Shock wave dynamics via symmetry-driven analysis of a two-phase flow with the Chaplygin pressure law

Abstract

This article investigates wave propagation in a two-phase flow with Chaplygin pressure law, an equation where pressure inversely depends on density. The study employs Lie symmetries and symmetry-driven analysis to derive one-dimensional optimal subalgebras using the adjoint transformation and the invariant functions. Symmetry reductions yield several new exact solutions, and their physical behavior is examined graphically. Further, solutions such as peak-on waves, kinks, and parabolic solitons are identified using traveling wave transformation. Next, a framework of non-locally related PDE, including potential system and inverse potential systems (IPS), is designed to classify non-local symmetries and discover more non-trivial exact solutions for the model. Then, novel conservation laws are constructed using the non-linear self-adjointness property of the model. Finally, the research explores the dynamic evolution of characteristic shock, weak discontinuity, and their interactions using one of the developed solutions. It contributes to understanding two-phase flow, offering practical implications for astrophysics, high-speed aerodynamics, and energy systems with unconventional pressure laws.