Discontinuous shear-thickening asymptotic for power-law systems related to compressible flows

TL;DR

This paper investigates the convergence of power-law models for compressible fluids to models with a maximum shear rate, extending previous elliptic and incompressible results to the compressible setting with applications to high-pressure flows.

Contribution

It extends convergence results of shear-thickening models from elliptic and incompressible cases to the compressible regime, including multi-dimensional and viscous stress extensions.

Findings

Convergence established for 1D non-stationary compressible power-law systems.

Results extended to semi-stationary multi-dimensional cases.

Includes an extension for viscous stress with singular shear rate dependence.

Abstract

In this paper we study the convergence of a power-law model for dilatant compressible fluids to a class of models exhibiting a maximum admissible shear rate, called thick compressible fluids. These kinds of problems were studied previously for elliptic equations, stating with the work of Bhattacharya, E. DiBenedetto and J. Manfredi [Rend. Sem. Mat. Univ. Politec. Torino 1989], and more recently for incompressible fluids by J.F. Rodrigues [J. Math. Sciences 2015]. Our result may be seen as an extension to the compressible setting of these previous works. Physically, this is motivated by the fact that the pressures generated during a squeezing flow are often large, potentially requiring the consideration of compressibility, see M. Fang and R. Gilbert [Z. Anal. Anwend 2004]. Mathematically, the main difficulty in the compressible setting concerns the strong hyperbolicparabolic coupling between the density and velocity field. We obtain two main results, the first concerning the one-dimensional non-stationary compressible power-law system while the second one concerns the semi-stationary multi-dimensional case. Finally, we present an extension in onedimension for a viscous Cauchy stress with singular dependence on the shear rate.