Computational modelling of membrane viscosity for immersed boundary simulations of red blood cell dynamics

Abstract

Although tremendous efforts have been devoted to modelling various membrane properties, few studies considered the membrane viscous effects. Meanwhile, immersed boundary method (IBM) has been a popular choice for simulating the motion of deformable cells in flow for the convenience of incorporating the flow-membrane interaction. Unfortunately, the direct implementation of membrane viscosity in IBM suffers severe numerical instability. In this thesis, three numerical schemes for implementing membrane viscosity in IBM are developed. Furthermore, the effects of membrane viscosity on the capsule dynamics in shear flow have been examined in detail. In Chapter 1, the biomechanical properties of red blood cells (RBCs) are introduced followed with a literature review. Also, the motivations and objectives, the structure of this thesis, and the contributions of the candidate are described. In Chapter 2, a finite-difference approach is proposed for implementing membrane viscosity in IBM. To improve the simulation stability, an artificial elastic element is added in series to the viscous component in the membrane mechanics. The detailed mathematical description and key steps for its implementation in immersed boundary programs are provided. Validation tests show a good agreement with analytical solutions and previous calculations. The accuracy dependence on membrane mesh resolution and simulation time step is also examined. In Chapter 3, two other schemes are proposed based on the convolution integral expression of the Maxwell viscoelastic element. Several carefully designed tests are conducted and the results show that the three schemes have nearly identical performances in accuracy,