4.1 Collective diffusivity using two methods at $Ca=0.05$

Figure 2 shows a typical case of an initially concentrated layer of viscous drops experiencing shear-induced diffusion with time. Figure 3(a) plots individual drop positions with time for a low capillary number of $Ca=0.05$ . The drops exhibit a random-walk-like movement. The concentration profile for this case is plotted in figure 3(b), fitting a parabolic profile in successive time instants (also see the movie in the supplemental material). In the inset of figure 3(b), we plot the concentration profiles scaled by $t^{\prime -1/3}$ versus $y^{\prime }/t^{\prime 1/3}$ to see that indeed they all collapse as they should according to the similarity profile $\unicode[STIX]{x1D713}(\unicode[STIX]{x1D702})$ described in (3.2). Plotting the width $\text{}\underline{w}$ on a log–log plot in figure 3(c) shows an eventual linear curve indicating the $1/3$ power-law dependence on time. The initial data ( $t^{\prime }<20$ ) are discarded; starting from $t_{0}^{\prime }=20$ , the remaining portion is subdivided into smaller time intervals (15 inverse shear units). The slope of $\text{}\underline{w}^{3}-\text{}\underline{w}_{o}^{3}$ in each subinterval and the corresponding value of $f_{2}$ there according to (3.3) is computed. The dynamics over these different time intervals are approximately uncorrelated (Zinchenko & Davis Reference Zinchenko and Davis2002). The mean of this average is reported (under the assumption of ergodicity, it is equal to an ensemble average). The corresponding standard deviation of the values from different time intervals is reported as the uncertainty of the estimation. We obtain $f_{2}=0.329\pm 0.02$ . In figure 3(d), the plot of the standard deviation $w$ of the layer according to (3.4) against time also shows a similar $1/3$ slope with time. An identical analysis is executed on this data to obtain the average $f_{2}$ according to the second method to obtain $f_{2}=0.327\pm 0.021$ , very similar to the one obtained by the other method. Note that there is minimal ‘noise’ in both curves in figure 3(c,d), demonstrating the robust nature of the $1/3$ exponent. Presence of noise detracts from the confidence of the fitting; which is why ensemble averaging of mean square displacements in the computation of self-diffusivity requires a significantly larger number of particles (Marchioro & Acrivos Reference Marchioro and Acrivos2001).

Figure 4.

Effect of the computational domain: the width of drop layer as a function of time for different widths of the computational domain.

4.2 Effects of domain size

Although we are interested in computing the collective diffusivity in an emulsion in a free shear flow, i.e. without any boundary, our code requires walls to generate the shear flow. Therefore, it is important to show that the domain size chosen is sufficiently large and that the results are independent of the physical dimensions of the domain. The pairwise interaction between drops in a shear flow with walls was shown to be exactly the same as the free-shear interaction for $L_{y}\geqslant 25a$ when the drops are at the centre of the domain (Singh & Sarkar Reference Singh and Sarkar2015). However, unlike that case, the layer of drops considered here spreads its thickness with time and eventually would be affected by the walls. Figure 4 shows $w^{3}-w_{o}^{3}$ increasing with time for three different separations $L_{y}$ between the walls. All three curves are linear for a substantial portion initially and all have the same slope. This shows that, in this linear region the walls do not affect the dynamics of the drops. For $L_{y}=21a$ , at approximately $t^{\prime }-t_{0}^{\prime }=200$ , the linearity breaks down due to the wall effects. For $L_{y}=28a$ , this happens at approximately $t^{\prime }-t_{0}^{\prime }=300$ and for $L_{y}=42a$ there is no noticeable deviation in the curve until approximately $t^{\prime }-t_{0}^{\prime }=400$ . Table 1 shows the values of the diffusivity calculated in each of these cases from the linear portion. We obtain a similar value for all three cases with no particular trend. We choose $L_{y}=28a$ for all subsequent simulations. For this wall separation, one can estimate the lateral velocity on a droplet induced by the bounding walls using a small deformation analysis (Chan & Leal Reference Chan and Leal1979):

(4.1) $$\begin{eqnarray}\frac{V_{lat}}{\dot{\unicode[STIX]{x1D6FE}}a}=-\frac{3(16+19\unicode[STIX]{x1D706})(54\unicode[STIX]{x1D706}^{2}+97\unicode[STIX]{x1D706}+54)}{4480(1+\unicode[STIX]{x1D706})^{3}}Ca\left(\frac{a}{L_{y}}\right)^{2}\left(\frac{y}{L_{y}}\right)\left(1+\frac{8}{(1-4y^{2}/L_{y}^{2})^{2}}\right).\end{eqnarray}$$

