Soft Mobility Theory

Abstract

Predicting how a deformable body moves and deforms in a viscous flow underlies problems ranging from microorganism locomotion to soft microrobotics, yet existing frameworks are either problem-specific or ill-suited to inverse design. We propose the soft mobility theory: applying the principle of virtual power and the Lorentz reciprocal theorem to a hyperelastic body in a background Stokes flow yields a configuration-dependent ordinary differential equation for the generalized coordinates of the body. This soft mobility equation extends classical rigid-body mobility theory in that the mobility, elastic, body-force, and flow-coupling tensors all depend explicitly on the instantaneous deformation. We specialize the framework to assemblies of hydrodynamically interacting spheres connected by elastic springs, using the Rotne-Prager-Yamakawa approximation to compute the mobility, and validate it on canonical problems spanning rigid and flexible bodies in quiescent and shear flows. An open-source JAX implementation makes entire simulations end-to-end differentiable. This allows efficient gradient-based inverse design: as proofs of concept, we recover the asymptotic optimum of a three-sphere swimmer and design a soft gyrotactic "surfer" that exploits passive deformation to ascend faster than its rigid counterpart in a Taylor-Green flow.