Existence and stability of steady flows of weakly viscoelastic fluids

Abstract

SynopsisWe consider steady flows of viscoelastic fluids for which the extrastress tensor is given by a differential constitutive equation and is such that the retardation time is large (weakly viscoelastic fluids).We show the existence of a unique viscoelastic steady flow close to a given Newtonian flow and investigate its linear stability.As an example, we consider the Bénard problem for viscoelastic fluids and we prove that there exists a nontrivial linearly stable flow of a weakly viscoelastic fluid in a container heated from below.

On Flows of Viscoelastic Fluids of Oldroyd Type with Wall Slip

“…Thus, the fluid is weakly viscoelastic, as in [7,9]. This restriction on the fluid model is a consequence of using equation (1.6) -which leads to a simple fixed point formulation of the problem -and is not necessary for solving the original problem.…”

Nonlinear stability of flows of Jeffreys fluids at low Weissenberg numbers

“…i,i,2 + ||e v T ||i,2 + ||e T || 2>2 + \\e g ||i, 2 ). (77) The lemma now follows by combining these estimates.…”

“…Guillope and Saut [1], [2] have given stability proofs for flows of Jeffreys fluids which are small perturbations of either the rest state or a stable Newtonian flow. The objective of the present study is somewhat different.…”

Asymptotic behaviour at infinity of three-dimensional steady viscoelastic flows

“…[7,8,22,23,32,26]. These studies, as most of the other related works on viscoelastic fluids of differential and rate type, see e.g., [2,28,9,3,25] and all the references quoted therein, are crucially based on the mixed structure of the equations of motion. In fact, the equations form a system of nonlinear PDE's of elliptichyperbolic type which can most naturally be studied by suitably decoupling the elliptic and hyperbolic parts of the equations.…”