For a sample case of $Ca=0.1$ , the half-width of the spreading layer reaches $y\sim 4.5a$ . With a local volume fraction of ${\sim}0.1$ , one obtains a Péclet number of $Pe=V_{lat}/(f_{2}\dot{\unicode[STIX]{x1D6FE}}a\unicode[STIX]{x1D719})\sim 0.02$ , indicating negligible effects of the walls. We also study the effects of $L_{x}$ , shown in table 2. In the interest of a reasonable computation time, $L_{x}=14a$ is chosen which seems sufficient for our purpose.

Table 1.

Effect of domain length in the $y$ direction, i.e. the distance between the walls, on the diffusivity.

Table 2.

Effect of domain length in the flow direction on the diffusivity.

Figure 5.

Width of the drop layer as a function of time for $Ca=0.05$ in the power law regime for different values of $N_{o}$ . Inset shows the same, scaled by $N_{o}$ , collapses onto a single curve.

4.3 Effects of $N_{o}$ (initial volume fraction in the layer) and initial configuration

In addition to the $t^{1/3}$ scaling of the width, the dependence of diffusivity on the quantity $N_{o}$ that appears in (3.3) and (3.5) shows that the diffusion is mediated by inter-particle interactions. In experiments at high volume fractions ( ${>}0.3$ ), the effective coefficient of diffusion ( $K$ in (3.3)) increased quadratically with $N_{o}$ , indicating a non-negligible contribution from three-particle interactions (Grandchamp et al. Reference Grandchamp, Coupier, Srivastav, Minetti and Podgorski2013). It results in a linear increase of $f_{2}$ with $N_{o}$ . If only pairwise interactions dominate, $f_{2}$ is expected to be independent of $N_{o}$ . To verify this and to ensure that the size of the system is sufficient, we perform simulations with different initial configurations and different $N_{o}$ . Different $N_{o}$ values represent different volume fractions in the central layer at $t^{\prime }=t_{0}^{\prime }$ (after the power law sets in as seen in figure 3 c,d) – e.g. the curves for $N_{o}=1.44$ and 3.28 correspond to 18 % and 31 % volume fraction. The curve for $N_{o}=2.44$ with an asterisk is for a volume fraction of 49 % obtained with an initial configuration with a smaller initial layer width. They also represent different layer widths from ${\sim}6$ to ${\sim}8$ drop radii as well as different initial random configurations. Figure 5 shows $w^{3}-w_{o}^{3}$ versus time for these various cases. As expected from (3.5) the slope increases with $N_{o}$ . The inset shows that upon scaling with $N_{o}$ all lines collapse onto a single line, indicating that the results are independent of $N_{o}$ as well as initial configurations.

Note that the relation $w^{3}\propto t$ arising from (3.2) is contingent on the assumption of the dominance of pair interactions in determining the shear-induced diffusion, that gave rise to $D_{c}\propto \unicode[STIX]{x1D719}$ . In contrast, if three-particle interactions were to dominate, the diffusivity would be proportional to $\unicode[STIX]{x1D719}^{2}$ giving rise to $w^{4}\propto t$ (Grandchamp et al. Reference Grandchamp, Coupier, Srivastav, Minetti and Podgorski2013). In general, at high enough concentrations, one would expect the higher-order interactions to be significant, and a clear scaling might not appear. In the initial part of the data in figure 3(c,d), when the local volume fraction is high in the closely packed layer, the deviation from the $1/3$ scaling – slopes are smaller – could be a signature of higher-order interactions in addition to the deviation from the parabolic profile. With time the local volume fraction continues to decrease and eventually higher than two-drop interactions become increasingly rare. Note that the time scale of drop deformation, the capillary time scale, is quite small – identical to $Ca$ in shear unit, i.e. 0.05–0.35 for the cases considered here. Therefore, the drops achieve their final deformed state long before the $1/3$ scaling behaviour sets in. Based on the scaling seen in figure 5 and the corresponding volume fractions (specifically the curve for $N_{o}=2.44$ ), we conclude that the $1/3$ scaling (and the relationship $D_{c}\propto \unicode[STIX]{x1D719}$ ) holds for $\unicode[STIX]{x1D719}\leqslant \sim 0.5$ .

