B. D. Pandey

Symmetry groups and similarity solutions for the system of equations for a viscous compressible fluid

Here, using Lie group transformations, we consider the problem of finding similarity solutions to the system of partial differential equations (PDEs) governing one-dimensional unsteady motion of a compressible fluid in the presence of viscosity and thermal conduction, using the general form of the equation of state. The symmetry groups admitted by the governing system of PDEs are obtained, and the complete Lie algebra of infinitesimal symmetries is established. Indeed, with the use of the entailed similarity solution the problem is transformed to a system of ordinary differential equations(ODEs), which in general is nonlinear; in some cases, it is possible to solve these ODEs to determine some special exact solutions.

Propagation of rapid pulses through a two-phase mixture of gas and dust particles

Abstract Small and finite amplitude pulses of an arbitrary initial profile propagating into a uniform region of a dusty gas have been studied. Solutions based on an asymptotic simple wave procedure are presented under the approximation that the characteristic length of the signal is much shorter than the characteristic length of the medium. The modulation of simple waves by the dissipative effects owing to the presence of solid particles, and the effect of nonlinearity leading to wave distortion and shock formation are discussed. In the small amplitude limit, a solution up to the second order is obtained, and numerical computations are carried out over one complete cycle for typical values of the physical parameters involved in the solution. The second order solution, in contrast to the first order, depends on the precursor wavelets and exhibits changes in those parameters which remained constant in the first order approximation. Finally, the second order solution is used to describe the far field behavior of weak shocks.

Non-linear analysis of a traffic flow

A nonlinear mathematical model, which takes into account the dissipative mechanism, is used to describe the signal transmission in a traffic flow. It is shown that dissipative mechanisms, under certain conditions, may produce attenuation effects against the typical nonlinear steepening of waves. An asymptotic analysis is carried out to discuss wave features when the governing hyperbolic system of equations is objective to different kinds of approximations.

Weak discontinuities in a non-equilibrium flow of a dusty gas

SummaryThe behavior of a discontinuity in flow gradients at the head of a disturbance propagating through a homogeneous mixture of gas and small dust particles has been studied. It is shown that the presence of dust particles results in increasing the shock formation time as compared with a similar pure-gas case. When the disturbance is arbitrarily small in amplitude, the solution to the first order in the whole disturbed domain is constructed and analysed. It is found that the concentration of solid particles has a decaying, effect on the shock strength as one, might expect.

Non-linear wave propagation in a hot-electron plasma

SummaryThe theory of relatively undistorted waves is used to study the finite-amplitude waves in a hot-electron model of plasma. A simple asymptotic expansion accounting for the non-linear effects has been used in the small amplitude limit to calculate the first and the second order solutions. The speed of propagation and the location of weak shocks has been determined through solutions of high-frequency waves.

Asymptotic solution of a non-linear wave motion in an electron plasma

The theory of high-frequency waves has been used to calculate first and second-order asymptotic solutions for the propagation of non-linear waves in a cylindrical symmetric flow of an electron plasma. The behaviour of acceleration waves and weak shock waves has been analysed through these solutions and Whithams rule for a weak shock wave on any wavelet has been confirmed through the first-order solution. The appearance of a weak shock wave on any wavelet has been determined and its strength, the location, and the speed of propagation have been found from the asymptotic solution presented in this paper.

Wave front analysis of weak nonlinear waves in a two-phase flow of gas and dust particles

The non-linear behavior of waves including the characteristic front, the expansion wave front and the shock front, in a mixture of gas and dust particles has been studied. Such waves are conceived of as produced by a piston moving with a small velocity as compared with the speed of sound. The trajectories of these waves and the particle paths in the physical plane are determined. The effect of solid particles and the adiabatic heat exponent on the wave propagation is also investigated.