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<p>The Cahn-Hilliard equation is increasingly popular in two-phase flow simulations due to its implicit capture of the interface and its easy extension to higher dimensions. However, it also has its drawbacks, for instance, the mass shrinkage of a small drop in a large computational domain. To that end, a Cahn–Hilliard equation with a degenerate mobility is proposed and coupled with the Navier–Stokes equation. To solve the system equations, a simple and efficient finite difference method is employed. The Laplacian of the chemical potential is discretized using a modified central difference scheme. It is this modification that lends the model to larger time steps. Moreover, the method is fully explicit. The model was tested on a number of cases and compared with the Cahn–Hilliard equation with a constant mobility. It was shown that the new model can conserve mass better, thus sustaining a small drop longer due to the eliminated bulk diffusion. The model was also compared with experimental and analytical outcomes, showing reasonable agreement.</p>

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A Cahn–Hilliard–Navier–Stokes system with degenerate mobilities: a finite difference solution

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年2月2日

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  • Mingguang Shen,
  • Ben Q. Li,
  • Huan Yang

摘要

The Cahn-Hilliard equation is increasingly popular in two-phase flow simulations due to its implicit capture of the interface and its easy extension to higher dimensions. However, it also has its drawbacks, for instance, the mass shrinkage of a small drop in a large computational domain. To that end, a Cahn–Hilliard equation with a degenerate mobility is proposed and coupled with the Navier–Stokes equation. To solve the system equations, a simple and efficient finite difference method is employed. The Laplacian of the chemical potential is discretized using a modified central difference scheme. It is this modification that lends the model to larger time steps. Moreover, the method is fully explicit. The model was tested on a number of cases and compared with the Cahn–Hilliard equation with a constant mobility. It was shown that the new model can conserve mass better, thus sustaining a small drop longer due to the eliminated bulk diffusion. The model was also compared with experimental and analytical outcomes, showing reasonable agreement.