scholarly journals Converging Cylindrical Symmetric Shock Waves in a Real Medium with a Magnetic Field

Symmetry ◽  
2019 ◽  
Vol 11 (9) ◽  
pp. 1177 ◽  
Author(s):  
Munesh Devi ◽  
Rajan Arora ◽  
Mustafa M. Rahman ◽  
Mohd Junaid Siddiqui
Keyword(s):  
Magnetic Field ◽  
Shock Waves ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Inertial Confinement Fusion ◽  
Matrix Form ◽  
Ideal Medium ◽  
Group Theoretic Method ◽  
Real Medium ◽  
Cylindrical Shock Waves

The topic “converging shock waves” is quite useful in Inertial Confinement Fusion (ICF). Most of the earlier studies have assumed that the medium of propagation is ideal. However, due to very high temperature at the axis of convergence, the effect of medium on shock waves should be taken in account. We have considered a problem of propagation of cylindrical shock waves in real medium. Magnetic field has been assumed in axial direction. It has been assumed that electrical resistance is zero. The problem can be represented by a system of hyperbolic Partial Differential Equations (PDEs) with jump conditions at the shock as the boundary conditions. The Lie group theoretic method has been used to find solutions to the problem. Lie’s symmetric method is quite useful as it reduces one-dimensional flow represented by a system of hyperbolic PDEs to a system of Ordinary Differential Equations (ODEs) by means of a similarity variable. Infinitesimal generators of Lie’s group transformation have been obtained by invariant conditions of the governing and boundary conditions. These generators involves arbitrary constants that give rise to different possible cases. One of the cases has been discussed in detail by writing reduced system of ODEs in matrix form. Cramer’s rule has been used to find the solution of system in matrix form. The results are presented in terms of figures for different values of parameters. The effect of non-ideal medium on the flow has been studied. Guderley’s rule is used to compute similarity exponents for cylindrical shock waves, in gasdynamics and in magnetogasdynamics (ideal medium), in order to set up a comparison with the published work. The computed values are very close to the values in published articles.

scholarly journals Propagation of Shock Waves in a Polytrope with a Toroidal Magnetic Field. II. Solution of Complete Differential Equations

1969 ◽  
Vol 22 (5) ◽  
pp. 605
Author(s):  
NK Sinha
Keyword(s):  
Magnetic Field ◽  
Shock Wave ◽  
Shock Waves ◽  
Differential Equations ◽  
Toroidal Magnetic Field ◽  
Initial Values ◽  
Made In

The differential equations for the shock parameters along shock rays in the case of propagation of a spherically developed shock wave in a polytrope with a toroidal magnetic field, obtained in Part I, have been integrated numerically for a particular set of initial values. The results are compared with the corresponding results in Part I obtained by neglecting certain small terms and it is found that the effect of this omission is not significant. This substantiates the results and justifies the simplification made in Part 1.

scholarly journals Propagation of Shock Waves in a Polytrope with a Toroidal Magnetic Field. I. Simplified Solution of Differential Equations

1969 ◽  
Vol 22 (5) ◽  
pp. 589
Author(s):  
NK Sinha
Keyword(s):  
Magnetic Field ◽  
Shock Wave ◽  
Approximate Solution ◽  
Partial Differential Equations ◽  
Shock Waves ◽  
Differential Equations ◽  
Toroidal Magnetic Field ◽  
Spherical Shock Wave ◽  
Simplified Solution ◽  
Spherical Shock

The propagation of an initially spherical shock wave in a polytrope with a magnetic field has been studied. The model chosen for the purpose was that of a poly trope with a toroidal magnetic field given previously by Sinha. Butler's method has been extended to transform the set of governing partial differential equations into a set of ordinary differential equations involving derivatives in the direction of propagation of the shock element at any point. An approximate solution is obtained and the effect of the toroidal magnetic field on the geometry of the front as well as on the effects brought about by the shock is discussed.

Numerical solution of the propagation of cylindrical shock waves in a magnetic field

1987 ◽  
Vol 27 (6) ◽  
pp. 879-883 ◽  
Author(s):  
N. A. Kudryashov ◽  
S. S. Kucherenko ◽  
A. I. Poberezhnyi
Keyword(s):  
Magnetic Field ◽  
Shock Waves ◽  
Numerical Solution ◽  
Cylindrical Shock Waves

scholarly journals Heat and Mass Transfer in Unsteady Boundary Layer Flow of Williamson Nanofluids

2020 ◽  
Vol 2020 ◽  
pp. 1-13 ◽  
Author(s):  
Tesfaye Kebede ◽  
Eshetu Haile ◽  
Gurju Awgichew ◽  
Tadesse Walelign
Keyword(s):  
Magnetic Field ◽  
Boundary Layer ◽  
Mass Transfer ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Thermal Radiation ◽  
Heat And Mass Transfer ◽  
Chemical Reaction ◽  
Stretching Sheet ◽  
Analytic Approximation

