Nonlinear Acoustics and Shock Formation in Lossless Barotropic Green--Naghdi Fluids

TL;DR

This paper investigates nonlinear acoustic wave propagation and shock formation in lossless Green--Naghdi fluids, deriving simplified models, analyzing hyperbolicity constraints, and numerically studying shock development in one-dimensional flows.

Contribution

It introduces a weakly nonlinear model for Green--Naghdi fluids and numerically examines shock formation, contrasting results with classical fluid models.

Findings

Hyperbolicity conditions restrict model parameters.

Shock formation via acceleration wave blowup is confirmed numerically.

The weakly nonlinear model accurately predicts shock development.

Abstract

The equations of motion of lossless compressible nonclassical fluids under the so-called Green--Naghdi theory are considered for two classes of barotropic fluids: (\textit{i}) perfect gases and (\textit{ii}) liquids obeying a quadratic equation of state. An exact reduction in terms of a scalar acoustic potential and the (scalar) thermal displacement is achieved. Properties and simplifications of these model nonlinear acoustic equations for unidirectional flows are noted. Specifically, the requirement that the governing system of equations for such flows remain hyperbolic is shown to lead to restrictions on the physical parameters and/or applicability of the model. A weakly nonlinear model is proposed on the basis of neglecting only terms proportional to the square of the Mach number in the governing equations, without any further approximation or modification of the nonlinear terms. Shock formation via acceleration wave blowup is studied numerically in a one-dimensional context using a high-resolution Godunov-type finite-volume scheme, thereby verifying prior analytical results on the blowup time and contrasting these results with the corresponding ones for classical (Euler) fluids.