Abstract:
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We present a Lagrangian phase-field method to study the low Reynolds number dynamics
of vesicles embedded in a viscous fluid. In contrast to previous approaches, where the field variables are the phase-field and the fluid velocity, here we exploit the fact that the phasefield tracks a material interface to reformulate the problem in terms of the Lagrangian motion of a background medium, containing both the biomembrane and the fluid. We discretize
the equations in space with maximum-entropy approximants, carefully shown to perform well in phase-field models of biomembranes in a companion paper. The proposed
formulation is variational, lending itself to implicit time-stepping algorithms based on
minimization of a time-incremental energy, which are automatically nonlinearly stable.
The proposed method deals with two of the major challenges in the numerical treatment
of coupled fluid/phase-field models of biomembranes, namely the adaptivity of the grid to resolve the sharp features of the phase-field, and the stiffness of the equations, leading to very small time-steps. In our method, local refinement follows the features of the phasefield as both are advected by the Lagrangian motion, and large time-steps can be robustly chosen in the variational time-stepping algorithm, which also lends itself to time adaptivity.
The method is presented in the axisymmetric setting, but it can be directly extended to
3D. |