Figure 6.

Snapshots of the drop layer at $t^{\prime }=\sim 17$ for three different $Ca$ .

Figure 7.

(a) Width at half-height of the concentration profile as a function of time for different $Ca$ . (b) Modified width of the drop layer as a function of time for different $Ca$ . Cube of both quantities has been plotted to show their linear scaling.

4.4 Effects of capillary number

Figure 6 shows the drops at three different capillary numbers at approximately the same time when the drops achieve their deformed shapes. We compute the non-dimensional collective diffusivity $f_{2}$ using both methods at different capillary numbers. Figure 7(a,b) shows that $\text{}\underline{w}^{3}-\text{}\underline{w}_{o}^{3}$ and $w^{3}-w_{o}^{3}$ both vary linearly and almost identically with $t^{\prime }-t_{0}^{\prime }$ at different $Ca$ values. Therefore, $f_{2}$ (the slopes) computed by the two methods are very similar at all values of $Ca$ , as has been tabulated in table 3. In figure 8, we plot $f_{2}$ versus $Ca$ using both methods. They resulted in very similar values. It shows a non-monotonic variation, as was also seen for the self-diffusivity (Loewenberg & Hinch Reference Loewenberg and Hinch1997; Tan et al. Reference Tan, Le and Chiam2012). An intuitive explanation for the non-monotonic variation can be obtained as arising from a competition between the effects of increasing deformation and decreasing drop orientation angle with increasing $Ca$ in pair interactions. The time variation of the average deformation and average inclination angle of the drops are plotted in figure 9. As noted before, the initially spherical drops quickly reach their equilibrium deformation over the capillary time scale $t_{capillary}^{\prime }=Ca$ . The inclination angle starts at the inclination of the extensional axis $45^{\circ }$ and decreases due to deformation (Taylor Reference Taylor1934). The fluctuations in their values are caused by the drop–drop interactions in the shear flow. Note that although the local volume fraction decreases due to diffusion of the drops, the average values of deformation and inclination angle do not change significantly over the time period considered.

Figure 8.

Dimensionless gradient diffusivity ( $f_{2}$ ), deformation parameter ( $D$ ) and drop inclination angle ( $\unicode[STIX]{x1D703}$ ) plotted as functions of $Ca$ . Symbols are the results from the simulation. The solid and dash-dotted lines are predictions of small deformation perturbative analysis for $\unicode[STIX]{x1D703}$ and $D$ respectively. The dashed line is a quadratic fit of the simulated $f_{2}$ values (solid triangles) obtained by standard deviation according to (4.2). The error bar corresponds to the $f_{2}$ values.

Figure 9.

Average deformation (a) and average orientation angle (b) with time for a simulation at $Ca=0.05$ . The shading shows the standard deviation.

In figure 9, we plot the Taylor deformation parameter $D=(L-B)/(L+B)$ ( $L$ and $B$ are the maximal and the minimal distances of the drop surface from the centroid) averaged over all drops and over the power-law regime shown in figure 3(c,d); it approximately matches with the small deformation perturbative result $D=Ca(19\unicode[STIX]{x1D706}+16)/(16\unicode[STIX]{x1D706}+16)$ ( $\unicode[STIX]{x1D706}$ is the viscosity ratio) (Taylor Reference Taylor1934) for the entire range of the $Ca$ considered here, the deviation also resulting from the interactions. The average drop inclination angle plotted in the same figure (figure 9) shows a decrease as also predicted by the small deformation perturbative analysis $\unicode[STIX]{x1D703}=\unicode[STIX]{x03C0}/4-Ca(2\unicode[STIX]{x1D706}+3)(19\unicode[STIX]{x1D706}+16)/(80\unicode[STIX]{x1D706}+80)$ (Chaffey & Brenner Reference Chaffey and Brenner1967).

Figure 10.

Schematic showing two approaching drops in three different regimes of $Ca$ to explain the non-monotonic dependence of $f_{2}$ on $Ca$ . The overlapping region increases initially and then decreases.

Table 3.

Tabulated values of $f_{2}$ using two methods, (i) from standard deviation of the droplet position and (ii) width of the concentration profile.

