Similarity solution for magnetogasdynamic shock wave in a perfectly conducting dusty gas with axial or azimuthal magnetic field in rotating medium under the influence of radiative and conductive heat fluxes
Introduction
The problem of shock waves is of immense significance from the stand point of both primary research and practical applications. Because of the deposition of large amount of energy in a very small region over small intervals, similar to the spark discharges in air or in explosion cases, the shock wave phenomenon takes place. The collisionless and collisional shock waves can come into view because of friction between wave-particle and the particles interactions respectively [1,2]. The shock waves frequently occur in environment due to the balance between wave damping dissipative and the wave breaking non-linear forces Zel'dovich and Raizer [3].
The problem formulation and examples describing the self-similar adiabatic motion of the gas model of stars in non-rotating medium are considered in Refs. [[3], [4], [5]]. The shock wave propagation through a gas having solid body rotation in the case of cylindrical geometry was studied by Chaturani [6]. The solutions were obtained in Ref. [6] by the use of similarity method adopted by Sakurai [7]. Vishwakarma et al. [8] have studied the magnetogasdynamic cylindrical shock waves propagating in a rotating non-ideal gas by using the similarity method. Vishwakarma and Nath [9] have studied the shock wave in a rotating mixture of non-ideal gas and small solid particles with radiation and conduction heat flux in cylindrical geometry. Levin and Skopina [10] have studied the propagation of detonation wave in rotational gas flows. Also, a three dimensional unsteady flow with a rotating detonation wave arising in an annular gap of an axially symmetric engine between two parallel planes perpendicular to its symmetry axis was studied by Levin et al. [11]. We refer the readers to the important works, within the context of shock waves propagation in rotating medium, carried out in Refs. [[12], [13], [14]] and the references cited therein.
The study of shock or blast wave in perfectly conducting mixture of non-ideal gas and suspended small solid particles is of immense important because of its various technological applications and for the understanding of many astrophysical phenomena such as planetary rings, asteroids, interstellar clouds, supernova remnants, cometary tails, lunar ash flow, nozzle flow, bomb blast, under-ground, coal-mine blast, cosmic and volcanic explosions, coma collision with a planet, metalized propellant rocket, in polluted air supersonic flight landing, in the description of star formation processes, acceleration of particle in shocks, in supernova explosions shock, dusty crystals formation, in power generator in magnetohydrodynamic (MHD), in nuclear reactors cooling, in nuclear processing, in dust entrainment in a cloud created at some stage in a nuclear explosion, in combustion, in lunar surface erosion by the exhaust of a landing vehicle and in a lot of many other problems in engineering (see Refs. [9,13,[15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29]] and the references cited therein).
Recently, the study of interaction between radiation and gas dynamic field has been given considerable attention. Using the similarity approach the blast wave problem has been extended by several authors by taking into account the effects of radiation (see, for instance, Refs. [[30], [31], [32], [33], [34]] and the references cited therein). Ghoneium et al. [35] have studied the spherical explosion problem using the similarity method by taking into consideration the effects of both radiation and conduction heat fluxes in the two limits of Planck radiative emission and Rosseland radiative diffusion. We refer the readers to the important works, within the context of shock wave propagation in rotating or non-rotating medium with conductive and radiative heat fluxes, carried out in Refs. [9,13,14,27,29] and the references cited therein.
At extreme condition that exist in the majority of the problems connected with the propagation of shock wave, the statement that the gas is ideal no more valid, in the case when the gas density is too low and the gas temperature is extremely high, thus in this condition the substitute of the perfect gas model i.e. a van der Waals type model of gas is required. For more detail we refer the readers to motivating works, related to the shock wave propagation in real gases Refs. [[36], [37], [38], [39], [40], [41]] and the references cited therein.
It is well known that the gas attain high temperature due to passes of shock waves through it and the gas get ionized at such high temperature, thus the electromagnetic effects becomes important in the study of shock wave propagation. The universe permeates the magnetic field and the magnetic field play the crucial roles in a number of astrophysical situations. The magnetic field most probably affects the all astrophysical plasmas. An important role is played by the magnetic field in energy and momentum transport and in rapidly release energy in flares. In current years, technological development in the diagnostics and laser technology have enabled convincing uses in high-energy density physics, because of the extreme densities, temperature and pressure (bar) easy to get with the help of intense optical drivers, it has become feasible to generate conditions pertinent to astrophysical phenomena (see Blandford and Eichler [42]) such as gamma-ray bursts and there afterglows, supernova remnants, and energetic antimatter conquered plasma flows experiments in laboratory. By using the shock wave formed with long-pulse (nanosecond-scale) laser drivers, outcomes were recently reported that provide insight into potential mechanisms for the generation of the proto-galactic magnetic field [43,44]. The self-similar solutions for the shock wave propagation with magnetic field have obtained by a number of researchers (see Refs. [5,8,12,14,[39], [40], [41],[45], [46], [47]] and many others).
