Rajan Arora - Academia.edu (original) (raw)

Papers by Rajan Arora

Physics of Fluids

In this paper, we investigate a system of quasilinear hyperbolic partial differential equations, ... more In this paper, we investigate a system of quasilinear hyperbolic partial differential equations, which describes the propagation of cylindrical shock waves in a rotating non-ideal gas with the effects of the gravitational field and the axial magnetic field. It is assumed that the flow is isothermal. The Lie group of transformations is used to generate the self-similar solutions of the considered problem in the medium of uniform density. The axial and azimuthal components of fluid velocity and magnetic field are supposed to be varying. We find the generators of the Lie group of transformations by employing the invariant surface criteria. We discovered four alternative solutions by selecting the arbitrary constants indicated in the generators' phrase. Only in three out of these four cases, the self-similar solutions exist. Two types of shock paths appear while solving the above cases. The power-law shock path appears in the first and third cases, while the exponential-law shock pa...

Physics of Fluids, 2022

In this paper, the authors study the problem of an imploding strong cylindrical/spherical shock w... more In this paper, the authors study the problem of an imploding strong cylindrical/spherical shock wave collapsing at the axis/center of a cylindrical/spherical piston that is filled with 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 a micrometric size and uniformly distributed in the mixture. A mathematical model using a system of hyperbolic partial differential equations is presented for the considered problem. The perturbation series method is used to solve the implosion problem, 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 values of all possible real similarity exponents and the corresponding amplitudes are determined in the vicinity of the shock collapse by extending the flow variables and shock location in the Taylor series in time t. Furthermore, the obtained values of similarity exponents have been compared with the existing results and numerical results obtained by the other methods. The effects of the adiabatic exponent c, the wavefront curvature a, and various dusty gas parameters such as r, K p , and G 0 on the shock trajectory and flow variables have been graphically analyzed.

Using the He's variational iteration method, it is possible to find the exact solutions or better... more Using the He's variational iteration method, it is possible to find the exact solutions or better approximate solutions of the partial differential equations. In this method, a correction functional is constructed by a general Lagrange multiplier, which can be identified via variational theory. In this paper, this method is used for solving a nonlinear partial differential equation, a three-dimensional linear parabolic partial differential equation and the one-dimensional parabolic-like equation with variable coefficients and with given initial conditions. The solutions obtained by VIM show the accuracy and efficiency of the method.

Physics of Fluids, 2021

In this article, the effect of the dust particles is studied on the propagation of a cylindrical ... more In this article, the effect of the dust particles is studied on the propagation of a cylindrical shock wave in rotational axisymmetric ideal gas under isothermal flow conditions with the magnetic field. Here, magnetic pressure, azimuthal fluid velocity, and axial fluid velocity are supposed to vary according to a power law in the undisturbed medium. With the help of Sakurai's technique, we obtain approximate solutions analytically by expanding the flow parameters in the form of a power series in / ¼ ð C V Þ 2. The power series method is used to derive the zeroth and the first-order approximations. The solutions for the zeroth-order approximation are constructed in analytical form. Distributions of the hydrodynamical quantities are analyzed graphically behind the shock front. Also, the effects of shock Cowling number ðc o Þ, mass fraction of the solid particles in the mixture ðk p Þ; adiabatic exponent ðcÞ, and rotational parameter (L) on the flow variables are discussed. It is investigated that the density and pressure near the line of symmetry in the case of isothermal flow become zero, and hence a vacuum is formed at the axis of symmetry when the flow is isothermal. The present work may be used to verify the correctness of the solution obtained by self-similarity and numerical methods. Furthermore, the results obtained in the present work are found to be in good agreement with those obtained from the study by Nath and Singh [Can. J. 98, 1077 (2020)].

Zeitschrift für Naturforschung A, 2021

In this work, we consider the system of partial differential equations describing one-dimensional... more In this work, we consider the system of partial differential equations describing one-dimensional (1D) radially symmetric (i.e., cylindrical or spherical) flow of a nonideal gas with small solid dust particles. We analyze the implosion of cylindrical and spherical symmetric strong shock waves in a mixture of a nonideal gas with small solid dust particles. An evolution equation for the strong cylindrical and spherical shock waves is derived by using the Maslov technique based on the kinematics of 1D motion. The approximate value of the similarity exponent describing the behavior of strong shocks is calculated by applying a first-order truncation approximation. The obtained approximate values of similarity exponent are compared with the values of the similarity exponent obtained from Whitham’s rule and Guderley’s method. All the above computations are performed for the different values of mass fraction of dust particles, relative specific heat, and the ratio of the density of dust par...

