Weakly deformable poroelastic particle in an unbounded Stokes flow

A framework is developed to study the deformation of a spherical poroelastic particle immersed in an unbounded, three-dimensional Stokes flow. The flow is driven by imposing a steady far-field condition, and the particle is modeled using a two-phase approach, where a deformable solid skeleton is fully saturated by the surrounding viscous fluid. Slip is permitted on the interface between the poroelastic particle and the surrounding Stokes flow. We consider the regime in which the ratio of typical viscous fluid stress to elastic stiffness is small, leading to small deformations and a decoupling of the fluid and solid problems. The traction exerted by the fluid on the particle and the Darcy pressure within are computed and used to formulate a purely solid-mechanics problem for the equilibrium particle deformation. To demonstrate the method, two example far-field flow profiles (shear flow and Poiseuille flow) are considered. Closed-form solutions for the translational velocity, rotation, and surface deformation of the particle are presented and analyzed as functions of the particle's permeability, slip, and Poisson's ratio. We show that the rotation of the particle is not impacted by its poroelasticity and that the Poisson's ratio plays a key role in selecting the dominant mechanism of particle deformation. For incompressible particles, the shear traction exerted by the fluid on the particle drives the deformation, causing the deformation to decrease with slip. For compressible particles, the Darcy pressure in the particle drives the deformation, and the deformation increases with slip.