumed to have variable axial and azimuthal velocity components and the initial density and the magnetic field are assumed to obey power laws. The thermal conductivity and the absorption coefficients are assumed to vary with temperature and the total energy of the wave to vary with time. Similarity solutions are obtained and the effects of variation of the initial density exponent and the non-idealness parameter on the flow field are investigated. Keywords: Ionizing Shock Wave, Non-Ideal Gas, xisymmetric Flow, Similarity Solution, Heat Transfer Effects INTODUCTION The experimental studies and astrophysical observations show that the outer atmosphere of the planets rotates due to rotation of the planets. Macroscopic motion with supersonic speed occurs in an interplanetary atmosphere and shock waves are generated. Thus, the rotation of planets or stars significantly affects the process taking place in their outer layers. Therefore, the questions connected with the explosions in the rotating gas atmospheres are of definite astrophysical interest. Shock wave often arises in nature because of a balance between wave breaking non-linear and wave damping disspative forces (Zel dovich Yab, aizer YwP). Chaturani (97) studied the propagation of cylindrical shock waves through a gas having solid body rotation and obtained the solution by a similarity method introduced by Sakurai (956). Nath et al., (999) obtained the similarity solutions for the flow behind a spherical shock wave propagating in a non-uniform rotating interplanetary atmosphere with increasing energy. Vishwarkarma and Vishwakarma (7) and Vishwakarma et al., (7) obtained similarity solutions for magnetogasdynamics cylindrical shock waves propagating in a rotating medium. They have taken the electrical conductivity of the initial medium as well as the medium behind the shock to be infinite. But, in many practical cases the medium may be of low conductivity which becomes highly conducting due to passage of shock. Such a shock wave is called a gas-ionizing shock or simply ionizing shock. The propagation of an ionizing shock has been studied by various authors namely Greenspan (96), Greigfinger and Cole (96), Christer (97), angarao and amana (97), Singh (98) for nonrotating medium. lso, the medium in which the shock is propagating is considered to be ideal gas. But at extreme conditions that prevail in most of the problems associated with shock waves, the assumption that the gas is ideal is no longer valid. In recent years, several studies have been performed concerning the problem of shock waves in non-ideal gas. nisimov and Spiner (97), investigated the problem of a point explosion in a non-ideal gas by taking the equation of state in a simplified form. oberts and Wu (996, ) studied the structure and stability of a spherical implosion by assuming the gas to obey a simplified van der Waals equation of state. Vishwakarma and Singh () investigated the cylindrical ionizing shock wave to a non-ideal gas in presence of radiation heat flux considering the van der Waals equation of state. Centre for Info Bio Technology (CIBTech) 5

2 International Journal of Physics and Mathematical Sciences ISSN: 77- (Online) 7 Vol. 7 () January-March, pp. 5-6/Nath and Talukdar esearch rticle Marshak (958) studied the effects of radiation on the shock propagation by introducing the radiation diffusion approximation. Using the same mode of radiation, Elliott (96) discussed the conditions leading to self-similarity with a specified functional form of the mean free path of radiation and obtained a solution for self- similar spherical explosions. Gretler and Wehle (99) studied the propagation of blast waves with exponential heat release by taking internal heat conduction and thermal radiation in a detonating medium. lso, bdel-aouf and Gretler (99) obtained the non-self-similar solution for the blast waves with internal heat transfer effects. Ghoniem et al., (98) obtained a self-similar solution for spherical explosions taking into account the effects of both conduction and radiation in the two limits of osseland radiative diffusion and Plank radiative emission. In these works, where both radiation and heat conduction are considered, the density of the ambient medium is taken to be constant. lso, the effects of presence of the magnetic field are omitted. Keeping in view of the above, in this paper we are considering the propagation of a gas ionizing shock wave in a rotating axisymmetric gas with heat conduction and radiation heat flux, in presence of an azimuthal magnetic field. The initial density of the gas and the initial azimuthal magnetic field are assumed to vary as some power of distance. The heat transfer fluxes are expressed in terms of Fourier s law for heat conduction and a diffusion radiation mode for an optically thick grey gas, which is typical of large scale explosions. The thermal conductivity and absorption co-efficient of the gas are assumed to be proportional to appropriate powers of temperature and density (Ghoneim et al., 98). Similarity, solutions are obtained and the effects of variation of the initial density exponent and the non-idealness parameter on the flow field are investigated. Equations of Motion and Boundary Conditions The fundamental equations governing the unsteady adiabatic axisymmetric rotational flow of a perfectly conducting non-ideal gas in which heat conduction and radiation heat flux are taken into account in presence of an azimuthal magnetic field, may be expressed as (Christer and Helliwell, 969; Gretler and Whele, 99; Levin and Skopina, 4; Singh and Nath, ) u u u () t r r r u u p h h v u h () t r r r r r h h u u h () t r r v uv u t r r (4) w w u r r (5) e e p u u Fr t r t r (6) r r Where, r and t are independent space and time co-ordinates, respectively, the density, p the pressure, h the azimuthal magnetic field, u, v and w are the radial, azimuthal and axial components of the fluid velocity q in the cylindrical coordinates r,,z, is the magnetic permeability, F is the heat flux and e is the internal energy per unit mass. The total heat flux F, which appears in the energy equation may be decomposed as F F F (7) C Where, F C is the conduction heat flux and F is the radiation heat flux. T ccording to the Fourier s law of heat conduction FC K (8) r Centre for Info Bio Technology (CIBTech) 6

