Large deformation of elastic capsules under uniaxial extensional flow

• Published in: Journal of fluid mechanics Vol. 1012
• Main Authors: Yariv, Ehud, Howell, Peter D., Stone, Howard A.
• Format: Journal Article
• Language: English
• Published: Cambridge, UK Cambridge University Press 09.06.2025
• Subjects: JFM Papers; membranes; slender-body theory; capsule/cell dynamics
• ISSN: 0022-1120; 1469-7645
• DOI: 10.1017/jfm.2025.10216

A spherical capsule (radius $R$ ) is suspended in a viscous liquid (viscosity $\mu$ ) 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 $\epsilon$ of the deformed capsule depends only upon $K/G$ and the elastic capillary number ${Ca}=\mu R E/G$ . We address the coupled flow–deformation problem in the limit of strong flow, ${Ca}\gg 1$ , where large deformation allows for the use of approximation methods in the limit $\epsilon \ll 1$ . 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 $\epsilon$ is proportional to ${Ca}^{-2/3}$ , with the hydrodynamic problem introducing a dependence of the proportionality prefactor upon $\ln \epsilon$ . 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 $\epsilon ^{2/3} {Ca}\approx 0.2753\ln (2/\epsilon ^2)$ for the relation between $\epsilon$ 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 us to rigorously discriminate between strain-softening and strain-hardening models.