Mathematical analysis of the stabilization of lamellar phases by a shear stress

V. Torri

ESAIM: Control, Optimisation and Calculus of Variations (2002)

  • Volume: 7, page 239-267
  • ISSN: 1292-8119

Abstract

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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.

How to cite

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Torri, V.. "Mathematical analysis of the stabilization of lamellar phases by a shear stress." ESAIM: Control, Optimisation and Calculus of Variations 7 (2002): 239-267. <http://eudml.org/doc/245326>.

@article{Torri2002,
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.},
author = {Torri, V.},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {stabilization; shear stress; Couette system; global solution; energy estimates; local existence},
language = {eng},
pages = {239-267},
publisher = {EDP-Sciences},
title = {Mathematical analysis of the stabilization of lamellar phases by a shear stress},
url = {http://eudml.org/doc/245326},
volume = {7},
year = {2002},
}

TY - JOUR
AU - Torri, V.
TI - Mathematical analysis of the stabilization of lamellar phases by a shear stress
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2002
PB - EDP-Sciences
VL - 7
SP - 239
EP - 267
AB - 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.
LA - eng
KW - stabilization; shear stress; Couette system; global solution; energy estimates; local existence
UR - http://eudml.org/doc/245326
ER -

References

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  1. [1] 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. 
  2. [2] A. Babin, B. Nicolaenko and A. Mahalov, Regularity and integrability of 3 D Euler and Navier–Stokes equations for rotating fluids. Asymptot. Anal. 15 (1997) 103-150. Zbl0890.35109
  3. [3] 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). 
  4. [4] A. de Bouard and J.C. Saut, Solitary waves of generalized Kadomtsev-Petviashvili equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 14 (1997) 211-236. Zbl0883.35103MR1441393
  5. [5] O. Diat, D. Roux and F. Nallet. J. Phys. II France 3 (1993) 1427. 
  6. [6] O. Diat and D. Roux. J. Phys. II France 3 (1993) 9. 
  7. [7] I. Gallagher, Asymptotic of solutions of hyperbolic equations with a skew-symmetric perturbation. J. Differential Equations 150 (1998) 363-384. Zbl0921.35095MR1658597
  8. [8] E. Grenier, Oscillatory perturbations of the Navier–Stokes equations. J. Math. Pures Appl. 76 (1997) 477-498. Zbl0885.35090
  9. [9] O.A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow. Gordon and Breach (1963). Zbl0121.42701MR155093
  10. [10] J.L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod (1969). Zbl0189.40603MR259693
  11. [11] J.L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications. Dunod (1968). Zbl0165.10801
  12. [12] J. Simon, Compact sets in the Spaces L p ( 0 , T ; B ) . Ann. Mat. Pura Appl. 146 (1987) 65-96. Zbl0629.46031MR916688
  13. [13] R. Temam, Infinite-dimensional dynamical systems in mechanics and physics. Springer-Verlag, Appl. Math. Sci. 68 (1997). Zbl0871.35001MR1441312

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