STUDY OF SHOCK WAVES IN GASEOUS MEDIUM AND A SHALLOW WATER WAVE MODEL

Abstract

Gas dynamics is a particular branch of uid dynamics, which is the science of uids, liquids, and gases. It developed around the end of the nineteenth century as a result of attempts to comprehend the principles of high-speed compressible ow theories and their applications. The equations of gas dynamics have traditionally been used to study shock wave propagation. A shock wave can be brie y described as a non-linear wave traveling faster than the sound speed in the medium. Shock waves are a common occurrence in everyday life, yet they go unnoticed most of the time. The bangs of an applauding crowd or an explosion, the sound of thunderstorms, earthquakes and volcanic eruptions are some examples of situations in which shock waves are noticeable. Physically, the emergence of a shock wave in a uid ow is always characterized by instant changes in the ow velocity, pressure and temperature. This change is not reversible; inside the shock wave, dissipation of energy occurs and the entropy increases. Shock waves have gained much importance due to instantaneous changes in velocity and pressure, and are being investigated for potential applications in a variety of elds. In the eld of medical science, shock waves are used in traumatology and orthopedics to treat various insertional tendinopathies (enthesopathies) and delayed unions and nonunions of fracture. Shock waves are bene cial in material synthesis and industrial elds too. Physically, the shock wave is very thin and the thickness of a shock is about 0.2 m (10􀀀5 in), or roughly four times the mean free path of the gas molecules for air at ambient conditions. The entire width of a shock wave therefore only contains a small number of particles in the longitudinal direction and thus the shock appears as nearly a singularity in the continuum model, yet the existence of shock waves was predicted by considering certain waves traveling in a uid governed by the Euler equations. The Euler equations are a non-linear hyperbolic system of partial di erential equations (PDEs) that are derived from the compressible Navier-Stokes equations by omitting the i e ects of viscosity and heat conduction. Mathematically, the fascinating aspect of such systems is that they admit shock waves in their solution; discontinuities in the solution that can form even from the smooth initial data. Since the superposition of the solutions is not possible for the system of non-linear PDEs, we still do not have a general approach for nding the solution of an initial or boundary value problem associated with a system of nonlinear PDEs. It is necessary to look for approximate analytical and numerical techniques whose objective is precisely to nd the solutions of the system of non-linear PDEs. The present thesis deals with the mathematical study of shock wave propagation in the gaseous medium by considering the e ects of dust particles and magnetic eld, and a shallow waterwave model of non-linear PDEs. Shallow water waves correspond to the ow beneath a horizontal pressure surface in a uid or the ow at the free surface of a body of shallow water under the pull of gravity. In atmospheric science, oceanography, and many other elds, the shallow water-wave models are of particular importance. The present thesis is organized into six chapters as brie y described below: Chapter 1: This is an introductory chapter. We present our objectives and the motivation behind them supported by a literature survey. Chapter 2: This chapter concerns the study of converging shock waves in a dusty gas of uniform density. The dusty gas is assumed to be a mixture of an ideal gas and a large number of dust particles. The dust particles are of the micrometric size and uniformly distributed in the mixture. The dusty gas is lled into a cylindrical/spherical piston, which then begins to compress at a faster pace than the medium's acoustic speed, generating a cylindrically/spherically symmetric shock wave within the piston. The position of the shock wave is unknown and has to be determined. In this chapter, the perturbation series method is used to solve the implosion problem in a dusty gas, providing a global solution and yielding accurately the results of Guderley's local similarity solution, which holds only in the neighborhood of the axis/center of implosion. The similarity exponents are determined together with the corresponding amplitudes in the vicinity of the shock collapse by extending the ow variables and shock position in the Taylor series in time t. A comparison is done between the computed values of the similarity exponents and the numerical results obtained by other methods. The shock position and ow variables are analyzed graphically in the region extending from the piston to the axis/center of collapse for di erent values of the adiabatic exponent ( ), the wavefront curvature ( ) and for various dusty gas parameters, namely the mass concentration of the dust particles (Kp), the ratio of the density of ii the dust particles to the initial density of the gas (G0), and the relative speci c heat ( ). Chapter 3: With the impact of an azimuthal magnetic eld, the strong converging cylindrically symmetric shock waves collapsing at the axis are investigated for a non-ideal gas. It is demonstrated in this chapter that the perturbation series approach, when applied to the shock implosion problem in non-ideal magnetogasdynamics, yields a global solution, in contrast to Guderley's asymptotic solution, which holds only in the immediate neighborhood of the axis of implosion. The similarity exponents and the corresponding amplitudes are found near the shock collapse. The re nement of the leading similarity exponents near the axis of implosion is also made. Figures depicting the distributions of the ow variables and shock trajectory for various values of the adiabatic exponent ( ), non-ideal parameter (b) and shock Cowling number (C0) have been presented. Chapter 4: This chapter demonstrates the study of the propagation of converging cylindrical shock waves in a non-ideal gas (van der Waals type) with the e ect of magnetic eld and isothermal ow conditions via the Lie group theoretic method. The ambient gas ahead of the shock is considered to be homogeneous. The Lie group of transformations is used to determine the entire class of self-similar solutions to the problem involving strong converging shock waves. The surface invariance conditions are used to determine the in nitesimal generators of the Lie group of transformations. Based on the arbitrary constants that occur in the expressions for the generators, two di erent cases of possible solutions with power-law shock path and exponential shock path are obtained. A particular case of the power-law shock path is worked out in detail. The similarity exponents are obtained numerically for di erent values of the non-ideal parameter (b) and shock Cowling number (C0), and then compared to the similarity exponents obtained using the \Guderley's approach". All the ow variables are graphically analyzed behind the shock for di erent values of the non-ideal parameter and shock Cowling number. Chapter 5: In this chapter, the propagation of cylindrical shock waves produced on account of a strong explosion in a non-ideal gas under the e ect of an azimuthal magnetic eld is studied. The problem can be expressed mathematically by a non-linear hyperbolic system of PDEs with jump conditions at the shock as the boundary conditions. The approximate analytic solutions to the considered problem are obtained using the power series method (Sakurai's approach) by extending the power series of the ow variables in terms of (a0=V )2, where a0 and V are the velocities of sound in the undisturbed medium and shock front, respectively. This chapter discusses the rst-order and second-order approximate soiii lutions and provides closed-form solutions for the rst-order approximation. The behavior of the ow variables is depicted via gures behind the shock front for the rst-order approximation along with the variation in the values of the non-ideal parameter (b) and shock Cowling number (C0). To verify the obtained results, numerical calculations are performed in the absence of the magnetic eld which showed that the existing results in the literature are recovered very well. Also, it is observed that these results are in good agreement with the results obtained by the Runge-Kutta method of fourth-order (RK4 method). Chapter 6: This chapter provides an analysis of a (2+1)-dimensional modi ed dispersive water-wave (MDWW) system of partial di erential equations, which describes the non-linear and dispersive long gravity waves traveling in two horizontal directions on shallow waters of uniform depth. The Lie group theoretic approach is employed in this chapter to nd the similarity reductions and analytic solutions of the (2+1)-dimensional MDWW system. The in nitesimal generators for the considered system are obtained under the invariance property of the Lie group of transformations. The one-dimensional optimal system of subalgebras is established. Finally, based on the optimal system, the similarity reductions and invariant solutions of the (2+1)-dimensional MDWW system are obtained. Moreover, the dynamical behaviors of the obtained solutions such as multi-soliton, doubly soliton, single soliton, solitary waves and stationary waves are graphically shown using 2D, 3D and corresponding contour plots.