V. D. Sharma

Quasilinear hyperbolic systems, compressible flows, and waves

Hyperbolic Systems of Conservation Laws Preliminaries Examples Scalar Hyperbolic Equations in One Dimension Breakdown of Smooth Solutions Entropy Conditions Revisited Riemann Problem for Nonconvex Flux Function Irreversibility Asymptotic Behavior Hyperbolic Systems in One Space Dimension Genuine Nonlinearity Weak Solutions and Jump Condition Entropy Conditions Riemann Problem Shallow Water Equations Evolution of Weak Waves in Hyperbolic Systems Waves and Compatibility Conditions Evolutionary Behavior of Acceleration Waves Interaction of Shock Waves with Weak Discontinuities Weak Discontinuities in Radiative Gasdynamics One-Dimensional Weak Discontinuity Waves Weak Nonlinear Waves in an Ideal Plasma Relatively Undistorted Waves Asymptotic Waves for Quasilinear Systems Weakly Nonlinear Geometrical Optics Far Field Behavior Energy Dissipated across Shocks Evolution Equation Describing Mixed Nonlinearity Singular Ray Expansions Resonantly Interacting Waves Self-Similar Solutions Involving Discontinuities and Their Interaction Waves in Self-Similar Flows Imploding Shocks in a Relaxing Gas Exact Solutions of Euler Equations via Lie Group Analysis Kinematics of a Shock of Arbitrary Strength Shock Wave through an Ideal Gas in 3-Space Dimensions An Alternative Approach Using the Theory of Distributions Kinematics of a Bore over a Sloping Beach Bibliography Index

Self-similar shocks in a dusty gas

Abstract A group theoretic method is used to establish the entire class of self-similar solutions to the problem of shock wave propagation through a dusty gas. Necessary conditions for the existence of similarity solutions for shocks of arbitrary strength as well as for strong shocks are obtained. It is found that the problem admits a self-similar solution only when the ambient medium ahead of the shock is of uniform density. Collapse of imploding cylindrical and spherical shocks is worked out in detail to investigate as to how the shock involution is influenced by the mass concentration of solid particles in the medium, the ratio of the density of solid particles to that of initial density of the medium, the relative specific heat and the amplification mechanism of the flow convergence.

Similarity solutions for converging shocks in a relaxing gas

Abstract In the present paper, we use the method of Lie group invariance to determine the class of self-similar solutions to a problem concerning plane and radially symmetric flows of a relaxing gas involving shocks of arbitrary strength. The ambient gas ahead of the shock is considered to be inhomogeneous. The method yields a general form of the relaxation rate for which the self-similar solutions are admitted. 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. In contrast to situations without relaxation, the inclusion of relaxation effects imply constraint conditions. A particular case of the collapse of an imploding shock is worked out in detail for radially symmetric flows. Numerical calculations have been performed to determine the effects of relaxation and the ambient density on the self-similar exponent and the flow patterns.

The progressive wave approach analyzing the decay of a sawtooth profile in magnetogasdynamics

An asymptotic approach is used to analyze the main features of weakly nonlinear waves propagating through an electrically conducting gas permeated by a transverse magnetic field. The analysis leads to an evolution equation, which characterizes the wave process in the high‐frequency domain. The growth equation for an acceleration front is recovered as a special case. The influence of the magnetic field on the decay behavior of a sawtooth profile, which is headed by a weak shock front and ends with a magnetosonic disturbance, is investigated. A remarkable difference between the plane and cylindrical wave profiles is noted; for instance, when the adiabatic index γ is 2, the field does not affect the decay behavior of plane waves but does affect cylindrical waves.

Uniformly valid analytical solution to the problem of a decaying shock wave

An explicit representation of an analytical solution to the problem of decay of a plane shock wave of arbitrary strength is proposed. The solution satisfies the basic equations exactly. The approximation lies in the (approximate) satisfaction of two of the Rankine-Hugoniot conditions. The error incurred is shown to be very small even for strong shocks. This solution analyses the interaction of a shock of arbitrary strength with a centred simple wave overtaking it, and describes a complete history of decay with a remarkable accuracy even for strong shocks. For a weak shock, the limiting law of motion obtained from the solution is shown to be in complete agreement with the Friedrichs theory. The propagation law of the non-uniform shock wave is determined, and the equations for shock and particle paths in the (x, t)-plane are obtained. The analytic solution presented here is uniformly valid for the entire flow field behind the decaying shock wave.

Similarity solutions for three dimensional Euler equations using Lie group analysis

A similarity analysis of the nonlinear three dimensional unsteady Euler equations of gas dynamics is presented using Lie group of transformations with commuting infinitesimal operators. Symmetry groups admitted by the governing system of partial differential equations (PDEs) are obtained, and the complete Lie algebra of infinitesimal symmetries is established. The symmetry generators are used for constructing similarity variables which lead to a reduced system of equations with one independent variable less at each step and eventually to a system of ordinary differential equations (ODEs); in some cases, it is possible to solve these equations exactly.

Convergence of strong shock in a Van der Waals gas

Strong cylindrical and spherical shock waves, collapsing at the center (or axis) of symmetry, are studied for a Van der Waals gas. The perturbation technique applied in this paper provides a global solution to the implosion problem, yielding the results for Guderley’s local self‐similar solution, which is valid only in the vicinity of the center/axis of implosion. The similarity exponents are found along with the corresponding amplitudes in the vicinity of the shock‐collapse. The flow parameters and the shock trajectory have been computed for different values of the adiabatic coefficient and the Van der Waals excluded volume.

On interaction of shock waves with weak discontinuities

This paper deals with the wave interaction problem, taking into account the jump in shock acceleration as a consequence of its interaction with a weak wave. The existence and uniqueness of the reflection and transmission coefficients is discussed. As an illustration, the flow properties of self—similar solutions for plane and radially symmetric flows are examined. The exact selfsimilar solution and the results of interaction theory are used to study the interaction between a weak discontinuity wave and a blast wave in plane and radially symmetric flows.

On the general behavior of acceleration waves

A systematic study is made of the general behaviour of solutions to the Bernoulli equation which governs the evolution of acceleration waves in nonlinear systems. The theorems obtained contain all the known results and, in some instances, when specialized to the case of an existing theorem they provide a sharper results obtained here for the Bernoulli equation, and the corresponding results for the propagation of weak discontinuity in a general quasilinear hyperbolic system

IMPLODING CYLINDRICAL AND SPHERICAL SHOCK WAVES IN A NON-IDEAL MEDIUM

Converging shock waves in an almost ideal medium are considered. The kinematics of one-dimensional motion have been applied to construct an evolution equation for strong cylindrical and spherical shock waves propagating into a low density gas at rest. The approximate value of the similarity parameter obtained from there is compared with those derived from Whithams Rule and the exact similarity solution at the instant of collapse of the shock wave. The above computation is carried out for different values of the parameter α, which depends on the internal volume of the gas molecules.