Mathematical existence results for the Doi-Edwards polymer model

In this paper, we present some mathematical results on the Doi-Edwards model describing the dynamics of flexible polymers in melts and concentrated solutions. This model, developed in the late 1970s, has been used and tested extensively in modeling and simulation of polymer flows. From a mathematical point of view, the Doi-Edwards model consists in a strong coupling between the Navier-Stokes equations and a highly nonlinear constitutive law. The aim of this article is to provide a rigorous proof of the well-posedness of the Doi-Edwards model, namely it has a unique regular solution. We also prove, which is generally much more difficult for flows of viscoelastic type, that the solution is global in time in the two dimensional case, without any restriction on the smallness of the data.Comment: 48 page

Global existence results for some viscoelastic models with an integral constitutive law

We provide a proof of global regularity of solutions of some models of viscoelastic flow with an integral constitutive law, in the two spatial dimensions and in a periodic domain. Models that are included in these results are classical models for flow memory: for instance some K-BKZ models, the PSM model or the Wagner model. The proof is based on the fact that these models naturally give a $L^\infty$-bound on the stress and that they allow to control the spatial gradient of the stress. The main result does not cover the case of the Oldroyd-B model

Roughness effect on the Neumann boundary condition

36 pagesInternational audienceWe study the effect of a periodic roughness on a Neumann boundary condition. We show that, as in the case of a Dirichlet boundary condition, it is possible to approach this condition by a more complex law on a domain without rugosity, called wall law. This approach is however different from that usually used in Dirichlet case. In particular, we show that this wall law can be explicitly written using an energy developed in the roughness boundary layer. The first part deals with the case of a Laplace operator in a simple domain but many more general results are next given: when the domain or the operator are more complex, or with Robin-Fourier boundary conditions. Some numerical illustrations are used to obtain magnitudes for the coefficients appearing in the new wall laws. Finally, these wall laws can be interpreted using a fictive boundary without rugosity. That allows to give an application to the water waves equation

The FENE model for viscoelastic thin film flows: Justification of new models and applications.

In this article, we rigorously determine an asymptotic model for viscoelastic flows of FENE type for thin domains. The proof presented here is based on existence and unicity results for a Fokker-Planck equation and for the limit problem when the ratio between height and width of the physical domain vanished. We finally show that the error between complete FENE constitutive law and the approximation suggested for thin films domains can be controlled. Some applications are given at the end of this article: in the fields of lubrication, phenomena of boundary layers, of the industry of the nanotechnology, of biology or Shallow-Water equations

Fokker-Planck equation in bounded domain

We study the existence and the uniqueness of a solution \fy to the linear Fokker-Planck equation -\Delta \fy + \div(\fy \F) = f in a bounded domain of $\R^d$ when \F is a "confinement" vector field acting for instance like the inverse of the distance to the boundary. An illustration of the obtained results is given within the framework of fluid mechanics and polymer flows

Convergence to the Reynolds approximation with a double effect of roughness

We prove that the lubrication approximation is perturbed by a non-regular roughness of the boundary. We show how the flow may be accelerated using adequate rugosity profiles on the bottom. We explicit the possible effects of some abrupt changes in the profile. The limit system is mathematically justified through a variant of the notion of two-scale convergence. Finally, we present some numerical results, illustrating the limit system in the three-dimensional case

A bi-projection method for Bingham type flows

International audienceWe propose and study a new numerical scheme to compute the isothermal and unsteady flow of an incompressible viscoplastic Bingham medium.The main difficulty, for both theoretical and numerical approaches, is due to the non-differentiability of the plastic part of stress tensor in regionswhere the rate-of-strain tensor vanishes. This is handled by reformulating the definition of the plastic stress tensor in terms ofa projection.A new time scheme, based on the classical incremental projection method for the Newtonian Navier-Stokes equations, is proposed. The plastictensor is treated implicitly in the first sub-step of the projection scheme and is computed by using a fixed point procedure. A pseudo-timerelaxation is added into the Bingham projection whose effect is to ensure a geometric convergence of the fixed point algorithm. This is akey feature of the bi-projection scheme which provides a fast and accurate computation of the plastic tensor.Stability and error analyses of the numerical scheme are provided. The error induced by the pseudo-time relaxation term is controlled bya prescribed numerical parameter so that a first-order estimate of the time error is derived for the velocity field.A second-order cell-centred finite volume scheme on staggered grids is applied for the spatial discretisation.The scheme is assessed against previously published benchmark results for both Newtonian and Bingham flows in a two-dimensional lid-drivencavity for Reynolds number equals 1 000.Moreover, the proposed numerical scheme is able to reproduce the fundamental property of cessation in finite time of a viscoplasticmedium in the absence of any energy source term in the equations.For a fixed value (100) of the Bingham number, various numerical simulations for a range of Reynolds numbers up to 200 000 were performedwith the bi-projection scheme on a grid with 1024x1024 mesh points. The effect of this (physical) parameter on the flow behaviour is discussed

Steady state solutions for a lubrication two-fluid flow

International audienceIn this paper, we describe possible solutions for a stationary flow of two superposed fluids between two close surfaces in relative motion. Physically, this study is within the lubrication framework, in which it is of interest to predict the relative positions of the lubricant and the air in the device. Mathematically, we observe that this problem corresponds to finding the interface between the two fluids, and we prove that it is equivalent to solve some polynomial equation. We solve this equation using an original method of polynomial resolution. First, we check that our results are consistent with previous work. Next, we use this solution to answer some physically relevant questions related to the lubrication setting. For instance, we obtain theoretical and numerical results enabling to predict the apparition of a full film with respect to physical parameters (fluxes, shear velocity, . . . ). In particular, we present a figure giving the number of stationary solutions depending on the physical parameters. Moreover, in the last part, we give some indications for a better understanding of the multi-fluid case