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Mathematical analysis of the stabilization of lamellar phasesby a shear stress

Published online by Cambridge University Press:  15 September 2002

V. Torri
V. Torri*
Affiliation:
Mathématiques Appliquées de Bordeaux, Université de Bordeaux 1 et UMR 5466 du CNRS, 351 cours de la Libération, 33405 Talence Cedex, France; torri@math.u-bordeaux.fr.
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Abstract

We consider a 2D mathematical model describing the motion of a solution of surfactants submitted to a high shear stress in a Couette-Taylor system. We are interested in a stabilization process obtained thanks to the shear. We prove that, if the shear stress is large enough, there exists global in time solution for small initial data and that the solution of the linearized system (controlled by a nonconstant parameter) tends to 0 as t goes to infinity. This explains rigorously some experiments.

Keywords

Stabilizationshear stressCouette systemglobal solution.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 7 , 2002 , pp. 239 - 267
Copyright
© EDP Sciences, SMAI, 2002

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References

O.V. Besov, V.P. Il'in and S.M. Nikol'skii, Integral representations of functions and embeddings theorems, Vol. 1. V.H. Winston and Sons.
Babin, A., Nicolaenko, B. and Mahalov, A., Regularity and integrability of 3D Euler and Navier-Stokes equations for rotating fluids. Asymptot. Anal. 15 (1997) 103-150.
A. Colin, T. Colin, D. Roux and A.S. Wunenburger, Undulation instability under shear: A model to explain the different orientation of a lamellar phase under shear. European J. Soft Condensed Matter (to appear).
de Bouard, A. and Saut, J.C., Solitary waves of generalized Kadomtsev-Petviashvili equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 14 (1997) 211-236. CrossRef
Diat, O., Roux, D. and Nallet, F.. J. Phys. II France 3 (1993) 1427. CrossRef
O. Diat and D. Roux. J. Phys. II France 3 (1993) 9.
Gallagher, I., Asymptotic of solutions of hyperbolic equations with a skew-symmetric perturbation. J. Differential Equations 150 (1998) 363-384. CrossRef
Grenier, E., Oscillatory perturbations of the Navier-Stokes equations. J. Math. Pures Appl. 76 (1997) 477-498. CrossRef
O.A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow. Gordon and Breach (1963).
J.L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod (1969).
J.L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications. Dunod (1968).
J. Simon, Compact sets in the Spaces L p (0,T;B). Ann. Mat. Pura Appl. 146 (1987) 65-96.
R. Temam, Infinite-dimensional dynamical systems in mechanics and physics. Springer-Verlag, Appl. Math. Sci. 68 (1997).