In this paper, we present a comprehensive study of the fingering phenomenon of a power-law non-Newtonian fluid falling down a cylinder surface. A theoretical analysis is firstly carried out and the governing equation describing the film thickness is established for the non-Newtonian fluid denoted by a power-law index . Using the lubrication theory with dimensionless variables, the partial differential equation for the film thickness is derived, and both two-dimensional flow and three-dimensional flow are investigated. The traveling wave characteristic of the two-dimensional flow is displayed by using the finite difference scheme associated with a Newton iteration technique. The effect of changing the radius of the cylinder, precursor-layer thickness and power-law index is then considered through examining the variation of the capillary waves. Furthermore in three-dimensional complex flow, the fingering patterns for different parameters are simulated. The linear stability analysis based on the traveling wave solutions is given to elucidate the influence of different physical parameters, explaining the physical mechanism of nonlinear flow behaviors in two dimension and three dimension. Results from linear stability analysis show that the power-law index reinforces the fingering instability whether the fluid is shear-thinning or shear-thickening. Moreover, the numerical results illustrate that increasing the radii of the cylinder expands the unstable region. The flow profiles for different power-law coefficients are consistent with the result of the linear stability analysis, proving the unstable role of the power-law coefficient.

中文翻译：

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幂律非牛顿流体沿圆柱体表面下落的非线性流动现象


在本文中，我们对幂律非牛顿流体从圆柱体表面下落的指法现象进行了全面的研究。首先进行了理论分析，建立了以幂律指数 表示的非牛顿流体描述膜厚的控制方程。利用无量纲变量的润滑理论，推导了油膜厚度的偏微分方程，并研究了二维流动和三维流动。二维流的行波特性通过使用与牛顿迭代技术相关的有限差分格式来显示。然后通过检查毛细管波的变化来考虑改变圆柱体半径、前体层厚度和幂律指数的影响。此外，在三维复杂流中，模拟了不同参数的指法模式。基于行波解的线性稳定性分析阐明了不同物理参数的影响，解释了二维和三维非线性流动行为的物理机制。线性稳定性分析结果表明，无论流体是剪切稀化还是剪切增稠，幂律指数都会增强指进不稳定性。此外，数值结果表明，增加圆柱体的半径会扩大不稳定区域。不同幂律系数的流动剖面与线性稳定性分析的结果一致，证明了幂律系数的不稳定作用。