A particle‐based moving interface method (PMIM) for modeling the large deformation of boundaries in soft matter systems

Abstract

SUMMARYThe mechanics of the interaction between a fluid and a soft interface undergoing large deformations appear in many places, such as in biological systems or industrial processes. We present an Eulerian approach that describes the mechanics of an interface and its interactions with a surrounding fluid via the so-called Navier boundary condition. The interface is modeled as a curvilinear surface with arbitrary mechanical properties across which discontinuities in pressure and tangential fluid velocity can be accounted for using a modified version of the extended finite element method. The coupling between the interface and the fluid is enforced through the use of Lagrange multipliers. The tracking and evolution of the interface are then handled in a Lagrangian step with the grid-based particle method. We show that this method is ideal to describe large membrane deformations and Navier boundary conditions on the interface with velocity/pressure discontinuities. The validity of the model is assessed by evaluating the numerical convergence for a axisymmetrical flow past a spherical capsule with various surface properties. We show the effect of slip length on the shear flow past a two-dimensional capsule and simulate the compression of an elastic membrane lying on a viscous fluid substrate.

Phoretic motion of soft vesicles and droplets: an XFEM/particle-based numerical solution

“…As discussed in [37], the force equilibrium for a fluid-fluid interface can generally be written in terms of the jump in the stress vector across Γ as: $$\left[\mathit{\sigma }\cdot \overline{\mathrm{boldn}}\right]=-2\gamma \mathscr{H}\overline{\mathbf{n}}$$with $\mathscr{H}$ the mean curvature of the interface and γ its surface tension. The normal projection of this equation yields the standard Laplace–Young relation [46]: $$\left[p\right]=-2\gamma \mathscr{H}$$while the tangential projection implies that the tangential stress vector ${\left(\mathit{\sigma }\cdot \overline{\mathrm{boldn}}\right)}^{║}$ is continuous across the interface.…”

“…The normal projection of this equation yields the standard Laplace–Young relation [46]: $$\left[p\right]=-2\gamma \mathscr{H}$$while the tangential projection implies that the tangential stress vector ${\left(\mathit{\sigma }\cdot \overline{\mathrm{boldn}}\right)}^{║}$ is continuous across the interface. Equation (6) is furthermore complemented with the tangential force balance between the membrane and the interface [37], or Navier condition that reads: $${\left(\mathit{\sigma }\cdot \overline{\mathrm{boldn}}\right)}^{║+}={\left(\frac{\mu }{\ell }\right)}^{+}\left\{{\left[\mathrm{boldv}\right]}^{║+}-{\mathbf{v}}_a\right\}$$ $${\left(\mathit{\sigma }\cdot \overline{\mathrm{boldn}}\right)}^{║-}={\left(\frac{\mu }{\ell }\right)}^{-}{\left[\mathrm{boldv}\right]}^{║-}$$where ℓ ± > 0 are the slip length parameters enforced on the convective flow (on each side of Γ ); the case ℓ = 0 corresponds to a no-slip condition while ℓ = ∞ corresponds to a free-slip condition at the interface. The quantity v a is the active tangential velocities created by the solute-interface interactions, the source of which will be discussed later in this section.…”

“…In the present paper, we propose a numerical method based on the extended finite element method (XFEM) and particle-based moving interface method (PMIM) [37,38] that can naturally describe the generation of active flow on particle surfaces and its mobility. The X-FEM is indeed ideal to model a variety of problems [39–43] where field discontinuities play an important role.…”

“…The X-FEM is indeed ideal to model a variety of problems [39–43] where field discontinuities play an important role. The PMIM further provides a flexible complementary method when boundaries move on the background mesh [37]. The proposed approach possesses three main features that are particularly important in the context of particle propulsion: (a) First, the discontinuous tangential velocity and pressure caused by the phoretic active flow is accounted by introducing enriched tangential degrees of freedom for the fluid velocity across the interface.…”

Mechanics and stability of vesicles and droplets in confined spaces

“…The global equilibrium of the vesicle can be easily derived by taking the difference between ρ 2 and ρ 1 in order to obtain $$\mathrm{normal\Delta }P=2\gamma \left({\rho }_2-{\rho }_1\right).$$Note that this expression is only valid for equilibrium or quasistatic systems in which the inner vesicle pressure is homogeneous and there is no fluid flow around the pore. A dynamic approach would require solving the Navier-Stokes equations coupled with the membrane governing equations [43]. By simple geometrical relations, one can show that the cap curvatures can be related to the pore geometry by ρ i = −cos( θ + β i )/ r i , where r i and β i = arctan[ r ′( z i )] are the radii and the signed tangent angle (with r′ = dr / dz ) of each contact lines (Fig.…”