Elastohydrodynamic repulsion between particles translating near an elastic sheet

The biological world can be extremely soft, and biological membranes are easily deformed by the motion of nearby particles. The motion of a single particle translating near a soft substrate is well-studied in recent years. In this work, we investigate, using theory and experiments, the interaction between two particles sedimenting side-by-side near a soft substrate. Experimentally, we measure the relative displacement of two millimetric-sized spheres settling under gravity alongside an elastic sheet suspended in silicone oil. We observe that the spheres drift apart from each other while deforming the elastic sheet as they settle, whereas no such drift is observed in the absence of the sheet. We rationalize this behavior using numerical solutions of elastohydrodynamic lubrication theory for the two-particle system, and show that the particles drift apart as a result of a symmetry-breaking deformation of the nearby sheet. The hydrodynamic pressure associated with the motion of each particle deforms the sheet, which, in turn, generates a secondary flow that repels the other particle in the direction normal to the direction of settling. We validate the numerical results by developing an analytic theory, based on the Lorentz reciprocal theorem, for small deformations of the sheet. Our analysis shows that this elastohydrodynamic interaction is long-ranged and depends on both the bending rigidity and tension of the sheet. We posit that these elastohydrodynamic interactions are relevant to the motion of suspensions in confined soft systems.