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

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 now 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.