Thursday, February 23, 2017 - 3:30pm
Giovanna Guidoboni, PhD Chair of Mathematics, LABEX IRMIA, Université de Strasbourg, France.
Adjunct Professor, Mathematical Sciences, Indiana University and Purdue University Indianapolis; Adjunct Professor, Ophthalmology, Indiana University School of Medicine.
Fluid flow through porous deformable media is relevant for many applications in biology, medicine and bioengineering, including the study of blood flow through tissues in the human body and fluid flow inside cartilages, bones and engineered tissue scaffolds. From the mathematical viewpoint, we are led to study the initial and boundary value problem for a nonlinear system of partial differential equations describing the motion of a fluid-solid mixture.
In this talk, we discuss the problem of existence of weak solutions in bounded domains, accounting for non-zero volumetric and boundary forcing terms. We investigate the influence of structural 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. Furthermore, blow-up in the solution is proved by exhibiting an explicit solution in the one-dimensional case.
Applications will be presented in the case of blood flow through ocular tissues. In this context, the theoretical findings led us to formulate a novel hypothesis concerning the causes of hemorrhages in the optic disc, namely that abrupt time variations in stress conditions, due to changes in intraocular pressure or cerebrospinal fluid pressure, combined with lack of structural viscoelasticity, due to ageing or disease, could lead to microstructural damage and rupture of blood vessels due to local fluid-dynamical alterations.