In this dissertation, we focus on developing numerical method to mimic vesicle rhe-ology dynamics through IB method. For two-dimensional space, we propose a frac-tional step IB method and uncondifrac-tionally energy stable method, then we extend to three-dimensional axial symmetric coordinate by using a nearly inextensible approach.
These works are based on our previous work [37, 22, 23]. The brief descriptions of numerical method are illustrated as below.
• Fractional step immersed boundary method. The governing equation is an incompressible Stokes equation with a suspension of inextensible interface (that is, the bending rigidity is neglected). We found that the spreading opera-tor of force generaopera-tor has skew-adjoint property to surface divergence operaopera-tor both in theoretical continuous and numerical discrete version. By taking advan-tage of this crucial property, the time-stepping numerical discretization involves solving a symmetric positive semi-definite matrix, and this linear system can be done efficiently by employing a preconditioned conjugate gradient method. We successfully reproduce vesicle-like dynamics such as tank-treading motion under shear flow. As a benchmark, we also measure the inclination angle and tank-treading frequency when reaching steady state. The numerical result shows a highly agreement with those in many other literatures.
• Unconditionally energy stable immersed boundary method. The fluid equation is governed by a Navier-Stokes equation with suspension of vesicle.
Rather than enforcing purely inextensible constrain, we adopt a nearly inexten-sible approach to simulate vesicle dynamics. The fluid equations are discretized by projection method so the pressure is decoupled. Again by using skew-adjoint property, it leads to solve a symmetric positive sparse linear system respect to intermediate fluid variable, this can be done efficiently by aggregation-based multigrid. Meanwhile, we prove that the developed scheme shows an uncondi-tional stability in energy sense, which means the total energy of fluid system decreases during time integration. This is a significant step to release restric-tion of numerical time step due to applying the penalized method. Again we successfully simulated vesicle dynamics, such as a free relaxation to equilibrium state, tank-treading motion under shear flow, the numerical cost is substantial saved by this scheme.
• Simulating three-dimensional axisymmetric vesicles. We extend previ-ous work to three-dimensional space to link the real world. An axisymmetric vesicle is suspended in incompressible Navier-Stokes flow in three-dimensional capillary. Again the nearly inextensible approximation in axisymmetric version is applied. We have shown that the modified interfacial force due to nearly inex-tensible approaching has exactly the same form to the original one. Besides, we
adopt a high accuracy spectral method to evaluate interfacial quantities such as mean curvature, Gaussian curvature and surface Laplace of mean curvature.
The fluid variables are solved numerically by traditional projection method in which fast Poisson solver can be applied to obtain fluid velocity filed and pres-sure increment. We investigated behaviors of axisymmetric vesicle in situation of freely suspended, under influence of gravity field and passing through capil-lary in Poiseuille flow. The numerical result shows a good agreement with what observed in experiment, and this is an evidence showing the reliability of our proposed scheme.
Chapter 2
Mathematical model of vesicle hydrodynamics
The first model used for vesicles can be tracked back to Helfrich (1973) [18], who inves-tigated the rheology dynamics of vesicle through a elasticity theory. In this chapter, we introduce more details about vesicle, such as vesicle mechanism, the vesicle sur-face energy and the the boundary forces on vesicle membrane. The mathematical formulation of hydrodynamical system for suspended vesicle is given.
2.1 Vesicle mechanics
A vesicle can be visualized as a liquid droplet within another liquid enclosed by a lipid membrane with the size about 100 nm to 10 µm. Such lipid membrane consists of tightly packed lipid molecules with hydrophilic heads facing the exterior and inte-rior fluids and with hydrophobic tails hiding in the middle and thus forms a bilayer phospholipid with thickness about 6 nm (see Figure 2.1).
Vesicle is widely used for mimicking model of biological cells and also gives many features observed by living cells. The shape of vesicle is deformable, in equilibrium state, the vesicle usual appears in shape of biconcave disk because of minimized surface energy (see Figure 2.2). Topological changes (fusion, division, etc.) of vesicle are possible but rarely happened in real world. In principle, the vesicle membrane mainly undergoes two basic deformation factors, which are inextensibility and bending effect.
At usual room temperature, since the phospholipidic molecules do not bind with each other, the membrane is thus regard as a liquid phase. In this scenario, phos-pholipids stay very close and tend to approximate a constant density, it would lead to local conservation of vesicle surface. Therefore, the membrane is considered as an inextensible surface. Moreover, because the permeability of vesicle membrane is very small and fluid inside vesicle is Newtonian incompressible fluid, this contributes to
Figure 2.1: A cartoon showing a vesicle and its molecular structure on membrane.
Picture is taken from the web site of Wikipedia.
Figure 2.2: Biconcave equilibrium state of vesicles.
total volume of vesicle is conserved as well. The phospholipid membrane is known to exhibit a resistance against bending, we have to take bending effect into account in our mathematical formulation.
In consequence, we list three features of vesicle as below.
• The total surface area of vesicle is conserved because of the inextensibility. This is the intrinsic feature of vesicle which is regarded as a simplified model of red blood cell. The mathematical formulation of inextensibility will be explained later in Section 2.3.
• The total volume of vesicle remains a constant since it contains incompressible Newtonian fluid. The mathematical statement is shown in Section 2.4.
• The bending effect should be considered in modeling vesicle dynamics. The explicit form of bending force is given in Section 2.2.
When it comes to vesicle dynamics, we must mention a dimensionless characteristic variable, the reduced volume, which measures the deflation of the vesicle, plays a significant role in the vesicle dynamics. It is defined as
ν3D = V0
4
3π(S0/4π)3/2,
where V0 and S0 represent the volume and the surface area of vesicle, respectively.
This dimensionless number is nothing but the volume ratio of the vesicle to a sphere with the same surface area, and thus is equal to 1 for a sphere. In two dimensions a parameter equivalent to reduced volume is defined by reduced area, which is defined by
ν2D = A0 π(L0/2π)2,
where A0 and L0 denote for the area and the arc-length of vesicle respectively. Again this definition follows the ratio of the vesicle to a circle with the same arc-length. In the rest of dissertation, we would use the notation ν to denote reduced volume (3D) or reduced area (2D) for convenience.