3.1 Velocity field

The governing equations, in conjunction with the corresponding boundary conditions, can be solved using Lamb’s general solutions (Hetsroni & Haber Reference Hetsroni and Haber1970; Haber & Hetsroni Reference Haber and Hetsroni1972; Lamb Reference Lamb1975; Happel & Brenner Reference Happel and Brenner1981; Kim & Karrila Reference Kim and Karrila1991). To this end, we closely follow the general methodology shown by Brenner (Reference Brenner1964), Hetsroni & Haber (Reference Hetsroni and Haber1970) and Happel & Brenner (Reference Happel and Brenner1981), dealing with the application of Lamb’s solution to particle and droplet motion. We further note that Haber & Hetsroni (Reference Haber and Hetsroni1972), in a later work, generalized their method for single droplets with surfactants residing in an arbitrary far-field flow. We will also follow the framework of the latter work, in an effort to evaluate the surfactant concentration and the velocity field. The detailed procedure for deriving the velocities has been described in appendix A. Here, for the sake of completeness we briefly show a representative example for the choice of the velocity profile and discuss the general solution methodology very quickly. The fluid velocity in the outer fluid, i.e. fluid 1, takes the form (Haber & Hetsroni Reference Haber and Hetsroni1972)

(3.1a,b ) $$\begin{eqnarray}\boldsymbol{u}_{1}=\boldsymbol{V}_{\infty }-\boldsymbol{U}_{2}+\boldsymbol{v}_{1}\quad \text{and}\quad p_{1}=P_{s}^{(\infty )}+\mathop{\sum }_{n=-\infty }^{\infty }p_{n}^{(\infty )}+\mathop{\sum }_{n=1}^{\infty }p_{-n-1}^{(1)},\end{eqnarray}$$

with

(3.2) $$\begin{eqnarray}\boldsymbol{v}_{1}=\mathop{\sum }_{n=1}^{\infty }\left[\unicode[STIX]{x1D735}\times (\boldsymbol{r}\unicode[STIX]{x1D712}_{-n-1}^{(1)})+\unicode[STIX]{x1D735}\unicode[STIX]{x1D6F7}_{-n-1}^{(1)}-\frac{n-2}{2n(2n-1)}r^{2}\unicode[STIX]{x1D735}p_{-n-1}^{(1)}+\frac{n+1}{n(2n-1)}\boldsymbol{r}p_{-n-1}^{(1)}\right].\end{eqnarray}$$

In (3.1), $\boldsymbol{v}_{1}$ is the disturbance velocity in the suspending medium. Since the disturbance velocity must vanish as we move away from the droplet, we should only include the decaying components present in Lamb’s solutions. We further note that in (3.2), $\unicode[STIX]{x1D712}_{n}$ , $p_{n}$ , $\unicode[STIX]{x1D6F7}_{n}$ , etc. are solid harmonics of order ‘ $n$ ’, and the accompanying superscript (e.g. the (1) at the top right) denotes the corresponding fluid (i.e. fluid 1, 2 or 3). The harmonics $p_{n}$ are given by Hetsroni & Haber (Reference Hetsroni and Haber1970)

(3.3) $$\begin{eqnarray}p_{n}^{(1)}=r^{-n-1}\mathop{\sum }_{m=0}^{n}[A_{-n-1,m}\cos (m\unicode[STIX]{x1D719})+\hat{A}_{-n-1,m}\sin (m\unicode[STIX]{x1D719})]P_{n,m}(\cos \unicode[STIX]{x1D703}).\end{eqnarray}$$

The expressions for the other solid harmonics (i.e. $\unicode[STIX]{x1D712}_{n},\unicode[STIX]{x1D6F7}_{n}$ ) have very similar forms and have been described in appendix A. In (3.3), $P_{n,m}(\cos \unicode[STIX]{x1D703})$ is the associated Legendre polynomial of degree  $n$ and  $m$ .

Likewise, one can also express the velocities of the outer and inner drops using spherical harmonics. The details can be found in appendix A. It should be noted that for the outer drop, both growing and decaying harmonics must be considered, whereas for the inner drop, only growing harmonics have to be included, since the velocity must remain bounded there. We shall denote $\boldsymbol{u}_{\infty }=\boldsymbol{V}_{\infty }-\boldsymbol{U}_{2}$ , which signifies the free stream velocity from the outer drop reference frame. The far-field velocity ( $\boldsymbol{u}_{\infty }$ ), in general, will consist of both decaying and growing harmonics. The detailed expression has been given in appendix A. Following Haber & Hetsroni (Reference Haber and Hetsroni1972), the surfactant concentrations can also be expressed in terms of linear combination of surface harmonics in the following form:

(3.4) $$\begin{eqnarray}\tilde{\unicode[STIX]{x1D6E4}}_{ij}=\mathop{\sum }_{n=1}^{\infty }[\unicode[STIX]{x1D6E4}_{ij,n,m}\cos (m\unicode[STIX]{x1D719})+\hat{\unicode[STIX]{x1D6E4}}_{ij,n,m}\sin (m\unicode[STIX]{x1D719})]P_{n,m}(\cos \unicode[STIX]{x1D703}).\end{eqnarray}$$

One important thing to note here is that the current forms of the boundary conditions (represented by (2.22)–(2.29)) are not linear in surface harmonics, which makes the determination of the constants $A_{n}$ , $B_{n}$ , $C_{n}\ldots$  , etc. difficult. Following Brenner (Reference Brenner1964), Hetsroni & Haber (Reference Hetsroni and Haber1970) and Happel & Brenner (Reference Happel and Brenner1981), we recast the boundary conditions as well as the surfactant transport equations on both of the interfaces by performing a set of six vector operations. These furnish a total of 14 linear algebraic equations for every value of $n$ and  $m$ . The linear equations can be solved for the constants $A_{n,m}$ , $\unicode[STIX]{x1D6E4}_{ij,n,m},\ldots$  , etc., which would finally yield the fluid velocity components. For details of the operations and the corresponding linear equations, the reader should refer to appendix A.

3.2 Droplet migration velocities

The droplet migration velocities can be deduced very easily, once the velocity field is completely known. To this end, we write the force balance equation for the outer and inner drops and note that, since the whole system is essentially force-free, the net drag force acting on the drops must be zero. Therefore, we write (Happel & Brenner Reference Happel and Brenner1981)

(3.5) $$\begin{eqnarray}\displaystyle & \displaystyle -\unicode[STIX]{x1D735}(r^{3}p_{-2}^{(1)})=\mathbf{0}\quad \text{for the outer drop}, & \displaystyle\end{eqnarray}$$

(3.6) $$\begin{eqnarray}\displaystyle & \displaystyle -\unicode[STIX]{x1D735}(r^{3}p_{-2}^{(2)})=\mathbf{0}\quad \text{for the inner drop.} & \displaystyle\end{eqnarray}$$

The foregoing equations can be recast in terms of the coefficients of the solid harmonics as follows (refer to appendix A for more details):

(3.7) $$\begin{eqnarray}\displaystyle & \displaystyle A_{-2,0}=A_{-2,1}=\hat{A}_{-2,1}=0\quad \text{for the outer drop}, & \displaystyle\end{eqnarray}$$

(3.8) $$\begin{eqnarray}\displaystyle & \displaystyle E_{-2,0}=E_{-2,1}=\hat{E}_{-2,1}=0\quad \text{for the inner drop.} & \displaystyle\end{eqnarray}$$

