Viscoelastic fluids in a thin domain

Abstract

Abstract. The present paper deals with viscoelastic flows in a thin domain. In particular, we derive and analyse the asymptotic equations of the Stokes-Oldroyd system in thin films (including shear effects). We present a numerical method which solves the corresponding problem and we present some related numerical tests which evidence the effects of the elastic contribution on the flow. Introduction.Much literature has been devoted to the research on non-Newtonian fluids, in a thin film, in both mathematical aspects and applications. It is well known that numerous biological fluids, blood or physiological secretions like tears or synovial fluids, show these non-Newtonian characteristics. In engineering applications people are interested in controling the flows characteristics to suit various requirements such as maintaining the fluid qualities in a wide range of temperatures and stresses. The introduction of additives leads to non-Newtonian behavior of the modern lubricant. Another application domain is linked to polymers, whose non-Newtonian characteristics appear in a wide range of applications such as the molding or injection processes.It is to be noticed that, in most practical applications, the geometry of the flow to be considered is anisotropic. A well-known case deals with the study of boundary layers for complex flows [6,7,14]. Another case, which is the subject of the present paper, is the lubrication problem in which the fluid is contained between two close surfaces in relative motion. These two applications lead to two very different mathematical models, essentially since the order of magnitude of the parameters in the approximation process is different. For example, in the boundary layer study, the Reynolds number is large

An analytic approach for the evolution of the static/flowing interface in viscoplastic granular flows

“…Note that d X θ ≡ dθ/dX is the curvature of the topography. We define the velocity components (U, W ) at M in the coordinates tangent and normal to the topography 9) and the new stress tensor …”

“…As a result, several attempts have been made to reduce the computational cost, essentially based on socalled thin-layer (or shallow water) depth-averaged models. They have been derived rather precisely for Newtonian flows, and studies for non-Newtonian flows have been proposed in [47,6,7,9,50,25,17,24,45,11,30,31,27,12]. In these works, the approach is most of the time to average the equations (1.1), (1.2) with respect to a variable normal to the bottom topography, and to close the equations by an assumption on the dependency on this normal variable, motivated by observations from natural or experimental flows.…”

The FENE Model for Viscoelastic Thin Film Flows

“…About this choice for the size of the Deborah number De, notice that, concerning the Oldroyd model, the same remark is essential if we are interested in the non-common effects of elasticity (see [3] for more explanations). If the Deborah number De is not correctly correlated with the small parameter ε then either the effects of elasticity are invisible (in the case where De is too small with respect to ε) or the effects are translated by additional viscous contributions (in the case where De is too large).…”

“…In lubrication problems, the elastic character of a fluid seems to play a considerable part. In this framework, viscoelastic models of thin film fluids were already studied by J. Tichy [44] for the Maxwell model of viscoelasticity and by G. Bayada et al [3] for models obeying a Oldroyd-B relation. ε ≪ 1 s Lubrication geometry.…”

“…For example in the field of lubrication, the Navier-Stokes equations can be rigorously approximated by the Reynolds equation (see [2]). For a viscoelastic fluid like Oldroyd-B fluid, recent results (see [3]) show that in a thin film the flow is managed by more simple equations (in particular from a numerical point of view) than the Navier-Stokes-Oldroyd models usually obtained.…”