Numerical Method for Interfacial Flows with Surfactant
Ming-Chih Lai
Department of Applied Mathematics, National Chiao Tung University

In this talk, a numerical method based on immersed boundary formulation is proposed for the simulation of two-dimensional fluid interfaces with insoluble surfactant. The governing equations are written in a mixture of Eulerian flow with Lagrangian interfacial variables and the linkage between these two set of variables is provided by the Dirac delta function. The immersed boundary force comes from the surface tension which is affected by the distribution of surfactant along the interface. By tracking the interface in a Lagrangian manner, a simplified surfactant transport equation is derived. A new symmetric discretization for the surfactant concentration equation is proposed that ensures the surfactant mass conservation numerically. By introducing an artificial tangential velocity to the Lagrangian markers, one can keep those markers uniformly distributed to have better resolution. The effect of surfactant on drop deformation in a shear flow and the moving contact line problems are investigated in detail.

Bio:
Prof Ming-Chih Lai received his PhD degree in Mathematics from Courant Institute of Mathematical Sciences, New York University in 1998, under the supervision of Prof. Charles S. Peskin. His PhD thesis was about introducing a new formally second-order numerical scheme with reduced numerical viscosity and validating the immersed boundary method by applying to a fluid benchmark problem which experimental data are available. His thesis won Kurt O. Friedrichs Prize for an outstanding dissertation in Mathematics at Courant Institute of NYU in 1999. After his graduation, he spent a year as a Research Associate in Physics department of Duke university. Then he returned to Taiwan and took an assistant professorship in National Chung Cheng University from August 1999 to July 2002. He moved to National Chiao Tung University in August 2002 and stays therein ever since. He is currently the chair of the Department of Applied Mathematics and the director of Institute of Mathematical Modeling and Scientific Computing in National Chiao Tung University. Prof Lai's research interests are mainly on numerical methods for PDEs and computational fluid mechanics. In particular, his work includes the improvements of immersed boundary and immersed interface methods, and their applications to fluids with interfaces. He received the Dean's Special Recognition Award in 2003 and Academic Research Award in 2005, both from National Chiao Tung University. He also received the Outstanding Research Award in 2003 from National Science Council of Taiwan.

Dynamics of Multicomponent Vesicles in a Viscous Fluid
Yu-Hau Tseng
Department of Applied Mathematic, National Chiao Tung University

We develop and investigate numerically a thermodynamically consistent model of two dimensional multicomponent vesicles in an incompressible viscous fluid. The model is derived using an energy variation approach that accounts for different lipid surface phases, the excess energy (line energy) associated with surface phase domain boundaries, bending energy, spontaneous curvature, local inextensibility and fluid flow via the Stokes equations. The equations are high-order (fourth order) nonlinear and nonlocal due to incompressibility of the fluid and the local inextensibility of the vesicle membrane. To solve the equations numerically, we develop a nonstiff, pseudo-spectral boundary integral method that relies on an analysis of the equations at small scales. The algorithm is closely related to that developed very recently by Veerapaneni et al. for homogeneous vesicles although we use a different and more efficient time stepping algorithm and a reformulation of the inextensibility equation. We present simulations of multicomponent vesicles in an initially quiescent fluid and investigate the effect of varying the average surface concentration of an initially unstable mixture of lipid phases. The phases then redistribute and alter the morphology of the vesicle and its dynamics. When an applied shear is introduced, a sufficiently elongated vesicle tumbles and the presence of different surface phases with different bending stiffnesses and spontaneous curvatures results in a dramatic evolution as the vesicle bends in regions where the bending stiffness and spontaneous curvature are small.

The Fictitious Domain Method for Simulation of Flow-Body Interactions
Qiaolin He
Department of Mathematics, The Hong Kong University of Science & Technology

The fictitious domain method is an effective method for simulating flow-body interaction. The method was developed by Glowinski for simulating the particulate flow with no slip boundary condition. In many applications, fluid slip at the solid surface becomes important. The generalization of the fictitious domain method to slip boundary condition is not trivial. As a first step in this direction, we discuss a new least-square/fictitious domain method for linear elliptic problem with Robin boundary conditions. This is a joint work with Roland Glowinski and XP Wang.