Onsager's Variational Principle for the Dynamics of a Vesicle in a Poiseuille Flow

We propose a systematic formulation of the migration behaviors of a vesicle in a Poiseuille flow based on Onsager's variational principle. Our model is described by a combination of the phase field theory for the vesicle and the hydrodynamics for the flow field. The time evolution equations for the phase field of the vesicle and the flow field are derived based on the Onsager's principle, where the dissipation functional is composed of viscous dissipation of the flow field, bending energy of the vesicle and the friction between the vesicle and the flow field. We performed a series of simulations on 2-dimensional systems by changing the bending elasticity of the membrane, and observed 3 types of steady states, i.e. those with bullet, snaking, and slipper shapes. We show that the transitions among these steady states can be quantitatively explained with use of the Onsager's principle, where the dissipation functional is dominated by the contribution from the friction between the vesicle and the flow field.