Figure 10 explains the effects of competition between increasing deformation and decreasing inclination with increasing $Ca$ . For small $Ca$ , increased deformation with increasing $Ca$ enhances the irreversible relative displacement between droplets upon interactions increasing the diffusivity. However, with larger increase in $Ca$ , as the angle of inclination decreases, the drops are more aligned, allowing sliding, and eventually this effect overcomes the other, decreasing the diffusivity above $Ca\sim 0.2$ . The non-monotonic variation in relative displacement in the pair collisions between drops in shear has been investigated in detail in the past (Olapade et al. Reference Olapade, Singh and Sarkar2009). Loewenberg & Hinch (Reference Loewenberg and Hinch1997) also found non-monotonicity in self-diffusivity versus $Ca$ , and attributed it to the subtle difference in pairwise interactions between initially closely spaced drops at low and high $Ca$ numbers. At low $Ca$ , initially closely spaced drops experience higher post-collision net displacements in their streamlines compared to drops which are initially far apart; the converse is true at higher $Ca$ . The authors argued that the balance between the decreasing near-field influence and the increasing far-field influence may have caused the non-monotonicity. Finally, we fitted the simulated values of $f_{2}$ for different $Ca$ by a quadratic curve and obtained a phenomenological relation

(4.2) $$\begin{eqnarray}f_{2}=-6.89Ca^{2}+2.57Ca+0.20.\end{eqnarray}$$

The relation has been plotted in figure 8.

4.5 Comparison with past experiments and theory

The only experimental measurement of $f_{2}$ available in the literature is the one reported by Hudson (Reference Hudson2003), who used drops of poly(propylene glycol) or poly(ethylene-alt-propylene) dispersed in poly(ethylene glycol). We ran simulations with material properties and a capillary number corresponding to this experiment and computed the diffusivity. The comparison with the experiment, shown in table 4, is very good at $Ca=0.23$ but shows some departure at $Ca=0.05$ . At the large value of $Ca=0.40$ , the simulation indicated drop break-up. In search of possible reasons for the difference between the computed results and measurements, one notes that the current simulation is for a purely monodisperse system and without any coalescence. However, Hudson noted that in their experiment the coalescence is rare and polydispersity is minor (the ratio between volume averaged and number averaged radii is ${\sim}1.2$ ).

Table 4.

Comparison of the diffusivity calculated from the present simulations compared to the experimental values reported by Hudson (Reference Hudson2003).

In the limit of $Ca\rightarrow 0$ , the collective diffusivity for an emulsion of spherical drops was inferred from two drop interactions to be ${\sim}0.20$ (Ramachandran, Loewenberg & Leighton Reference Ramachandran, Loewenberg and Leighton2010), which matches with the value obtained by extrapolating the present numerical result. There have been no other computations of $f_{2}$ in the literature. Grandchamp et al. (Reference Grandchamp, Coupier, Srivastav, Minetti and Podgorski2013) reported a value of ${\sim}0.77$ (rescaled by particle volume) for red blood cells. Loewenberg & Hinch (Reference Loewenberg and Hinch1997) used boundary element simulation of pair collisions to obtain self-diffusivity 8–9 times smaller than the gradient diffusivity calculated here. As noted before, for emulsions, effort to compute the gradient diffusivity by integrating all possible pair collisions has been frustrated due to divergence of the integrals. However, for rough spheres pair collisions can be used to predict both diffusivities and the ratio of gradient to self-diffusivity has been found to be ${\sim}6$ (da Cunha & Hinch Reference da Cunha and Hinch1996), similar to what has been found here.

4.6 Collective diffusivity computed using dynamic structure factor

As noted in the Introduction, Leshansky & Brady (Reference Leshansky and Brady2005) provided an elegant means of computing the collective diffusivity of a rigid sphere suspension from the simulation of a statistically homogeneous suspension in simple shear. It is based on the fact that even in a homogeneous system, stochastic fluctuations appear spontaneously and their decay can be used to compute self- as well as collective diffusivities (Morris & Brady Reference Morris and Brady1996). However, collective diffusivity intrinsically is a property of a non-homogeneous system with concentration gradients. Therefore, it will be of interest to apply any methodology for computing collective diffusivity to a non-homogeneous system. Here, we apply the method proposed by the above authors to our system, get an estimate of the collective diffusivity, qualitatively compare with the values obtained in the previous section and discuss the applicability of the results.

