Department of Physics, West University of Timişoara, 300223 Timişoara, Romania.
Department of Physics, North Dakota State University, Fargo, North Dakota 58108, USA.
Phys Rev E. 2019 Dec;100(6-1):063306. doi: 10.1103/PhysRevE.100.063306.
We develop and implement a finite difference lattice Boltzmann scheme to study multicomponent flows on curved surfaces, coupling the continuity and Navier-Stokes equations with the Cahn-Hilliard equation to track the evolution of the binary fluid interfaces. The standard lattice Boltzmann method relies on regular Cartesian grids, which makes it generally unsuitable to study flow problems on curved surfaces. To alleviate this limitation, we use a vielbein formalism to write the Boltzmann equation on an arbitrary geometry, and solve the evolution of the fluid distribution functions using a finite difference method. Focusing on the torus geometry as an example of a curved surface, we demonstrate drift motions of fluid droplets and stripes embedded on the surface of the torus. Interestingly, they migrate in opposite directions: fluid droplets to the outer side while fluid stripes to the inner side of the torus. For the latter we demonstrate that the global minimum configuration is unique for small stripe widths, but it becomes bistable for large stripe widths. Our simulations are also in agreement with analytical predictions for the Laplace pressure of the fluid stripes, and their damped oscillatory motion as they approach equilibrium configurations, capturing the corresponding decay timescale and oscillation frequency. Finally, we simulate the coarsening dynamics of phase separating binary fluids in the hydrodynamics and diffusive regimes for tori of various shapes, and compare the results against those for a flat two-dimensional surface. Our finite difference lattice Boltzmann scheme can be extended to other surfaces and coupled to other dynamical equations, opening up a vast range of applications involving complex flows on curved geometries.
我们开发并实现了一个有限差分格子玻尔兹曼方法来研究曲面上的多组分流动，通过将连续性方程、纳维-斯托克斯方程与卡亨-希尔方程耦合，来追踪二元流体界面的演化。标准的格子玻尔兹曼方法依赖于规则的笛卡尔网格，这使得它通常不适合研究曲面上的流动问题。为了缓解这一限制，我们使用 Vielbein 形式主义在任意几何形状上编写玻尔兹曼方程，并使用有限差分方法来求解流体分布函数的演化。我们以环面几何形状作为曲面的一个例子，演示了嵌入在环面表面上的液滴和条纹的漂移运动。有趣的是，它们的迁移方向相反：液滴向环面的外侧迁移，而条纹向环面的内侧迁移。对于后者，我们证明了对于小条纹宽度，全局最小配置是唯一的，但对于大条纹宽度，它变得双稳定。我们的模拟结果还与流体条纹的拉普拉斯压力的解析预测以及它们在接近平衡配置时的阻尼振荡运动相吻合，捕捉到了相应的衰减时间尺度和振荡频率。最后，我们在各种形状的环面的流体动力学和扩散区域模拟了相分离二元流体的粗化动力学，并将结果与二维平面的结果进行了比较。我们的有限差分格子玻尔兹曼方法可以扩展到其他曲面，并与其他动力学方程耦合，为涉及复杂曲面流动的广泛应用开辟了道路。