In (3.7)–(3.8), we have neglected the effects of gravity. However, we note that gravity can be easily introduced within the present framework by simply modifying (3.5) and (3.6). The procedure to obtain the drop velocities has previously been applied in a handful of studies (Hetsroni & Haber Reference Hetsroni and Haber1970; Haber & Hetsroni Reference Haber and Hetsroni1972; Mandal et al. Reference Mandal, Bandopadhyay and Chakraborty2015), albeit for single drop systems. For compound drops, based on the boundary conditions (2.22)–(2.29) as well as the analysis of appendix A, we can easily infer that the constants $A_{-2,0},E_{2,0},\ldots$  , etc. can be expressed as follows:

(3.9) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}A_{-2,0}=\unicode[STIX]{x1D6F1}_{2,z}(\unicode[STIX]{x1D706}_{2},\unicode[STIX]{x1D706}_{3},R,\unicode[STIX]{x1D714}_{2})U_{2,z}+\unicode[STIX]{x1D6F1}_{3,z}(\unicode[STIX]{x1D706}_{2},\unicode[STIX]{x1D706}_{3},R,\unicode[STIX]{x1D714}_{2})U_{3,z}+\unicode[STIX]{x1D6F1}_{4,z}(\unicode[STIX]{x1D706}_{2},\unicode[STIX]{x1D706}_{3},R,\unicode[STIX]{x1D714}_{2}),\\ E_{2,0}=\unicode[STIX]{x1D6E5}_{2,z}(\unicode[STIX]{x1D706}_{2},\unicode[STIX]{x1D706}_{3},R,\unicode[STIX]{x1D714}_{2})U_{2,z}+\unicode[STIX]{x1D6E5}_{3,z}(\unicode[STIX]{x1D706}_{2},\unicode[STIX]{x1D706}_{3},R,\unicode[STIX]{x1D714}_{2})U_{3,z}+\unicode[STIX]{x1D6E5}_{4,z}(\unicode[STIX]{x1D706}_{2},\unicode[STIX]{x1D706}_{3},R,\unicode[STIX]{x1D714}_{2}).\end{array}\right\}\end{eqnarray}$$

In other words, based on the fact that the constants $A_{-2,0},E_{2,0},\ldots$  , etc. must be linearly dependent on the migration velocities, we can decompose them into the forms given in (3.9). Now, using the first equations of (3.7) and (3.8) along with (3.9), we can form a set of two independent linear algebraic equations involving $U_{2,z}$ and $U_{3,z}$ . These can be easily solved to determine the drop migration velocities in the $z$ direction. One can now follow the same steps to determine the velocities in the $x$ and $y$ directions as well. The functions $\unicode[STIX]{x1D6F1}_{2,z}$ , $\unicode[STIX]{x1D6F1}_{2,x}$ , etc. have very complicated algebraic expressions and hence we do not give them in this paper. The MAPLE files containing the detailed expressions are given as supplementary information available at http://dx.doi.org/10.1017/jfm.2016.497.

3.3 Interface deformation

The deformation for the outer interface can be found from the normal stress balance equation (2.25). Enforcing the spherical solid harmonics for the stresses and the spherical surface harmonics for the surfactant concentration, we deduce that the deformed shape must have the form (Hetsroni & Haber Reference Hetsroni and Haber1970)

(3.10) $$\begin{eqnarray}f_{12}=\mathop{\sum }_{n=2}^{\infty }[f_{12,n,m}\cos (m\unicode[STIX]{x1D719})+\hat{f}_{12,n,m}\sin (m\unicode[STIX]{x1D719})]P_{n,m}(\cos \unicode[STIX]{x1D703}).\end{eqnarray}$$

Similarly, for the inner interface, one can deduce the first correction to the shape of the interface by applying the normal stress balance at the inner surface, i.e. (2.29). The function $f_{23}$ also has a very similar expression to the one for $f_{12}$ , given by

(3.11) $$\begin{eqnarray}f_{23}=\mathop{\sum }_{n=2}^{\infty }[f_{23,n,m}\cos (m\unicode[STIX]{x1D719})+\hat{f}_{23,n,m}\sin (m\unicode[STIX]{x1D719})]P_{n,m}(\cos \unicode[STIX]{x1D703}).\end{eqnarray}$$

The constants $f_{12,n,m},f_{23,n,m},\ldots$  , etc. can be easily found by plugging in the stresses in the normal stress balance equations (2.25) and (2.29). It should be noted that in these equations the term involving $Ca_{S}^{-1}$ is balanced by the static pressure jump across the interface ( $P_{S}^{(\infty )}$ and $P_{S}^{(2)}$ ), while the terms involving the $n=1$ harmonics (the first harmonic) cancel out on account of the force balance equations, as mentioned in § 3.2. The detailed expressions for the foregoing constants are given in appendix A. For further details regarding the normal stress balance equations, one can refer to the works of Cox (Reference Cox1969), Hetsroni & Haber (Reference Hetsroni and Haber1970), Ramachandran & Leal (Reference Ramachandran and Leal2012) and Mandal et al. (Reference Mandal, Bandopadhyay and Chakraborty2015).

3.4 Representative examples – deformation and migration velocity

Having derived the leading-order deformation and velocity of the drops in an arbitrary flow (subject to the relevant assumptions, as mentioned in § 2), we now explicitly consider three representative cases. First, we study the drop deformation in a uniaxial extensional flow. Second, we focus on drop deformation in simple shear flows. It should be noted that both kinds of flows result in no droplet motion. Then, we derive expressions for the migration velocity of the droplets when the far-field velocity is a quadratic Poiseuille type flow. As repeatedly mentioned earlier, the two drops move with different velocities, which goes against our assumptions of concentric drops. We discuss the condition for the concentric drop assumption to remain valid in the forthcoming subsection. Later, in § 5, we validate the foregoing postulate about the drop velocities.

3.4.1 Uniaxial extensional flows

For a uniaxial extensional flow, the far-field velocity in dimensionless form is represented by (Leal Reference Leal2007)

(3.12) $$\begin{eqnarray}\boldsymbol{V}_{\infty }=\unicode[STIX]{x1D640}\boldsymbol{\cdot }\boldsymbol{x}.\end{eqnarray}$$

Here, $\unicode[STIX]{x1D640}$ is the rate-of-strain tensor, given by

(3.13) $$\begin{eqnarray}\unicode[STIX]{x1D640}=\frac{1}{2}\left[\begin{array}{@{}ccc@{}}-1 & 0 & 0\\ 0 & -1 & 0\\ 0 & 0 & 2\end{array}\right].\end{eqnarray}$$

It then follows from appendix A and the work of Hetsroni & Haber (Reference Hetsroni and Haber1970) and Haber & Hetsroni (Reference Haber and Hetsroni1972) that $\unicode[STIX]{x1D702}_{2,0}=1$ , $\unicode[STIX]{x1D702}_{1,0}=-U_{2,z}$ , $\unicode[STIX]{x1D702}_{1,1}=-U_{2,x}$ and $\hat{\unicode[STIX]{x1D702}}_{1,1}=-U_{2,y}$ . Therefore, by solving the force balance equations, (3.7) and (3.8), we deduce $\boldsymbol{U}_{2}=\boldsymbol{U}_{3}=\mathbf{0}$ . The resulting deformations can be evaluated to be of the form (refer to appendix A)

