Slightly deformable Darcy drop in linear flows

Abstract

The small-deformation of a poroelastic drop due to an external linear flow field in the suspending medium is investigated. A two-phase flow model is developed to study the small-deformation of such a poroelastic drop under linear flows. Inside the drop a deformable porous network with a bending rigidity and a viscosity is fully immersed in a viscous fluid. When the viscous dissipation of the interior fluid phase is negligible (compared to the friction between the fluid and the skeleton), the two-phase flow is reduced to a poroelastic Darcy fluid with a deformable porous network. At the interface between the poroelastic drop and the exterior Stokesian fluid, a novel set of boundary conditions are derived by the free energy dissipation principle. Both slip and permeability are taken into account and the permeating flow induces dissipation that depends on the elastic stress of the interior solid. Assuming that the porous network is slightly deformed from its original spherical shape due to its large bending rigidity, a small-deformation analysis is conducted. A steady equilibrium is computed for two different linear flows: a uniaxial extensional flow and a planar shear flow. By exploring the interfacial slip, permeability and network elasticity various flow patterns are found at equilibrium of these slightly deformed poroelastic drops. Linear dynamics of the small-amplitude deviation of the poroelastic drop from the spherical shape is governed by a nonlinear eigenvalue problem, and the eigenvalues are determined numerically, and asymptotically in certain limits. These results lay the foundation for studying the rheology of a suspension of poroelastic spherical particles, and give insight to possible flow patterns of a system of selfpropelling swimmers with porous flow (such as intracellular cytosol) inside.

An arbitrary Lagrangian-Eulerian method for simulating interfacial dynamics between a hydrogel and a fluid

“…We adopt the poroelastic model for a fluid-permeated hydrogel as presented by Young et al [41], together with recent developments of boundary conditions for the fluid-gel interface [42].…”

“…Recently, Young et al [41,42] took a thermodynamically based approach to the question of the BCs on a fluid-gel interface. First, the fluid mass balance and total traction balance on the interface Γ yield two generally valid BCs:…”

“…A poroelastic model for the gel itself is sometimes called "two-phase" for the solvent and solid skeleton inside the gel [37]. The boundary condition between a porous medium and a clear fluid has long been a point of debate [39,40,32,41,42]. The crux is that an additional stress condition is needed because of the two momentum equations posed inside the gel for the network and interstitial fluid.…”

“…Recently, Young et al [41,42] took a new approach to these two difficulties. They invoked an energy dissipation argument to pose boundary conditions to ensure positive entropy production in the gel-fluid two-phase system.…”

“…This led to velocity discontinuities at the interface, both between the gel and the exterior fluid and between the solid and solvent inside the gel, that are driven by stresses at the interface. Furthermore, Young et al [41] applied such boundary conditions to solve the problem of a Darcy gel particle deforming slightly in an external flow field. The analytical solution was derived using perturbation in the limit of small deformations.…”