The authors have obtained the results of interaction process for interstellar molecular clouds with strong shock by supernova in 3D using numerical simulation in [48]. The hydrodynamical flow emergence, fragmentation, contraction processes and development of turbulent flow in the cloud and nearby media were simulated by them without considering the rotational effect. Also, the parallel algorithms for astrophysical problems were developed in Ref. [49]. The combustion onset in inhomogeneous dispersed mixture was studied by Smirnov et al. [50]. Smirnov et al. [50] and Smirnov and Nikitin [51] have shown that the energy released due to turbulence and chemical kinetics increase after ignition, which brings to the generation of accelerating strong shock wave, which can be finally result in detonation. A number of researches have been devoted to the study of the behavior of molecular dusty clouds and the study of their role in the formation of stars [28,29,52,53] and references cited therein. In pioneering work by Rybakin et al. [53], the results of the 3D molecular simulation of molecular clouds collision processes was presented. During process of molecular clouds collide, the gas density in their compression region increases by many orders of magnitude. This leads to the emergence of super dense areas, fragmentation and rearrangement of clouds. As a result of such processes, a system of dense filaments and clumps is formed. Star formation process occurs in dense nuclei in filaments (see Rybakin et al. [54]).
Vishwakarm et al. [55] have studied the propagation of cylindrical shock wave in a weekly conducting mixture of small solid particles and perfect gas and obtained the similarity solutions. Recently, Vishwakarma and Lata [56,57] have discussed the shock wave propagation in a perfectly conducting mixture of perfect gas and chemically inert small solid particles with magnetic field using similarity method.
In all of the above studies, the propagation of shock wave in non-ideal gas and small solid particles conducting mixture for increasing energy with conductive and radiative heat fluxes in the presence of magnetic field is not studied by any of the authors. Here, we assume that the shock wave is driven out by a moving cylindrical piston or by an inner expanding surface. In this study, we have generalized the solution of the problem studied by Vishwakarma and Nath [13] in a rotational axisymmetric dusty gas (a mixture of non-ideal gas and small solid particles), with heat conduction and radiation heat flux by taking in to account the effect of magnetic field. Also, we have considered the effects of variation of adiabatic index on the shock wave and on the flow variables.
Section snippets
Formation of the problem
The mixture is assumed to be perfectly conducting mixture of non-ideal gas and inert small solid particles of micro or nano size. In order to obtain the significant features of the blast/shock wave propagation, the small solid particles are assumed to be of micro size and chemically inert and its behavior like a pseudo-fluid, and the conducting mixture with a constant specific heats ratio is in velocity and temperature equilibrium Pai [15]. For this gas and particle conducting mixture to be
Flow-field total energy
The total energy of the flow-field behind the cylindrical shock wave is taken to be time dependent and varying according as (see Refs. [64,65])where ‘’ and are constants. The values of ‘’ should be positive for the class in which the total energy of flow-field increases with time. For physical explanation readers are refer to Refs. [13,64].
Also, the total energy of the disturbance is given bywhere is the radius of
The self-similarity transformations
To discuss the similarity solutions, we introduce the similarity variable , and seek the solution of the basic equations (3)–(8) in the form (see Refs. [13,29,39])where , , , , , and are functions of only, in terms of which the differential equations are to be formulated.
By applying the similarity transformations (25) in the relation (24), we havewhere
Numerical results and discussion
To obtain the similarity solution of the considered problem, we have the relation between the constants , and as given below
To obtain the distribution of the flow variables in the flow field region behind the cylindrical shock front, we integrate the system of equations (35), (36), (37), (38), (39), (40), (45) with the help of boundary conditions (33) and (34) using the Runge-Kutta method of fourth order. The values of the problem parameters are taken to be (see Pai et al.
Concluding remarks
The current study investigates the cylindrical shock wave propagation in a perfectly conducting dusty gas in rotating medium in the presence of radiative and conductive heat fluxes with azimuthal or axial magnetic field. The drawn figures show the flow variables variations in the flow-field region and the tables display the global variation of the piston position for different value of the parameters in rotational axi-symmetric perfectly conducting mixture. The possible applications of the
Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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