(3.14) $$\begin{eqnarray}f_{12,2,0}^{(\mathit{ext.})}=\frac{\unicode[STIX]{x1D6EC}_{0}+(\unicode[STIX]{x1D6EC}_{11}\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D6EC}_{12})\unicode[STIX]{x1D714}_{2}+\unicode[STIX]{x1D6EC}_{2}\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D714}_{2}^{2}}{\unicode[STIX]{x1D6F9}_{0}+(\unicode[STIX]{x1D6F9}_{11}\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D6F9}_{12})\unicode[STIX]{x1D714}_{2}+\unicode[STIX]{x1D6F9}_{2}\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D714}_{2}^{2}}\end{eqnarray}$$

and

(3.15) $$\begin{eqnarray}f_{23,2,0}^{(\mathit{ext.})}=\frac{\unicode[STIX]{x1D6EC}_{3}+\unicode[STIX]{x1D6EC}_{4}\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D714}_{2}}{\unicode[STIX]{x1D714}_{1}\{\unicode[STIX]{x1D6F9}_{3}+(\unicode[STIX]{x1D6F9}_{41}\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D6F9}_{42})\unicode[STIX]{x1D714}_{2}+\unicode[STIX]{x1D6F9}_{5}\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D714}_{2}^{2}\}}.\end{eqnarray}$$

All other $f_{12,n,m}$ and $f_{23,n,m}$ terms vanish. The constants $\unicode[STIX]{x1D6EC}_{0}$ , $\unicode[STIX]{x1D6EC}_{11}$ , $\unicode[STIX]{x1D6F9}_{0}$ , $\unicode[STIX]{x1D6F9}_{11}$ , $\unicode[STIX]{x1D6EC}_{3},\ldots$  , etc. appearing in (3.14) and (3.15) are described in appendix B. The foregoing expressions for the deformation explicitly show the effects of surface active agents on the inner as well as the outer drop surface. It should be noted that in (3.14) both the denominator and the numerator have quadratic terms in $\unicode[STIX]{x1D714}_{2}$ , whereas in (3.15) only a linear term involving $\unicode[STIX]{x1D714}_{2}$ is present in the numerator. Since the quadratic term in $\unicode[STIX]{x1D714}_{2}$ is multiplied by $\unicode[STIX]{x1D6FC}$ , we conclude that it is the direct effect of the presence of surfactants on the inner surface. One can also obtain the single-drop limit, by applying $\lim _{R\rightarrow 0}f_{12}$ . It can be easily verified that, after application of the foregoing limit, the deformation magnitude for the outer drop reads

(3.16) $$\begin{eqnarray}\lim _{R\rightarrow 0}f_{12}^{(\mathit{ext.})}=\frac{5}{8}\left[\frac{16+19\unicode[STIX]{x1D706}_{2}+4\unicode[STIX]{x1D714}_{2}}{5(1+\unicode[STIX]{x1D706}_{2})+\unicode[STIX]{x1D714}_{2}}\right].\end{eqnarray}$$

This matches exactly to the expression previously derived by Stone & Leal (Reference Stone and Leal1990b ) pertaining to a single drop with surfactants residing in a uniaxial extensional flow. Another interesting point to note from the present study is that, as $\unicode[STIX]{x1D6FD}\rightarrow 1$ , or equivalently $\unicode[STIX]{x1D714}_{2}\rightarrow \infty$ , i.e. when the effects of surfactants are maximum, the deformations become

(3.17a,b ) $$\begin{eqnarray}\lim _{\unicode[STIX]{x1D714}_{2}\rightarrow \infty }f_{12}^{(\mathit{ext.})}=\frac{\unicode[STIX]{x1D6EC}_{2}}{\unicode[STIX]{x1D6F9}_{2}}=\frac{5}{2}\quad \text{and}\quad \lim _{\unicode[STIX]{x1D714}_{2}\rightarrow \infty }f_{23}^{(\mathit{ext.})}=0.\end{eqnarray}$$

Equation (3.17) shows that the inner drop starts to behave like a particle when $\unicode[STIX]{x1D6FD}$ approaches 1. Following Stone & Leal (Reference Stone and Leal1990a ), we define the deformation index in the following way:

(3.18a,b ) $$\begin{eqnarray}\mathscr{D}_{O}=\frac{r_{O}(\unicode[STIX]{x1D703}=0)-r_{O}(\unicode[STIX]{x1D703}=\unicode[STIX]{x03C0}/2)}{r_{O}(\unicode[STIX]{x1D703}=0)+r_{O}(\unicode[STIX]{x1D703}=\unicode[STIX]{x03C0}/2)},\quad \mathscr{D}_{I}=\frac{r_{I}(\unicode[STIX]{x1D703}=0)-r_{I}(\unicode[STIX]{x1D703}=\unicode[STIX]{x03C0}/2)}{r_{I}(\unicode[STIX]{x1D703}=0)+r_{I}(\unicode[STIX]{x1D703}=\unicode[STIX]{x03C0}/2)}.\end{eqnarray}$$

In (3.18), $r_{O}$ and $r_{I}$ denote the outer and inner interface positions, i.e. $r_{O}=(1+Ca_{S}\,f_{12})$ and $r_{I}=R(1+Ca_{S}\,f_{23})$ .

3.4.2 Shear flows

Here, we consider a far-field velocity profile of the form

(3.19) $$\begin{eqnarray}\boldsymbol{V}_{\infty }=x\hat{\boldsymbol{e}}_{\boldsymbol{z}}.\end{eqnarray}$$

The constants in the spherical harmonics representing the far-field flow thus become $\unicode[STIX]{x1D702}_{2,1}=1/3$ , $\unicode[STIX]{x1D702}_{1,0}=-U_{2,z}$ , $\unicode[STIX]{x1D702}_{1,1}=-U_{2,x}$ , $\hat{\unicode[STIX]{x1D702}}_{1,1}=-U_{2,y}$ and $\hat{\unicode[STIX]{x1D713}}_{1,1}=-1$ . Solving the force balance equations finally yields $\boldsymbol{U}_{2}=\boldsymbol{U}_{3}=\mathbf{0}$ . The deformations also have very similar forms to those of extensional flow and have the following expressions (based on appendix A):

(3.20a,b ) $$\begin{eqnarray}f_{12,2,1}^{(\mathit{shear})}={\textstyle \frac{1}{3}}\,f_{12,2,0}^{(\mathit{ext.})}\quad \text{and}\quad f_{23,2,1}^{(\mathit{shear})}={\textstyle \frac{1}{3}}\,f_{23,2,0}^{(\mathit{ext.})}.\end{eqnarray}$$

All other constants in the expressions for the deformations vanish. It should be noted that the various limiting cases as discussed for extensional flow (i.e. $\unicode[STIX]{x1D6FD}\rightarrow 1$ , $R\rightarrow 0$ , etc.) can also be applied here, and the results are quite similar to those in (3.16) and (3.17). Therefore, for the sake of brevity, these limits have not been reproduced here. Although shear flows are fully three-dimensional in nature, following Ramachandran & Leal (Reference Ramachandran and Leal2012) we can still define the deformation for shear flows in the same way as done in (3.18).

