Micro-swimmer locomotion and hydrodynamics in Brinkman flows

Micro-swimmer locomotion in heterogeneous media is central to biological physics, as microorganisms often move through complex environments. The Brinkman fluid—a model incorporating a sparse matrix of stationary obstacles via a linear resistance term in the momentum equation—captures key features of such porous media. We study two models for swimming and flow generation in this setting. First, we analyze a dumbbell swimmer: two spheres connected by a spring and actuated by a flagellar force. We derive its exact velocity as a function of Brinkman resistance and show it decreases monotonically with increasing drag. In the zero-resistance limit, the model recovers the classical Stokes dipole, while finite resistance introduces hydrodynamic screening that attenuates long-range interactions. We also derive the far-field flow of a Brinkmanlet force dipole and show that it closely matches the dumbbell swimmer's flow field in the far-field regime. These analytical results provide insights into locomotion in complex fluids and lay groundwork for studying collective behavior in active suspensions within porous or structured environments.