3 International Journal of Physics and Mathematical Sciences ISSN: 77- (Online) 7 Vol. 7 () January-March, pp. 5-6/Nath and Talukdar esearch rticle Where, K is the coefficient of thermal conductivity of the gas and T is the absolute temperature. ssuming local thermo dynamical equilibrium and using the radiative diffusion model for an optically thick grey gas (Pomraning, 97) the term F, which represents the radiative heat flux, may be obtained from the differential approximation of the radiation-transport equation in the diffusion limit as 4 4 T F, (9) r Where, is the Stefan-Boltzman constant and is the osseland mean absorption co-efficient. lso, v r () Where, is the angular velocity of the medium at a radial distance r from the axis of symmetry. In this case the vorticity vector curlq has the components w r,, z r v. () r r r The above system of equations should be supplemented with an equation of state. To discover how deviation from the ideal gas can affect the solutions, we adopt a simple model. We assume that the gas obeys a simplified van der Waals equation of state of the form (oberts and Wu, 996, ; Nath, 7; Singh and Nath, ) T p b p, e CVT () b Where, is the gas constant, CV is the specific heat at constant volume and is the ratio of specific heats. The constant b is the van der Waals excluded volume; it places a limit max, on the b density of the gas. The thermal conductivity K and the absorption coefficient are assumed to vary with temperature and density. These can be written in the form of power laws, namely (Ghonien et al., 98). C C T T K K, T T () Where, subscript denotes a reference state. The exponents in above equations should satisfy the similarity requirements if a self-similar solution is sought. We assume that a cylindrical shock wave is propagating outwards from the axis of symmetry in a rotating non-ideal gas with variable density which has a zero radial velocity, a variable azimuthal and axial velocity and negligible electrical conductivity in presence of an azimuthal magnetic field. The flow variables immediately ahead of the shock front are u,, h d D m h, m m m h d B D p d m, v B, w a G, F (Laumbach and Probstein, 97),, where d m, m (4) Where, is the shock radius, a, d, m, h, D, B, G are constants. From equations () and (4), we find the initial angular velocity varies as B, it decreases as the distance from the axis increases if head of the shock, the components of the vorticity vector, therefore vary as r, a Ga, z B. Centre for Info Bio Technology (CIBTech) 7

4 International Journal of Physics and Mathematical Sciences ISSN: 77- (Online) 7 Vol. 7 () January-March, pp. 5-6/Nath and Talukdar esearch rticle Due to passage of the shock, the gas is highly ionized and its electrical conductivity becomes infinitely large. The conditions across such a gas ionizing shock are (Singh & Srivastava, 99; Viswakarma & Pandey, 6) (V u ) V m s p p m u (5) s p F p e (V u ) e V v h h, v v,w w,t T where, the subscripts and refer to the values just ahead and just behind the shock respectively and V denotes the shock velocity. From equations (5) we get u ( )V p ( ) V M (6) V b F b M M where b, and bb is the parameter of non-idealness of the gas. Here, M is M the shock-mach number referred to the frozen speed of sound p Following Levin and Skopina (4) and Nath () we obtain the jump conditions for the components of the vorticity vector across the shock as, z z The total energy of the flow field behind the shock is not constant, but assumed to be time dependent and varying as (ogers, 958; Freeman, 968). s E Et s where, E and s are constants. 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 fluid by an inner expanding surface (a contact surface or a piston). Similarity Solutions We introduce the following similarity transformations to reduce the equations of motion into ordinary differential equations u VU, v V, w VW, g, p V P, F V Q, h VH (7) Where, U,, W, g, P, H, Q are functions of the non-dimensional parameter only and r, t rp Evidently p at the inner expanding surface and at the shock front The total energy of the gas behind the shock is given by Centre for Info Bio Technology (CIBTech) 8