The dynamic structure factor approach to obtaining the collective diffusivity stemmed from the theory of dynamic light scattering, where the scattered response of a monochromatic beam of a laser from a scattering volume containing multiple scatterers (large macromolecules such as DNA, proteins, amino acids, viruses and bacteria) is measured. The scattered field fluctuates due to thermal fluctuations of the molecules and can provide structural and dynamic information about the system (Berne & Pecora Reference Berne and Pecora1976). For an infinitely dilute system of non-interacting scatterers, the autocorrelation of the fluctuation decays exponentially and the decay time is inversely proportional to the diffusivity. For concentrated systems, the fluctuating particle motions are however affected by their hydrodynamic interactions and the experimental observations require careful analysis (Altenberger & Deutch Reference Altenberger and Deutch1973; Altenberger Reference Altenberger1979; Russel & Glendinning Reference Russel and Glendinning1981) and proper interpretation, which then could furnish the collective or self-diffusivity (Rallison & Hinch Reference Rallison and Hinch1986) in appropriate limits. Leshansky & Brady (Reference Leshansky and Brady2005) carefully described the theory and applied it to shear-induced diffusion. Here, we briefly sketch the argument. Note that the analysis would lose its validity in the case of drop coalescence or break-up which are not considered here. In dynamic light scattering, the scattered light corresponding to the scattered wavenumber $\boldsymbol{k}$ (non-dimensionalized by $a$ ) from $N$ scatterers located at $\boldsymbol{x}_{\unicode[STIX]{x1D6FC}}^{\prime }(t^{\prime })$ , $\unicode[STIX]{x1D6FC}=1,2,\ldots ,N$ is proportional to the intermediate scattering function

(4.3) $$\begin{eqnarray}F(\boldsymbol{k},t^{\prime })=\frac{1}{N}\left\langle \mathop{\sum }_{\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD}=1}^{N}\text{e}^{\text{i}\boldsymbol{k}\boldsymbol{\cdot }(\boldsymbol{x}_{\unicode[STIX]{x1D6FC}}^{\prime }(t^{\prime })-\boldsymbol{x}_{\unicode[STIX]{x1D6FD}}^{\prime }(0))}\right\rangle .\end{eqnarray}$$

Note that by the property of the Dirac delta function, the number density of the scatterers (here droplets) and its spatial Fourier transform can be written as

(4.4a,b ) $$\begin{eqnarray}n(\boldsymbol{x}^{\prime },t^{\prime })=\mathop{\sum }_{\unicode[STIX]{x1D6FC}=1}^{N}\unicode[STIX]{x1D6FF}(\boldsymbol{x}^{\prime }-\boldsymbol{x}_{\unicode[STIX]{x1D6FC}}^{\prime }),\quad \hat{n}(\boldsymbol{k},t^{\prime })=\mathop{\sum }_{\unicode[STIX]{x1D6FC}=1}^{N}\text{e}^{\text{i}\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{x}_{\unicode[STIX]{x1D6FC}}^{\prime }}.\end{eqnarray}$$

Therefore, $F(\boldsymbol{k},t^{\prime })=1/N\langle \hat{n}(\boldsymbol{k},t^{\prime })\hat{n}^{\ast }(\boldsymbol{k},0)\rangle$ may be regarded as measuring the autocorrelation of the fluctuation $n^{\prime }(\boldsymbol{x}^{\prime },t^{\prime })$ (where $n(\boldsymbol{x}^{\prime },t)=n_{o}+n^{\prime }(\boldsymbol{x}^{\prime },t^{\prime })$ ) at wavenumber $\boldsymbol{k}$ for a statistically homogeneous system, as the constant background $n_{o}$ would not contribute to the autocorrelation. Our system is not homogeneous, but evolves from a non-homogeneous initial condition with a strong concentration gradient. However, one can note the following. Leshansky & Brady (Reference Leshansky and Brady2005) showed that the theory assumes a model of the number density satisfying an advection diffusion equation in a shear flow $\boldsymbol{U}+\dot{\unicode[STIX]{x1D71E}}\boldsymbol{\cdot }\boldsymbol{x}$ ( $\boldsymbol{U}$ is the average flow and $\dot{\unicode[STIX]{x1D71E}}$ is the velocity gradient tensor):

(4.5) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}n}{\unicode[STIX]{x2202}t^{\prime }}+(\boldsymbol{U}+\dot{\unicode[STIX]{x1D71E}}\boldsymbol{\cdot }\boldsymbol{x})\boldsymbol{\cdot }\unicode[STIX]{x1D735}n=\boldsymbol{D}_{c}\unicode[STIX]{x1D6FB}^{2}n.\end{eqnarray}$$

