Isogeometric analysis of hydrodynamics of vesicles using a monolithic phase-field approach

Abstract

In this paper, we present an isogeometric analysis for studying the dynamical behavior of inextensible vesicles under an external fluid flow with inertial forces. We consider a phase-field model Aland, et al. (2014) for the coupled fluid–vesicle problem which enforces global area and volume constraints using a Lagrange multiplier method and employs an extra equation for enforcing local inextensibility condition. Full Navier–Stokes equations are considered and their finite element formulation is presented based on a residual-based variational multiscale method while a standard Galerkin finite element framework is employed for the rest of partial differential equations in the model. We solve the system of PDEs using an implicit, monolithic scheme based on the generalized-α time integration method. Compared to the system of equations considered in Aland, et al. (2014), we reduce the number of equations to be solved by leveraging high continuity of NURBS functions. We also extend the algorithm of the phase-field method to three-dimensional problems. A number of two-dimensional numerical examples which model the dynamics of a vesicle in a quiescent fluid, in a shear flow, and in plane Poiseuille flow with and without obstructions are studied. The resistive immersed surface method is employed for dealing with obstructions. We also consider a 3D example where we study the dynamics of a vesicle in a constricted channel which resembles the situation that a vesicle experiences in a stenosed microchannel.