Large deformation of elastic capsules under uniaxial extensional flow

A spherical capsule (radius R) is suspended in a viscous liquid (viscosity μ) and exposed to a uniaxial extensional flow of strain rate E. The elasticity of the membrane surrounding the capsule is described by the Skalak constitutive law, expressed in terms of a surface shear modulus G and an area dilatation modulus K. Dimensional arguments imply that the slenderness ε of the deformed capsule depends only upon K/G and the elastic capillary number Ca=μRE/G. We address the coupled flow-deformation problem in the limit of strong flow, Ca>>1, where large deformation allows for the use of approximation methods in the limit ε<<1. 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 ε is proportional to Ca^-2/3, with the hydrodynamic problem introducing a dependence of the proportionality pre-factor upon ln ε. 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 K/G; in particular, we find the approximation ε^2/3Ca≈0.2753 ln(2/ε²) for the correlation between ε and Ca. 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 to rigorously discriminate between strain-softening and strain-hardening models.