Fluid Vesicles in Shear Flow

The dynamical behavior of fluid vesicles in simple shear flow has been stodied experimentally [190-193], theoretically [194-201], numerically with the boundary-integral technique [202,203] or the phase-field method [203,204], and with meso-scale solvents [37,180,205]. The vesicle shape is now determined by the competition of the curvature elasticity of the membrane, the constraints of constant volume V and constant surface area S, and the external hydrodynamic forces. 

Shear flow is characterized (in the absence of vesicles or cells) by the flow field V = jyCx, where Cx is a unit vector, compare Sect. 10.4. The control parameter of shear flow is the shear rate f, which has the dimension of an inverse time. Thus, a dimensionless, scaled shear rate y = ft can be defined, where T is a characteristic relaxation time of a vesicle. Here, t = rjoRpkBT is used, where rjo is the solvent viscosity, Rq the average radius [206]. For 1, the internal vesicle dynamics is 

One of the difficulties in theoretical studies of the hydrodynamic effects on vesicle dynamics is the no-slip boundary condition for the embedding fluid on the vesicle surface, which changes its shape dynamically under the effect of flow and curvature forces. In early studies, a fluid vesicle was therefore modeled as an ellipsoid with fixed shape [194]. This simplified model is still very useful as a reference for the interpretation of simulation results. 

The theory of Keller and Skalak [194] describes the hydrodynamic behavior of vesicles of fixed ellipsoidal shape in shear flow, with the viscosities t)in and qo of the internal and external fluids, respectively. Despite of the tqtproximations needed to derive the equation of motion for the inclinalion angle 6, which measures the deviation of the symmetry axis of the ellipsoid from the flow direction, this theory describes vesicles in flow surprisingly well. It has been generalized later [197] to describe the effects of a membrane viscosity Tjmb- 

The main result of the theory of Keller and Skalak is the equation of motion for the inclination angle [194], 

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H. Noguchi and G. Gompper, Dynamics of fluid vesicles in shear flow effect of membrane viscosity and thermal fluctuations, Phys. Rev. E 72, 011901 (2005). 

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