General solutions of poroelastic equations with viscous stress

TL;DR

This paper extends Biot's poroelastic theory by incorporating viscous stresses, deriving general solutions for displacement fields, and demonstrating applications in microrheology to distinguish material properties.

Contribution

It introduces a two-phase model with viscous stresses, derives closed-form solutions in spherical coordinates, and applies these to particle motion in poroelastic media.

Findings

Network compressibility causes slow relaxation of particles.

Non-monotonic network displacements occur over time.

Results enable differentiation of PE and VE materials in microrheology.

Abstract

Mechanical properties of cellular structures, including the cell cytoskeleton, are increasingly used as biomarkers for disease diagnosis and fundamental studies in cell biology. Recent experiments suggest that the cell cytoskeleton and its permeating cytosol, can be described as a poroelastic (PE) material. Biot theory is the standard model used to describe PE materials. Yet, this theory does not account for the fluid viscous stress, which can lead to inaccurate predictions of the mechanics in the dilute filamentous network of the cytoskeleton. Here, we adopt a two-phase model that extends Biot theory by including the fluid viscous stresses in the fluid's momentum equation. We use generalized linear viscoelastic (VE) constitutive equations to describe the permeating fluid and the network stresses and assume a constant friction coefficient that couples the fluid and network displacement fields. As the first step in developing a computational framework for solving the resulting equations, we derive closed-form general solutions of the fluid and network displacement fields in spherical coordinates. To demonstrate the applicability of our results, we study the motion of a rigid sphere moving under a constant force inside a PE medium, composed of a linear elastic network and a Newtonian fluid. We find that the network compressibility introduces a slow relaxation of the sphere and a non-monotonic network displacements with time along the direction of the applied force. These novel features cannot be predicted if VE constitutive equation is used for the medium. We show that our results can be applied to particle-tracking microrheology to differentiate between PE and VE materials and to independently measure the permeability and VE properties of the fluid and the network phases.