3.4.3 Poiseuille flow

The Poiseuille flow profile in non-dimensional form in the far field is given by

(3.21) $$\begin{eqnarray}\boldsymbol{V}_{\infty }=\left(1-\frac{\unicode[STIX]{x1D70C}^{2}}{R_{0}^{2}}\right)\boldsymbol{e}_{z}.\end{eqnarray}$$

In (3.21), $\unicode[STIX]{x1D70C}$ is the distance from the principal axis or centreline of the flow and $R_{0}$ is the location where the velocity is zero. We assume that the droplet centre is located at a distance $b$ from the flow centreline, i.e. at the location of the outer drop centre, $\unicode[STIX]{x1D70C}=b$ . It then follows from appendix A that (Hetsroni & Haber Reference Hetsroni and Haber1970; Haber & Hetsroni Reference Haber and Hetsroni1972)

(3.22) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle \unicode[STIX]{x1D701}_{1,0}=-\frac{2}{5R_{0}^{2}},\quad \unicode[STIX]{x1D702}_{1,0}=1-\frac{b^{2}}{R_{0}^{2}}-U_{2,z},\quad \unicode[STIX]{x1D702}_{2,1}=-\frac{2b}{3R_{0}^{2}},\\ \displaystyle \unicode[STIX]{x1D702}_{3,0}=\frac{2}{5R_{0}^{2}},\quad \unicode[STIX]{x1D702}_{1,1}=-U_{2,x},\quad \hat{\unicode[STIX]{x1D702}}_{1,1}=-U_{2,y}.\end{array}\right\}\end{eqnarray}$$

One can therefore evaluate the resulting drop velocities by solving the coupled equations (3.7) and (3.8). These velocities are given by

(3.23) $$\begin{eqnarray}\displaystyle & \displaystyle U_{2,z}=1-\frac{b^{2}}{R_{0}^{2}}+\left(\frac{1}{R_{0}^{2}}\right)\left[\frac{2\ell _{1}\unicode[STIX]{x1D714}_{2}^{2}+\ell _{2}\unicode[STIX]{x1D714}_{2}+\ell _{3}}{-3\ell _{1}\unicode[STIX]{x1D714}_{2}^{2}-3(2\ell _{1}+\ell _{2}/2)\unicode[STIX]{x1D714}_{2}+\ell _{4}}\right],\quad \text{for the outer drop}, & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

(3.24) $$\begin{eqnarray}\displaystyle \displaystyle U_{3,z} & = & \displaystyle 1-\frac{b^{2}}{R_{0}^{2}}+\left(\frac{1}{R_{0}^{2}}\right)\left[\frac{2\ell _{1}\unicode[STIX]{x1D714}_{2}^{2}+\{\ell _{2}+\unicode[STIX]{x1D6FC}(4R^{5}-10R^{2}+6)\}\unicode[STIX]{x1D714}_{2}+\ell _{5}}{-3\ell _{1}\unicode[STIX]{x1D714}_{2}^{2}-3(2\ell _{1}+\ell _{2}/2)\unicode[STIX]{x1D714}_{2}+\ell _{4}}\right],\nonumber\\ \displaystyle & & \displaystyle \text{for the inner drop}.\end{eqnarray}$$

The expressions for the constants $\ell _{1}$ , $\ell _{2},\ldots$  , etc. are given in appendix B. We note that the relative velocities of the drops, as compared with the imposed flow velocity at the drop centre, have very similar mathematical forms to those of the interface deformation derived in the previous section. We again note that the presence of surfactant on the inner drop surface leads to a quadratic term involving $\unicode[STIX]{x1D714}_{2}$ . We define the relative velocity between the two drops as $W=|(U_{2,z}-U_{3,z})/U_{2,z}|$ . It is quite clear that, in general, the two drops move with different velocities and therefore, for the concentric drop approximation to remain valid, one must have $e\ll 1$ , $e$ being the eccentricity. However, the eccentricity in the double emulsion system would still increase at a linear rate as follows: $e(t)=(U_{3,z}-U_{2,z})t=WU_{2,z}t$ . Therefore, even with $W\ll 1$ and $U_{2,z}\sim O(1)$ , it is quite natural that the eccentricity would become large for $t\sim O(W^{-1})$ . Therefore, we can conclude that the concentric theory can be expected to furnish approximately correct results only for low values of the eccentricity ( $e$ ), or, in other words, for $t\ll W^{-1}$ . Later, we verify this claim quantitatively. Despite the limited applicability of (3.23) and (3.24), they serve two important purposes. First, these velocities can be compared with the corresponding solution from the eccentric drop configuration (see § 4), which will give us a quantitative idea about the accuracy of the concentric drop theory, even when the drops are not concentric. Second, as we shall see later, it is rather difficult to obtain closed form analytical expressions based on eccentric drop theory, in contrast to the analysis carried out in this section. Therefore, we can look into several interesting limiting cases (like the single-drop solution, $\unicode[STIX]{x1D6FD}\rightarrow 1$ , etc.) based on the expressions given above. These limits are discussed below.

The drop velocities without the effects of surfactant can be easily recovered by letting $\unicode[STIX]{x1D714}_{2}=0$ in (3.23) and (3.24). We further note that, as $\unicode[STIX]{x1D6FD}$ approaches 1, or, equivalently, as $\unicode[STIX]{x1D714}_{2}\rightarrow \infty$ , both outer and inner drop velocities tend to a constant, given by

(3.25) $$\begin{eqnarray}\lim _{\unicode[STIX]{x1D714}_{2}\rightarrow \infty }U_{2,z}=\lim _{\unicode[STIX]{x1D714}_{2}\rightarrow \infty }U_{3,z}=1-\frac{b^{2}}{R_{0}^{2}}-\frac{2}{3}\left(\frac{1}{R_{0}^{2}}\right).\end{eqnarray}$$

This is the same as the velocity of a solid particle in a Poiseuille flow, centred at the same location as that of the compound drop. Therefore, we can conclude that as $\unicode[STIX]{x1D6FD}\rightarrow 1$ , both of the drops behave like solid particles, which is in perfect agreement with the results of the previous section. The velocity of a single drop can be deduced from (3.23) by enforcing $R\rightarrow 0$ . The single-drop velocity has the following expression:

(3.26) $$\begin{eqnarray}\lim _{R\rightarrow 0}U_{2,z}=U_{2,z}^{(sd)}=\left[1-\frac{b^{2}}{R_{0}^{2}}-\frac{2\unicode[STIX]{x1D706}_{2}}{(2+3\unicode[STIX]{x1D706}_{2})R_{0}^{2}}\right]-\underbrace{\left[\frac{4\unicode[STIX]{x1D714}_{2}}{3(2+3\unicode[STIX]{x1D706}_{2})(2+3\unicode[STIX]{x1D706}_{2}+\unicode[STIX]{x1D714}_{2})R_{0}^{2}}\right]}_{\text{effect}\;\text{of}\;\text{surfactant}}.\end{eqnarray}$$

In (3.26), as clearly stated, the last term explicitly denotes the effect of surfactant on the velocity of a single drop. Since this last term is always positive, we conclude that the presence of surfactant always slows down the streamwise motion of a single drop.

3.5 Effective viscosity of a dilute emulsion of compound drops

We now aim to derive the effective viscosity of a dilute suspension of compound droplets, with surfactants, when the system is subjected to linear flows. To this end, we note that the effective viscosity (non-dimensional) of a dilute suspension of particles or drops can be represented in the following way (Barthès-Biesel Reference Barthès-Biesel2012):

(3.27) $$\begin{eqnarray}\langle \unicode[STIX]{x1D749}\rangle =-\langle p\rangle \unicode[STIX]{x1D644}+2\unicode[STIX]{x1D706}_{eff}\langle \unicode[STIX]{x1D640}\rangle .\end{eqnarray}$$