Nonlinear Engineering, 2021

Studies on Non-linear evolutionary equations have become more critical as time evolves. Such equa... more Studies on Non-linear evolutionary equations have become more critical as time evolves. Such equations are not far-fetched in fluid mechanics, plasma physics, optical fibers, and other scientific applications. It should be an essential aim to find exact solutions of these equations. In this work, the Lie group theory is used to apply the similarity reduction and to find some exact solutions of a (3+1) dimensional nonlinear evolution equation. In this report, the groups of symmetries, Tables for commutation, and adjoints with infinitesimal generators were established. The subalgebra and its optimal system is obtained with the aid of the adjoint Table. Moreover, the equation has been reduced into a new PDE having less number of independent variables and at last into an ODE, using subalgebras and their optimal system, which gives similarity solutions that can represent the dynamics of nonlinear waves.

Mathematical Methods in the Applied Sciences, 2021

In this article, a system of partial differential equations governing the one‐dimensional motion ... more In this article, a system of partial differential equations governing the one‐dimensional motion of inviscid, self‐gravitating, spherical dusty gas cloud is considered. The evolutionary behavior of spherical shock waves of arbitrary strength in an interstellar dusty gas cloud is examined. By utilizing the method based on the kinematics of the one‐dimensional motion of shock waves, we derived an infinite set of transport equations governing the strength of shock waves and induced discontinuity behind it. By applying the truncation procedure to the infinite set of transport equations, we get an efficient system of finite number ordinary differential equations describing shock propagation, which can be regarded as a good approximation of the infinite hierarchy of the system. The truncated equations describing the shock strength and the induced discontinuity are used to analyze the growth and decay behavior of shock waves of arbitrary strength in the dusty gas medium. We considered the first two truncation approximations and the obtained results for the exponent from the successive approximation and compared our results with Guderley's exact similarity solution and the characteristic rule.

The European Physical Journal Plus, 2020

The present paper demonstrates the analysis of converging shock wave in an ideal relaxing gas wit... more The present paper demonstrates the analysis of converging shock wave in an ideal relaxing gas with dust particles via Lie group theoretic method. The Lie group of transformations is used to determine the whole range of self-similar solutions to a problem involving both planar and non-planar flows in an ideal, relaxing and dusty gas involving strong converging shocks. For the strong shock waves, all the necessary group invariance properties associated with ambient gas are presented and the general form of rate of relaxation for which the self-similar solutions exist is determined. By using the invariance surface conditions, we determine the infinitesimal generators of Lie group of transformations associated with the system of partial differential equations and on the basis of arbitrary constants occurring in the expressions for the generators, four different possible cases involving self-similar solutions are reckoned. For the different values of dust parameters, the similarity exponents are obtained numerically and comparison is made with the similarity exponents obtained from the characteristic rule (or CCW method). The effects of mass fraction of dust particle, relative specific heat and ratio of density of dust particle to density of gas, on the flow variables and shock formation, have been shown. The patterns of all flow variables behind the shock are analyzed graphically.

The Quarterly Journal of Mechanics and Applied Mathematics, 2020

Summary The converging problem of cylindrically or spherically symmetric strong shock wave collap... more Summary The converging problem of cylindrically or spherically symmetric strong shock wave collapsing at the axis/centre of symmetry, is studied in a non-ideal inhomogeneous gaseous medium. Here, we assume that the undisturbed medium is spatially variable and the density of a gas is decreasing towards the axis/centre according to a power law. In the present work, we have used the perturbation technique to the implosion problem and obtained a global solution that also admits Guderley’s asymptotic solution in a very good agreement which holds only in the vicinity of the axis/centre of implosion. The similarity exponents together with their corresponding amplitudes are determined by expanding the flow parameters in powers of time. We also refined the leading similarity exponents near the axis/centre of convergence. We compared our calculated results with the already existing results and found them in good agreements up to two decimal places. Shock position and flow parameters are analy...

International Journal of Mathematical, Engineering and Management Sciences, 2020

In this paper, we first describe the methodology of the Homotopy Analysis Method (HAM) which is a... more In this paper, we first describe the methodology of the Homotopy Analysis Method (HAM) which is an analytical technique and then employ it to some of the non-linear problems which are used in different fields of sciences like plasma physics, fluid dynamics, laser optics, biology, chemical kinetics, nucleation kinetics, physiology, etc. Approximate series solutions have been obtained and the results are compared with the closed form solutions of the equations, which show that this technique gives high accurate results. HAM is a reliable technique, easy to use and is widely applicable to a large class of non-linear differential equations. MATHEMATICA software package has been used for computations.

