Simulation of the dynamics of bio-membranes in a viscous fluid with a phasefield variational Lagrangian approach

Bio-membranes are the basic separation structure in animal cells. Their complex behaviour, rich physical properties, formation and dynamics have been the object of experimental and theoretical investigation for biologists, chemists and physicists for many years. Bio-membranes are made out of several kinds of lipids self-assembled in a fuid bilayer, which presents a fluid behaviour in-plane and solid out-of-plane (curvature elasticity). Vesicles are closed fluid membranes, which play an important role in biophysical processes such as transfer of proteins, antibodies or drug delivery into the cells. Vesicles serve as simplified models of more complex cell membranes, as well as the basis for bio-mimetic engineered systems. Bio-membranes only exist in solution and intimately interact with the surrounding fluid, which owing to the characteristic sizes and velocities, can be modeled with the incompressible Stokes equations. The aim of our work is to simulate the dynamics of the interaction between a bio-membrane and the fluid media surrounding. We take as basis our previous work on bio-membrane simulations [1], in which the solution of the fourth order PDE governing the bending elasticity of a vesicle is tackled with a phase-field or diffuse interface approach. The nonlinear, fourth-order PDE governing the phase field are conveniently solved using the local maximum-entropy (LME) approximants, a type of meshfree shape functions [2]. We merge the phase field model with the Stokes fluid media to treat naturally the coupling between the viscous forces in the fluid, the elastic forces due to the membrane, and the various constraints in the problem. The dynamics arise from a variational principle, and dictate the Lagrangian motion of the particles, convecting the phase field.