  Many parts of biological organisms are comprised of deformable porous media.
The biological media is both pliable enough to deform in response to an outside
force and can deform by itself using the work of an embedded muscle. For
example, the recent work (Ludeman et al., 2014) has demonstrated interesting
'sneezing' dynamics of a freshwater sponge, when the sponge contracts and
expands to clear itself from surrounding polluted water. We derive the
equations of motion for the dynamics of such an active porous media (i.e., a
deformable porous media that is capable of applying a force to itself with
internal muscles), filled with an incompressible fluid. These equations of
motion extend the earlier derived equation for a passive porous media filled
with an incompressible fluid. We use a variational approach with a Lagrangian
written as the sum of terms representing the kinetic and potential energy of
the elastic matrix, and the kinetic energy of the fluid, coupled through the
constraint of incompressibility. We then proceed to extend this theory by
computing the case when both the active porous media and the fluid are
incompressible, with the porous media still being deformable, which is often
the case for biological applications. For the particular case of a uniform
initial state, we rewrite the equations of motion in terms of two coupled
telegraph-like equations for the material (Lagrangian) particles expressed in
the Eulerian frame of reference, particularly suitable for numerical
simulations, formulated for both the compressible media/incompressible fluid
case and the doubly incompressible case. We derive interesting conservation
laws for the motion, perform numerical simulations in both cases and show the
possibility of self-propulsion of a biological organism due to particular
running wave-like application of the muscle stress.