5 International Journal of Physics and Mathematical Sciences ISSN: 77- (Online) 7 Vol. 7 () January-March, pp. 5-6/Nath and Talukdar esearch rticle p b E u v w rdr E t h s rp Where, rp is the radius of the inner expanding surface. pplying similarity transformations (7), the relation (8) becomes s Et J v (9). P bg H J gu W d where p For similarity transformations, the shock-mach number and the lfven-mach number constant. Therefore we have m d, V L Where, L is a constant. From equation () we get L t lso from equation (9) we get s E d d d t J L D Comparing equations () and () and using relation () we obtain m s This show that, for s,, m B m m From equation L M M Where, M V h is the lfven-mach number Then the shock conditions (7) are transformed into m m G U() ( ), g(), (), W(), H() M, M M L b P() ( ), M Centre for Info Bio Technology (CIBTech) 9 (8) M must be b Q (4) M M Where, a is taken in order to make the shock conditions consistent for similarity solutions. In addition to the above mentioned shock conditions, the condition to be satisfied at the inner boundary surface is that the velocity of the fluid is equal to the velocity of the inner boundary itself. This kinematic condition can be written as U p p Using transformations (7), the equations of motion () to (6) takes the form () () () ()

6 International Journal of Physics and Mathematical Sciences ISSN: 77- (Online) 7 Vol. 7 () January-March, pp. 5-6/Nath and Talukdar esearch rticle dg du U U g gd d d du dp H dh H U bu d g d g d g dh du d U H H d d U b d U dw bw d d U Q dp P dg dq d U (U ) d d g bg d bg d bg bg Using equations (8), (9) and (7) in (7), we get K c 6 c c T F T T T c c T r Using equations () and (7) in the equation () we get d c c c c d Kg P bg D c c V d c c c c T L bg dp P dg Q d c c g d g d 6T g P bg D 5 d V d 4 L which shows that the similarity solution of the present problem exists only when d c c, 5 d Therefore, equation () becomes bg dp p dg Q X g d g d Where, Here, C and The parameters d d d c c c d c X g p bg g p bg are the conductive and radiative non-dimensional heat transfer parameters respectively. and C (5) (6) (7) (8) (9) () () () () depend on the thermal conductivity K and the mean free path of the radiation respectively and also on the exponents d and and are given by d c dc KD L C d d c c c T Centre for Info Bio Technology (CIBTech)

7 International Journal of Physics and Mathematical Sciences ISSN: 77- (Online) 7 Vol. 7 () January-March, pp. 5-6/Nath and Talukdar esearch rticle d 5 d d 6T D L d T By solving equations (5) to () and () we have dg g du U d d U d dh H du d b d U d d U b d U dw bw d U gu dp H du H d H g bug d U d U H du d U H g U H bug U dq bg bg g d d PH d gu d Q P d bgg bg bg H bg d U bu bg du gu gu d (4) P H gu bg P U Q bg d gu X lso applying the similarity transformations on equation (), we obtain the non dimensional components r z of the vorticity vectors l r, l, l z, in the flow field behind the shock as V V V U d aw lr, l, lz U U Now the ordinary differential equations (4) to (4) may be integrated numerically with the boundary conditions (4) and appropriate values of the constant parameters to obtain the solutions. For exhibiting the numerical solution it is convenient to write the flow variables in the following form as u U u U, w W, w W v v Centre for Info Bio Technology (CIBTech) g p P,,, g p P h H h H, F Q F Q, (4) (5) (6) (7) (8) (9)

