Rajan Arora

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.

Similarity Solutions for Strong Shocks in a Non-Ideal Gas

A group theoretic method is used to obtain an entire class of similarity solutions to the problem of shocks propagating through a non-ideal gas and to characterize analytically the state dependent form of the medium ahead for which the problem is invariant and admits similarity solutions. Different cases of possible solutions, known in the literature, with a power law, exponential or logarithmic shock paths are recovered as special cases depending on the arbitrary constants occurring in the expression for the generators of the transformation. Particular case of collapse of imploding cylindrically and spherically symmetric shock in a medium in which initial density obeys power law is worked out in detail. Numerical calculations have been performed to obtain the similarity exponents and the profiles of the flow variables behind the shock, and comparison is made with the known results.

Evolutionary behavior of weak shocks in a non-ideal gas

Except some empirical methods, which have been developed in the past, no analytical method exists to describe the evolutionary behavior of a shock wave without limiting its strength. In this paper, we have derived a system of transport equations for the shock strength and the induced continuity. We generate a completely intrinsic description of plane, cylindrical, and spherical shock waves of weak strength, propagating into a non-ideal gas. It is shown that for a weak shock, the disturbance evolves like an acceleration wave at the leading order. For a weak shock, we may assume that p=Oϵ,0<ϵ≪1.. We have considered a case when the effect of the first order-induced discontinuity or the disturbances that overtook the shock from behind are strong, i.e., [px] = O(1). The evolutionary behavior of the weak shocks in a non-ideal gas is described using the truncation approximation.

Non-planar shock waves in a magnetic field

The method of multiple time scale is used to obtain the asymptotic solution to the spherically and cylindrically symmetric flow into a perfectly conducting gas permeated by a transverse magnetic field. The transport equations for the amplitudes of resonantly interacting high frequency waves are also found. The evolutionary behavior of non-resonant wave modes culminating into shock waves is studied.

Similarity Solutions of Cylindrical Shock Waves in Magnetogasdynamics With Thermal Radiation

In the present work we have taken one-dimensional unsteady flow of non-ideal gas with magnetic effect under the presence of thermal radiation. The system is hyperbolic in nature and solved by similarity method using Lie Group of Transformations under the assumption that the system is constantly conformally invariant under the transformations. The similarity solutions are investigated behind a cylindrical shock which is a consequence of a sudden explosion or produced by an expanding piston. The shock is assumed to be strong and propagating into the medium which is at rest, with uniform density. The total energy of the shock is assumed to be time dependent and obeying the power law. By means of similarity method our system of PDEs transformed into the system of ordinary differential equations (ODEs), which in general are nonlinear. The effects of thermal radiation on the the flow variables velocity, density, pressure and magnetic field are investigated behind the shock.

Wave Interaction and Resonance in a Non-Ideal Gas

The method of multiple time scales is used to obtain the asymptotic solutions to the planar and non-planar flows into a non-ideal gas. The transport equations for the amplitudes of resonantly interacting high frequency waves are also found. Furthermore, the evolutionary behavior of non-resonant wave modes culminating into shock waves is studied.

Application of HAM to seventh order KdV equations

In this paper, the homotopy analysis method (HAM) is applied to obtain the approximate solutions of KdV equations of seventh order, which are Sawada Kotera Ito equation, Lax equation, and Kaup–Kuperschmidt equation, respectively. The convergence of the homotopy analysis method is discussed with the help of auxiliary parameter h, which controls the convergence of the method and also called as the convergence control parameter. The results obtained by the HAM are compared with the exact solutions, by fixing the value of arbitrary constant. It is found that HAM is very robust and elegant method, and by choosing a suitable value of h we can get the approximate solution in very few iteration. Computations are performed with the help of symbolic computation package MATHEMATICA.

Propagation of non-linear waves in a non-ideal relaxing gas

ABSTRACT We have derived an evolution equation governing the far-field behaviour of small amplitude waves in a non-ideal relaxing gas for planar and converging flow. Asymptotic expansions of the flow variables for small amplitude waves have been used to derive the evolution equation. This equation turns out to be a generalized Burgers equation. The numerical solution of this equation is obtained by using the homotopy analysis method (HAM) proposed by Liao with two different initial conditions. Using the HAM, we have studied the effect of relaxation and nonlinearity. The convergence control parameter enables us to find a good approximate solution for such a complex flow problem. This method also confirms the capabilities and usefulness of convergence control parameter and HAM for complex and highly non-linear problems.

Analytical and Numerical Solutions of Two-Dimensional Brusselator System by Modified Variational Iteration Method

The aim of this paper is to study the modification of He’s variational iteration method (VIM), i.e., the modified variational iteration method (MVIM). The study demonstrates the power of the MVIM over the standard VIM. It investigates the exactness of the MVIM by showing that the results obtained by it are far nearer to the exact solutions than those obtained by the VIM. The study also reveals that the MVIM has the capability of reducing the size of calculations. Two-dimensional Brusselator system is solved using the MVIM. The numerical results obtained by these methods are compared with the known closed-form solutions.

Numerical simulation of coupled MKdV equation by reduced differential transform method

The main goal of this article is to demonstrate the use of the Reduced Differential Transform Method (RDTM). This method has been applied directly without using bilinear forms, Wronskian, or inverse scattering method. Also it is worth pointing out that if the terms of the series increase, the RDTM provides better convergence to the analytical solution. In this paper, this method is used for solving nonlinear coupled MKdV equation with given initial conditions having arbitrary constants. The numerical solutions obtained by RDTM are compared with the known exact solutions by fixing the arbitrary constants and with Adomian Decomposition Method (ADM) in tables as well as in figures.