On the Generalization of the Hébraud-Lequeux Model to Multidimensional Flows

In this article we build a model for multidimensional flows based on the idea of Hébraud and Lequeux for soft glassy materials. Care is taken to build a frame indifferent multi-dimensional model. The main goal of this article is to prove that the methodology we have developed to study the well-posedness and the glass transition for the original Hébraud-Lequeux model can be successfully generalized. Thus this work may be used as a starting point for more sophisticated studies in the modeling of general flows of glassy materials

Wolfgang von Ohnesorge

This manuscript got started when one of us (G.H.M.) presented a lecture at the Institute of Mathematics and its Applications at the University of Minnesota. The presentation included a photograph of Rayleigh and made frequent mention of the Ohnesorge number. When the other of us (M.R.) enquired about a picture of Ohnesorge, we found out that none were readily available on the web. Indeed, little about Ohnesorge is available from easily accessible public sources. A good part of the reason is certainly that, unlike other “numbermen” of fluid mechanics, Ohnesorge did not pursue an academic career. The purpose of this article is to fill the gap and shed some light on the life of Wolfgang von Ohnesorge. We shall discuss the highlights of his biography, his scientific contributions, their physical significance, and their impact today

Symmetric factorization of the conformation tensor in viscoelastic fluid models

The positive definite symmetric polymer conformation tensor possesses a unique symmetric square root that satisfies a closed evolution equation in the Oldroyd-B and FENE-P models of viscoelastic fluid flow. When expressed in terms of the velocity field and the symmetric square root of the conformation tensor, these models' equations of motion formally constitute an evolution in a Hilbert space with a total energy functional that defines a norm. Moreover, this formulation is easily implemented in direct numerical simulations resulting in significant practical advantages in terms of both accuracy and stability.Comment: 7 pages, 5 figure

Glass Transition Seen Through Asymptotic Expansions

18 pagesInternational audienceSoft glassy materials exhibit the so-called glassy transition which means that the behavior of the model at low shear rate changes when a certain parameter (which we call the glass parameter) crosses a critical value. This behavior goes from a Newtonian behavior to a Herschel-Bulkley behavior through a power-law-type behavior at the transition point. In a previous paper we rigorously proved that the Hébraud-Lequeux model, a Fokker-Planck-like description of soft glassy material, exhibits such a glass transition. But the method we used was very specific to the one dimensional setting of the model and as a preparation for generalizing this model to take into account multidimensional situations, we look for another technique to study the glass transition of this type of model. In this paper we shall use matched asymptotic expansions for such a study. The difficulties encountered when using asymptotic expansions for the \Heb-Lequeux are that multiple ansaetze have to be used even though the initial model is unique, due to the glass transition. We shall delineate the various regimes and give a rigorous justification of the expansion by means of an implicit function argument. The use of a two parameter expansion plays a crucial role in elucidating the reasons for the scalings which occur

Well-posedness of boundary layer equations for time-dependent flow of non-Newtonian fluids

We consider the flow of an upper convected Maxwell fluid in the limit of high Weissenberg and Reynolds number. In this limit, the no-slip condition cannot be imposed on the solutions. We derive equations for the resulting boundary layer and prove the well-posedness of these equations. A transformation to Lagrangian coordinates is crucial in the argument

Finite element approximation of surfactant spreading on a thin film

Published versio

Surface Gap Soliton Ground States for the Nonlinear Schr\"{o}dinger Equation

We consider the nonlinear Schr\"{o}dinger equation $(-\Delta +V(x))u = \Gamma(x) |u|^{p-1}u$, $x\in \R^n$ with $V(x) = V_1(x) \chi_{\{x_1>0\}}(x)+V_2(x) \chi_{\{x_1<0\}}(x)$ and $\Gamma(x) = \Gamma_1(x) \chi_{\{x_1>0\}}(x)+\Gamma_2(x) \chi_{\{x_1<0\}}(x)$ and with $V_1, V_2, \Gamma_1, \Gamma_2$ periodic in each coordinate direction. This problem describes the interface of two periodic media, e.g. photonic crystals. We study the existence of ground state $H^1$ solutions (surface gap soliton ground states) for $0<\min \sigma(-\Delta +V)$. Using a concentration compactness argument, we provide an abstract criterion for the existence based on ground state energies of each periodic problem (with $V\equiv V_1, \Gamma\equiv \Gamma_1$ and $V\equiv V_2, \Gamma\equiv \Gamma_2$) as well as a more practical criterion based on ground states themselves. Examples of interfaces satisfying these criteria are provided. In 1D it is shown that, surprisingly, the criteria can be reduced to conditions on the linear Bloch waves of the operators $-\tfrac{d^2}{dx^2} +V_1(x)$ and $-\tfrac{d^2}{dx^2} +V_2(x)$.Comment: definition of ground and bound states added, assumption (H2) weakened (sign changing nonlinearity is now allowed); 33 pages, 4 figure

Takens-Bogdanov bifurcation on the hexagonal lattice for double-layer convection

Abstract In the Bénard problem for two-fluid layers, Takens-Bogdanov bifurcations can arise when the stability thresholds for both layers are close to each other. In this paper, we provide an analysis of bifurcating solutions near such a Takens-Bogdanov point, under the assumption that solutions are doubly periodic with respect to a hexagonal lattice. Our analysis focusses on periodic solutions, secondary bifurcations from steady to periodic solutions and heteroclinic solutions arising as limits of periodic solutions. We compute the coefficients of the amplitude equations for a number of physical situations. Numerical integration of the amplitude equations reveals quasiperiodic and chaotic regimes, in addition to parameter regions where steady or periodic solutions are observed