In this paper, a non-dimensional unsteady adiabatic flow of a plane or cylindrical strong shock wave propagating in plasma is studied. The plasma is assumed to be an ideal gas with infinite electrical conductivity permeated by a transverse magnetic field. A self-similar solution of the problem is obtained in terms of density, velocity and pressure in the presence of magnetic field. We use the method of Lie group invariance to determine the class of self-similar solutions. The arbitrary constants, occurring in the expressions of the generators of the local Lie group of transformations, give rise to different cases of possible solutions with a power law, exponential or logarithmic shock paths. A particular case of the collapse of an imploding shock is worked out in detail. Numerical calculations have been performed to obtain the similarity exponents and the profiles of flow variables. Our results are found in good agreement with the known results. All computational work is performed by using software package MATHEMATICA.
Keywords:Lie group; similarity solutions; magnetogasdynamics; shock waves
The spread of shock waves under the control of strong magnetic field is a problem of great interest to researchers in a variety of fields such as nuclear science, geophysics, plasma physics and astrophysics. Hunter [], Guderley [], Greifinger and Cole [] studied the problem of blast wave propagation in homogeneous and inhomogeneous media. Most recently, van Dyke and Guttmann [], Sharma and Radha [, ], Madhumita and Sharma [], Pandey et al. [], Sharma and Arora [], Arora et al. [, ] presented high accuracy results and alternative approaches for the investigation of blast wave by using the self-similar solutions method. In the same decade, a number of analytical solutions for the blast wave propagations have been obtained by Sachdev [], Chisnell [] and Singh et al. [, ]. Chisnell [] provided analytical solutions to the problem of converging shock waves by using the singular points method. Singh et al. [, ] used the method of Lie group of transformations to obtain an exact solution for unsteady equation of non-ideal gas and magnetogasdynamics.
The magnetic fields have important roles in a variety of astrophysical situations. Complex filamentary structures in molecular clouds, shapes and the shaping of planetary nebulae, synchrotron radiation from supernova remnants, magnetized stellar winds, galactic winds, gamma-ray bursts, dynamo effects in stars, galaxies and galaxy clusters as well as other interesting problems all involve magnetic fields. When the internal disturbances accompanied by an increase in pressure take place in the central region of a star, a shock wave is formed. It travels from the central region to the periphery and emerges at the surface of the star. In the present paper, we consider the problem of propagation of a one-dimensional unsteady flow of an inviscid ideal gas permeated by a transverse magnetic field with infinite electrical conductivity as it approaches the surface of a star. It is assumed that mass density distribution in the medium follows a power law of the radial distance from the point of explosion.
In flows with imploding shocks, conditions of very high temperature and pressure can be produced near the center (axis) of implosion on account of the self-amplifying nature of imploding shocks. As a result of high temperatures attained by gases in motion, the effects of nonequilibrium thermodynamics on the dynamic motion of a converging shock wave can be important.
In this paper, we use the method of Lie group invariance under infinitesimal point transformations [–] to study the problem of propagation of strong shock waves in a radiating and electrically conducting gas permeated by a transverse magnetic field. The arbitrary constants, occurring in the expressions for the generators of the local Lie group of transformations, give rise to different cases of possible solutions with a power law, exponential or logarithmic shock paths. The Lie symmetry approach does not necessarily take into account the boundary and initial conditions unless the same are invariant under the change of variable transformations.
The basic equations describing the one-dimensional unsteady non-planar motion in which the direction of magnetic field is orthogonal to the trajectories of gas particles and electrical conductivity is infinite can be written as follows (Whitham []):]):
Similarity analysis by invariance groups
Here, we suppose that there exists a solution of system (1) along a family of curves, called similarity curves, for which system (1) of partial differential equations reduces to a system of ordinary differential equations; this type of solution is called a similarity solution. In order to obtain the similarity solutions of system (1), we derive its symmetry group such that the system is invariant under this group of transformations. The idea of the calculation is to find a one-parameter infinitesimal group of transformations (see Sharma and Arora []):
In continuation, we shall use the summation convention and introduce the notation , , , , , and , where and .
The system of basic equations (1), which is represented as
System (1) implies
Substitution of from (7) into (8) yields a polynomial equations in the . Setting the coefficients of and to zero yields a system of first order, linear partial differential equations in the generators ψ, χ, S, U, P and E. This system, which is called the system of determining equations, is given as follows:
The arbitrary constants, which appear in the expressions for the infinitesimals of the invariant group of transformations, yield different cases of possible solutions as discussed below.
Case I. When and , the change of variables from to , defined as
The set of equations (16) together with (15) yields on integration the following forms of the flow variables:
The functions , , and depend only on the dimensionless form of the similarity variable ξ, which is determined as follows:
Using (24), we rewrite equations (17) as follows:
Case II. When and , the change of variables from to is defined as
Case III. When and , the study reveals that this condition cannot be obtained in an axially symmetric () flow as it does not permit for the existence of a similarity solution in such a flow pattern. However, this condition can arise in a plane () flow where the change of variables from to , defined as
Case IV. When and , this situation is similar to the previous case in the sense that it does not permit for the existence of a self-similar solutions in an axially symmetric flow. However, the plane flow involving a shock wave moving at constant speed admits a self-similar solution. Accordingly, the similarity variable and the similarity solutions for the flow variables follow from (13), and can be expressed in the following form:
Substituting (39) in the equations in system (1) for , and using (2), we obtain the following system of ordinary differential equations in , , and , which on suppressing the asterisk sign becomes:
Here, we consider Case I of an imploding strong shock in the neighborhood of implosion. For the problem of a converging shock collapsing at the axis, the origin of time t is taken to be the instant at which the shock reaches the axis so that in (28). Therefore, the definition of the similarity variable is slightly modified by setting
Numerical results and discussion
We integrate equations (45) from the shock to the singular point by choosing a trial value of δ, and compute the values of U, S, P, E and at ; the value of δ is corrected by successive approximations in such a way that for these values, the determinant vanishes at . The values of the similarity exponent δ, obtained from the numerical calculations for different values of , m and θ are given in Table 1.
Table 1. Similarity exponentδfor planar and cylindrically symmetric flows and the density exponentθwith
Figure 1. Flow patterns: (a) density, (b) pressure, (c) temperature, (d) velocity for(planar flow) andand.
Figure 2. Flow patterns: (a) density, (b) pressure, (c) temperature, (d) velocity for(cylindrically symmetric flow) andand.
The typical flow profiles show that the density, pressure, temperature and velocity increase behind the shock wave with the increase in the value of θ; this is because a gas particle passing through the shock is subjected to a shock compression. Indeed, this increase in pressure and density behind the shock may also be attributed to the geometrical convergence or area contraction of the shock wave. Figures 1 and 2 show that the growth of the flow variables is slower in cylindrical symmetry as compared with that in planar symmetry. Figures 1 and 2 also confirm the generation of higher pressure near the axis of symmetry, i.e., near . The difference between flow profiles in cylindrical waves and those in planar waves is attributed to the fact that for planar waves, the flow distribution is relatively less influenced by the interaction between the gasdynamic phenomena as compared to cylindrical waves.
In the present investigation a self-similar method is used to study the flow pattern behind an exponential shock driven by a piston in ideal magnetogasdynamics. The general behavior of density, velocity and pressure profiles remains unaffected due to presence of magnetic field in ideal gas. However, there is a decrease in values of density, velocity and pressure in the case of magnetogasdynamics as compared to non-magnetic case. It may be noted that the effect of magnetic field on the flow pattern is more significant in the case of isothermal flow as compared to that of adiabatic flow.
The authors declare that they have no competing interests.
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
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