流體和Lagrangian介面的變數的混合物，被寫成一個常用的immersed boundary公式化。然而，兩個集合變數之間的連結是由某些假設架構出來的Dirac delta function。接下來，我們也會比較一個新型態的delta function和原本型態的差異。我們使用的數值方法是一個半隱性二階的投影方法去處理有黏滯性不可壓縮的數學方程式，然後藉由流體的速度內插得到介面的速度去移動介面。在數值結果方面，我們首先證實沉浸邊界法的幾個論據，然後假設一個水泡侵入在一個不可壓縮的流體，隨著不同的Capillary number，我們觀察一個水泡在二向流的形變。另外，我們利用等價分佈的技術去控制Lagrangian markers均勻分佈。正如所料，隨著markers的控制，那數值的實驗在面積維持方面比原本沒均勻分布的markers有較好的成果。

In this thesis, we introduce the fundamental concepts of the immersed boundary method and also apply it to the simulation of two-dimensional interfacial flows. The governing equations are written in a usual immersed boundary formulation where a mixture of Eulerian flow and
Lagrangian interfacial variables are used, and the linkage between these two set of variables is provided by the Dirac delta function which is constructed under certain postulates. A new type of smooth delta functions is compared with the original ones. The incompressible
viscous Navier-Stokes equations are solved by a
semi-implicit second-order projection method, and the interface moves by the velocity which is interpolated from the fluid velocity. In numerical results, we first verify several facts of the immersed boundary method and then consider a bubble immersed in an two-dimensional
incompressible fluid. We observe the deformation of a bubble with different Capillary number in a shear flow. Moreover, we take the advantage of an equi-distributed technique to control the distribution of the Lagrangian markers uniformly. As expected, the numerical experiments
with marker control technique have better performance in the area preservation than the case without it.

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