E ect of rugosity on a ow governedby Navier { Stokes

We study the ow of a uid occupying an innnite horizontal domain bounded by a rugose plate which is at rest and by a plane plate which moves with a constant velocity. The rugose plate is covered with periodically distibuted asperities of size ". We prove that, outside a neighbourhood of the rugose region, the ow behaves asymptotically as a Couette ow, as " ! 0, up to an exponentially small error. Innuence de la rugosit e sur un ecoulement gouvern e par les equations de Navier{Stokes R esum e. On etudie l' ecoulement d'un uide connn e entre une plaque rugueuse qui est au repos et une plaque plane qui se d eplace a vitesse constante. La plaque rugueuse est couverte d'asp erit es de taille " r eparties p eriodiquement. On montre qu'en dehors d'un voisinage des asp erit es, l' ecoulement se comporte asymptotiquement comme un ecoulement de Couette, quand " ! 0, avec une erreur exponentiellement petite. Abstract. We study the ow of a uid occupying an innnite horizontal domain bounded by a rugose plate which is at rest and by a plane plate which moves with a constant velocity. The rugose plate is covered with periodically distibuted asperities of size ". We prove that, outside a neighbourhood of the rugose region, the ow behaves asymptotically as a Couette ow, as " ! 0, up to an exponentially small error.