We first establish a variable density and viscosity, volume-conserved, hydrodynamically coupled phase-field variable elastic bending energy model for lipid vesicles, and then construct an efficient time-discrete scheme for solving it. The numerical scheme combines the penalty method for solving the Navier–Stokes equation, the explicit-IEQ (invariant energy quadratization) method for the nonlinear potentials, and the operator-splitting method. Hence it is not only fully decoupled but also owns some desired properties of linearity and unconditional energy stability. The feature of full decoupling is achieved by introducing some auxiliary variables and designing additional ordinary differential equations which are used for discretizing the coupled and nonlinear terms. The solvability and the unconditional energy stability of the numerical scheme have been further rigorously and numerically proven. Several numerical examples are carried out on the sedimentation process of the vesicle cell to show the effectiveness of the model and scheme.

我们首先建立了脂质囊泡的变密度和粘度、体积守恒、流体动力学耦合的相场可变弹性弯曲能模型，然后构建了一个有效的时间离散方案来求解它。数值方案结合了求解 Navier-Stokes 方程的惩罚方法、非线性势的显式 IEQ（不变能量二次化）方法和算子分裂方法。因此，它不仅是完全解耦的，而且还具有一些所需的线性和无条件能量稳定性的特性。通过引入一些辅助变量并设计附加的常微分方程来实现完全解耦的特点，这些常微分方程用于离散耦合和非线性项。数值方案的可解性和无条件能量稳定性得到了进一步严格和数值证明。对囊泡细胞的沉降过程进行了几个数值算例，证明了模型和方案的有效性。