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Ondes non-linéaires à une et deux dimensions
dans une mince couche de fluide

Résumé
Les ondes hydrothermales constituent un système modèle d'ondes non-linéaires dont nous étudions expérimentalement la transition vers le chaos spatio-temporel. A une dimension d'espace, par comparaison entre une cellule annulaire périodique et une cellule rectangulaire de taille finie, nous montrons que l'instabilité primaire en ondes ne survient dans ce dernier cas que lorsque la transition convectif /absolu est franchie. Nous isolons de même pour l'instabilité secondaire d'Eckhaus les régimes d'instabilité convective et absolue. A deux dimensions d'espace dans une cellule cylindrique, nous étudions la structuration de l'écoulement de base en rouleaux corotatifs, puis les différents modes d'instabilités en fonction de la différence de température appliquée et de la hauteur de fluide : spirales d'Archimède, cibles, fleurs et rayons apparaissent chacuns par bifurcation de Hopf supercritique. Deux modes d'ondes hydrothermales sont alors distingués. Nous tentons d'expliquer l'existence de ces deux modes différents par une étude théorique et numérique : nous quantifions les effets de la courbure sur les instabilités à deux dimensions.

One and two-dimensionnal non-linear waves
in a thin liquid layer

Abstract
Hydrothermal waves constitute an ideal non-linear waves system which we are studying experimentaly the transition to spatio-temporal chaos. When the system is unidimensional, we compare an annular cell and a rectangular cell of finite extent and show that the primary instability into travelling waves occurs in the rectangular case only when the convective/absolute threshold is reached. With the same arguments, we distinguish two regimes for Eckhaus secondary instability in finite geometry : a convective and an absolute one. When the system is two-dimensional and the geometry is cylindrical, we study first the basic flow structuration, then the different instability modes depending on fluid depth and temperature difference across the cell : Archimedean spirals, targets, flowers and radial lines each appear through a supercritical Hopf bifurcation. Two hydrothermal waves modes are isoled. We present a theoretical and numerical study of the curvature effects in the two-dimensional geometry which may explain some of the differences between the two modes.