A theoretical study on the capillary rise of non-Newtonian power-law fluids

Abstract

The phenomena of capillary rise with the effect of displaced fluids are ubiquitous in both science and engineering, and the mathematical models and their analytical solutions of this problem have also received increasing attention. In this paper, a theoretical study on the rising dynamics of non-Newtonian power-law fluids in a capillary is performed, and the classical Lucas–Washburn equation is generalized to a nonlinear second-order differential equation in which the effects of the power-law index and displaced fluids are included. We analyze the imbibition behaviors of power-law fluids under the influence of displaced fluids in details, and also present some analytical solutions of different special cases and different time stages. The results show that for different special cases, it takes shorter time for shear thickening fluid to reach the equilibrium height. For different time stages, however, the rising phenomena of power-law fluids are more complex, and the average time for the shear-thinning fluid to reach the equilibrium height is longer compared to shear-thickening fluid, but an opposite phenomenon is observed for the case of μ1,0/μ2,0=100 (here μ1,0/μ2,0 is the viscosity ratio of two power-law fluids) in viscous time stage. In inertial time stages, the density ratio of the absorbed fluid to the displaced fluid also has a significant effect on the rising dynamics of imbibing fluid. Furthermore, the effect of dynamic contact angle is also included in the governing equation and analytical solutions. Through a comparison between the theoretical results and experimental data, a good agreement is observed. These results can be used as a priori for liquid absorption in industrial applications, including oil recovery, fuel cells and water collection of artificial silk.