We propose a systematic formulation of the migration behaviors of a vesiclein a Poiseuille flow based on Onsager's variational principle. Our model isdescribed by a combination of the phase field theory for the vesicle and thehydrodynamics for the flow field. The time evolution equations for the phasefield of the vesicle and the flow field are derived based on the Onsager'sprinciple, where the dissipation functional is composed of viscous dissipationof the flow field, bending energy of the vesicle and the friction between thevesicle and the flow field. We performed a series of simulations on2-dimensional systems by changing the bending elasticity of the membrane, andobserved 3 types of steady states, i.e. those with bullet, snaking, and slippershapes. We show that the transitions among these steady states can bequantitatively explained with use of the Onsager's principle, where thedissipation functional is dominated by the contribution from the frictionbetween the vesicle and the flow field.
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