Analytical, experimental, and numerical study of capillary rise dynamics from inertial to viscous flow

Abstract

Fundamental understanding of capillary rise dynamics and precise evaluation of imbibition processes should be considered in many natural and industrial phenomena. The assumptions considered to solve the capillary rise motion usually neglect specific forces which limit the reliability of the derived solutions. In the present study, the dominant forces and regimes involved in the initial moments of the capillary rise imbibition process in a tube were investigated analytically, experimentally, and numerically. Analytical solutions available in the literature were discussed, and then, their validity was verified by comparing them to the experimental observations and numerical results. Comparing the capillary rise behavior at the initial stages revealed significant differences between the theoretical models and the numerical lattice Boltzmann method. This behavior is attributed to simplifying assumptions and ignoring the entrance effect, dynamic contact angle, and the inertial term in the theoretical model. By removing these assumptions in numerical formulations, closer results to the experimental records were observed. In the present study, for the first time, capillary rise dynamics were divided into five steps: (1) transition regime with h~t2; (2) purely inertial (stage one) with h~t; (3) viscous-inertial or crossover (stage two) with h~log10(t); (4) purely viscous (stage three) with h~t1/2; and (5) gravitational-viscous with a constant h. It was known that stage one was purely dominated by the inertial forces, then the influence of viscosity increased (viscous-inertial flow), and finally, the effect of inertia faded and the flow became purely viscous and approached the Lucas-Washburn law.

Fundamentals and Applications of Surface Wetting

“…Evaporation of the liquid, fiber swelling, tortuosity, dynamic contact angle, and the gravitational effect play significant roles in these deviations. Researchers have identified different regimes of liquid spreading during wicking. ,− The first regime is the inertial regime that is dominated by the inertial forces; in the second regime, the viscosity effect increases and is called the viscous–inertial regime; the viscous flow governs the third regime, and in the last regime, the gravitational and viscous forces dominate.…”

Dynamics of Passive Filling of Fluids in Cylindrical Capillaries

“…The model is obtained by mainly neglecting gravity as well as fluid’s inertia effects, and the forces controlling the fluid’s flow are mainly capillary driving force and viscous force, whose fluid’s rising height H is proportional to t 0.5 . On this basis, the researchers have further studied the capillary phenomenon and have revised the Lucas–Washburn model by taking into account the neglected gravity, momentum effects, viscosity losses of the liquid in the tube, and momentum losses associated with the inlet effect. − Later, Quéré proved that in the early stage of the fluid’s filling process, there is an inertial regime between the filling distance and the filling time, and the mathematical model of this stage is as follows: H = 2 γ cos θ ρ R t where ρ denotes the fluid’s density and the liquid’s rising height H in the capillary is linearly related to the filling time t . From the perspective of dynamic balance, the capillary force is the driving force for the rising of fluid, and the viscous force, gravity, and inertia force are all the resistances that hinder the fluid from rising.…”