Dynamics of strain-hardening and strain-softening capsules in strong planar extensional flows via an interfacial spectral boundary element algorithm for elastic membranes

Abstract

In the present study we investigate the dynamics of initially spherical capsules (made from elastic membranes obeying the strain-hardening Skalak or the strain-softening neo-Hookean law) in strong planar extensional flows via numerical computations. To achieve this, we develop a three-dimensional spectral boundary element algorithm for membranes with shearing and area-dilatation tensions in Stokes flow. The main attraction of this approach is that it exploits all the benefits of the spectral methods (i.e. high accuracy and numerical stability) but without creating denser systems. To achieve continuity of the interfacial geometry and its derivatives at the edges of the spectral elements during the interfacial deformation, a membrane-based interfacial smoothing is developed, via a Hermitian-like interpolation, for both the interfacial shape and the membrane elastic forces. Our numerical results show that no critical flow rate exists for both Skalak and neo-Hookean capsules in the moderate and strong planar extension flows considered in the present study. As the flow rate increases, both capsules reach elongated ellipsoidal steady-state configurations; the cross-section of the Skalak capsule preserves its elliptical shape, while the neo-Hookean capsule becomes more and more lamellar. The curvature at the pointed edges of these elongated steady-state shapes shows a very fast increase with the flow rate. The large interfacial deformations are accompanied with the development of strong membrane tensions especially for the strain-hardening Skalak capsule; the computed increase of the membrane tensions with the flow rate or the shape extension can be used to predict rupture of a specific membrane (with known lytic tension) due to excessive tensions. The type of the experiment imposed on the capsule as well as the applied flow rate affect dramatically the time evolution of the capsule edges owing to the interaction of the hydrodynamic forces with the membrane tensions; when a spherical Skalak capsule is let to deform in a strong flow, very large edge curvatures (with respect to the steady-state value) are developed during the transient evolution.

Semi-analytical solutions for two-dimensional elastic capsules in Stokes flow

“…Figure 2 shows the steady-state curvature at the capsule tip versus Ca as a solid line. The tip curvature grows rapidly with Ca as Ca → Ca c , which also occurs for three-dimensional capsules in the same linear strain flow (see Dodson & Dimitrakopoulos 2009). We see below in §4d that, when Ca > Ca c , the capsule 'bursts'-that is, it elongates without bound.…”

An unstructured solver for simulations of deformable particles in flows at arbitrary Reynolds numbers

“…This reduces the fluid-structure interaction (FSI) to a problem where the only unknowns are the membrane coordinates over time. Boundary integral method is thus a very popular technique to compute flows of capsules [4,5,[7][8][9][10][11], vesicles [12][13][14][15][16] and red blood cells [17][18][19], because of its precision and its relatively moderate computational cost.…”

Effects of membrane hardness and scaling analysis for capsules in planar extensional flows

“…where n is the interfacial unit normal pointing into the surrounding fluid, and the tensors S and T are the fundamental solutions for the velocity and stress for the three-dimensional Stokes equations, respectively (Pozrikidis 2001;Lac et al 2004;Dodson & Dimitrakopoulos 2009). Owing to the no-slip condition at the interface, the time evolution of the material points x of the membrane may be determined via the kinematic condition at the interface…”

“…In this work, we consider elastic membranes with shearing and area-dilatation resistance but negligible bending resistance. Our membrane description is based on the well-established continuum approach and the theory of thin shells which consider the membrane as a two-dimensional continuum with in-plane isotropy (Pozrikidis 2003;Lac et al 2004), as described in detail in § 2.2 of our earlier publication (Dodson & Dimitrakopoulos 2009). We emphasize that the study of capsules or cells via the continuum approach and the theory of thin shells is now a rather classical problem.…”

“…In this equation, the τ αβ | α notation denotes covariant differentiation, t β = ∂ x/∂θ β are the tangent vectors on the capsule surface described with arbitrary curvilinear coordinates θ β , and b αβ is the surface curvature tensor (Pozrikidis 2003;Lac et al 2004;Dodson & Dimitrakopoulos 2009). The in-plane stress tensor τ is described by constitutive laws that depend on the material composition of the membrane.…”