[Paper Review] Deformation and orientation of a capsule with viscosity contrast in linear flows: a theoretical study

Paul Regazzi, Marc Leonetti|arXiv (Cornell University)|Feb 17, 2026

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TL;DR

The paper develops a perturbation theory to characterize the shape and orientation of an initially spherical capsule with viscosity contrast in linear flows, up to second order in deformation, including surface tension and bending rigidity, and validates results against boundary-integral simulations.

ABSTRACT

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.

Motivation & Objective

• Motivate understanding of capsule mechanics in linear flows as a model for cells and microcapsules.
• Develop a perturbation framework to compute shape and orientation up to second order in deformation.
• Incorporate viscosity contrast, surface tension, and bending rigidity via elastic constitutive laws (Hookean, Neo-Hookean, Skalak).
• Derive the generalized orientation (Chaffey–Brenner type) relation for capsules and compare with numerical results.

Proposed method

• Use perturbation theory with a small deformation expansion around a sphere (r = 1 + F^(1) + F^(2)).
• Model membrane elasticity with Hookean, Neo-Hookean, and Skalak constitutive laws through a Cauchy stress tensor and surface energy function w(I1,I2).
• Include bending and surface tension via Helfrich bending energy and a surface tension term, leading to f_el, f_kappa, and f_sigma forces at the interface.
• Non-dimensionalize with Ca = eta dot{epsilon} R / G_s and include viscosity contrast lambda, elastocapillary Sigma = sigma / G_s, and bending ratio B = kappa/(G_s R^2).
• Solve the Stokes flow with boundary-integral-like harmonic expansions (solid harmonics) up to second order to obtain deformation tensors F^(1), K^(1) and the orientation angle.
• Provide explicit analytic expressions for deformation and orientation in shear flow under each constitutive law and validate against boundary-integral numerical results.

Figure 1 : Comparison between the numerical results and the theoretical equation ( 88 ) in the case of a capsule obeying to the Skalak constitutive law and with a viscosity contrast $\lambda$ : deviation from $\pi/4$ of the angle $\phi_{TT}$ of the capsule’s orientation with the direction of the she

Experimental results

Research questions

• RQ1 How does a viscosity contrast (lambda) affect capsule deformation and orientation in linear flows beyond the leading order?
• RQ2 How do surface tension (sigma) and bending rigidity (kappa) interact with elastic membrane properties to modify the deformation and the orientation angle?
• RQ3 What is the second-order correction to the orientation angle of a capsule in shear flow (analog of Chaffey–Brenner for capsules) under different constitutive laws?
• RQ4 Do the analytical predictions agree with high-fidelity boundary-integral simulations across relevant parameter regimes (Ca, lambda, Sigma, B)?
• RQ5 Can the general framework recover known results in limiting cases (e.g., Barthes-Biesel and Rallison 1981) and extend them to more general membrane physics?

Key findings

• At leading order, the deformation (Taylor parameter) is proportional to Ca and is independent of the internal viscosity contrast lambda (recovering Barthes-Biesel and Rallison 1981 results).
• At the next order, the deformation remains independent of lambda and becomes governed by Sigma = sigma/G_s and B = kappa/(G_s R^2).
• The orientation angle in shear flow (phi_TT) is given by a second-order expression (e.g., Eq. (88) for Skalak) and reduces to known results in appropriate limits; orientation deviates from pi/4 with a Ca-dependent correction that depends on lambda, C, B, Sigma.
• Deformation components and the orientation predicted by the analytical theory show excellent agreement with numerical boundary-integral simulations across the studied parameter space (valid for lambda Ca << 1).
• The Neo-Hookean, Hooke, and Skalak constitutive laws yield consistent first-order deformations and orientation corrections, with explicit formulas provided for each case (e.g., E_ij-based expressions in shear).
• The framework yields explicit expressions for semi-axes L, S, W and cross-deformations D_12, D_13, D_23 in planar shear, and confirms the generalized Chaffey–Brenner type relation for capsules.

Figure 2 : Comparison between the numerical results and the theoretical equation ( 113 ) of the deformation of a Neo-Hookean capsule with the surface elastocapillary number $\Sigma$ under shear flow. The results are determined in the limit $\lambda\,Ca\,<<\,1$ . The range of $\Sigma$ is chosen to va