Nonlinear flow phenomenon of a power-law non-Newtonian fluid falling down a cylinder surface

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 $n$. 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.