Predicting droplet velocity in a Hele-Shaw cell

We study the motion of low viscous nonwetting 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 nonlinearly 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, 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.

Figure 1

(a) Design of the microfluidic chip: droplets/bubbles are generated at the T-junction and flow toward the Hele-Shaw cavity. To avoid hydrodynamical interactions between successive droplets, external phase is added through the comb shape channels located aside of the main channel. (b) Pancake-shaped droplet/bubble moving in a Hele-Shaw cell: $2H_o$ is the cell thickness, $U_f$ and $U_d$ are, respectively, the external fluid and the droplet/bubble velocity, $h_{\infty }$ is the lubrication film thickness.

Figure 2

Mobility $U_d/U_f$ as a function of Ca for different radii. (a) bubbles (System 1) with $R=160\mu \mathrm{m}$ ($\circ$), $R=100\mu \mathrm{m}$ ($\times$) and $R=77\mu \mathrm{m}$ ($★$). The horizontal line is Taylor and Saffman prediction ${(U_d/U_f)}_{\mathrm{TS}}=2$ [6]. (b) droplets (System 2) with $R=100\mu \mathrm{m}$ ($★$) and $R=80\mu \mathrm{m}$ ($\blacksquare$). The horizontal line is Gallaire et al. prediction ${(U_d/U_f)}_{\mathrm{TSV}}=2/(1+\frac{{\eta }_i}{{\eta }_o})$ with ${\eta }_i/{\eta }_o=0.64/2.5=0.26$ [25].

Figure 3

Cross-section of the bubble/droplet for $z=H_o$. The energy balance is performed within the cylindrical control volume $\mathrm{\Omega }$ of radius $(R+H_o)$ and height $2H_o$.

Figure 4

Transverse cross-section of the droplet for $y=0$, with the different domains and notations used in the text. The central part of the film and the dynamical meniscus constitute the lubrication film, in which most of the dissipation occurs. The size of the dynamical meniscus is exaggerated for the sake of clarity.

Figure 5

Scheme of the droplet/bubble, of equatorial radius $R\gg H_o$ (pancake shape), moving at the velocity $U_d{\mathbf{e}}_x$ in the laboratory frame of reference.

Figure 6

Numerical topography of the lubrication film obtained by joining the profiles calculated in the different dynamical menisci with the central domain for $\text{Ca}=7.2\times {10}^{-4}$.

Figure 7

Relative importance of all dissipative terms in System 1 as a function of the capillary number. $D_{\mathrm{men}}^f$, $D_{\mathrm{men}}^r$, and $D_{\mathrm{lat}}$ are, respectively, the dissipations in the front [Eq. (24)], the rear [Eq. (25)], and the lateral [Eq. (26)] dynamical meniscus.

Figure 8

Mobility $U_d/U_f$ as a function of Ca for bubbles (System 1) of different radii with $R=160\mu \mathrm{m}$ ($\circ$), $R=100\mu \mathrm{m}$ ($\times$) and $R=77\mu \mathrm{m}$ ($★$). The colored lines are our theoretical predictions: the dashed lines include every source of dissipation (front, rear and lateral meniscus: $D_{\mathrm{men}}^{f,r}$ and $D_{\mathrm{lat}}$) whereas the solid lines do not take into account the contribution from the lateral meniscus shown to be negligible. The horizontal line is Taylor and Saffman prediction ${(U_d/U_f)}_{\mathrm{TS}}=2$ [6].

Figure 9

Relative importance of all dissipative terms in System 2 as a function of the capillary number. $D_{\mathrm{drop}}$, $D_{\mathrm{men}}^f$ and $D_{\mathrm{th}}$ are, respectively, the dissipations in the drop [Eq. (27)], in the front dynamical meniscus [Eq. (24)] and in the thickening region [Eq. (32)]. $D_{\mathrm{osc},\mathrm{Pois}}$ and $D_{\mathrm{osc},\mathrm{sh}}$ are the Poiseuille [Eq. (37)] and shear [Eq. (38)] contributions to the dissipation in the oscillating region.

Figure 10

(a) Oscillation at the rear meniscus. Longitudinal ($y=0$) thickness profile for $\text{Ca}=1.3\times {10}^{-3}$. The film thickness at large $x$ is $h_{\infty }^{\mathrm{BF}}$ and the local maximum is $h_{\max }$. The dotted line shows the film profile deduced using the surfactant mass transport model with $\text{Ma}=1.6$. The solid line shows the theoretical prediction from Eq. (34) using $U_i\left(B\right)=2U_d(1-h_{\infty }^{\mathrm{BF}}/h_{\max })$; (b) interfacial velocity in the thickening region normalized by the droplet velocity for $\text{Ca}=1.3\times {10}^{-3}$. The solid line is the interfacial velocity obtained from the surfactant mass transport model with $\text{Ma}=1.6$. (Extracted from Ref. [13].)

Figure 11

Mobility $U_d/U_f$ of droplets (System 2) of different radii, with $R=100\mu \mathrm{m}$ ($★$) and $R=80\mu \mathrm{m}$ ($\blacksquare$). The colored lines are our theoretical predictions without $D_{\mathrm{lat}}$: solid lines take into account the stiffening of the interface at the rear of the droplet. The dashed lines corresponds to the full stress-free model.