Research On The Immersed Boundary Method And Its Application On Cell Mechanics

In nature as well as in engineering applications, especially in biological systems, there exist abundant examples where a flexible structure is immersed in a viscous incompressible fluid. The immersed boundary method is an effective technique for modeling and simulating this type of fluid-structure interactions. The immersed boundary method considers the structure as an immersed boundary, which can be represented by a singular force in the Navier-Stokes equations. The main characteristic of the immersed boundary method is that the simple Eulerian mesh is used, which avoids difficulties associated with moving boundaries faced by conventional methods.The immersed boundary has a great advantage on simulating such a system with small Reynolds number and large deformation. The cell motion lies within that scope. Cells are the basic unit of life. Deep study on cells is the key point to reveal the mystery of life, transform its status and conquer diseases. In most situations, cells change their shapes to resist the external forces. Therefore, cells may undergo large deformation during their movement. Study of cell deformation is of great significance on the clinical and bio-mechanical area.The entire work in this dissertation consists of the research on the immersed boundary method itself and its application on cell mechanics, including the introduction of this method, the stability and accuracy analyses, development of the corresponding computational code for the elastic membrane, simulation on the cell motion in simple shear flow, and mathematical model set-up for osmotic membrane.The primary researching contents and innovations are as follows:1. By analyzing the engivalues of the jump and smooth mathematical models under different modes, it was firstly proved that the solutions of the fluid-structures problem with the bending membrane obtained by the immersed boundary method was verified to be at least linearly stable. Meantime, several conclusions could be presented as follows. The immersed boundary problem was stiff and the immersed boundary method reduced the stiffness due to the regularization (smoothing) of the singular force term. The membrane-fluid system with bending resistance was stiffer than the system with elastic resistance, and the computations for the coupled membrane-fluid with bending resistance were more difficult than those with elastic tension, especially on a relative fine grid. For membranes with both in-plane elastic tension and bending rigidity, the bending effect was dominant for high frequency modes while the in-plane tension was more important for the low frequency modes. The high frequency modes had larger decay rate and highly oscillatory, and implicit time stepping scheme was highly desirable for bending system.2. The order of accuracy of the immersed boundary method was analyzed by the method of manufactured solutions. Based on the proof of the correctness of the computational program, the application of the method of manufactured solutions was extended. By analyzing the calculated order of a jump pressure filed, it could be concluded that even though four different versions of regularized Dirac delta function were used, the order of accuracy of the immersed boundary method kept one order. This conclusion matched the theoretical judge by Peskin on the basis of its mathematical model. The inconsistency with other works was explained. The study results showed that using various versions of regularized Dirac delta function could not change the order of accuracy of the immersed boundary method, but could help improving the discretization values.3. The computational software for the immersed boundary method used to simulate the cell motion was developed. It implemented the computation which could solve the domains inside and outside the membranes simultaneously. Such simulating program, neglecting the thickness of the membrane and ignoring the bending force, contained four constitutive laws such as Hookean, Neo-Hookean, Skalak and Evans-Skalak laws. The software could be used to simulate the cell motion in two- or three-dimensional flow and obtain variety of information on the cell membrane like velocity, displacement, strain and stress.4. The cell motion in simply shear flow was simulated, the characteristic of such motion was studied and several parameter effects were analyzed. The simulation results presented the process of the cell deformation: the cell elongated along the major axis until it achieved its steady shape, and then the cell rotated about the cytoplasm in a fashion called"tank-treading". The numerical results showed that the circularity of the initial shape and the capillary number had positive effects on the Taylor deformation parameter and had negative effects on the inclination at steady state. With the increase of the circularity, the periods of tank-treading motion got larger. As the capillary number increased, the effect of the constitutive laws got more obvious.5. The mathematical model for the transport of nonelectrolyte across the membrane was set up. Such model, consisting of both the solute transport and the movement of water, extended the application of the immersed boundary method in cell mechanics. The osmosis membrane was taken as the immersed boundary and the equations for the concentration and solute flux density were obtained. These equations, verified by one-dimensional steady case, were the mathematical model of the immersed boundary method for nonelectrolyte. Because of the fact that the cell volume remaining unchanged during motion, the original immersed boundary method was used for the simulation of the water motion. The new form of the immersed boundary method for the mass transport included water motion equations, solute diffusion transport equations and membrane deformation equations.