Reciprocal theorem and Faxen's laws for particles in linear two-phase materials

Many soft and biological materials, including the cell cytoskeleton and polymer gels, are composed of a filamentous network that is permeated by a fluid, and they often contain particles of various shape, sizes and mechanical properties. In many applications we are interested in computing the net force/toque and velocity of these particles in response to external and active forces. The net force and velocity of inclusions in continuum scale are conventionally computed by solving the governing equations and integrating over the particle's surface. Here we present a reciprocal theorem for linear two-fluid models, which formulates the governing equations of the elastic network and the viscous fluid in terms of integrals of their traction and displacement fields over the particle's surfaces. This formulation allows for direct calculations of the net variables without the need to solve the governing equations. To demonstrate its utility, we use the reciprocal theorem to develop Faxen's laws for calculating the force on a rigid sphere in a general two-phase background deformation field. Faxen's laws can be used to develop fast particle simulation methods such as Stokesian Dynamics. Finally, we use the reciprocal theorem to develop a boundary integral formulation of the governing equations.