In the first part of this thesis, we discuss the convergence of Poisson equations with source terms defined on embedded interface in the computational domain. We provide one-dimensional proof and two-one-dimensional numerical tests to judge the results. For the source terms as derivatives of delta functions, such as pressure problem in Stokes equations, or indicator function in front tracking methods, the overall accuracy is first-order accurate in L1norm, half-order accurate in L2 norm, but has O(1) error in L∞norm. For the singular source as delta functions, the overall convergent rate is second-order accurate in L1 norm, one and half-order accurate in L2 norm, and first-order accurate in L∞ norm. We also give some applications to solve second-order elliptic equations with piecewise-constant coefficients or Stokes problems by using the solution of Poisson equations we obtained.
In the second part of this thesis, we consider the surfactant, an amphiphilic molecular, in the multi-phase fluids. Due to the particle structure, it usually favor the presence in the fluid interface. We take solubility of the surfactant in one subdivision of the domain into account, and discuss the interactions between bulk domain and interface such as adsorption and desorption. These form a coupled surface-bulk interaction system of convection-diffusion equations. In order to reduce difficulties in the calculations, we rewrite the bulk concentration equation into a regular domain by using the indicator function introduced in first part. The concentration flux across the interface is treated as a singular source term in the equation. Based on immersed boundary formulation, we
propose a numerical scheme to solve this coupled surface-bulk concentration equations with providing the conservation of total surfactant mass. We use a series of examples to validate the proposed scheme, and combine with Navier-Stokes solver to extend our previous works.
In the present thesis, all studies are done in one-dimension or two-dimension, we will try to expand our work into three-dimensional cases in the future. The challenges are much harder, for instance, how to set a good grid on the complex surface? how to modify the grid when it has large deformation, especially under the flow? how to solve convection-diffusion equations on this grid? and how to maintain the mass conservation property in the computation? Such problems are the major issues that we need to conquer.
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