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Deformation and orientation of a capsule with viscosity contrast in linear flows: a theoretical study

Paul Regazzi, Marc Leonetti

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.

Deformation and orientation of a capsule with viscosity contrast in linear flows: a theoretical study

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 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 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 contrary to the case of a viscous droplet. In our more general model, the deformation does no longer depend on at the upper order. Now, the Taylor parameter also depends on two other dimensionless numbers: the surface elastocapillary ratio and the dimensionless bending rigidity . 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.
Paper Structure (29 sections, 142 equations, 3 figures)

This paper contains 29 sections, 142 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Problem statement and numerical procedure
  3. Notations
  4. Problem statement
  5. Two-dimensional elastic constitutive laws
  6. Bending force and capillary force: bending rigidity, spontaneous curvature and surface tension
  7. Dimensionless quantities and relevant numbers
  8. Numerical procedure
  9. Analytic procedure
  10. Expansion to solid harmonics
  11. Geometric quantities
  12. Resolution at first order
  13. Second order
  14. Results and discussion
  15. Deformation and orientation under shear flow
  16. ...and 14 more sections

Figures (3)

  • Figure 1: Comparison between the numerical results and the theoretical equation (\ref{['eq:phiSk']}) 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 shear flow. The results are valid in the limit $\lambda\,Ca\,<<\,1$.
  • Figure 2: Comparison between the numerical results and the theoretical equation (\ref{['eq:DNH']}) 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 valid fully the agreement.
  • Figure 3: Comparison between the numerical results and the theoretical equation (\ref{['eq:DNH']}) of the deformation of a Neo-Hookean capsule with dimensionless bending rigidity $B$ under shear flow. The results are determined in the limit $\lambda\,Ca\,<<\,1$. The range of $B$ is chosen to valid fully the agreement.