In this paper, we establish a new hydrodynamically coupled phase-field model for three immiscible fluid components system. The model consists of the Navier–Stokes equations and three coupled nonlinear Allen–Cahn type equations, to which we add nonlocal type Lagrange multipliers to conserve the volume of each phase accurately. To solve the model, a linear and energy stable time-marching method is constructed by combining the stabilized-Invariant Energy Quadratization (S-IEQ) approach and the projection method. The well-posedness of the scheme and its unconditional energy stability are rigorously proved. Several numerical simulations in 2D and 3D are carried out, including spinodal decomposition, dynamical deformations of a liquid lens and rising liquid drops, to validate the model and demonstrate the efficiency and energy stability of the proposed scheme, numerically.

中文翻译：

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一种新的流体动力耦合三分量体积守恒 Allen-Cahn 相场模型的高效、二阶时间和能量稳定方案

在本文中，我们为三个不混溶的流体组分系统建立了一种新的流体动力学耦合相场模型。该模型由 Navier-Stokes 方程和三个耦合的非线性 Allen-Cahn 型方程组成，我们在其中添加了非局部型 Lagrange 乘子以准确地保存每个相的体积。为求解该模型，将稳定不变能量二次化（S-IEQ）方法与投影方法相结合，构建了一种线性且能量稳定的时间推进方法。严格证明了该方案的适定性及其无条件的能量稳定性。进行了几个 2D 和 3D 数值模拟，包括旋节线分解、液体透镜的动态变形和上升的液滴，