Viscous Taylor droplets in axisymmetric and planar tubes: from Bretherton’s theory to empirical models

Abstract

The aim of this study is to derive accurate models for quantities characterizing the dynamics of droplets of non-vanishing viscosity in capillaries. In particular, we propose models for the uniform-film thickness separating the droplet from the tube walls, for the droplet front and rear curvatures and pressure jumps, and for the droplet velocity in a range of capillary numbers, Ca, from 10 −4 to 1 and inner-to-outer viscosity ratios, λ , from 0, i.e. a bubble, to high viscosity droplets. Theoretical asymptotic results obtained in the limit of small capillary number are combined with accurate numerical simulations at larger Ca. With these models at hand, we can compute the pressure drop induced by the droplet. The film thickness at low capillary numbers (Ca < 10 −3 ) agrees well with Bretherton's scaling for bubbles as long as λ < 1. For larger viscosity ratios, the film thickness increases monotonically, before saturating for λ > 10 3 to a value 2 2/3 times larger than the film thickness of a bubble. At larger capillary numbers, the film thickness follows the rational function proposed by Aussillous & Quéré [5] for bubbles, with a fitting coefficient which

How to measure the thickness of a lubrication film in a pancake bubble with a single snapshot?

“…Nevertheless, as mentioned earlier, the correlating function f(m) is already valid for inviscid pancake droplets. Practically, Balestra et al 16 have demonstrated that the shape of droplets for k Շ 10 2 is almost identical to the shape obtained for k ¼ 0, as it has been considered in this work. This said, it is worth mentioning that the ratio of refractive indices m being lower for droplets than for bubbles, the sensitivity of the method would also be lower, as inferred from Fig.…”

“…To this purpose, the new shape function should be constructed depending additionally on the viscosity ratio, denoted k ¼ l d /l, between the viscosity of the drop, l d , and the one of the surrounding liquid. 16 Updating the ray tracing algorithm with these new shapes should allow one to extend the correlating function in Eq. (5) for non-zero viscosity ratio, i.e., f(m, k).…”

Predicting droplet velocity in a Hele-Shaw cell

“…In eq. (34) it is assumed that the surface tension variation, ensuring the uniform interfacial velocity, remains small in comparison with the equilibrium surface tension value. In that case, the Laplace pressure is determined from the non modified surface tension γ, which is the classical Bretherton approximation, see eq.…”

Motion of Viscous Droplets in Rough Confinement: Paradoxical Lubrication

“…More generally, the transport of droplets in confined environment is crucial for medical and chemical engineering applications, as illustrated by the fast development of digital microfluidics during the last two decades [7,8]. This problem motivated theoretical, numerical and experimental works, with the aim of understanding the dynamics of confined drops in ideal geometries [9][10][11][12][13][14][15][16][17]. Although they demonstrated the major contribution of lubricating films, these studies mainly remained focused on smooth confining walls.…”