Collective diffusivity in a sheared viscous emulsion: Effects of viscosity ratio

The shear-induced collective or gradient diffusivity in an emulsion of viscous drops, specifically as a function of viscosity ratio, was computed using a fully resolved numerical method. An initially randomly packed layer of viscous drops spreading due to drop-drop interactions in an imposed shear has been simulated. The collective diffusivity coefficient was computed using a self-similar solution of the drop concentration profile. We also obtained the collective diffusivity (the collective diffusivity coefficient multiplied by the average drop volume fraction), computing the dynamic structure factor from the simulated drop positions—an analysis typically applied only to homogeneous systems. The two quantities computed using entirely different methods are in broad agreement, including their predictions of nonmonotonic variations with increasing capillary number and viscosity ratio. The computed values were also found to match with past experimental measurements. The collective diffusivity coefficient computed here, as expected, is 1 order of magnitude larger than the self-diffusivity coefficient for a dilute emulsion previously computed using pairwise simulation of viscous drops in shear. The collective diffusivity coefficient computed here shows a nonmonotonic variation with viscosity ratio, in contrast to self-diffusivity computed using pairwise computation. The difference might point to an intrinsic difference in physics underlying the two diffusivities. Alternatively, it also might arise from drops not reaching equilibrium deformation in the period after one interaction and before the next—an effect absent in the pairwise simulation used for the computation of self-diffusivity. We offer a qualitative explanation of the nonmonotonic variation by relating it to average nonmonotonic drop deformation with increasing viscosity ratio. We also provide empirical correlations of the collective diffusivity as a function of viscosity ratio and capillary number.

Figure 1

A schematic of the layer of randomly placed drops in simple shear flow.

Figure 2

(a) Cube of the width of the layer of drops plotted as a function of time grows linearly with time. Snapshots of the evolving drop configurations at various time instants. See Supplemental Material [31] for a movie of drop evolution. (b) Concentration profile of the drops at various time instants, showing a parabolic profile broadening with time. Inset shows a collapse of the same when plotted against the scaled similarity variable according to Eq. (2).

Figure 3

Cube of width of the drop layer vs time for various viscosity ratios for $\mathrm{Ca}=0.05\left(\mathrm{a}\right)$ and $\mathrm{Ca}=0.30\left(\mathrm{b}\right)$ showing the expected linear trend.

Figure 4

(a) Dimensionless coefficient of diffusion $f_2$ vs viscosity ratio ${\lambda }_{\mu }$ for different capillary numbers. (b) Main figure shows mean drop deformation parameter and inset shows drop orientation angle.

Figure 5

(a) Relative displacement of two drops interacting in shear flow comparing our results (solid line) with those obtained from boundary element method (dotted line) by Cristini et al. [52] and experimental measurements (open circles) by Guido and Simeone [51]. Main figure is with $\mathrm{Ca}=0.135$, ${\lambda }_{\mu }=1.37$. Inset is with $\mathrm{Ca}=0.13$ and ${\lambda }_{\mu }=1.4$. (b) Final net relative displacement of two drops, initially separated $0.5a$ in the gradient direction for different viscosity ratios ${\lambda }_{\mu }$ at $\mathrm{Ca}=0.30$. Present results (diamonds) compare reasonably well with boundary element method results of Loewenberg and Hinch [4].

Figure 6

(a) Main figure shows the linear growth in $-lnF(\mathbf{k},t^{\prime })/k^2$ with time for various wave numbers for $\mathrm{Ca}=0.20$ and ${\lambda }_{\mu }=5.0$. Inset shows $D_{yy}^c/\dot{\gamma }{a}^2$ vs wave number $k$ for a few different viscosity ratios. (b) $D_{yy}^c/\dot{\gamma }{a}^2$ vs viscosity ratio for different Ca values.

Figure 7

(a) Dimensionless gradient diffusivity $f_2$ and (b) average drop deformation parameter, orientation angle vs Ca for different viscosity ratios. (c) $D_{yy}^c/\dot{\gamma }{a}^2$ vs Ca for different viscosity ratios. (d) Surface plot showing $f_2$ as a function of capillary number Ca and viscosity ratio ${\lambda }_{\mu }$. The correlation describing the surface is given in Eq. (9)).