8 International Journal of Physics and Mathematical Sciences ISSN: 77- (Online) 7 Vol. 7 () January-March, pp. 5-6/Nath and Talukdar esearch rticle ESULTS ND DISCUSSION In order to get physical insight into the problem considered the non-dimensional flow variables v, w,, h, p, v w h p F the boundary conditions (4). F Centre for Info Bio Technology (CIBTech) u, u are obtained by integrating the equations (4) to (4) numerically with Throughout our discussion, the values of the constant parameters are taken as 5 ; M.; G M.5,.;.; Q m.; d.5,.; C ; ; C ; and b,.5,.. For 5 a fully ionized gas, and therefore it is applicable to stellar medium. The value b, corresponds to the case of perfect gas. The results obtained are discussed thoroughly in graphs (Figure to 9) and in tabular form. Figures to 7 show the variation of the flow variables for different values of the parameters and figure 8 and 9 show the variation of non-zero and non-dimensional azimuthal and axial components l and lz of the vorticity vector. Figure and 5 shows that as we move towards the inner expanding surface, the non-dimensional radial velocity and the magnetic field increases for the case when d.5 and decreases when d. It can be observed from figures, and 7 that the non-dimensional azimuthal velocity, the axial velocity and the total heat flux decreases as we move inward from the shock front. Figure 4 shows that the non-dimensional density decreases slightly but increases rapidly near the inner expanding surface. Figure 6 shows that the non-dimensional pressure increases for the case when d whereas decreases rapidly near the inner expanding surface for the case when d.5. Figures 8 and 9 show that the profiles of the non-dimensional vorticity vector increases near the inner expanding surface except for the case when d where l z exhibits no visible change. Table shows the variation of p for different values of M, d and b. Table shows that the variation of the density ratio for different values of b. The Effects of an Increase in the Value of the Non-Idealness Parameter are (i) to increase the density ratio i.e. to decrease the shock strength. (See table ) (ii) to increase the distance of the inner expanding surface from the shock front. Thus, the non-idealness of the gas has a decaying effect on the shock strength. (See table ) (iii) to decrease the reduced radial velocity, pressure, magnetic field where as to increase the axial and azimuthal velocity component and density. To decrease the non-dimensional reduced heat flux for the case when d.5 and to increase when d.. (iv) To decrease the non-dimensional reduced axial component of vorticity vector in the case when d.5 The Effects of Increase in the Index for Initial Density are (i) to increase the distance of the inner expanding surface from the shock front i.e. the gas behind the shock is less compressed. (See table ) (ii) to decrease the non-dimensional reduced radial velocity, azimuthal velocity, magnetic field, density, total heat flux where as to increase the non-dimensional reduced axial velocity and pressure. (iii) to decrease the reduced axial component of the vorticity vector z l and to increase the reduced azimuthal component of the vorticity vector l.

9 International Journal of Physics and Mathematical Sciences ISSN: 77- (Online) 7 Vol. 7 () January-March, pp. 5-6/Nath and Talukdar esearch rticle Figure : Variation of the educed adial Component of Fluid Velocity in the Flow Field behind the Shock Figure : Variation of educed zimuthal Component of Fluid Velocity in the Flow Field behind the Shock Figure : Variation of the educed xial Component of Fluid Velocity in the Flow Field behind the Shock Figure 4: Variation of the educed Density in Flow Field behind the Shock Centre for Info Bio Technology (CIBTech)

10 International Journal of Physics and Mathematical Sciences ISSN: 77- (Online) 7 Vol. 7 () January-March, pp. 5-6/Nath and Talukdar esearch rticle Centre for Info Bio Technology (CIBTech) 4

11 International Journal of Physics and Mathematical Sciences ISSN: 77- (Online) 7 Vol. 7 () January-March, pp. 5-6/Nath and Talukdar esearch rticle Table : Position of the Inner Expanding Surface 5 and, M., m., c, and G. Q p for Different Values of M, d and b M d b p Table : Density atio for Different Values of b and and G. Q b , M., m., c, EFEENCE bdel-auof M and Gretler W (99). Quasi-similar solutions for blast wave with internal heat transfer effects. Fluid Dynamics esearch Christer H and Helliwell JB (969). Cylindrical shock and detonation waves in magnetogasdynamics. Journal of Fluid Mechanics Chaturani P (97). Strong cylindrical shocks in a rotating gas. pplied Scince esearch 97-. Christer H (97). Self-similar cylindrical ionizing shock and detonation waves. ZMM-Journal of pplied Mathematics and Mechanics 5 -. Director MN and Dabora EK (977). n experimental investigation of variable energy blast waves. cta stronautica Elliott L (96). Similarity methods in radiation hydrodynamics. Proceedings of oyal Society, London Freeman (968). Variable energy blast waves. Journal of Physics D Greenspan HP (96). Similarity solution for a cylindrical shock magnetic field interaction. Physics of Fluids Greinfinger C and Cole JD (96). Similarity solution for cylindrical magnetohydrodynamic blast waves. Physics of Fluids Centre for Info Bio Technology (CIBTech) 5

