Large deformation of elastic capsules under uniaxial extensional flow

A spherical capsule (radius) is suspended in a viscous liquid (viscosity) and exposed to a uniaxial extensional flow of strain rate. The elasticity of the membrane surrounding the capsule is described by the Skalak constitutive law, expressed in terms of a surface shear modulus and an area dilatation modulus. Dimensional arguments imply that the slenderness of the deformed capsule depends only upon and the elastic capillary number. We address the coupled flow-deformation problem in the limit of strong flow, where large deformation allows for the use of approximation methods in the limit. The key conceptual challenge, encountered at the very formulation of the problem, is in describing the Lagrangian mapping from the spherical reference state in a manner compatible with hydrodynamic slender-body formulation. Scaling analysis reveals that is proportional to, with the hydrodynamic problem introducing a dependence of the proportionality prefactor upon. Going beyond scaling arguments, we employ asymptotic methods to obtain a reduced formulation, consisting of a differential equation governing a mapping field and an integral equation governing the axial tension distribution. The leading-order deformation is independent of the ratio; in particular, we find the approximation for the relation between and. A scaling analysis for the neo-Hookean constitutive law reveals the impossibility of a steady slender shape, in agreement with existing numerical simulations. More generally, the present asymptotic paradigm allows us to rigorously discriminate between strain-softening and strain-hardening models.