Abstract In this paper, we consider numerical approximations for a hydrodynamics coupled Cahn–Hilliard phase-field binary fluid-surfactant model. By adding a quartic form of the gradient potential, we first modify the total free energy for the commonly used phase-field surfactant model into a form which is bounded from below and establish the energy law for the new system. Then we combine the Invariant Energy Quadratization approach for the nonlinear potentials, the projection method for the Navier–Stokes equations, and a subtle implicit–explicit treatment for the stress and convective terms, to arrive at a linear and second-order time marching scheme for solving this system. Meanwhile, to enhance the stability, two crucial linear stabilization terms are added into the scheme thus large time steps are allowed in computations. We further prove the well-posedness of the linear system, and its unconditional energy stability rigorously. Various 2D and 3D numerical experiments are performed to validate the accuracy and energy stability of the proposed scheme.

中文翻译：

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一种新的流体动力学耦合二元相场表面活性剂系统的高效、二阶精确和无条件能量稳定的数值方案

摘要 在本文中，我们考虑了流体动力学耦合 Cahn-Hilliard 相场二元流体表面活性剂模型的数值近似。通过添加梯度势的四次形式，我们首先将常用的相场表面活性剂模型的总自由能修改为一种从下方有界的形式，并建立新系统的能量定律。然后我们结合非线性势的不变能量二次方方法、纳维-斯托克斯方程的投影方法以及对应力和对流项的隐式-显式处理，得出线性和二阶时间推进方案解决这个系统。同时，为了增强稳定性，该方案中添加了两个关键的线性稳定项，因此在计算中允许大的时间步长。我们进一步严格证明了线性系统的适定性及其无条件能量稳定性。进行了各种 2D 和 3D 数值实验以验证所提出方案的准确性和能量稳定性。