Spherical capsules in three-dimensional unbounded Stokes flows: effect of the membrane constitutive law and onset of buckling

Abstract

The dynamic response of an initially spherical capsule subject to different externally imposed flows is examined. The neo-Hookean and Skalak et al. (Biophys. J., vol. 13 (1973), pp. 245-264) constitutive laws are used for the description of the membrane mechanics, assuming negligible bending resistance. The viscosity ratio between the interior and exterior fluids of the capsule is taken to be unity and creeping-flow conditions are assumed to prevail. The capillary number ε is the basic dimensionless number of the problem, which measures the relative importance of viscous and elastic forces. The boundary-element method is used with bi-cubic B-splines as basis functions in order to discretize the capsule surface by a structured mesh. This guarantees continuity of second derivatives with respect to the position of the Lagrangian particles used for tracking the location of the interface at each time step and improves the accuracy of the method. For simple shear flow and hyperbolic flow, an interval in ε is identified within which stable equilibrium shapes are obtained. For smaller values of ε, steady shapes are briefly captured, but they soon become unstable owing to the development of compressive tensions in the membrane near the equator that cause the capsule to buckle. The post-buckling state of the capsule is conjectured to exhibit small folds around the equator similar to those reported by Walter et al. Colloid Polymer Sci. Vol. 278 (2001), pp. 123-132 for polysiloxane microcapsules. For large values of ε, beyond the interval of stability, the membrane has two tips along the direction of elongation where the deformation is most severe, and no equilibrium shapes could be identified. For both regions outside the interval of stability, the membrane model is not appropriate and bending resistance is essential to obtain realistic capsule shapes. This pattern persists for the two constitutive laws that were used, with the Skalak et al. law producing a wider stability interval than the neo-Hookean law owing to its strain hardening nature.

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

“…By way of contrast, in a simple shear flow, the capsule major axis rotates to a steady limiting value, and the deformation slightly overshoots its steady value before relaxing back to it. This feature is also observed in the three-dimensional capsule computations of Lac et al (2004). In the steady state, the capsule 'tank-treads'-that is, the capsule membrane rotates with non-zero tangential velocity.…”

“…However, subsequent studies (Navot 1998;Ramanujan & Pozrikidis 1998) found steady deformation for a range of capillary number Ca and viscosity ratio l, and instead suggested that there is no critical capillary number for bursting. The three-dimensional computations of time-dependent evolution by Lac et al (2004) attain a stable steady deformation only when the capillary number lies in a range Ca ∈ [Ca L , Ca H ], which depends on the specific membrane constitutive law, viscosity ratio and initial membrane stress. For values of the capillary number with Ca < Ca L or Ca > Ca H , compressive stresses develop in at least one of the two principal elastic tensions on the interface (see also Barthès-Biesel 1980), and when this occurs the evolution problem becomes highly unstable and the numerical method fails.…”

Dynamics of a viscous vesicle in linear flows

“…Larger deformations from sphericity can be explored by numerical simulations. A body of work exists on capsule dynamics, mostly considering elastic membranes with no bending resistance 40,41,42 . To our knowledge there are only a couple of numerical studies that include bending resistance 43,44 .…”

Wrinkling, tumbling and swinging microcapsules in simple shear flow

“…We noticed that the folding process always appeared parallel to the main axis of the capsules (Fig. 7) [12]. This phenomenon might be induced by the presence of negative membrane tensions.…”

“…These folding processes, which occur in shear flow, were first observed by A. Walter et al for polyamide and polysiloxane membranes [10]. It is still under debate in which degree osmotic pressures, membrane pre-stresses or bending elasticity's are responsible for these buckling instabilities [9,11,12].…”