Affiliations: LATP, CMI, UMR 6632, Université Aix‐Marseille 1, 39 rue Joliot‐Curie, 13453 Marseille cedex 13, France E‐mail: [email protected]‐mrs.fr | Mathématiques Appliquées de Bordeaux, UMR 5466, Université Bordeaux 1, 351 cours de la Libération, 33405 Talence cedex, France E‐mail: [email protected]‐bordeaux.fr
Abstract: We consider a diphasic system in a high shear regime when separation of phases occurs. One can observe that the two phases organize themselves into numerous bands, parallel to the flow direction. Mathematically, these are solutions of a certain system of equations depending only upon the transversal variable (1D solutions). We study the stability of these 1D solutions with respect to 2D perturbations. The mathematical model used in our analysis is a coupling between a Cahn–Hilliard equation and the Navier–Stokes equations in two dimensions. We show that a small 2D perturbation of a given 1D solution persists for significant times. We give the precise size of such a perturbation and its time of persistence. Moreover, we obtain an asymptotic expansion of the solution in the considered cases. Note that, for a mathematical model to be realistic one has to take into account the fact that in experiments the high shear regime is obtained in elongated domains (a very thin Couette cell for example). Therefore, we perform the mathematical analysis of this problem in a stretched domain.