12 International Journal of Physics and Mathematical Sciences ISSN: 77- (Online) 7 Vol. 7 () January-March, pp. 5-6/Nath and Talukdar esearch rticle Ghoniem F, Kamel MM, Berger S and Oppenheim K (98). Effects of internal heat transfer on the structure of self-similar blast wave. Journal of Fluid Mechanics Gretler W and Wehle P (99). Propagation of blast waves with exponential heat release and internal heat conduction and thermal radiation. Shock Waves Gretle W and Whele P (99). Propagation of blast waves with exponential heat release and internal heat conduction and thermal radiation. Shock Waves Laumbach DD and Probstein F (97). point explosion in a cold exponential atmosphere- part : adiating flow. Journal of Fluid Mechanics Levin V and Skopina G (4). Detonation wave propagation in a rotational gas flows. Journal of pplied Mechanics and Technical Physics Marshak E (958). Effects of radiation on shock wave behavior. Physics of Fluids 4-9. Nath O, Ojha SN and Takhar HS (999). Propagation of a shock wave in a rotating interplanetary atmosphere with imcreasing energy. Journal of MHD and Plasma esearch Nath G (7). Shock wave generated by a piston moving in a non-ideal gas in the presence of a magnetic field: Isothermal flow. South East sian Journal of Mathematics and Mathematical Science Pomraning GC (97). The equations of radiation hydrodynamics. International Series of Monographs in Natural Philosophy 54, (Pergamon Press, Oxford, UK). angarao MP and amana BV (97). Self-similar cylindrical magnetogasdynamics and ionizing shock waves. International Journal of Engineering Science 7-5. oberts PH and Wu CC (996). Structure and stability of a spherical implosion. Physics. Letters 5-64 oberts PH and Wu CC (). The shock wave theory of sonoluminescence. In Shock Focussing Effects in Medical Science and Sonoluminescence, Edited by Srivastava C, Leutloff D, Takayama K and Groning H, (Springer-Verlag, Heidelberg, Germany). Sakurai (956). Propagation of spherical waves in stars. Journal of Fluid Mechanics Singh JB (98). Self-similar ionizing shock waves with radiation heat flux. Indian Journal of Technology 5-8. Singh JB and Srivastava SK (99). Cylindrical ionizing shock waves in a non-ideal gas. Journal of Purvanchal cademy of Science, Jaunpur Singh KK and Nath B (). Self-similar flow of a non-ideal gas with increasing energy behind a magnetogasdynamics shock wave under a gravitational field. Journal of Theoretical and pplied Mechanics 49() 5-5. Singh KK and Nath B (). Self-similar cylindrical ionizing shock wave in a rotating axisymmetric gas flow. Indian Journal of Science and Technology (Special Issue) 6(S) 7-5. Vishwakarma JP and Pandey SN (6). Converging cylindrical shock waves in a non-ideal gas with an axial magnetic field. Defence Science Journal Vishwakarma JP and Vishwakarma S (7). Magnetogasdynamic cylindrical shock wave in a rotating gas with variable density. International Journal of pplied Mechanics and Engineering Vishwakarma JP, Maurya K and Singh KK (7). Self-similar adiabatic flow headed by a magnetogasdynamics cylindrical shock wave in a rotating non-ideal gas. Geophysical strophysical Fluid Dynamics () Vishwakarma JP and Singh M (). Self similar cylindrical ionizing shock waves in a non-ideal gas with radiation heat-flux. pplied Mathematics -7. Zel dovich Ya B and aizer Yu P (967) Physics of Shock Waves and High Temperature Hydrodynamic Phenomena, (cademic Press, New York, US). Centre for Info Bio Technology (CIBTech) 6