Predicting droplet velocity in a Hele-Shaw cell

Abstract

We study the motion of low viscous non-wetting droplet in a Hele-Shaw cell, while it is pushed by an external phase of imposed flow rate, at low capillary numbers. In this regime, the droplet's mobility, defined as the ratio between the droplet velocity and the external phase mean velocity, evolves non linearly with the capillary number, a signature of the different dissipation mechanisms at play. Experiments are performed with surfactant free air bubbles in fluorinated oil, and with surfactant laden fluorinated oil droplets in water. We propose a model based on a power balance which takes into account the dissipation in the thin wetting film trapped between the bubble (or the drop) and the channel wall. The full topography of this thin film is obtained theoretically for the bubble case. By contrast, the presence of surfactants in the drop case induces uncontrolled boundary conditions at the interface, thus imposing to use the experimental topography measured in the previous paper [Reichert et al., J. Fluid. Mech., 850 p.708 (2018)]. Remarkably, the model reproduces the experimental velocities and shows that the velocity can be strongly affected by a stagnant cap effect at the rear of the drop, even if localized in less than a few percents of the total film area.

Local origin of the visco-elasticity of a millimetric elementary foam

“…This domain C is far enough from the meniscus for the Laplace pressure to be negligible, but close enough from it so that the boundary condition difference on both interfaces is not screened. Such behaviour has already been observed and quantified for a meniscus in contact with a solid wall (Cantat 2011; Reichert, Cantat & Jullien 2019), and was conjectured in Petit (2014) for a free meniscus.…”

“…It is, however, usually observed that the velocity may vanish in such cases, this effect being known as the stagnant cap limit (Cuenot, Magnaudet & Spennato 1997; Cantat 2011; Reichert et al. 2019). In this limit, and in steady state, the whole flux advected on interface 2 must diffuse to interface 1, which imposes (as depicted in figure 15).…”

Local origin of the visco-elasticity of a millimetric elementary foam

“…This domain C is far enough from the meniscus for the Laplace pressure to be negligible, but close enough from it so that the boundary condition difference on both interfaces is not screened. Such behavior has already been observed and quantified for a meniscus in contact with a solid wall [32,33], and was conjectured in [34] for a free meniscus.…”

“…In compression, the surface excess Γ 2 (s m ) is larger than its equilibrium value and can not vanish. It is however usually observed that the velocity may vanish in such cases, this effect being known as the stagnant cap limit [32,33,40]. In this limit, and in steady state, the whole flux Φ 2 = u ∞ Γ ∞ advected on interface 2 must diffuse to interface 1, which imposes Γ ∞ u ∞ ∼ j ∼ −D δΓ 1 /(h ∞ h Γ ) (as depicted in Fig.…”

Circular bubbles in a Hele-Shaw channel: a Hele-Shaw Newton's cradle

“…The effective boundary condition used by Meiburg (1989) was improved by Burgess & Foster (1990), both to capture correctly the Bretherton pressure drop at the rear interface of a moving bubble, and to analyse inner regions where the liquid flow is approximately tangent to the bubble interface and the Park & Homsy (1984) model breaks down. Reichert, Cantat & Jullien (2019) included the Bretherton pressure drop in their model for an isolated circular bubble in a Hele-Shaw cell with a uniform background flow. Reyssat (2014) also included the Bretherton drag force in his model for a bubble in a Hele-Shaw cell whose walls are slightly inclined to form a thin wedge, and observed that the bubble migrates out of the wedge to reduce its surface area.…”