Numerical and dynamical analysis of undulation instability under shear stress

This article presents a numerical and dynamical study of a system of partial differential equations, describing the motion of a lamellar phase in a solution of surfactants in a Couette-Taylor system. It has been shown that, at high shear rate, a stabilization of the system occurs. We show, under a hypothesis on the spectrum, that this system has a local center manifold. This hypothesis on the spectrum is verified numerically, by using a finite difference method. The numerical results show that a Hopf bifurcation occurs at some shear rate. The velocity of the layers at the Hopf bifurcation corresponds to the one when the layers break themselves in the physical case. In addition, an instability result at low shear rate is proved.