Dissipative waves in real gases

Abstract

In this paper, we characterize a class of solutions to the unsteady 2-dimensional flow of a van der Waals fluid involving shock waves, and derive an asymptotic amplitude equation exhibiting quadratic and cubic nonlinearities including dissipation and diffraction. We exploit the theory of nonclassical symmetry reduction to obtain some exact solutions. Because of the nonlinearities present in the evolution equation, one expects that the wave profile will eventually encounter distortion and steepening which in the limit of vanishing dissipation culminates into a shock wave; and once shock is formed, it will propagate by separating the portions of the continuous region. Here we have shown how the real gas effects, which manifest themselves through the van der Waals parametersã andb influence the wave characteristics, namely the shape, strength, and decay behavior of shocks.

Codimension two Lie invariant solutions of the modified Khokhlov–Zabolotskaya–Kuznetsov equation

“…In this section, the modified KZK equation is derived 38 which describes the evolution of signal with dissipative effects and the variations in direction transverse to the characteristic rays. We use the method of multiple scale [44][45][46] to derive the evolution equation (modified KZK equation).…”

“…Gupta and Sharma 38 derived a transport equation from unsteady two‐dimensional Navier–Stokes equations by using multiple scales method reads as where Γ, Λ, and c 0 are real constants. This model () is studied in Güngör and Özemir 39 for Γ ≠ 0 and which is reduced to the well‐known KZK equation by rescaling of the dependent variable h and alternating its sign, called also the dispersive Khokhlov–Zabolotskaya equation, in (2+1)‐ dimension.…”

“…Further, optimal classification for higher dimensional Lie algebra was proposed by Ovsiannikov 36 and these ideas were widely used in various physically relevant models. 37 Gupta and Sharma 38 derived a transport equation from unsteady two-dimensional Navier-Stokes equations by using multiple scales method reads as…”