In (3.27), $\langle p\rangle$ is the volume averaged pressure, defined as (Barthès-Biesel Reference Barthès-Biesel2012) $\langle p\rangle =(\unicode[STIX]{x0394}V)^{-1}\int _{\unicode[STIX]{x0394}V}p\,\text{d}V$ , and $\langle \unicode[STIX]{x1D749}\rangle$ and $\langle \unicode[STIX]{x1D640}\rangle$ are the volume averaged stress and rate-of-strain tensors respectively, defined as $\langle \unicode[STIX]{x1D749}\rangle =(\unicode[STIX]{x0394}V)^{-1}\int _{\unicode[STIX]{x0394}V}\unicode[STIX]{x1D749}\,\text{d}V$ and $\langle \unicode[STIX]{x1D640}\rangle =(\unicode[STIX]{x0394}V)^{-1}\int _{\unicode[STIX]{x0394}V}\unicode[STIX]{x1D640}\,\text{d}V$ . In the integration, $\unicode[STIX]{x0394}V$ represents the volume of integration, which includes a sufficient number of particles or drops. It can be shown that the effective viscosity can finally be represented in the form

(3.28) $$\begin{eqnarray}\unicode[STIX]{x1D706}_{eff}=\frac{\unicode[STIX]{x1D707}_{eff}}{\unicode[STIX]{x1D707}_{1}}=1+\frac{1}{2}\unicode[STIX]{x1D72B}^{(p)}\boldsymbol{ : }\langle \unicode[STIX]{x1D640}\rangle ^{-1}.\end{eqnarray}$$

In (3.28), $\unicode[STIX]{x1D72B}^{(p)}$ is the excess contribution to the volume averaged stress from the particles or the drops and has the following expression in the present case (Barthès-Biesel Reference Barthès-Biesel2012):

(3.29) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D72B}^{(p)} & = & \displaystyle \frac{3\unicode[STIX]{x1D711}}{4\unicode[STIX]{x03C0}}(\unicode[STIX]{x0394}V)^{-1}\int _{S_{12}}\left[\frac{1}{2}\left\{(\unicode[STIX]{x1D749}_{1}\boldsymbol{\cdot }\boldsymbol{n}_{12})\boldsymbol{x}+\boldsymbol{x}(\unicode[STIX]{x1D749}_{1}\boldsymbol{\cdot }\boldsymbol{n}_{12})-\frac{2}{3}(\unicode[STIX]{x1D749}_{1}\boldsymbol{\cdot }\boldsymbol{n}_{12})\boldsymbol{\cdot }\boldsymbol{x}\unicode[STIX]{x1D644}\right\}\right.\nonumber\\ \displaystyle & & \displaystyle -\left.\{\boldsymbol{u}_{1}\boldsymbol{n}_{12}+(\boldsymbol{u}_{1}\boldsymbol{n}_{12})^{T}\}\vphantom{{\displaystyle \frac{2}{3}}}\right]\text{d}S.\end{eqnarray}$$

It should be noted that, in (3.29), $\unicode[STIX]{x1D711}$ denotes the volume fraction of the drops, while the integration is performed over the surface area of a single particle or drop, assuming the interface to remain spherical. For a detailed derivation of the above equation, one can refer to the book by Barthès-Biesel (Reference Barthès-Biesel2012). We now plug in the expressions for stresses as deduced from Lamb’s solution and perform the integration to obtain the following expression for the effective viscosity:

(3.30) $$\begin{eqnarray}\unicode[STIX]{x1D706}_{eff}=1+\unicode[STIX]{x1D711}\left(\frac{\unicode[STIX]{x1D717}_{2}\unicode[STIX]{x1D714}_{2}^{2}\unicode[STIX]{x1D6FC}+(\unicode[STIX]{x1D717}_{11}\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D717}_{12}d)\unicode[STIX]{x1D714}_{2}+\unicode[STIX]{x1D717}_{0}d}{{\textstyle \frac{2}{5}}\unicode[STIX]{x1D717}_{2}\unicode[STIX]{x1D714}_{2}^{2}\unicode[STIX]{x1D6FC}+(\unicode[STIX]{x1D717}_{31}\unicode[STIX]{x1D6FC}+{\textstyle \frac{2}{5}}\unicode[STIX]{x1D717}_{12}d)\unicode[STIX]{x1D714}_{2}+\unicode[STIX]{x1D717}_{4}d}\right).\end{eqnarray}$$

In (3.30), the expressions for the constants $\unicode[STIX]{x1D717}_{1}$ , $\unicode[STIX]{x1D717}_{2},\ldots$  , etc. are given in appendix B. It should be noted that both shear as well as extensional flows furnish the same expression for the effective viscosity, as given in (3.30). The mathematical form of the effective viscosity is very similar to that of the outer drop deformation. We note that the presence of surfactant in the inner drop leads to a quadratic term involving $\unicode[STIX]{x1D714}_{2}$ in both the numerator and the denominator of (3.30). It can be easily verified that in the limit $R\rightarrow 0$ , with $\unicode[STIX]{x1D714}_{2}=0$ , we recover the effective viscosity relation for a dilute single drop suspension (Taylor Reference Taylor1932), $\lim _{R\rightarrow 0}\unicode[STIX]{x1D706}_{eff}=(\unicode[STIX]{x1D717}_{11}/\unicode[STIX]{x1D717}_{31})=1+\unicode[STIX]{x1D711}((5/2\unicode[STIX]{x1D706}_{2}+1)/(\unicode[STIX]{x1D706}_{2}+1))$ . We further note that, in the limit $\unicode[STIX]{x1D714}_{2}\rightarrow \infty$ , or, equivalently, $\unicode[STIX]{x1D6FD}\rightarrow 1$ , we recover the Einstein viscosity formula (Leal Reference Leal2007), i.e. $\lim _{\unicode[STIX]{x1D714}_{2}\rightarrow \infty }\unicode[STIX]{x1D706}_{eff}=1+(5/2)\unicode[STIX]{x1D711}$ . Therefore, by maximizing the effects of surfactant, one can make the suspension of drops behave like a suspension of rigid particles. We have further compared our solutions with the effective viscosity derived by Stone & Leal (Reference Stone and Leal1990a ) for a compound droplet system, without the presence of surfactants ( $\unicode[STIX]{x1D714}_{2}=0$ ), and have, as expected, found an exact match between the two, as is easily verifiable.

It must be noted that the effective viscosity, as given in (3.30), suffers from a number of shortcomings (Leal Reference Leal2007; Pal Reference Pal2007). Most notably, this relation ignores the interactions between the drops, which restricts its validity to infinitely dilute solutions with $\unicode[STIX]{x1D711}\rightarrow 0$ . A number of previous researchers (Mooney Reference Mooney1951; Krieger & Dougherty Reference Krieger and Dougherty1959; Pal Reference Pal2007, Reference Pal2011) have suggested a way around this limitation, using the differential effective medium approach or DEMA (Pal Reference Pal2007). This method was devised by Bruggeman (Pal Reference Pal2007) to evaluate the effective permittivity of suspensions. Many other researchers including Pal (Reference Pal2007) and Krieger & Dougherty (Reference Krieger and Dougherty1959) have used modified versions of this approach to compute the viscosity of relatively concentrated drop suspensions. According to the DEMA, a concentrated suspension of emulsion can be conceptually constructed by successively adding small amount of droplets with an infinitesimally small volume fraction ( $\text{d}\unicode[STIX]{x1D711}$ ) to a pre-existing suspension. The effective viscosity in (3.30) is then recast as

(3.31) $$\begin{eqnarray}\unicode[STIX]{x1D707}_{eff}=1+\left[{\textstyle \frac{5}{2}}-{\textstyle \frac{3}{2}}(\unicode[STIX]{x1D608}_{1}+\unicode[STIX]{x1D608}_{2}\unicode[STIX]{x1D707}_{2}/\unicode[STIX]{x1D707}_{1})/\{\unicode[STIX]{x1D609}_{1}+\unicode[STIX]{x1D609}_{2}\unicode[STIX]{x1D707}_{2}/\unicode[STIX]{x1D707}_{1}+\unicode[STIX]{x1D609}_{3}(\unicode[STIX]{x1D707}_{2}/\unicode[STIX]{x1D707}_{1})^{2}\}\right]\unicode[STIX]{x1D711}.\end{eqnarray}$$

