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.

Three constitutive laws (Skalak et al.'s law extended to area-compressible interfaces, Hooke's law and the Mooney–Rivlin law) commonly used to describe the mechanics of thin membranes are presented and compared. A small-deformation analysis of the tension–deformation relation for uniaxial extension and for isotropic dilatation allows us to establish a correspondence between the individual material parameters of the laws. A large-deformation analysis indicates that the Mooney–Rivlin law is strain softening, whereas the Skalak et al. law is strain hardening for any value of the membrane dilatation modulus. The large deformation of a capsule suspended in hyperbolic pure straining flow is then computed for several membrane constitutive laws. A capsule with a Mooney–Rivlin membrane bursts through the process of continuous elongation, whereas a capsule with a Skalak et al. membrane always reaches a steady state in the range of parameters considered. The small-deformation analysis of a spherical capsule embedded in a linear shear flow is modified to account for the effect of the membrane dilatation modulus.

An analysis is presented of the dynamics of a small deformable capsule freely suspended in a viscous fluid undergoing shear. The capsule consists of an elastic membrane which encloses another viscous fluid, and it deforms in response to the applied external stresses and the elastic forces generated within the membrane. Equations are derived which give its time-dependent deformation in the limit that the departure of the shape from sphericity is small. The form of the shear flow is arbitrary and a general (two-dimensional) elastic material is considered. Limiting forms are obtained for highly viscous capsules and for membranes which are area-preserving, and earlier results for surface tension droplets and incompressible isotropic membranes are derived as particular cases. Results for the viscosity of a dilute suspension of capsules are also given.The theoretical prediction for the relaxation rate of the shape is derived for an interface which has elastic properties appropriate for a red-blood-cell membrane, and is compared with experimental observations of erythrocytes.

This article reviews the mechanical behavior of a capsule under the influence of viscous deforming forces due to a flowing fluid. It focuses on artificial capsules and vesicles with an internal liquid core enclosed by a very thin membrane with different constitutive laws. The recent modeling strategies are outlined together with their respective advantages and limitations. I then consider the motion and deformation of a single, initially spherical capsule freely suspended in a simple shear or plane hyperbolic flow and discuss the effect of the membrane constitutive law, initial prestress, membrane buckling, and bulk or membrane viscosity. Finally, I consider the flow of spherical capsules in small pores and show how numerical models can be used to evaluate the mechanical properties of the membrane.

SUMMARYWe introduce a new numerical method to model the fluid-structure interaction between a microcapsule and an external flow. An explicit finite element method is used to model the large deformation of the capsule wall, which is treated as a bidimensional hyperelastic membrane. It is coupled with a boundary integral method to solve for the internal and external Stokes flows. Our results are compared with previous studies in two classical test cases: a capsule in a simple shear flow and in a planar hyperbolic flow. The method is found to be numerically stable, even when the membrane undergoes in-plane compression, which had been shown to be a destabilizing factor for other methods. The results are in very good agreement with the literature. When the viscous forces are increased with respect to the membrane elastic forces, three regimes are found for both flow cases. Our method allows a precise characterization of the critical parameters governing the transitions.

A theoretical method is presented for predicting the deformation and the conditions for breakup of a liquid droplet freely suspended in a general linear shear field. This is achieved by expanding the solution to the creeping-flow equations in powers of the deformation parameter ε and using linear stability theory to determine the onset of bursting. When compared with numerical solutions and with the available experimental data, the theoretical results are generally found to be of acceptable accuracy although, in some cases, the agreement is only qualitative.

Compression experiments between two parallel plates are performed on a series of biocompatible HSA-alginate capsules with two different membrane thicknesses. The capsule geometry and size as well as the average membrane thickness are first measured. The compression set-up is fitted with a sensitive force transducer that allows measurement of the compression force as a function of plate separation. The response of the capsule is analyzed by assuming different constitutive models for the membrane, where the shear and surface dilatation effects are accounted. An apparent area dilatation modulus is then computed for different values of the plate separation and required to remain constant as the capsule deformation increases. This allows identification of plausible constitutive laws for the membrane material.

Red blood cells or artificial vesicles may be conveniently represented by capsules, i.e. liquid droplets surrounded by deformable membranes. The aim of this paper is to assess the importance of viscoelastic properties of the membrane on the motion of a capsule freely suspended in a viscous liquid subjected to shear flow. A regular perturbation solution of the general problem is obtained when the particle is initially spherical and undergoing small deformations. With a purely viscous membrane (infinite relaxation time) the capsule deforms into an ellipsoid and has a continuous flipping motion. When the membrane relaxation time is of the same order as the shear time, the particle reaches a steady ellipsoidal shape which is oriented with respect to streamlines at an angle that varies between 45° and 0°, and decreases with increasing shear rates. Furthermore it is predicted that the deformation reaches a maximum value, which is consistent with experimental observations of red blood cells.