Coupling Stokes flow with inhomogeneous poroelasticity

Abstract

We investigate the behaviour of a system where a single phase fluid domain is coupled to a biphasic poroelastic domain. The fluid domain consists of an incompressible Newtonian viscous fluid while the poroelastic domain consists of a linear elastic solid filled with the same viscous fluid. The properties of the poroelastic domain, i.e. permeability and elastic parameters, depend on the inhomogeneous initial porosity field. The theoretical framework highlights how the heterogeneous material properties enter the linearised governing equations for the poroelastic domain. To couple flows through this domain with a surrounding Stokes flow, we show case a numerical implementation based on a new mixed formulation where the equations in the poroelastic domain are rewritten in terms of three fields: displacement, fluid pressure and total pressure.Coupling single phase and multiphase flow problems are ubiquitous in many industrial and biological applications, and here we consider an example from in-vitro tissue engineering. We consider a perfusion system, where a flow is forced to pass from the single phase fluid to the biphasic poroelastic domain. We focus on a simplified two dimensional geometry with small aspect ratio, and perform an asymptotic analysis to derive analytical solutions for the displacement, the pressure and the velocity fields. Our analysis advances the quantitative understanding of the role of heterogeneous material properties of a poroelastic domain on its mechanics when coupled with a fluid domain. Specifically, (i) the analytical analysis gives closed form relations that can be directly used in the design of slender perfusion systems; (ii) the numerical method is validated by comparing its result against selected theoretical solutions, opening towards the possibility to investigate more complex geometrical configurations.

The Biot-Stokes coupling using total pressure: formulation, analysis and application to interfacial flow in the eye

“…(2.4) Likewise, it is also possible to assume heterogeneity of the Lamé constants, as in [47]. In such a case, we need to assume that there exist constants λ min and λ max such that 0 < λ min ≤ λ ≤ λ max < ∞.…”

“…Examples include the interaction between soft permeable tissue and blood flow, or the study of the spatial growth of biofilm in fluids. When the interaction with a free fluid is considered, the mechanics of the fluid and poroelastic domains are coupled through balance of forces and continuity conditions that adopt diverse forms depending on the expected behaviour in the specific application (see, e.g., [23,46,47,48] and the references therein). The particular problem we consider in this paper as motivation for the design of the finite element formulation is the interfacial flow of aqueous humour between the anterior chamber and the trabecular meshwork (which is a deformable porous structure) in the eye, and how such phenomenon relates to early stages of glaucoma.…”

“…Another advantage of the present approach, inherited from [41] is that the formulation is free of poroelastic locking, meaning that it is robust with respect to the modulus of dilation of the poroelastic structure and with respect to permeability, which is of particular importance when we test variations of the flow response to changes in the material properties of the skeleton and when the solid approaches the incompressibility limit. The present work also stands as an extension of the formulation recently employed in [47] (where only the case of intrinsic incompressible constituent in the poroelastic region were considered) to obtain approximate solutions for heterogeneous poroelasticity coupled with Stokes flow in channels (and using also heterogeneous elastic moduli); while the PDE analysis, numerical aspects, and applicability of the formalism to more realistic scenarios have not yet been addressed.…”