We further define $\unicode[STIX]{x1D706}_{32}=\unicode[STIX]{x1D706}_{3}/\unicode[STIX]{x1D706}_{2}$ and $\unicode[STIX]{x1D706}_{eff}=\unicode[STIX]{x1D707}_{eff}/\unicode[STIX]{x1D707}_{1}$ to non-dimensionalize (3.31). At the ( $n+1$ )th stage, a small increment to the volume fraction of the double emulsion in the suspension is made, while the already existing suspension (after the $n$ th stage) is considered to be the effective medium. The corresponding increment in the effective viscosity ( $\text{d}\unicode[STIX]{x1D706}_{eff}$ ) can then be calculated from (3.31), where $\unicode[STIX]{x1D707}_{1}$ is replaced with $\unicode[STIX]{x1D706}_{eff}$ (in non-dimensional form), which is now treated as a variable. The detailed expressions for the constants $\unicode[STIX]{x1D608}_{1},\unicode[STIX]{x1D608}_{2},\unicode[STIX]{x1D609}_{1},\ldots$  , etc. are given in appendix B. As per the modified methods of Krieger & Dougherty (Reference Krieger and Dougherty1959) and Pal (Reference Pal2007), the effective increase in the volume fraction of double emulsions is represented by $\text{d}\unicode[STIX]{x1D711}/(1-\unicode[STIX]{x1D711}/\unicode[STIX]{x1D711}_{max})$ , where $\unicode[STIX]{x1D711}_{max}$ is the maximum possible volume fraction of the droplets in the suspension. The above method furnishes the following differential equation for the effective viscosity:

(3.32) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle \frac{\text{d}\unicode[STIX]{x1D706}_{eff}}{\text{d}\unicode[STIX]{x1D711}}=\frac{F(\unicode[STIX]{x1D706}_{eff})}{1-\unicode[STIX]{x1D711}/\unicode[STIX]{x1D711}_{max}},\\ \displaystyle \text{with}~F(\unicode[STIX]{x1D706}_{eff})=\left[\frac{5}{2}-\frac{3}{2}\left.\left(\unicode[STIX]{x1D608}_{1}+\unicode[STIX]{x1D608}_{2}\frac{\unicode[STIX]{x1D706}_{2}}{\unicode[STIX]{x1D706}_{eff}}\right)\right/\left\{\unicode[STIX]{x1D609}_{1}+\unicode[STIX]{x1D609}_{2}\frac{\unicode[STIX]{x1D706}_{2}}{\unicode[STIX]{x1D706}_{eff}}+\unicode[STIX]{x1D609}_{3}\left(\frac{\unicode[STIX]{x1D706}_{2}}{\unicode[STIX]{x1D706}_{eff}}\right)^{2}\right\}\right].\end{array}\right\}\end{eqnarray}$$

We can integrate the above equation to evaluate the effective viscosity for any given droplet concentration, subject to the initial condition $\unicode[STIX]{x1D706}_{eff}(\unicode[STIX]{x1D711}=0)=1$ . Since it is quite difficult to find the explicit functional dependence of $\unicode[STIX]{x1D706}_{eff}$ on $\unicode[STIX]{x1D711}$ , we choose to integrate the above equation numerically. On a related note, Pal (Reference Pal2007), in his analysis, referred to this model as ‘model 4’. Later, in § 5.3.1, we compare the effective viscosity evaluated in (3.32) with the experimental results of Pal (Reference Pal2007).

4.1 Evaluation of the flow field

The stream function in the $i$ th fluid in bipolar coordinate reads (Sadhal & Oguz Reference Sadhal and Oguz1985)

(4.4) $$\begin{eqnarray}S_{i}=(\cosh \unicode[STIX]{x1D709}-\unicode[STIX]{x1D701})^{-3/2}\mathop{\sum }_{n=0}^{\infty }W_{n}^{(i)}(\unicode[STIX]{x1D709})\unicode[STIX]{x1D60A}_{n+1}^{-1/2}(\unicode[STIX]{x1D701}).\end{eqnarray}$$

In (4.4), $\unicode[STIX]{x1D60A}_{n+1}^{-1/2}(\unicode[STIX]{x1D701})$ denotes the Gegenbauer polynomial of degree $n+1$ and order $-1/2$ . The expression for $W_{n}^{(1)}(\unicode[STIX]{x1D709})$ has the following form (Sadhal & Oguz Reference Sadhal and Oguz1985):

(4.5) $$\begin{eqnarray}\displaystyle W_{n}^{(1)}(\unicode[STIX]{x1D709}) & = & \displaystyle A_{n}\exp \left\{\!\left(n-{\textstyle \frac{1}{2}}\right)\unicode[STIX]{x1D709}\right\}+B_{n}\exp \left\{\left(n+{\textstyle \frac{3}{2}}\right)\unicode[STIX]{x1D709}\right\}\nonumber\\ \displaystyle & & \displaystyle +\,C_{n}\exp \left\{-\left(n-{\textstyle \frac{1}{2}}\right)\unicode[STIX]{x1D709}\right\}+D_{n}\exp \left\{-\left(n+{\textstyle \frac{3}{2}}\right)\unicode[STIX]{x1D709}\right\}.\end{eqnarray}$$

The expressions for $W_{n}^{(2)}$ and $W_{n}^{(3)}$ have very similar forms and we have given them in appendix C. In bipolar coordinates, the relevant velocity components are  $u_{\unicode[STIX]{x1D709}}$ and  $u_{\unicode[STIX]{x1D702}}$ , and these are related to the stream function as follows (Subramanian & Balasubramaniam Reference Subramanian and Balasubramaniam2005):

(4.6a,b ) $$\begin{eqnarray}u_{\unicode[STIX]{x1D709}}=\frac{(\cosh \unicode[STIX]{x1D709}-\unicode[STIX]{x1D701})^{2}}{c^{2}\sqrt{1-\unicode[STIX]{x1D701}^{2}}}\frac{\unicode[STIX]{x2202}S}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}},\quad u_{\unicode[STIX]{x1D702}}=-\frac{(\cosh \unicode[STIX]{x1D709}-\unicode[STIX]{x1D701})^{2}}{c^{2}\sqrt{1-\unicode[STIX]{x1D701}^{2}}}\frac{\unicode[STIX]{x2202}S}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}}.\end{eqnarray}$$

Here, we look for far-field flows, for which the stream function can be expressed in terms of a series involving Gegenbauer polynomials in the following way (Sadhal & Oguz Reference Sadhal and Oguz1985):

(4.7) $$\begin{eqnarray}S_{1,\infty }=\mathop{\sum }_{n=1}^{\infty }\left[\tilde{c}_{n}\exp \left\{-\left(n-\frac{1}{2}\right)\unicode[STIX]{x1D709}\right\}+\tilde{d}_{n}\exp \left\{-\left(n-\frac{1}{2}\right)\unicode[STIX]{x1D709}\right\}\right]\unicode[STIX]{x1D60A}_{n+1}^{-1/2}(\unicode[STIX]{x1D701}).\end{eqnarray}$$

In (4.7), $S_{1,\infty }$ represents the stream function in fluid 1 far away from the droplets. We briefly discuss the procedure to convert a given far-field flow to the form (4.7), when we focus on the specific example of Poiseuille flow, in § 4.3. We now recast the boundary conditions as per the bipolar coordinate systems. The new conditions are given by Sadhal & Oguz (Reference Sadhal and Oguz1985).

