A reciprocal theorem for poroelastic materials

In studying the transport of particles and inclusions in multi-phase systems we are often interested in integrated quantities such as the net force and the net velocity of the inclusions. In the reciprocal theorem, the known solution to the first and typically easier boundary value problem is used to compute the integrated quantities, such as the net force, in the second problem without the need to solve that problem. Here, we derive a reciprocal theorem for biphasic materials which are composed of a linear compressible viscoelastic solid phase, permeated by a viscous fluid. As an example, we analytically calculate the time-dependent net force on a rigid sphere in response to point-forces applied to the elastic network and the Newtonian fluid phases of the biphasic material. We find that when the point-force is applied to the fluid phase, the net force on the sphere evolves over timescales that are independent of the distance between the point-force and the sphere; in comparison, when the point-force is applied to the elastic phase the timescale for force development increases quadratically with the distance, in line with the scaling of poroelastic relaxation time. Finally, we discuss some applications of these solutions, including their utility in developing pseudo-analytical solutions, analogous to Stokes flow microhydronamics, to a mixture of spherical particles in a poroelastic material.