70 research outputs found
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
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
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
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
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
A Milstein scheme for SPDEs
This article studies an infinite dimensional analog of Milstein's scheme for
finite dimensional stochastic ordinary differential equations (SODEs). The
Milstein scheme is known to be impressively efficient for SODEs which fulfill a
certain commutativity type condition. This article introduces the infinite
dimensional analog of this commutativity type condition and observes that a
certain class of semilinear stochastic partial differential equation (SPDEs)
with multiplicative trace class noise naturally fulfills the resulting infinite
dimensional commutativity condition. In particular, a suitable infinite
dimensional analog of Milstein's algorithm can be simulated efficiently for
such SPDEs and requires less computational operations and random variables than
previously considered algorithms for simulating such SPDEs. The analysis is
supported by numerical results for a stochastic heat equation and stochastic
reaction diffusion equations showing signifficant computational savings.Comment: The article is slightly revised and shortened. In particular, some
numerical simulations are remove
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