On the outer drop surface:

(4.8) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}u_{\unicode[STIX]{x1D709}}^{(1)}=u_{\unicode[STIX]{x1D709}}^{(2)}=0,\\ u_{\unicode[STIX]{x1D702}}^{(1)}=u_{\unicode[STIX]{x1D702}}^{(2)},\\ \displaystyle \unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D709}\unicode[STIX]{x1D702}}^{(1)}-\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D709}\unicode[STIX]{x1D702}}^{(2)}=-\unicode[STIX]{x1D714}_{2}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6E4}_{12}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D701}}h_{1}\sqrt{1-\unicode[STIX]{x1D701}^{2}}.\end{array}\right\}\end{eqnarray}$$

On the inner drop surface:

(4.9) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}u_{\unicode[STIX]{x1D709}}^{(2)}=u_{\unicode[STIX]{x1D709}}^{(3)}=\tilde{\boldsymbol{U}}_{3}\boldsymbol{\cdot }\boldsymbol{e}_{\unicode[STIX]{x1D709}},\\ u_{\unicode[STIX]{x1D702}}^{(2)}=u_{\unicode[STIX]{x1D702}}^{(3)},\\ \displaystyle \unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D709}\unicode[STIX]{x1D702}}^{(2)}-\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D709}\unicode[STIX]{x1D702}}^{(3)}=-\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D714}_{2}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6E4}_{23}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D701}}h_{1}\sqrt{1-\unicode[STIX]{x1D701}^{2}}.\end{array}\right\}\end{eqnarray}$$

It should be noted that, in (4.8) and (4.9), $h_{1}$ is the scale factor associated with the coordinate $\unicode[STIX]{x1D702}$ and is given by $h_{1}=(\cosh (\unicode[STIX]{x1D709})-\unicode[STIX]{x1D701})/c$ . In (4.9), $\tilde{\boldsymbol{U}}_{3}$ is the relative velocity of the inner drop, given by $\tilde{\boldsymbol{U}}_{3}=(U_{3}-U_{2})\hat{\boldsymbol{e}}_{\boldsymbol{z}}$ , where $U_{2}$ and $U_{3}$ represent the respective $z$ components of the outer and inner drop velocities. We further note that, in bipolar coordinates, the surface gradient operator on a spherical surface is given by $\unicode[STIX]{x1D735}_{s}=-\hat{\boldsymbol{e}}_{\boldsymbol{\unicode[STIX]{x1D702}}}h_{1}\sqrt{1-\unicode[STIX]{x1D701}^{2}}(\unicode[STIX]{x2202}/\unicode[STIX]{x2202}\unicode[STIX]{x1D701})$ . The surfactant transport equations take the following form:

(4.10a,b ) $$\begin{eqnarray}\unicode[STIX]{x1D735}_{s}\boldsymbol{\cdot }[\boldsymbol{u}_{1}]_{S_{12}}=\unicode[STIX]{x1D6FB}_{s}^{2}\tilde{\unicode[STIX]{x1D6E4}}_{12},\quad \frac{1}{d}\unicode[STIX]{x1D735}_{s}\boldsymbol{\cdot }[\boldsymbol{u}_{3}]_{S_{23}}=\unicode[STIX]{x1D6FB}_{s}^{2}\tilde{\unicode[STIX]{x1D6E4}}_{23}.\end{eqnarray}$$

In bipolar coordinates the surface divergence of any vector $\boldsymbol{a}$ of the form $\boldsymbol{a}=a_{\unicode[STIX]{x1D709}}\hat{\boldsymbol{e}}_{\unicode[STIX]{x1D743}}+a_{\unicode[STIX]{x1D702}}\hat{\boldsymbol{e}}_{\boldsymbol{\unicode[STIX]{x1D702}}}$ has the following expression (Happel & Brenner Reference Happel and Brenner1981):

(4.11) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D735}_{s}\boldsymbol{\cdot }\boldsymbol{a} & = & \displaystyle h_{1}h_{2}h_{3}\left[\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}}\left(\frac{a_{\unicode[STIX]{x1D709}}}{h_{1}h_{3}}\right)-\sqrt{1-\unicode[STIX]{x1D701}^{2}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D701}}\left(\frac{a_{\unicode[STIX]{x1D702}}}{h_{2}h_{3}}\right)\right]\nonumber\\ \displaystyle & & \displaystyle +\,\sqrt{1-\unicode[STIX]{x1D701}^{2}}h_{1}h_{2}a_{\unicode[STIX]{x1D702}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D701}}\left(\frac{1}{h_{2}}\right)-h_{2}\frac{\unicode[STIX]{x2202}a_{\unicode[STIX]{x1D709}}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}}.\end{eqnarray}$$

In (4.11), $h_{1}$ , $h_{2}$ and $h_{3}$ are scale factors, given by Happel & Brenner (Reference Happel and Brenner1981): $h_{1}=h_{2}=(\cosh (\unicode[STIX]{x1D709})-\unicode[STIX]{x1D701})/c$ , $h_{3}=(\cosh (\unicode[STIX]{x1D709})-\unicode[STIX]{x1D701})/(c\sqrt{1-\unicode[STIX]{x1D701}^{2}})$ . The far-field conditions are given by Sadhal & Oguz (Reference Sadhal and Oguz1985):

(4.12) $$\begin{eqnarray}S_{1}\rightarrow S_{1,\infty },\quad \text{as}~\unicode[STIX]{x1D709}\rightarrow 0~\text{and}~\unicode[STIX]{x1D702}\rightarrow 0.\end{eqnarray}$$

The far-field conditions suggests that the constants $C_{n}$ and $D_{n}$ in (4.5) are simply given by $C_{n}=\tilde{c}_{n}$ , $D_{n}=\tilde{d}_{n}$ , where $\tilde{c}_{n}$ and $\tilde{d}_{n}$ are known constants. We are yet to determine the form of the surfactant distribution on the droplet interfaces. We can work it out by plugging the expressions for the surface velocities ( $\boldsymbol{u}_{3}$ , $\boldsymbol{u}_{1}$ , etc.) into the left-hand side of (4.10). Use of the expression (4.11) along with the normal velocity conditions (first of (4.8) and (4.9)) eventually furnishes that (note that $\unicode[STIX]{x1D6FB}_{s}^{2}\tilde{\unicode[STIX]{x1D6E4}}_{ij}=\unicode[STIX]{x1D735}_{s}\boldsymbol{\cdot }[\unicode[STIX]{x1D735}_{s}\tilde{\unicode[STIX]{x1D6E4}}_{ij}]$ ) $\unicode[STIX]{x2202}\tilde{\unicode[STIX]{x1D6E4}}_{12}/\unicode[STIX]{x2202}\unicode[STIX]{x1D701}$ and $\unicode[STIX]{x2202}\tilde{\unicode[STIX]{x1D6E4}}_{23}/\unicode[STIX]{x2202}\unicode[STIX]{x1D701}$ must have the following functional forms:

