Spindles, Cusps, and Bifurcation for Capsules in Stokes Flow

Dodson, W. R.; Dimitrakopoulos, P.

doi:10.1103/physrevlett.101.208102

Cited by 34 publications

(60 citation statements)

References 21 publications

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“…We believe that our findings are relevant to three-dimensional capsules in examples where there is a rapid growth in tip curvature and motion of the interface towards a 'pinched' cusp shape, as in Dodson & Dimitrakopoulos (2008), because, locally, the flow near a three-dimensional emerging cusp is quasi-twodimensional. Our results, therefore, provide evidence that the local flow conditions in three dimensions are conducive to finite-time singularity formation.…”

Section: Introductionmentioning

confidence: 92%

“…This is similar to cusped profiles observed by Taylor (1934) for low viscosity ratio drops or bubbles. The first numerical investigations to show development of cusp-like shapes in the evolution of strained three-dimensional capsules that are similar to the experiments of Barthès-Biesel (1991) are those of Dodson & Dimitrakopoulos (2008). The membrane in the capsule experiments eventually breaks at the tips where the curvature is greatest.…”

Section: Introductionmentioning

confidence: 99%

“…The greater stability of the finite-element method is thought to be due to numerical regularization. Three-dimensional spectral element computations (Dodson & Dimitrakopoulos 2008) for a capsule in a planar far-field strain flow show steady-state shapes with tips that change from rounded to spindled to cusped as the strain rate is increased. It was not determined analytically if there is an upper bound on the capillary number for steady states, or if true cusp singularities that are associated with a blow-up of curvature occur.…”

Section: Introductionmentioning

confidence: 99%

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

2012

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Elastic capsules occur in nature in the form of cells and vesicles and are manufactured for biomedical applications. They are widely modelled, but there are few analytical results. In this paper, complex variable techniques are used to derive semi-analytical solutions for the steady-state response and time-dependent evolution of two-dimensional elastic capsules with an inviscid interior in Stokes flow. This provides a complete picture of the steady response of initially circular capsules in linear strain and shear flows as a function of the capillary number Ca. The analysis is complemented by spectrally accurate numerical computations of the time-dependent evolution. An imposed nonlinear strain that models the far-field velocity in Taylor's four-roller mill is found to lead to cusped steady shapes at a critical capillary number Ca c for Hookean capsules. Numerical simulation of the time-dependent evolution for Ca > Ca c shows the development of finite-time cusp singularities. The dynamics immediately prior to cusp formation are asymptotically self-similar, and the similarity exponents are predicted analytically and confirmed numerically. This is compelling evidence of finite-time singularity formation in fluid flow with elastic interfaces.

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Section: Introductionmentioning

confidence: 92%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

2012

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“…This problem has been studied for the past three decades and different techniques have been considered to derive numerical solutions (as summarized in Table I). Many studies have used a coupling strategy based on the boundary integral method to solve the Stokes flow and the membrane elasticity equations on the same mesh [6][7][8][9]. The velocity field at any position within the fluid domain is given by surface integrals calculated on the geometric boundaries.…”

Section: Introductionmentioning

confidence: 99%

Coupling of finite element and boundary integral methods for a capsule in a Stokes flow

Walter¹,

Salsac²,

Barthès-Biesel³

et al. 2010

Numerical Meth Engineering

139

155

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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.

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“…Soft deformable objects such as liquid droplets [1,2], lipid vesicles , red blood cells (RBCs) [25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41], and synthetic capsules [42][43][44][45][46][47][48][49][50][51] exhibit rich behaviors in flows. In recent years, they have received growing attention experimentally, theoretically, and numerically.…”

Section: Introductionmentioning

confidence: 99%

Dynamic modes of microcapsules in steady shear flow: Effects of bending and shear elasticities

Noguchi

2010

Phys. Rev. E

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The dynamics of microcapsules in steady shear flow was studied using a theoretical approach based on three variables: The Taylor deformation parameter αD, the inclination angle θ, and the phase angle φ of the membrane rotation. It is found that the dynamic phase diagram shows a remarkable change with an increase in the ratio of the membrane shear and bending elasticities. A fluid vesicle (no shear elasticity) exhibits three dynamic modes: (i) Tank-treading (TT) at low viscosity ηin of internal fluid (αD and θ relaxes to constant values), (ii) Tumbling (TB) at high ηin (θ rotates), and (iii) Swinging (SW) at middle ηin and high shear rateγ (θ oscillates). All of three modes are accompanied by a membrane (φ) rotation. For microcapsules with low shear elasticity, the TB phase with no φ rotation and the coexistence phase of SW and TB motions are induced by the energy barrier of φ rotation. Synchronization of φ rotation with TB rotation or SW oscillation occurs with integer ratios of rotational frequencies. At high shear elasticity, where a saddle point in the energy potential disappears, intermediate phases vanish, and either φ or θ rotation occurs. This phase behavior agrees with recent simulation results of microcapsules with low bending elasticity.

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Spindles, Cusps, and Bifurcation for Capsules in Stokes Flow

Cited by 34 publications

References 21 publications

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

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

Coupling of finite element and boundary integral methods for a capsule in a Stokes flow

Dynamic modes of microcapsules in steady shear flow: Effects of bending and shear elasticities

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