One can use the same governing equation here, but unlike the homogeneous case considered by the previous authors, the initial condition for (4.5) here is different and represents the droplets concentrated in the central layer. Therefore, whereas the concentration variation in the previous case results from the fluctuations caused by the interactions of moving drops, here we introduce an initial inhomogeneous concentration variation. But (4.5) is linear and therefore, the different wavenumber components $\hat{n}(\boldsymbol{k},t^{\prime })$ are independent and the theory remains valid. One possible concern is the value of the diffusivity obtained being dependent on the strength of the variation and, in principle, one has to calculate for different strengths and the results extrapolated to zero as was also argued by Marchioro & Acrivos (Reference Marchioro and Acrivos2001) for one of their proposed methods for the computation of collective diffusivity. (Note that (4.5) is slightly different from the one in Leshansky & Brady (Reference Leshansky and Brady2005), where a non-local Fick’s law of diffusion was used to mathematically derive the expression governing the wavenumber dependent diffusivity in the Fourier domain. Alternatively one can postulate an independent linear Fick’s law in the Fourier domain (Russel & Glendinning Reference Russel and Glendinning1981).)

In spite of the advection terms in (4.5), in a simple shear due to the orthogonality of the $\boldsymbol{k}(=k\hat{\boldsymbol{y}})$ vector to the velocity field, one obtains a simple relation for the diffusivity in the gradient direction (Leshansky & Brady Reference Leshansky and Brady2005):

(4.6) $$\begin{eqnarray}D_{c,yy}=-\frac{1}{k^{2}}\frac{\text{d}(\ln F)}{\text{d}t^{\prime }}.\end{eqnarray}$$

Note that instead of $F$ , if one uses the self-intermediate scattering function $F_{s}$ defined by

(4.7) $$\begin{eqnarray}F_{s}(\boldsymbol{k},t^{\prime })=\frac{1}{N}\left\langle \mathop{\sum }_{\unicode[STIX]{x1D6FC}=1}^{N}\text{e}^{\text{i}\boldsymbol{k}\boldsymbol{\cdot }(\boldsymbol{x}_{\unicode[STIX]{x1D6FC}}^{\prime }(t)-\boldsymbol{x}_{\unicode[STIX]{x1D6FC}}^{\prime }(0))}\right\rangle ,\end{eqnarray}$$

one obtains long-term self-diffusivity $D_{s,yy}^{\infty }$ from the same equation (4.6) (Morris & Brady Reference Morris and Brady1996). We compute $F(\boldsymbol{k},t^{\prime })$ according to (4.3) using the simulated drop evolution data positions after eliminating the initial part ( $t_{0}^{\prime }=20$ ). The resulting intermediate scattering functions are averaged over overlapping intervals to obtain a smooth time evolution curve as was also executed by Leshansky & Brady (Reference Leshansky and Brady2005). In figure 11(a), we plot $-\!\ln F(\boldsymbol{k}^{\prime },t)/k^{2}$ for $Ca=0.05$ for different wavenumbers normalizing their initial values. They show a linear growth with time and the curves tend to converge to a single curve in the small $k$ limit. Accordingly, the slope of the curves asymptotes to a value as $k\rightarrow 0$ (large length scale limit), as shown in figure 11(c) for different $Ca$ values. Figure 11(d) plots this asymptotic value of the slope, i.e. the collective diffusivity $D_{c,yy}$ (non-dimensionalized by $a$ and $\dot{\unicode[STIX]{x1D6FE}}$ ) as a function of $Ca$ . Similar to $f_{2}$ in figure 8, $D_{c,yy}$ shows a non-monotonic behaviour with $Ca$ . Due to the spatial variation and continuous decrease in the local volume fraction $\unicode[STIX]{x1D711}(y^{\prime },t^{\prime })$ with time, one cannot directly compare the two values. However, the approximate ratio $D_{c,yy}/f_{2}\sim 0.1$ can be recognized as an average $\unicode[STIX]{x1D711}\sim 0.1$ starting with a maximum local value of 0.25 (see figure 3 b).

Figure 11.

(a) The dependence of $-\!\ln F(\boldsymbol{k},t^{\prime })/k^{2}$ on time for various wavenumbers obtained from individual particle positions. (b) The same computed using the autocorrelation function of the fitted parabolic concentration profile. (c) Slope of $-\!\ln F(\boldsymbol{k},t^{\prime })/k^{2}$ as a function of the wavenumber for both the previous calculations showing similar values. (d) Slope of $-\!\ln F(\boldsymbol{k},t^{\prime })/k^{2}$ at the lowest wavenumber obtained from both calculations. The dependence on $Ca$ is similar to what was obtained for $f_{2}$ versus $Ca$ (figure 8).

