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

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, starting with the work of Bhattacharya et al (1989 Rend. Sem. Mat. Univ. Politec. Torino 47 15–68), and more recently for incompressible fluids by Rodrigues (2015 J. Math. Sci. 210 11). 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 Fang and Gilbert (2004 Z. Anal. Anwend. 23 783–807). Mathematically, the main difficulty in the compressible setting concerns the strong hyperbolic-parabolic 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 one-dimension for a viscous Cauchy stress with singular dependence on the shear rate.