Impact of shear and dilational membrane viscosities on capsule shape and dynamics under shear flow: a theoretical study

In artificial (micro)capsules or cells such as red blood cells for example, the two-dimensional (2D) rheology of the membrane (or the shell) is viscoelastic and not only elastic. The deformation and inclination of such a capsule are theoretically studied under shear and extensional flows. The membrane mechanics is described by the Kelvin-Voigt model: the 2D dissipation with the Boussinesq-Scriven law associated to specific dilational and shear Boussinesq numbers $B{q}_d$ and $B{q}_d$ respectively while the elastic constitutive law is either Hookean, strain-softening (neoHookean) or strain-hardening (Skalak). The dynamics depends on the elastic capillary number $Ca$, the Poisson ratio (or $C$ in Skalak law), the viscosity ratio $\lambda$, the nonlinearities included in the constitutive law and both boussinesq numbers. Three dynamics are identified in the limit of weak deformations: a tank-treading motion in the limit of small elastic capillary number with a characteristic time depending on $\lambda$, $B{q}_s$ and $B{q}_d$, an oscillation without any elasticity and a damped oscillation leading to a tank-treading motion in the limit of large Boussinesq numbers. In all cases, both shear and dilational 2D dissipations contribute differently leading to non-trivial results. In the case of a Skalak law with $C=1$, $Bq=B{q}_s=B{q}_d$ and $Bq>>\lambda$, the deformations in the limits $Ca<<1$, $Bq=O(1)$ and $Ca=O(1)$, $Bq>>1$ determined in the seminal work of \cite{barthes1985role} are recovered.