In this paper, analytic approximation to the heat and mass transfer characteristics of a two-dimensional time-dependent flow of Williamson nanofluids over a permeable stretching sheet embedded in a porous medium has been presented by considering the effects of magnetic field, thermal radiation, and chemical reaction. The governing partial differential equations along with the boundary conditions were reduced to dimensionless forms by using suitable similarity transformation. The resulting system of ordinary differential equations with the corresponding boundary conditions was solved via the homotopy analysis method. The results of the study show that velocity, temperature, and concentration boundary layer thicknesses generally decrease as we move away from the surface of the stretching sheet and the Williamson parameter was found to retard the velocity but it enhances the temperature and concentration profiles near the surface. It was also found that increasing magnetic field strength, thermal radiation, or rate of chemical reaction speeds up the mass transfer but slows down the heat transfer rates in the boundary layer. The results of this study were compared with some previously published works under some restrictions, and they are found in excellent agreement.

Split Mode Method for the Elliptic 2D Sine-Gordon Equation: Application to Josephson Junction in Overlap Geometry

1998 ◽  
Vol 09 (02) ◽  
pp. 301-323 ◽  
Author(s):  
Jean-Guy Caputo ◽  
Nikos Flytzanis ◽  
Yuri Gaididei ◽  
Irene Moulitsa ◽  
Emmanuel Vavalis
Keyword(s):  
Magnetic Field ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Josephson Junction ◽  
Gordon Equation ◽  
Splitting Method ◽  
Two Dimensional ◽  
Sine Gordon Equation ◽  
Dimensional Solution ◽  
One Dimensional

We introduce a new type of splitting method for semilinear partial differential equations. The method is analyzed in detail for the case of the two-dimensional static sine-Gordon equation describing a large area Josephson junction with overlap current feed and external magnetic field. The solution is separated into an explicit term that satisfies the one-dimensional sine-Gordon equation in the y-direction with boundary conditions determined by the bias current and a residual which is expanded using modes in the y-direction, the coefficients of which satisfy ordinary differential equations in x with boundary conditions given by the magnetic field. We show by direct comparison with a two-dimensional solution that this method converges and that it is an efficient way of solving the problem. The convergence of the y expansion for the residual is compared for Fourier cosine modes and the normal modes associated to the static one-dimensional sine-Gordon equation and we find a faster convergence for the latter. Even for such large widths as w=10 two such modes are enough to give accurate results.

The theory of inoizing shock waves in a magnetic field. Part 1. Skew and oblique shock waves, boundary conditions and ionization stability

1981 ◽  
Vol 26 (1) ◽  
pp. 29-53 ◽  
Author(s):  
M. A. Liberman ◽  
A. L. Velikovich
Keyword(s):  
Magnetic Field ◽  
Shock Wave ◽  
Shock Waves ◽  
Electric Field ◽  
Boundary Conditions ◽  
Wave Front ◽  
Shock Wave Front ◽  
Neutral Gas ◽  
Supersonic Shock ◽  
Photo Ionization

The general theory of ionizing shock waves in a magnetic field has been constructed. The theory takes into account precursor ionization of a neutral gas ahead of the shock wave front, caused by photo-ionization, as well as by the impact ionization with electrons accelerated by a transverse electric field induced by the shock front in the incident flow of a neutral gas. The concept of shock wave ionization stability, being basic in the theory of ionizing shock waves in a magnetic field, is introduced. An additional equation for the electric field in the shock wave is obtained. This equation, together with the investigation of the singular point in the downstream flow behind the shock wave front, provides all the information required for solving the problem. For example, this provides two additional boundary conditions for the shock waves of type 2, determining the value and direction of the electric field in the incident flow. One additional boundary condition determines a relation between the value and direction of the electric field for supersonic shock waves of type 3. There are no additional boundary conditions for supersonic shock waves of type 4. The electric field ahead of the shock front has two degrees of freedom. As well as for shocks of other types, its value is less than that of the transverse electric field at which an ionization wave could be emitted by the shock wave front (the ionization stability condition). The additional relationship for supersonic waves of type 4 determines the onset of an isomagnetic (viscous) jump in the structure of the shock wave front. The boundary conditions and ionizing shock wave structures, considered earlier by the authors of the present paper in the ‘limit of electrostatic breakdown’, as well as the structural determination of the electric field, considered earlier by Leonard, are limiting cases in the theory developed here. The ionizing shock wave structures are shown to transform from the GD regime at a low shock velocity to the MHD regime at an enhanced intensity of the shock wave. The abruptness of such a transition (e.g. the transition width on the Mach number scale) is determined by precursor photo-ionization.

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