几类界面问题的相场建模及其数值方法

       相场方法起源于热力学，具有不需要显式追踪界面的优势，被广泛应用于各类界面问题的建模和数值模拟。相场模型使用一组连续变量来描述物质微观结构，这些变量基于系统自由能递减，在时间与空间上演化，且可以通过求解偏微分方程研究这类动力学。本文将运用相场模型及其数值方法对表面活性剂吸附、液滴撞击及细菌趋化这三类界面问题进行研究，主要由以下三个部分组成。

       第一部分运用相场模型研究表面活性剂动力学问题。该模型由两个分别控制界面动力学和活性剂浓度演化的Cahn-Hilliard型方程组成，并带有接触线动力学边界条件。基于Flory-Huggins势的对数奇异性和凸分裂技术，构造了一类无条件能量稳定的保界数值格式，其是解耦的、唯一可解的和质量守恒的。一阶全离散格式被严格证明具备上述所有性质；二阶格式也是如此，除能量稳定性外。通过数值实验不仅验证了所期望的性质，还确定了表面活性剂对接触线动力学的影响。

       第二部分通过将相场表面活性剂模型与流体动力学耦合，进行了液滴撞击固壁的数值研究。该两相流系统的相场建模通过引入Navier-Stokes方程实现，且广义Navier边界条件被应用于移动接触线之上。基于凸分裂和压力稳定化技术，提出了该系统的能量稳定数值格式，且一阶格式的原始能量耗散律被严格证明。通过在准三维下实现算法，在数值实验中对干净液滴和污染液滴的撞击动力学系统地研究。一般而言，污染液滴的撞击动力学产生的耗散小于干净液滴，且污染液滴更容易发生拓扑形变。此外，数值实验还获得了与真实实验一致的结果。

       最后一部分通过基于相场的扩散域方法来研究复杂几何区域中的趋化-流体系统。该模型将对流趋化系统与Navier-Stokes方程相耦合，来描述流体环境中的生物趋化现象。本文为所研究的趋化-流体系统提供了一种新的保正的高分辨率方法。该方法基于扩散域方法导出了新的趋化-流体扩散域模型，其中复杂几何区域被嵌入到更大的矩形域中，原始边界被具有有限厚度的扩散界面所代替。经分析表明，随着扩散界面的厚度收缩到零，新模型渐近收敛到原趋化-流体模型。二阶混合有限体积-有限差分格式被用来求解新模型。大量数值实验证实了方法的性能，并展示了不同形状液滴中的趋化现象。

Phase-field methods originate from thermodynamics and have the advantage of implicitly tracking the interface between phases, so it is widely used in modelling and numerical simulations for various interface problems. Phase-field models describe the microstructure of matter using a set of continuous variables that evolve in time and space based on free energy dissipation, and dynamics of this microstructure can be studied by solving partial differential equations. In this paper, phase-field models and numerical methods will be used to study three interface problems: surfactant adsorption, droplet impact and bacterial chemotaxis. This paper consists of the following three parts.

In the first part, a phase-field model is used to study the problem of the surfactant dynamics. This model consists of two Cahn-Hilliard type equations, governing the dynamics of interface and surfactant concentration, with dynamical boundary condition for contact lines. Based on the logarithmic singularity of Flory-Huggins potential and the convex splitting technique, we propose a set of unconditionally energy stable and bound-preserving schemes for the new model. The proposed schemes are decoupled, uniquely solvable, and mass conservative. These properties are rigorously proved for the first-order fully discrete scheme. In addition, we prove that the second-order scheme satisfies all these properties except for the energy stability. We numerically validate the desired properties for both schemes and present numerical results to systematically study the influence of surfactants on contact line dynamics.

The second part is devoted to the study of droplet impact on solid substrate by coupling the phase-field surfactant model with hydrodynamics. We introduce the Navier-Stokes equations for the phase-field modeling of two-phase flows. The generalized Navier boundary condition are used to account for moving contact lines. Based on the convex splitting and pressure stabilization technique, we develop unconditionally energy stable schemes for this model. The energy dissipation law for the original energy is rigorously proved for the first-order scheme. Using the proposed methods implemented in quasi-three-dimensional coordinates for this model, we systematically study the impact dynamics of both clean and contaminated droplets in a series of numerical experiments. In general, the dissipation in the impact dynamics of a contaminated drop is smaller than that in the clean case, and topological changes are more likely to occur for contaminated drops. Moreover, some quantitative agreements with experiments are also obtained.

In the final part, we investigate a chemotaxis-fluid system in complex geometry using a phase-field based diffuse-domain approach. This model describes the biological chemotaxis phenomenon in the fluid environment and couples a convective chemotaxis system with the Navier-Stokes equations. We develop a new positivity preserving and high-resolution method for the studied chemotaxis-fluid system. Our method is based on the diffuse-domain approach, which we use to derive a new chemotaxis-fluid diffuse-domain (cf-DD) model. The complex geometric domain is imbedded into a larger rectangular domain, and the original boundary is replaced by a diffuse interface with finite thickness. We show that the cf-DD model converges to the original system asymptotically as the width of the diffuse interface shrinks to zero. We numerically solve the resulting cf-DD system by a second-order hybrid finite-volume finite-difference method and demonstrate the performance of the proposed approach on a number of numerical experiments that showcase several chemotactic phenomena in sessile drops of different shapes.

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