Microfluidic breakups of confined droplets against a linear obstacle: The importance of the viscosity contrast

Louis Salkin, Laurent Courbin, and Pascal Panizza

Phys. Rev. E 86, 036317 – Published 21 September 2012

Abstract

Combining experiments and theory, we investigate the break-up dynamics of deformable objects, such as drops and bubbles, against a linear micro-obstacle. Our experiments bring the role of the viscosity contrast $Δ η$ between dispersed and continuous phases to light: the evolution of the critical capillary number to break a drop as a function of its size is either nonmonotonic ( $Δ η > 0$ ) or monotonic ( $Δ η \leq 0$ ). In the case of positive viscosity contrasts, experiments and modeling reveal the existence of an unexpected critical object size for which the critical capillary number for breakup is minimum. Using simple physical arguments, we derive a model that well describes observations, provides diagrams mapping the four hydrodynamic regimes identified experimentally, and demonstrates that the critical size originating from confinement solely depends on geometrical parameters of the obstacle.

Received 15 September 2011

DOI:https://doi.org/10.1103/PhysRevE.86.036317

Authors & Affiliations

Louis Salkin, Laurent Courbin^*, and Pascal Panizza^†

IPR, UMR CNRS 6251, Campus Beaulieu, Université Rennes 1, 35042 Rennes, France

^*laurent.courbin@univ-rennes1.fr
^†pascal.panizza@univ-rennes1.fr

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Issue

Vol. 86, Iss. 3 — September 2012

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Images

Figure 1
(a) Schematic of the setup near the obstacle and top-view images of the flow defining the ten governing parameters at play. (b)–(e) Typical flow behaviors of slugs having the same size meeting an obstacle at different speeds. For large enough speeds, breakup occurs without (b) or with (c) the retraction of an interface in the narrow gap. The slug does not break at lower speeds; this occurs with the invasion and subsequent retraction of an interface of the narrow gap (d) or without the invasion of this gap (e). $ℓ_{i}$ denotes the position of an interface in the $i$ th gap, $i$ $=$ 1 or 2. White arrows indicate the flow direction in both gaps. Scale bars 100 $μ$ m.Reuse & Permissions
Figure 2
Experimental diagrams characterizing the dynamical behavior for (a) $η_{d} > η_{c}$ and (b) $η_{d} < η_{c}$ as a function of $L_{d}$ and $v$ . A slug may (orange and red symbols) or may not (blue and cyan symbols) break against the obstacle. These regimes can occur with (orange and cyan symbols) or without the retraction of a two-fluid interface in gap (2) (red and blue symbols). The dashed lines are guides for the eyes and indicate the transition between breakup and no breakup regimes. The length of the obstacle is $L$ $=$ 300 $μ$ m. (a) Viscous water-glucose mixture (44 wt. $%$ glucose) in hexadecane and $W$ $=$ $0.5$ . (b) Water in a viscous silicone oil and $W$ $=$ $0.6$ .Reuse & Permissions
Figure 3
Diagrams mapping the break-up dynamics as a function of $L_{d} / L$ and $C$ for (a) $Δ η_{}$ $>$ 0 and (b) $Δ η_{}$ $<$ 0. The solid lines are predictions of our model for the transitions between regimes that are compared to experimental data of Figs. 2a and 2b. Symbols are identical to those of Fig. 2. The dimensionless quantities are (a) $W$ $=$ $0.5$ , $\frac{w}{h}$ $=$ 3, $\frac{w_{2}}{h} = 0.7$ , $Z$ $=$ $2.4 \times 10^{- 3}$ , and (b) $W$ $=$ $0.6$ , $\frac{w}{h}$ $=$ 3, $\frac{w_{2}}{h}$ $=$ $0.8$ , $Z$ $=$ $2.1 \times 10^{- 3}$ . The two free parameters are (a) $c$ $=$ $0.9$ and $Δ η_{}$ $=$ 8, and (b) $c$ $=$ $0.6$ and $Δ η_{}$ $=$ $- 0.2$ . The model predicts the critical capillary numbers above which an interface enters the gap (2): $\frac{C_{☆}}{1 + Δ η_{}}$ for $Δ η_{}$ $>$ 0 and $α$ $\geq$ 1, $\frac{C_{☆}}{1 + α Δ η_{}}$ , for $Δ η_{}$ $>$ 0 and $α$ $\leq$ 1, and $C_{☆}$ for $Δ η_{}$ $<$ 0. When $Δ η_{}$ $>$ 0, we find that $η_{d}^{e f f}$ $>$ $η_{c}$ , which is consistent with the literature [21]. By contrast, when $Δ η$ $<$ 0, we determine that $η_{d}$ $<$ $η_{d}^{e f f}$ $<$ $η_{c}$ . Similar results have been reported and discussed in [22].Reuse & Permissions
Figure 4
Evolution of $C_{c r}$ with $C_{☆}$ for $Δ η_{}$ $>$ 0. For both fluid systems, the experimental data collapse onto a single line whose slope is 1 $/$ $(Δ η_{}$ $+$ 1). Inset: Evolution of the critical slug size $L_{d}^{c r}$ with $\frac{L w_{1}}{w}$ . The fluid systems are ( $▪$ ) water in hexadecane, ( $⧫$ ) a viscous water-glucose (28 wt. $%$ glucose) mixture in hexadecane, and ( $•$ ) a viscous water-glucose mixture (44 wt. $%$ glucose) in hexadecane. The ratio of the widths of the gaps is $W$ $=$ $0.48$ . Each data point corresponds to a value of $Z = 0.8$ –3.8 $\times 10^{- 3}$ . The dashed line stands for the linear fit $L_{d}^{c r} = \frac{L w_{1}}{w} + 0.9 w$ .Reuse & Permissions

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