We develop a perturbation theory to study the shape and the orientation of an initially spherical capsule of radius R with a viscosity contrast, a surface tension σ and a bending rigidity $κ$ in linear flows. The elastic mechanical response of membrane to deformations is described by three elastic constitutive law which are either Hookean, Neohookean or Skalak type leading to the introduction of a surface shear elastic modulus $G_s$ and the Poisson ratio (or analog quantities). At the leading order, the deformation, i.e. the so-called Taylor parameter is proportional to the elastic capillary number Ca which evaluates the ratio between the external viscous stress and the elastic membrane response. In this linear regime, the results do not depend on the elastic constitutive law as expected. Without surface tension and bending rigidity, we recover the results of Barthes-Biesel & Rallison (1981) and notably the fact that the Taylor parameter does not depend on the viscosity contrast $λ$ contrary to the case of a viscous droplet. In our more general model, the deformation does no longer depend on $λ$ at the upper order. Now, the Taylor parameter also depends on two other dimensionless numbers: the surface elastocapillary ratio $σ/G_s$ and the dimensionless bending rigidity $B= κ/G_sR^2$. At the further order, the angle of inclination of the capsule with the direction of the shear flow, the analog of the Chaffey and Brenner equation for droplets is determined in each case. The results are in excellent agreement with the numerical ones performed with a code based on the boundary integral method providing an useful method to valid numerical developments.
Conveying, protecting and delivering active molecules are general goals encountered in many fields leading to the building of new smart multiphasic materials which can be passive or active. These ones often mimic nature. Indeed, migrating cells, organelles or cells carried by living fluids offered some guides to pave the way to such new structures. One of them, the capsule or the microcapsule has attracted much attention from several decades (Bah et al. (2020)) as a specific elastic body or as a biomimetic model of cells, Red Blood Cells in particular (Becic et al. (2025); Misbah (2012); Sui et al. (2008); Yazdani and Bagchi (2011)), but also due to its many analogs in cosmetics, health and food science/design/manufacturing (Sagis (2015)) and building materials (Tyagi et al. (2011)). Capsules belong to the large family of coated drops, whose the mechanical properties mainly depend on those of the interface. Other examples are liposomes, vesicles (Faizi et al. (2022); Has and Pan (2021)), polymersomes (Dionzou et al. (2016); Discher and Ahmed (2006)), cells (Dabagh et al. (2020); Lu et al. (2025)), droploon (Ginot et al. (2022)) and so on.
Capsules are droplets bounded by thin elastic shells (Barthes-Biesel (2016); Dupré de Baubigny et al. ( 2017)) that preserve their integrity and their ability to be a (bio-)chemical mini-reactor and may avoid any contamination from outside. More generally, the mechanic response of the shell to deformations and stress depends on its material properties driving by strainhardening or strain-softening elasticity (De Loubens et al. (2015)) and twodimensional (2D) dissipation characterized by surface viscosities (De Loubens et al. (2016); Gires et al. (2016)). Some capsules exhibit a viscoplastic behavior (Xie et al. (2017)). Several experimental configurations have been studied: capsules in constrictions (Chen et al. (2023); Chu et al. (2011); Le Goff et al. (2017); Risso et al. (2006); Rorai et al. (2015)), capsules in shear flow (Chang and Olbricht (1993a)), capsules in extensional flow (Chang and Olbricht (1993b); De Loubens et al. ( 2014)), capsules under compression and capsules under electric field (Karyappa et al. (2014)). In some cases, the rupture of the capsule is observed, an essential characteristic linked to the material properties and the applied mechanical tension along the shell (Chachanidze et al. (2023); Chang and Olbricht (1993b);El Mertahi et al. (2024); Feng et al. (2024); Husmann et al. (2005); Joung et al. (2020)). In shear and extensional flows, the shell can be under compression leading to buckling or wrinkling instabilities (Rehage et al. (2002); Walter et al. (2001)) if the bending resistance is not too high. Moreover, as shown in other elastocapillary systems (Bico et al. (2018); Style et al. (2017)) and wrinkling instability, surface tension can play an essential role in the emergence of such patterns.
Pioneering theoretical works have been developed to gain insight in the understanding on how deforms a capsule in linear flows in the limit of weak deformations. They determined the deformation at the leading order, i.e. proportional to the capillary number Ca which compares the viscous hydrodynamic stress to the shear elastic response: Barthes-Biesel et al. (2002); Barthes-Biesel and Rallison (1981). All these developments have been validated further by numerical simulations and they have been essential to determine the shear elastic modulus by comparison with experiments in the linear regime of deformations.
In this paper, a theoretical study is performed on initially spherical capsules under linear flows with additional properties compared to previous studies and up to the second order: viscosity contrast different from 1, bending rigidity and the surface tension. The analysis is done in the limit of weak deformation and in the regime of Stokes flow. The physical quantities are developed according to the capillary number up to the second order to allow the determination of the orientation of the capsule in a shear flow. This is the analog of the Chaffey-Brenner relation for a droplet (Chaffey and Brenner (1967)) but for a capsule with additional properties to be the most general. Moreover, the Hooke, NeoHookean and Skalak constitutive laws are considered. At the leading order, the expressions of the deformations of Barthes-Biesel and Rallison (1981) and Barthes-Biesel et al. (2002) are recovered without bending energy and surface tension. In chapter 2, all the basic equations in the bulk and at the interface are recalled. In chapter 3, the main analytic steps are provided in a generalized way. In chapter 4, the results (lengths of the equivalent ellipsoid, the deformations in each plane, the angle of orientation) are calculated and discussed. Our results are in excellent agreement with numerical ones.
2 Problem statement and numerical procedure
The summation convention is adopted that is each repeated indices in the equations are summed over.
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