Drop formation from a capillary tube: Comparison of one-dimensional and two-dimensional analyses and occurrence of satellite drops
Physics of Fluids, 14 (8):
2606--2621 (2002)Abstract
The axisymmetric formation of drops of Newtonian liquids from a vertical capillary into air is governed by the three-dimensional but axisymmetric Navier–Stokes system and appropriate boundary and initial conditions. Algorithms for obtaining accurate solutions of the resulting two-dimensional (2D) system of equations have recently been developed by Wilkes et al. Phys. Fluids 11, 3577 (1999), but are computationally intensive. A one-dimensional (1D) model based on simplification of the governing 2D system through the use of the slender-jet approximation has gained popularity in recent years Eggers, Rev. Mod. Phys. 69, 865 (1997). Such 1D algorithms not only result in great computational savings but appear to capture well the physics of drop formation as has been learned through a somewhat limited number of studies Eggers and Dupont, J. Fluid Mech. 262, 205 (1994); Brenner et al., Phys. Fluids 9, 1573 (1997). Indeed, existing 1D analyses Eggers and Dupont, J. Fluid Mech. 262, 205 (1994); Brenner et al., Phys. Fluids 9, 1573 (1997) have considered solely those situations in which the drops are grown quasi-statically prior to the onset of the instability and hence have ignored the effect of a finite flow rate or nonzero Weber number We, which measures the relative importance of inertial to capillary force, on the dynamics. In this paper, the accuracy of 1D algorithms is critically evaluated by comparing predictions made with them to those obtained with a recently developed 2D algorithm Wilkes et al., Phys. Fluids 11, 3577 (1999) based on the finite element method (FEM) over large ranges of the governing parameter space spanned by the Weber and Ohnesorge numbers, where the latter group, Oh, measures the relative importance of viscous to capillary force. When capillarity dominates, the predictions of the 1D algorithm are demonstrated to deviate no more than a couple of percent from those of the 2D algorithm. When inertial or viscous force is large compared to surface tension force, the 1D model can be in error by up to 15\%. The limitations of and errors incurred by the 1D model due to its inability to predict interface overturning for low Oh fluids are also discussed. Because the 2D algorithm Wilkes et al., Phys. Fluids 11, 3577 (1999) may take hours to a day to simulate the dynamics of a single drop up to the instant in time t = td at which the drop first breaks, and the 1D algorithm can perform the same task in a few minutes, the 2D FEM algorithm has not been used to date beyond the instant of first breakup. In this paper, calculations are continued beyond t = td with the 1D algorithm as in some previous works Brenner et al., Phys. Fluids 9, 1573 (1997), but with the following distinctions. First, for the first time regions in parameter space are identified over which satellite droplets are formed and ones over which formation of satellites are suppressed. The computed limit delineating the formation of satellites from suppression of satellites is compared when Oh<<1 to that determined by Zhang J. Coll. Int. Sci. 212, 107 (1999) for water-like liquids. Second, formation of several drops in a sequence is studied to determine the number of drops required to reach a state of steady dripping.