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

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.