A stable and accurate immersed boundary method for simulating vesicle dynamics via spherical harmonics

In this paper, we improve our previous immersed boundary (IB) method for 3D triangulated vesicle in unsteady Navier-Stokes flow (Seol et al., 2016 [31]) from several aspects. Firstly, we adopt spherical harmonic representation for approximating vesicle configuration. By applying spectral differentiation, we are able to obtain high accuracy of geometric quantities such as the mean and Gaussian curvatures, and the surface Laplacian of mean curvature, which is not achievable via triangulation. The vesicle membrane (interface) immersed in 3D Newtonian fluid ensures the surface incompressibility constraint; thus, an unknown elastic tension acting as Lagrange multiplier must be introduced along the interface. To efficiently solve the problem, a logarithmic formulation of approximate elastic tension is explicitly utilized in a nearly incompressible interface approach. Then in computing the elastic tension force, we propose to use the divergence form instead of the commonly used non-divergence one. By doing so, we find that numerical stability can be improved significantly during vesicle relaxation and its transient motions. Moreover, to maintain the interfacial mesh quality, a mesh control technique via filtering of interfacial tangential velocity is coupled within the nearly incompressible interface approach. Upon these improvements, a series of numerical tests on the present scheme is performed to verify numerical accuracy, stability, and convergence of our method. As for practical experiments, the tank-treading and tumbling motions of prolate vesicle in shear flow are extensively studied by varying some dimensionless parameters such as the reduced volume, bending capillary number, viscosity contrast, and the Reynolds number. We further study three types of vesicle shapes, namely, bullet, parachute, and croissant in rectangular Poiseuille flow.