Viscoelastic flows in a rough channel: A multiscale analysis

Abstract

In this paper, we consider viscoelastic flows in a rough domain (with typical roughness patterns of size ε ≪ 1). We present and rigorously justify an asymptotic expansion with respect to ε, at any order, based upon the definition of elementary problems: Oldroyd-type problems at the global scale defined on a smoothened domain and boundary-layer corrector problems. The resulting analysis guarantees optimality with respect to the truncation error.

Boundary Layer for a Non-Newtonian Flow Over a Rough Surface

“…In this setting, no boundary layer analysis is needed, and the author succeeds to describe the limit asymptotics by the unfolding method. Finally, we point out the very recent paper [11], where an Oldroyd fluid is considered in a rough channel. In this setting, no nonlinearity is associated to the boundary layer, which satisfies a Stokes problem.…”

The Cauchy problem for an inviscid Oldroyd-B model in three dimensions: global well posedness and optimal decay rates

“…However, the centre-of-mass diffusion can be physically justified to model the shear and vorticity banding phenomena [6, 9, 10, 13, 32, 37, 42], although it is small. In this case, some interesting works have been achieved.…”

Optimal time-decay estimates for an Oldroyd-B model with zero viscosity

“…As showed by Málek et al ([43]), the stress diffusion term can be interpreted either as a consequence of a nonlocal energy storage mechanism or as a consequence of a nonlocal entropy production mechanism. Thus the diffusive Oldroyd-B system has attracted much attention and been studied extensively, see [1,2,12,13,14,18,19,33,42,49].…”