Singularity-free approximate analytical solution of capillary rise dynamics

28 Aug 2018

Capillary rise is one of the most well-known and vivid il- lustrations of capillarity (shown in Figure 1). Knowledge of capillarity laws is important in oil recovery, civil engineering, dyeing of textile fabrics, ink printing, and a variety of other fields. It is capillarity that brings water to the upper layer of soils, drives sap in plants, or lays the basis for the operation of pens [1-10]. Washburn [1] developed an equation to describe the rate of liquid penetration into small cylindrical capillaries based on the Poiseuille flow profile, when neglecting the air resis- tance, a commonly used form of Lucas-Washburn?s equation is 8 hdh dt + gha2 sin = 2a cos . When the gravitational force is negligible, a well-known Washburn?s law can be ob- tained with the initial condition h(0) = 0: h = ? a cos 2 t, which predicts burst-like behavior with the velocity being in- finite at zero time. This singularity of the solution highlights a deep inconsistency of the above equation [2]. Taking into ac- count the liquid?s momentum in the tube and end-effect drag on the fluid entering the tube, Brittin [3] derived a more rigor- ous formulation of the Lucas-Washburn equation, as follows: hd2h dt2 + 54 ( dh dt )2 + 8 a2 hdh dt + gh = 2 a cos , where is the viscosity. The most popular equation is its cosmetic format hd2h dt2 +( dh dt )2 + 8 a2 hdh dt +gh = 2 a cos , however, it has a sin- gularity of t = 0, namely ?h = dh dt ? ? as t ? 0, which would lead to an ill-posed problem. This equation cannot even deal with the natural initial conditions h(0) = ?h(0) = 0. Hence, *Corresponding author (email: sunb@cput.ac.za) formal remedy has to be taken ?h(0) = ? 2 cos =( a) [4], neglecting such a logical drawback as the acceleration of the liquid front at zero time, which is infinite.