Mapping vesicle dynamics onto that of a rigid sphere in five dimensions

International audienceVesicles are sacs made of a phospholipid bilayer, and they mimic the cytoplasmic membrane of real cells, for which red blood cells constitute a canonical example. Vesicles deform under flow, such as a shear flow. Under a linear shear flow, they are known to exhibit several motions that combine orientation and shape deformation (such as tank treading, vacillating breathing, and so on). It is shown here that the equations of motion of a vesicle under shear flow in the weak deformation regime can be mapped onto those of a loaded (or heavy bottom) rigid sphere in five dimensions in fictitious gravitational and shear fields. Based on our previous exact analytical solutions for vesicles (which we extend here to out-of-shear-plane motions), we provide hitherto unrevealed exact explicit solution for the rigid sphere problem. We explain how deformation of a vesicle in real space can be extracted from a rigid body dynamics in five dimensions upon appropriate projection onto a lower dimension. This study offers a framework where rigid spheres and deformable vesicles are recast into the same universality class in which both systems are described by the same formal equations differing only by the space dimension