Shear dynamics of confined membranes

We model the nonlinear response of a lubricated contact composed of a two-dimensional lipid membrane immersed in a simple fluid between two parallel flat and porous walls under shear. The nonlinear dynamics of the membrane gives rise to a rich dynamical behavior depending on the shear velocity. In quiescent conditions (i.e., absence of shear), the membrane freezes into a disordered labyrinthine wrinkle pattern. We determine the wavelength of this pattern as a function of the excess area of the membrane for a fairly general form of the confinement potential using a sine-profile ansatz for the wrinkles. In the presence of shear, we find four different regimes depending on the shear rate. Regime I. For small shear, the labyrinthine pattern is still frozen, but exhibits a small drift which is mainly along the shear direction. In this regime, the tangential forces on the walls due to the presence of the membrane increase linearly with the shear rate. Regime II. When the shear rate is increased above a critical value, the membrane rearranges, and wrinkles start to align along the shear direction. This regime is accompanied by a sharp drop of the tangential forces on the wall. The membrane usually reaches a steady-state configuration drifting with a small constant velocity at long times. However, we also rarely observe oscillatory dynamics in this regime. Regime III. For larger shear rates, the wrinkles align strongly along the shear direction, with a set of dislocation defects which assemble in pairs. The tangential forces are then controlled by the number of dislocations, and by the number of wrinkles between the two dislocations within each dislocation pairs. In this dislocation-dominated regime, the tangential forces in the transverse direction most often exceed those in the shear direction. Regime IV. For even larger shear, the membrane organizes into a perfect array of parallel stripes with no defects. The wavelength of the wrinkles is still identical to the wavelength in the absence of shear. In this final regime, the tangential forces due to the membrane vanish. These behaviors give rise to a non-linear rheological behavior of lubricated contacts containing membranes.