6 Aprile, 2016 16:00

Dipartimento di Matematica del Politecnico di Milano

Nonlinear poro-elastic and poro-visco-elastic models: analysis and applications

Giovanna Guidoboni, PhD, Associate Professor, Mathematical Sciences, Indiana University and Purdue University at Indiana

Sala Consiglio - 7° piano - Dipartimento di Matematica

We consider the initial and boundary value problem for a system of partial differential equations describing the motion
of a fluid-solid mixture under the assumption of full saturation. The ability of the fluid phase to flow within the solid
skeleton is described by the permeability tensor, which is assumed here to be a multiple of the identity and to depend
nonlinearly on the volumetric solid strain.
In particular, we study the problem of existence of weak solutions in bounded domains, accounting for non-zero volumetric
and boundary forcing terms. We investigate the influence of viscoelasticity on the solution functional setting and on the
regularity requirements for the forcing terms. The theoretical analysis shows that different time regularity requirements
are needed for the volumetric source of linear momentum and the boundary source of traction depending on whether or not
viscoelasticity is present.
The theoretical results are further investigated via numerical simulations based on a novel dual mixed hybridized finite
element discretization. When the data are sufficiently regular, the simulations show that the solutions satisfy the energy
estimates predicted by the theoretical analysis.
Interestingly, the simulations also show that, in the purely elastic case, the Darcy velocity and the related fluid energy
might become unbounded if indeed the data do not enjoy the time regularity required by the theory.
Applications will be discussed in the case of ocular perfusion.