本文采用基于WENO格式的有限体积法,发展了包括铁磁流体的两相流相场方法。流体流动采用了不可压缩Navier-Stokes方程进行描述,利用Cahn-Hilliard方程来捕捉两相流界面运动,并采用Maxwell方程描述磁场分布。同时,在流体流动控制方程方程中加入了开尔文力和表面张力以实现磁场对界面动力学行为的描述。我们将四阶Cahn-Hilliard方程拆分为两个Helmholtz类型方程,从而克服四阶非线性项的离散和高精度计算带来的难题。我们采用五阶WENO格式对控制方程的对流项进行统一离散处理,从而提高数值计算的精度性,同时避免产生数值振荡。Zalesak’s圆盘问题数值模拟的结果表明,本文发展方法的相界面捕捉精度高于参考文献中所报道的离散方法,与高精度相场方法的精度相当；对剪切流中的液滴变形问题的数值模拟揭示我们数值方法可以更真实地捕捉到更多个卫星液滴。此外,针对磁场影响下较低毛细数条件的铁磁流体液滴剪切变形研究显示：当外加磁场方向与液滴水动力学变形方向一致时,磁场的作用会放大液滴变形,进一步增加磁场强度会诱发液滴分裂；而当外加磁场方向与液滴水动力学变形方向垂直时,较低强度的磁场作用能够改变液滴变形方向,而较高强度的磁场则会使液滴直接呈现出沿磁场方向变形。

In this paper, a finite volume method based on the WENO scheme is used to develop a two-phase flow phase field method including ferrofluids. The governing equations include the Navier-Stokes equations for incompressible flow, the Cahn-Hilliard equation for interfacial dynamics, and the Maxwell equation for the distribution of the external magnetic field. To overcome the challenges posed by the fourth-order nonlinear diffusion terms, we decompose the Cahn-Hilliard equation into two Helmholtz equations. The fifth-order WENO scheme is employed to handle the convection term of the governing equation, aiming to enhance accuracy and prevent numerical oscillations. The Zalesak"s disk problem shows that the proposed method has a higher phase interface capture accuracy compared to references, and is comparable to the high-precision phase field method. The proposed method is applied to the droplet shear deformation problem, and it is observed that the current method can capture more satellite droplets. Additionally, it is noted that the magnetic interfacial force favors droplet deformation when the direction of the external magnetic field aligns closely with the direction of droplet hydrodynamic deformation. Furthermore, an increase in the magnetic field intensity leads to droplet splitting. On the contrary, when the magnetic field is nearly perpendicular to the direction of droplet hydrodynamic deformation, a lower intensity magnetic field alters the direction of droplet deformation, while a higher intensity magnetic field directly deforms the droplet along the direction of the magnetic field.

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国家自然科学基金(12102071,12172070)；重庆市博士直通车项目(No.CSTB2022BSXM-JCX0086).