Limits...
Flow behind an exponential shock wave in a rotational axisymmetric perfect gas with magnetic field and variable density

View Article: PubMed Central - PubMed

ABSTRACT

A self-similar model for one-dimensional unsteady isothermal and adiabatic flows behind a strong exponential shock wave driven out by a cylindrical piston moving with time according to an exponential law in an ideal gas in the presence of azimuthal magnetic field and variable density is discussed in a rotating atmosphere. The ambient medium is assumed to possess radial, axial and azimuthal component of fluid velocities. The initial density, the fluid velocities and magnetic field of the ambient medium are assumed to be varying with time according to an exponential law. The gas is taken to be non-viscous having infinite electrical conductivity. Solutions are obtained, in both the cases, when the flow between the shock and the piston is isothermal or adiabatic by taking into account the components of vorticity vector. The effects of the variation of the initial density index, adiabatic exponent of the gas and the Alfven-Mach number on the flow-field behind the shock wave are investigated. It is found that the presence of the magnetic field have decaying effects on the shock wave. Also, it is observed that the effect of an increase in the magnetic field strength is more impressive in the case of adiabatic flow than in the case of isothermal flow. The assumption of zero temperature gradient brings a profound change in the density, non-dimensional azimuthal and axial components of vorticity vector distributions in comparison to those in the case of adiabatic flow. A comparison is made between isothermal and adiabatic flows. It is obtained that an increase in the initial density variation index, adiabatic exponent and strength of the magnetic field decrease the shock strength.

No MeSH data available.


Related in: MedlinePlus

Variation of the reduced flow variables in the region behind the shock front in the case of adiabatic flow: a radial component of fluid velocity , b azimuthal component of fluid velocity , c axial component of fluid velocity , d density , e pressure , f azimuthal magnetic field , g non-dimensional azimuthal component of vorticity vector , h non-dimensional axial component of vorticity vector : 1. , , ; 2. , , ; 3. , , ; 4. , , ; 5. , , ; 6. , , ;    7. , , ;    8. ,  , ; 9. , , ; 10. , , ; 11. , , ; 12. , ,
© Copyright Policy - OpenAccess
Related In: Results  -  Collection

License
getmorefigures.php?uid=PMC5016518&req=5

Fig2: Variation of the reduced flow variables in the region behind the shock front in the case of adiabatic flow: a radial component of fluid velocity , b azimuthal component of fluid velocity , c axial component of fluid velocity , d density , e pressure , f azimuthal magnetic field , g non-dimensional azimuthal component of vorticity vector , h non-dimensional axial component of vorticity vector : 1. , , ; 2. , , ; 3. , , ; 4. , , ; 5. , , ; 6. , , ;    7. , , ;    8. , , ; 9. , , ; 10. , , ; 11. , , ; 12. , ,

Mentions: The distribution of the flow variables between the shock front and the inner expanding surface or piston is obtained by the numerical integration of Eqs. (38)–(42) for isothermal flow, and from Eqs. (58)–(63) for adiabatic flow with the boundary conditions (48) and (49) by the Runge–Kutta method of the fourth order. The values of the constant parameters, for the determination of numerical integration, are taken to be (Rosenau and Frankenthal 1976; Nath 2011) . For fully ionized gas and for relativistic gases , which are applicable to interstellar medium. These two values of mark the most general range of values seen in real stars. For stars, the stability is related with the value of the adiabatic index in its interior that has to be larger than (Onsi et al. 1994; Casali and Menezes 2010). The stability of a star depends on the value of in the core being larger than , collapse beginning when falls below . However, as nuclear densities are approached in the core will rise above again, with the result that the collapse will come rapidly to a halt, and be reversed into a bounce that may lead to a supernova explosion (Onsi et al. 1994). So, the above values of are taken for calculations in the present problem. The above values of are taken for calculations in the present problem because Rosenau and Frankenthal (1976) have shown that the effects of magnetic field on the flow-field behind the shock are significant when . The non-magnetic case is represented by . In the present problem, we have taken initial density variation index for numerical calculations i. e. initial density of the ambient medium is assumed to be decreasing. There is astrophysical evidence for the existence of shocks propagating in regions of variable density. In a stellar explosion, the shock wave is expected to accelerate through the outer stellar layers where the density is decreasing rapidly with height. A similar situation may occur for an explosion in the gaseous atmosphere of a galaxy. Self-similar solutions provide an excellent description of the shock propagation because the accelerating shock structure becomes independent of the nature of initial explosion (Chevalier 1990). The present work is the extension to the work of Rao and Ramana (1976) by taking into account the rotation of the medium and the azimuthal magnetic field with variable density (see Figs. 1b, c, e–g, 2b, c, f–h).Fig. 1


