Self-Similar Shock Wave Behavior for the Inviscid Burgers Equation in Various Geometries

Abstract

In this research, we will investigate closed-form self-similar shock wave solutions for the inviscid Burgers equation in planar, cylindrical, and spherical geometries. We will derive the self-similar forms of the equations, examine the corresponding ordinary differential equations, and discuss the physical implications of these solutions in different geometric contexts. The approach employed is based on Lee's method for deriving self-similar solutions to the Euler equations in compressible fluid dynamics. This includes two types of self-similarity: one constrained by integral relations and the other by the need for solution regularity along limiting characteristics. The findings highlight the theoretical basis for Taylor-Sedov blast waves (first kind) and Guderley implosion problems (second kind). This thorough analysis aims to enhance our understanding of shock wave behavior and offer a unified framework for exploring-similar phenomena across various geometries.