The combined singular limits of compressible Oldroyd-B model at low Mach and Weissenberg numbers

Abstract

This paper is concerned with an initial-boundary value problem of the compressible Oldroyd-B (OB) model on 3D bounded and smooth domain subject to Navier's slip boundary conditions. The combined singular limits at low Mach and Weissenberg numbers are justified for the global smooth solutions with ill-prepared initial data and non-small coupling parameter. It is shown that as the Mach number and the Weissenberg number tend to zero, the solution of the compressible OB model for viscoelastic fluids converges to that of the incompressible Navier–Stokes equations for Newtonian fluids. The proofs are based on some subtle weighted estimates in Sobolev spaces. The different weights of various norms need to be chosen carefully such that the large singular operators can be well balanced and the linear interactions between the deformation and the divergence of extra stress tensor can be mutually cancelled.