International Journal of Mathematical, Engineering and Management Sciences, 2019

Self-similar solutions of the system of non-linear partial differential equations are obtained us... more Self-similar solutions of the system of non-linear partial differential equations are obtained using the Lie group of invariance technique. The system of equations governs the one dimensional and unsteady motion for the isothermal flow of an ideal gas. The medium has been taken the uniform. From the expressions of infinitesimal generators involving arbitrary constants, different cases arise as per the choice of the arbitrary constants. In this paper, the case of a collapse of an implosion of a cylindrical shock wave is shown in detail along with the comparison between the similarity exponent obtained by Guderley's method and by Crammer's rule. Also, the effects of the adiabatic index and the ambient density exponent on the flow variables are illustrated through the figures. The flow variables are computed behind the leading shock and are shown graphically.

Journal of Theoretical and Applied Physics, 2013

Except some empirical methods, which have been developed in the past, no analytical method exists... more 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 ½ ¼ O ϵ ð Þ; 0 < ϵ ≪1:. We have considered a case when the effect of the first orderinduced discontinuity or the disturbances that overtook the shock from behind are strong, i.e., [p x ] = O(1). The evolutionary behavior of the weak shocks in a non-ideal gas is described using the truncation approximation.

Boundary Value Problems, 2014

In this paper, a non-dimensional unsteady adiabatic flow of a plane or cylindrical strong shock w... more 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 b...

Journal Applied Mathematics, 2012

In this paper, the exact traveling wave solutions of thegeneralized forms B(n, 1) and B(-n, 1) of... more In this paper, the exact traveling wave solutions of thegeneralized forms B(n, 1) and B(-n, 1) of Burgers' equation are obtained by using (G`/G)-expansion method. It has been shown that the (G`/G)-expansion method, with the help of computation, provides a very effective and powerful tool for solving non-linear partial differential equations

![Research paper thumbnail of The (G`/G)-Expansion Method for Traveling Wave Solutions of Burgers’ Kdv and Generalization of Huxley Equations](https://attachments.academia-assets.com/98226855/thumbnails/1.jpg)

American Journal of Computational and Applied Mathematics, 2012

The (G`/G)-expansion method is used for determining the exact traveling wave solutions of the Bur... more The (G`/G)-expansion method is used for determining the exact traveling wave solutions of the Burgers-KdV and generalization of Huxley equations. The obtained solutions are compared with the solutions found by Wazwaz[18]. The (G`/G)-method is very powerful and easy tool for solving non-linear partial differential equations

Mathematical Modelling and Analysis, 2009

Using the weakly non‐linear geometrical acoustics theory, we obtain the small amplitude high freq... more Using the weakly non‐linear geometrical acoustics theory, we obtain the small amplitude high frequency asymptotic solution to the basic equations governing one dimensional unsteady planar, spherically and cylindrically symmetric flow in a vibrationally relaxing gas with Van der Waals equation of state. The transport equations for the amplitudes of resonantly interacting waves are derived. The evolutionary behavior of non‐resonant wave modes culminating into shock waves is also studied.

Mathematical Modelling and Analysis, 2012

A group theoretic method is used to obtain an entire class of similarity solutions to the problem... more 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.

International Journal of Applied and Computational Mathematics, 2018

In this paper, we obtained the exact and solitary wave solutions of modified equal width wave equ... more In this paper, we obtained the exact and solitary wave solutions of modified equal width wave equation by using Lie symmetry method. With the help of MAPLE software we obtained infinitesimal generators and commutation table. Lie symmetry transformation has been used for converting nonlinear partial differential equation into nonlinear ordinary differential equation. Then, we used tanh method and power series method for solving reduced nonlinear ordinary differential equations. Convergence of power series solution has also been shown.

Advanced Science, Engineering and Medicine, 2013

Advances in Intelligent Systems and Computing, 2014

This work is concerned with the analytical and numerical solutions of linear and nonlinear two-di... more This work is concerned with the analytical and numerical solutions of linear and nonlinear two-dimensional general rate models (2D-GRMs) describing the transport of single-solute and multi-component mixtures through chromatographic columns of cylindrical geometry packed with core-shell particles. The finite Hankel and Laplace transformations are successively applied to derive analytical solutions for a single-solute model considering linear adsorption isotherms and two different sets of boundary conditions. Moreover, analytical temporal moments are derived from the Laplace domain solutions. The process is further analyzed by numerically approximating the nonlinear 2D-GRM for core-shell particles considering multi-component mixtures and nonlinear Langmuir isotherm. A high resolution finite volume scheme is extended to solve the considered 2D-model equations. Several case studies of single-solute and multi-component mixtures are considered. The derived analytical results are validated against the numerical solutions of a high resolution finite volume scheme. Typical performance criteria are utilized to analyze the performance of the chromatographic process. The results obtained are considered to be useful to support further development of liquid chromatography.