Fall and break-up of viscous miscible drops in a Hele-Shaw cell

We investigate the fall in a Hele-Shaw cell of viscous miscible drops in a more viscous ambient fluid with negligible diffusivity compared to the kinematic viscosity of the fluids. We measure the drop shape, thickness, and velocity for various viscosity and density ratios. The elongation of the drops is found to increase quadratically with the ratio of the drop and ambient fluid viscosity. The thickness of the drop is shown to be approximately half the thickness of the gap of the Hele-Shaw cell. We develop a theoretical expression of the drop falling velocity, taking into account the elongation and thickness of the drop. This agrees well with observations after multiplying by an empirically determined constant. In some circumstances the drops break up, either because of defects advected downwards within the drop or by instability of the shape of the drop.

Figure 1

Schematic view of a drop with a tail of fluid falling into the experimental cell (a) ($x,z$) and (b) ($y,z$) plane. (c) Sketch of the velocity profiles in the cross section of half the cell (in the ($y,z$) plane). We denote by a circled 1 an area of the cell far from the drop, and by a circled 2 an area between the drops and the walls.

Figure 2

(a) Plot of the density of the ambient and drop fluid vs their viscosity. The black dashed lines correspond to fits ${\rho }_a=1075+30{log}_{10}{\mu }_a$ and ${\rho }_d=14000+125{log}_{10}{\mu }_d$. (b) Parameters $\lambda$ and $\mathrm{\Delta }\rho /{\rho }_a$ of all the experiments performed. The symbols indicate the width of the gap.

Figure 3

(a) Raw image and (b) measurement of fluid thickness $w/H$ of a drop of $({\mu }_d,{\rho }_d)=(1.57\times {10}^{-2}\mathrm{Pa}{\mathrm{s}}^{-1},1186\mathrm{k}\mathrm{g}{\mathrm{m}}^{-3})$ falling in an ambient fluid of $({\mu }_a,{\rho }_a)=(1.44\mathrm{Pa}{\mathrm{s}}^{-1},1078\mathrm{k}\mathrm{g}{\mathrm{m}}^{-3}$. On (c) the contour (red line) and centroid (red cross) of the detected plume are highlighted. On (d), the colormap shows measurements of $w/H$, the green cross highlights where the profiles of Fig. 5 are measured on the drop, and the blue square shows where the average of the fluid thickness, $w_{*}$, is computed (see Sec. 4a).

Figure 4

Plot of the plume elongation ratio $\alpha$ vs the viscosity ratio $\lambda$. Symbols discriminate between different values of $\mathrm{\Delta }\rho /{\rho }_a$. The black curve shows the quadratic fit $\alpha =1.17\times {10}^{-5}{\lambda }^2-6.40\times {10}^{-4}\lambda +1.11$. The insets show the shape of drop for experiments on the left with $({\mu }_a,{\mu }_d)=$ ($1.10,3.43\times {10}^{-1}$) $\mathrm{Pa}{\mathrm{s}}^{-1}$, $\mathrm{\Delta }\rho /{\rho }_a=$ $2.12\times {10}^{-1}$ and on the right with $({\mu }_a,{\mu }_d)=$ ($1.10,2.40\times {10}^{-3}$) $\mathrm{Pa}{\mathrm{s}}^{-1}$, $\mathrm{\Delta }\rho /{\rho }_a=$ $5.50\times {10}^{-3}$.

Figure 5

(a) Profiles of drop thickness versus the position $x$ or $z$ centered on the centroid of the drop. This is the same drop as that shown in Fig. 3 [$({\mu }_d,{\rho }_d)=$ ($1.57\times {10}^{-2}\mathrm{Pa}{\mathrm{s}}^{-1}$, $1186\mathrm{k}\mathrm{g}{\mathrm{m}}^{-3}$), $({\mu }_a,{\rho }_a)=$ ($1.44\mathrm{Pa}{\mathrm{s}}^{-1}$, $1078\mathrm{k}\mathrm{g}{\mathrm{m}}^{-3}$)], and the profiles are taken along the blue lines. (b) Plot of the average drop thickness over gap width $w_{*}/H$ for different values of $H$.

Figure 6

Plot of the tail (a) width (in the $x$ direction) scaled by the width of the drop, with a typical error bar showing error linked to a detection error of about one pixel from each side of the tail in the legend box (b) aspect ratio vs the viscosity ratio $\lambda$, for $H$=$5\mathrm{m}\mathrm{m}$ (open symbols) and $H$=$2\mathrm{m}\mathrm{m}$ (filled symbols).

Figure 7

Plot of the variation of the total volume of drop fluid in the cell, scaled by that of the drop at $t=0$, $(V+V_{\mathrm{tail}})/V(t=0)$, as a function of the time scaled by the total falling time, $t_{\max }$, $t/t_{\max }$ for 10 different drops. The measurement error on the volume of the drop is of about 10%.

Figure 8

Plot of the drop falling velocity $v_z$ versus (a) the viscosity ratio, $\lambda$ (b) $U$, given by (11). The mean drop equivalent radius is $R=1.05\pm 0.1\mathrm{cm}$ for $H=5\mathrm{mm}$ (open symbols), and $R=0.91\pm 0.3\mathrm{cm}$ for $H=2\mathrm{mm}$ (filled symbols). The symbols discriminate the points based on their behavior. The black dashed curve shows $v_z=1.18U$.

Figure 9

Successive snapshots of drops with (a) ${\mu }_d=$ $1.66\times {10}^{-2}\mathrm{Pa}{\mathrm{s}}^{-1}$, $\mathrm{\Delta }\rho /{\rho }_a=$ $0.11$, (b) ${\mu }_d=$ $5.20\times {10}^{-3}\mathrm{Pa}{\mathrm{s}}^{-1}$, $\mathrm{\Delta }\rho /{\rho }_a=$ $1.33\times {10}^{-2}$, (c) ${\mu }_d=$ $2.53\times {10}^{-1}\mathrm{Pa}{\mathrm{s}}^{-1}$, $\mathrm{\Delta }\rho /{\rho }_a=$ $0.21$. ${\mu }_a=$ $0.53\mathrm{Pa}{\mathrm{s}}^{-1}$ for (a), ${\mu }_a=$ $0.51\mathrm{Pa}{\mathrm{s}}^{-1}$ for (b), and $1.41\mathrm{Pa}{\mathrm{s}}^{-1}$ for (c). The break-up type is Type I for (b) and Type II for (c). The time interval between two snapshots is $0.8\mathrm{s}$ for (a), $5.0\mathrm{s}$ for (b), and $2.5\mathrm{s}$ for (c). The drop radius $R$ is $1.04\mathrm{c}\mathrm{m}$ for (a), $0.99\mathrm{c}\mathrm{m}$ for (b), and $1.07\mathrm{c}\mathrm{m}$ for (c). The colormap shows $w/H$. The vertical and horizontal scales are identical.

Figure 10

Map of the experiments performed on the space of parameters $({\mu }_a,{\mu }_d)$. The isocontours show values of $\mathrm{\Delta }\rho /{\rho }_a$. Symbols show the type of behavior observed. The green dashed line shows the separation between $\lambda <18$ (above the line) and $\lambda >18$ (below the line).