Abstract In this paper, we establish a binary fluid surfactant model by coupling two mass-conserved Allen–Cahn equations and the Navier–Stokes equations and consider numerical simulations of the developed model. Due to a large number of nonlinear and nonlocal coupling terms in the model, it is very challenging to design an efficient and accurate numerical scheme, especially the full decoupling scheme with second-order time accuracy. We solve this challenge by developing a novel fully-decoupled approach, where the key idea achieving the full decoupling structure is to introduce an ordinary differential equation to deal with the nonlinear coupling terms that satisfy the so-called “zero-energy-contribution” property. In this way, we can easily discretize the coupled nonlinear terms in a fully explicit way, while still maintaining unconditional energy stability. By combining with the projection type method and quadratization approach, at each time step, we only need to solve several fully-decoupled linear elliptic equations with constant coefficients. We strictly prove the solvability of the scheme, prove that the scheme satisfies the unconditional energy stability, and give various 2D and 3D numerical simulations to show its stability and accuracy numerically. As far as the author knows, this is the first fully-decoupled and second-order time-accurate scheme of the flow-coupled phase-field type model.

中文翻译：

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守恒的 Allen-Cahn 型流动耦合二元表面活性剂模型的一种新的完全解耦、二阶和能量稳定的数值方案

摘要 在本文中，我们通过耦合两个质量守恒的 Allen-Cahn 方程和 Navier-Stokes 方程建立二元流体表面活性剂模型，并考虑对所开发模型的数值模拟。由于模型中存在大量非线性和非局部耦合项，设计一个高效、准确的数值方案非常具有挑战性，尤其是具有二阶时间精度的全解耦方案。我们通过开发一种新颖的完全解耦方法来解决这一挑战，其中实现完全解耦结构的关键思想是引入一个常微分方程来处理满足所谓的“零能量贡献”特性的非线性耦合项. 通过这种方式，我们可以轻松地以完全显式的方式离散耦合非线性项，同时仍然保持无条件的能量稳定。结合投影式方法和二次化方法，在每个时间步长，我们只需要求解几个全解耦的常系数线性椭圆方程。我们严格证明了该方案的可解性，证明该方案满足无条件能量稳定性，并给出了各种2D和3D数值模拟，以数值方式展示其稳定性和准确性。据笔者所知，这是流耦合相场型模型的第一个全解耦二阶时间精确方案。证明该方案满足无条件能量稳定性，并给出了各种2D和3D数值模拟，以数值显示其稳定性和精度。据笔者所知，这是流耦合相场型模型的第一个全解耦二阶时间精确方案。证明该方案满足无条件能量稳定性，并给出了各种2D和3D数值模拟，以数值显示其稳定性和精度。据笔者所知，这是流耦合相场型模型的第一个全解耦二阶时间精确方案。