Flow behind an exponential shock wave in a rotational axisymmetric perfect gas with magnetic field and variable density
Variation of the reduced flow variables in the region behind the shock front in the case of adiabatic flow: a radial component of fluid velocity , b azimuthal component of fluid velocity , c axial component of fluid velocity , d density , e pressure , f azimuthal magnetic field , g non-dimensional azimuthal component of vorticity vector , h non-dimensional axial component of vorticity vector : 1. , , ; 2. , , ; 3. , , ; 4. , , ; 5. , , ; 6. , , ;    7. , , ;    8. ,  , ; 9. , , ; 10. , , ; 11. , , ; 12. , ,
© Copyright Policy - OpenAccess
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC5016518&req=5

Fig2: Variation of the reduced flow variables in the region behind the shock front in the case of adiabatic flow: a radial component of fluid velocity , b azimuthal component of fluid velocity , c axial component of fluid velocity , d density , e pressure , f azimuthal magnetic field , g non-dimensional azimuthal component of vorticity vector , h non-dimensional axial component of vorticity vector : 1. , , ; 2. , , ; 3. , , ; 4. , , ; 5. , , ; 6. , , ;    7. , , ;    8. , , ; 9. , , ; 10. , , ; 11. , , ; 12. , ,
Mentions: The distribution of the flow variables between the shock front and the inner expanding surface or piston is obtained by the numerical integration of Eqs. (38)–(42) for isothermal flow, and from Eqs. (58)–(63) for adiabatic flow with the boundary conditions (48) and (49) by the Runge–Kutta method of the fourth order. The values of the constant parameters, for the determination of numerical integration, are taken to be (Rosenau and Frankenthal 1976; Nath 2011) . For fully ionized gas and for relativistic gases , which are applicable to interstellar medium. These two values of mark the most general range of values seen in real stars. For stars, the stability is related with the value of the adiabatic index in its interior that has to be larger than (Onsi et al. 1994; Casali and Menezes 2010). The stability of a star depends on the value of in the core being larger than , collapse beginning when falls below . However, as nuclear densities are approached in the core will rise above again, with the result that the collapse will come rapidly to a halt, and be reversed into a bounce that may lead to a supernova explosion (Onsi et al. 1994). So, the above values of are taken for calculations in the present problem. The above values of are taken for calculations in the present problem because Rosenau and Frankenthal (1976) have shown that the effects of magnetic field on the flow-field behind the shock are significant when . The non-magnetic case is represented by . In the present problem, we have taken initial density variation index for numerical calculations i. e. initial density of the ambient medium is assumed to be decreasing. There is astrophysical evidence for the existence of shocks propagating in regions of variable density. In a stellar explosion, the shock wave is expected to accelerate through the outer stellar layers where the density is decreasing rapidly with height. A similar situation may occur for an explosion in the gaseous atmosphere of a galaxy. Self-similar solutions provide an excellent description of the shock propagation because the accelerating shock structure becomes independent of the nature of initial explosion (Chevalier 1990). The present work is the extension to the work of Rao and Ramana (1976) by taking into account the rotation of the medium and the azimuthal magnetic field with variable density (see Figs. 1b, c, e–g, 2b, c, f–h).Fig. 1

View Article: PubMed Central - PubMed

ABSTRACT

A self-similar model for one-dimensional unsteady isothermal and adiabatic flows behind a strong exponential shock wave driven out by a cylindrical piston moving with time according to an exponential law in an ideal gas in the presence of azimuthal magnetic field and variable density is discussed in a rotating atmosphere. The ambient medium is assumed to possess radial, axial and azimuthal component of fluid velocities. The initial density, the fluid velocities and magnetic field of the ambient medium are assumed to be varying with time according to an exponential law. The gas is taken to be non-viscous having infinite electrical conductivity. Solutions are obtained, in both the cases, when the flow between the shock and the piston is isothermal or adiabatic by taking into account the components of vorticity vector. The effects of the variation of the initial density index, adiabatic exponent of the gas and the Alfven-Mach number on the flow-field behind the shock wave are investigated. It is found that the presence of the magnetic field have decaying effects on the shock wave. Also, it is observed that the effect of an increase in the magnetic field strength is more impressive in the case of adiabatic flow than in the case of isothermal flow. The assumption of zero temperature gradient brings a profound change in the density, non-dimensional azimuthal and axial components of vorticity vector distributions in comparison to those in the case of adiabatic flow. A comparison is made between isothermal and adiabatic flows. It is obtained that an increase in the initial density variation index, adiabatic exponent and strength of the magnetic field decrease the shock strength.

No MeSH data available.


Related in: MedlinePlus