As noted before, unlike in Leshansky & Brady (Reference Leshansky and Brady2005), the results here are obtained from the evolution of an initially prescribed inhomogeneity instead of spontaneously occurring fluctuations. The scaled parabolic evolution (3.2) of the imposed initial condition also explains the increasing trend of the slope of $-\!\ln F(\boldsymbol{k},t^{\prime })/k^{2}$ with $k$ in figure 11(c). To further elucidate this fact, we interrogate the parabolic concentration profile $\bar{n}(y^{\prime },t^{\prime })=3\unicode[STIX]{x1D711}(y^{\prime },t^{\prime })/4\unicode[STIX]{x03C0}a^{3}$ fitted at successive time instants from the simulated droplet positions (as in figure 3 b). We compute fast Fourier transform (FFT) of the data $\hat{\bar{n}}(k,t^{\prime })$ and find the corresponding autocorrelation $\bar{F}(k,t^{\prime })=1/N\langle \hat{\bar{n}}(k,t^{\prime })\hat{\bar{n}}^{\ast }(k,0)\rangle$ . The time evolution was averaged following a procedure identical to the one for $F(\boldsymbol{k},t^{\prime })$ . For $Ca=0.05$ in figure 11(b), $-\!\ln \bar{F}(k,t^{\prime })/k^{2}$ shows behaviour very similar to that of $-\!\ln F(\boldsymbol{k},t^{\prime })/k^{2}$ shown in figure 11(a) derived directly from the discrete particle positions. The slopes at different $k$ and $Ca$ from the fitted parabolic profiles plotted in figure 11(c) also follow the same curve. In figure 11(d), the dashed line from this procedure matches the one computed from discrete positions directly. Therefore, we emphasize that the increasing trend of the slope of $-\!\ln F(\boldsymbol{k},t^{\prime })/k^{2}$ in figure 11(c) arises from the nonlinear diffusion of the imposed initial condition. For the same reason, interpretation of the values of $-\!\ln F(\boldsymbol{k},t^{\prime })/k^{2}$ for higher $k$ as wavenumber dependent diffusivity is not possible here. Note that in the limit of $k\rightarrow \infty$ , in (4.3) due to the large variations in the phase factors for terms corresponding to $\unicode[STIX]{x1D6FC}\neq \unicode[STIX]{x1D6FD}$ results in $F(\boldsymbol{k},t^{\prime })\rightarrow F_{s}(\boldsymbol{k},t^{\prime })$ (Rallison & Hinch Reference Rallison and Hinch1986; Leshansky & Brady Reference Leshansky and Brady2005). Therefore, the slope of $-\!\ln F(\boldsymbol{k},t^{\prime })/k^{2}$ in this limit gives rise to self-diffusivity. However, with the small system size (which becomes more acute for the single summation in (4.7)), the values of self-diffusivity computed using (4.7) were too noisy to be quantitatively useful, although they were of the right magnitude (smaller by a factor of 5 compared to the gradient diffusivity). In any event, the above analysis demonstrates that the procedure for computation of dynamic structure factor and correspondingly the collective diffusivity can also be applied to inhomogeneous systems. We also note the following about our method vis-à-vis the dynamic structure factor method. In a statistically homogeneous or non-homogeneous (as in here) system, the accuracy of the results obtained from a particular analysis, to which one subjects the experimental or the simulated data (e.g. the droplet positions here) depends on the validity of the assumptions of the theory underlying the analysis. The dynamic structure factor theory assumes a linear governing equation (4.5), whereas the results obtained before based on the observed ${\sim}t^{1/3}$ scaling indicates a nonlinear diffusion (3.2) with $D_{c,yy}$ linear in $\unicode[STIX]{x1D711}$ or  $n$ . One can apply dynamic structure factor theory to simulations performed at different volume fractions, and obtain volume fraction dependence of diffusivity, as was executed by Leshansky & Brady (Reference Leshansky and Brady2005). However, the implicit assumption is that during the time of analysis the variation in the fluctuation remains sufficiently small so that the linear theory remains. Nonetheless, we argue that different analytical procedures can provide fruitful information about a system as long as their underlying assumptions are kept in mind.