Effect of Marangoni stress on the bulk rheology of a dilute emulsion of surfactant-laden deformable droplets in linear flows

In the present study, we analytically investigate the deformation and bulk rheology of a dilute emulsion of surfactant-laden droplets suspended in linear flows. We use an asymptotic approach to determine the effect of surfactant distribution on the deformation of an isolated droplet as well as the effective shear and extensional viscosity for a dilute emulsion. The nonuniform distribution of surfactants due to the bulk flow results in the generation of a Marangoni stress, which affects both the deformation as well as the bulk rheology of the emulsion. The present analysis is done for the limiting case when the surfactant transport is dominated by the surface diffusion relative to surface convection. As an example, we have used two commonly encountered bulk flows, namely, uniaxial extensional flow and simple shear flow. With the assumption of negligible inertial forces present in either of the phases, we show that both the surfactant concentration on the droplet surface as well as the ratio of viscosity of the droplet phase with respect to the suspending fluid has a significant effect on the droplet deformation as well as the bulk rheology. It is seen that increase in the nonuniformity in surfactant distribution on the droplet surface results in a higher droplet deformation and a higher effective viscosity for either of the linear flows considered. The effect of surfactant distribution on effective viscosity is insignificant for highly viscous droplets. For the case of simple shear flow, surfactant distribution is found to have no effect on the inclination angle. However, a higher viscosity ratio predicts the droplet to be more aligned toward the direction of flow. First and second normal stress differences are present for the case of a simple shear flow, of which the former is found to increase with nonuniformity in surfactant distribution, whereas the later remains unaffected.

Figure 1

Schematic of a surfactant-laden droplet of radius $a$ suspended in a linear flow. As an example, we have shown the background flow to be a simple shear flow. Both the spherical $\left(\overline{r},\theta ,\phi \right)$ as well as the Cartesian coordinates $\left(\overline{x},\overline{y},\overline{z}\right)$ are shown.

Figure 2

Schematic of a surfactant-laden droplet suspended in a uniaxial extensional flow with $\overline{z}$ being the axis of extension. Both the imposed flow field as well as non-uniform distribution of surfactants are responsible for the deformation of the droplet.

Figure 3

Variation of deformation parameter $\left(D_{\mathrm{fe}}\right)$ with Ca is shown. For (a) $\lambda =0.1$, (b) $\lambda =1$, and (c) $\lambda =10$. In each of these figures numerical data from the work done by Milliken et al. [26] is shown along with $O\left(\mathrm{Ca}\right)$ and $O\left({\mathrm{Ca}}^2\right)$ solutions obtained from our theory. The value of other parameter in these plots are $\beta =0.5$, and $k=0.1$.

Figure 4

Contour plot showing the variation of deformation parameter $\left(D_{\mathrm{fe}}\right)$ with $\beta$ and $k$. For (a) $\lambda =0.1$, while in (b) $\lambda =10.$ The values of the deformation parameter corresponding to different values of $\beta$ and $k$ are also provided in the plot above. The value of capillary number for the above plot is taken to be $\mathrm{Ca}=0.1.$

Figure 5

Deformed shape of the droplet at different orders of perturbation. The different parameters involved in this plot are $\beta =0.5,k=0.1,\lambda =1,$ and $\mathrm{Ca}=0.1$.

Figure 6

Variation of $O\left(\mathrm{Ca}\right)$ and $O\left({\mathrm{Ca}}^2\right)$ solution for (a) $L$, (b) $B$, and (c) $W$ with Ca. The circular and the square points indicate the experimental and numerical data points, respectively, as obtained from Fig. 4 of Feigl et al. [37]. The values of the other parameters are $\beta =0.8,\lambda =0.335$, and $k=1$.

Figure 7

Variation of $O\left(\mathrm{Ca}\right)$ and $O\left({\mathrm{Ca}}^2\right)$ solution for (a) $L$, (b) $B$, and (c) $W$ with Ca. The circular and the square points indicate the experimental and numerical data points, respectively, as obtained from Fig. 4 of Feigl et al. [37]. The values of the other parameters are $\beta =0.8,\lambda =0.335.$ Each of the above plots are drawn for $k=0,5$.

Figure 8

Contour plot showing the variation of deformation parameter $\left(D_{\mathrm{fl}}\right)$ with $\beta$ and $k$. For (a) $\lambda =0.1$, while in (b) $\lambda =10.$ The values of the deformation parameter corresponding to different values of $\beta$ and $k$ are labeled in the plot above. The value of capillary number for the above plot is taken to be $\mathrm{Ca}=0.1.$

Figure 9

Variation of inclination angle with $\mathrm{Ca}$. (a) A series approximation for the inclination angle is found out [Eq. (40)] and plotted against Ca. The different parameters involved are $k=10,\lambda =1$, and $\beta =0.1$. The marker points indicate numerical data taken from the work of Li and Pozrikidis [28]. (b) A more accurate approach is used to calculate the angle of inclination directly from the solution for the deformed droplet shape. The “green” square points denote the experimental data points from the work done by Feigl et al. [37]. The other parameters are $\lambda =6.338,\beta =0.5$, and $k=5.$

Figure 10

Variation of the inclination angle of the droplet with $\mathrm{Ca}$ for different values of $\lambda (=0.1,1,10)$. The other parameters values are $\beta =0.5,k=5,\lambda =1$, and $\theta =\pi /\phantom{\pi 2}2$.

Figure 11

Shape of the surfactant laden-droplet when suspended in a simple shear flow. Both the shapes due to $O\left(\mathrm{Ca}\right)$ as well as $O\left({\mathrm{Ca}}^2\right)$ corrections are shown. The different parameters used for this plot are $\beta =0.1,k=5,\lambda =1$, and $\mathrm{Ca}=0.15.$

Figure 12

Variation of normalized effective extensional viscosity with $\lambda$ for different values of $k\left(=0,2.5,5\right)$. Variation of the normalized effective extensional viscosity with $\lambda$ for a clean droplet $\left(k=0\right)$ has been shown by Ramachandran and Leal. The value of the capillary number is taken as $\mathrm{Ca}=0.1$ and $\beta =0.5.$

Figure 13

Variation of normalized effective extensional viscosity with $\lambda$ at different orders of perturbation. The value of other important parameters are $\beta =0.3,k=1$, and $\mathrm{Ca}=0.1.$

Figure 14

(a) Variation of first normal stress difference $\left(N_1\right)$ with $\mathrm{Ca}$. (b) Variation of the second normal stress difference $\left(N_2\right)$ with $\mathrm{Ca}$. The circular points in the plot indicate the numerical result as obtained by Li and Pozrikidis [28]. The different parameters involved in either of the plots are $\beta =0.1,k=10$, and $\lambda =1.$

Figure 15

(a) Variation of normalized first normal stress difference $\left(N_1/\phantom{N_1\varphi }\varphi \right)$ with $\lambda$. (b) Variation of the normalized second normal stress difference $\left(N_2/\phantom{N_2\varphi }\varphi \right)$ with $\lambda$. Each of the plots are drawn for different values of $k(=0,1,10)$. The different parameters involved in either of the plots are $\beta =0.1$ and $\mathrm{Ca}=0.1$.