(4.13) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\displaystyle \frac{\unicode[STIX]{x2202}\tilde{\unicode[STIX]{x1D6E4}}_{12}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D701}}=\frac{(\cosh \unicode[STIX]{x1D709}_{1}-\unicode[STIX]{x1D701})^{-1/2}}{c^{2}(1-\unicode[STIX]{x1D701}^{2})}\mathop{\sum }_{n=0}^{\infty }Z_{n}^{(12)}\unicode[STIX]{x1D60A}_{n+1}^{-1/2}(\unicode[STIX]{x1D701}),\\ \displaystyle \frac{\unicode[STIX]{x2202}\tilde{\unicode[STIX]{x1D6E4}}_{23}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D701}}=\frac{(\cosh \unicode[STIX]{x1D709}_{2}-\unicode[STIX]{x1D701})^{-1/2}}{c^{2}(1-\unicode[STIX]{x1D701}^{2})}\mathop{\sum }_{n=0}^{\infty }Z_{n}^{(23)}\unicode[STIX]{x1D60A}_{n+1}^{-1/2}(\unicode[STIX]{x1D701})+\frac{(\cosh \unicode[STIX]{x1D709}-\unicode[STIX]{x1D701})^{-3/2}}{c^{2}(1-\unicode[STIX]{x1D701}^{2})}\mathop{\sum }_{n=0}^{\infty }\hat{Z}_{n}^{(23)}\unicode[STIX]{x1D60A}_{n+1}^{-1/2}(\unicode[STIX]{x1D701}).\end{array}\right\} & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

Therefore, the explicit surfactant distribution can be evaluated by integrating the above equations once with respect to $\unicode[STIX]{x1D701}$ . In (4.13), $Z_{n}^{(12)}$ , $Z_{n}^{(23)}$ , etc. are constants, yet to be determined. Fortunately, we note from the boundary conditions that for determining the velocity profiles, we do not need the explicit functional form of $\tilde{\unicode[STIX]{x1D6E4}}_{ij}$ , but only need its gradient. Finally, we note that the stresses in the bipolar coordinate can be expressed as follows (Rushton & Davies Reference Rushton and Davies1973):

(4.14) $$\begin{eqnarray}\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D709}\unicode[STIX]{x1D702}}^{(i)}=\frac{\unicode[STIX]{x1D706}_{i}(\cosh (\unicode[STIX]{x1D709})-\unicode[STIX]{x1D701})^{2}}{c^{3}(1-\unicode[STIX]{x1D701}^{2})^{1/2}}\left\{\begin{array}{@{}l@{}}\displaystyle (1-\unicode[STIX]{x1D701}^{2})\left[(\cosh \unicode[STIX]{x1D709}-\unicode[STIX]{x1D701})\frac{\unicode[STIX]{x2202}^{2}S_{i}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D701}^{2}}-3\frac{\unicode[STIX]{x2202}S_{i}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D701}}\right]\\ \quad \displaystyle -\left[(\cosh \unicode[STIX]{x1D709}-\unicode[STIX]{x1D701})\frac{\unicode[STIX]{x2202}^{2}S_{i}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{2}}+3\sinh \unicode[STIX]{x1D709}\frac{\unicode[STIX]{x2202}S_{i}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}}\right]\end{array}\right\}.\end{eqnarray}$$

We can now plug the expressions for velocities, stream functions, surfactant concentrations and stresses based on (4.6), (4.4), (4.13) and (4.14) into (4.8), (4.9) and (4.10), which yields 11 independent linear equations for each value of $n$ , involving the constants $A_{n}$ , $B_{n}$ , $Z_{n}^{(12)},\ldots$  , etc. To improve the readability of this paper, we have given the equations in appendix C. It should be noted that Tsemakh et al. (Reference Tsemakh, Lavrenteva and Nir2004) have previously derived similar equations pertaining to the motion of eccentric compound drop systems in the presence of soluble surfactants. Upon reviewing these equations (see appendix C), we note that they are not analytically solvable. These equations can be solved numerically by simply truncating the series after a certain number of terms. For the present study, we have considered the first 15 terms of the series.

4.2 Evaluation of droplet velocities

The streamwise migration velocities of the droplets can be determined for the eccentric configuration in much the same way as done for the concentric cases. To this end, we note from the boundary conditions that the individual constants (i.e.  $A_{1}$ , $B_{1}$ , $Z_{1}^{(12)}$ , $E_{1}$ , $F_{1},\ldots$  , etc.) can also be written as linear combinations of drop velocities. Out of the set of 15 constants, we will only focus on $A_{n}$ , $B_{n}$ , $,E_{n}$ and $F_{n}$ , since we only need these to evaluate the body forces on the droplets. We can express the foregoing constants as follows:

(4.15) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}A_{n}=\unicode[STIX]{x1D610}_{2}^{(n)}(\unicode[STIX]{x1D706}_{2},\unicode[STIX]{x1D706}_{3},e,R,\unicode[STIX]{x1D714}_{2})U_{2}+\unicode[STIX]{x1D610}_{3}^{(n)}(\unicode[STIX]{x1D706}_{2},\unicode[STIX]{x1D706}_{3},e,R,\unicode[STIX]{x1D714}_{2})U_{3}+\unicode[STIX]{x1D610}_{4}^{(n)}(\unicode[STIX]{x1D706}_{2},\unicode[STIX]{x1D706}_{3},e,R,\unicode[STIX]{x1D714}_{2}),\\ B_{n}=\unicode[STIX]{x1D611}_{2}^{(n)}(\unicode[STIX]{x1D706}_{2},\unicode[STIX]{x1D706}_{3},e,R,\unicode[STIX]{x1D714}_{2})U_{2}+\unicode[STIX]{x1D611}_{3}^{(n)}(\unicode[STIX]{x1D706}_{2},\unicode[STIX]{x1D706}_{3},e,R,\unicode[STIX]{x1D714}_{2})U_{3}+\unicode[STIX]{x1D611}_{4}^{(n)}(\unicode[STIX]{x1D706}_{2},\unicode[STIX]{x1D706}_{3},e,R,\unicode[STIX]{x1D714}_{2}),\\ E_{n}=\unicode[STIX]{x1D612}_{2}^{(n)}(\unicode[STIX]{x1D706}_{2},\unicode[STIX]{x1D706}_{3},e,R,\unicode[STIX]{x1D714}_{2})U_{2}+\unicode[STIX]{x1D612}_{3}^{(n)}(\unicode[STIX]{x1D706}_{2},\unicode[STIX]{x1D706}_{3},e,R,\unicode[STIX]{x1D714}_{2})U_{3}+\unicode[STIX]{x1D612}_{4}^{(n)}(\unicode[STIX]{x1D706}_{2},\unicode[STIX]{x1D706}_{3},e,R,\unicode[STIX]{x1D714}_{2}),\\ F_{n}=\unicode[STIX]{x1D613}_{2}^{(n)}(\unicode[STIX]{x1D706}_{2},\unicode[STIX]{x1D706}_{3},e,R,\unicode[STIX]{x1D714}_{2})U_{2}+\unicode[STIX]{x1D613}_{3}^{(n)}(\unicode[STIX]{x1D706}_{2},\unicode[STIX]{x1D706}_{3},e,R,\unicode[STIX]{x1D714}_{2})U_{3}+\unicode[STIX]{x1D613}_{4}^{(n)}(\unicode[STIX]{x1D706}_{2},\unicode[STIX]{x1D706}_{3},e,R,\unicode[STIX]{x1D714}_{2}).\end{array}\right\} & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

The velocities of the individual drops can be computed via an overall force balance. These force balance equations on the drops can be written as (Sadhal & Oguz Reference Sadhal and Oguz1985)

(4.16a,b ) $$\begin{eqnarray}\unicode[STIX]{x1D60D}_{outer}=\frac{4\sqrt{2}\unicode[STIX]{x03C0}}{c}\mathop{\sum }_{n=0}^{\infty }(A_{n}+B_{n})=0,\quad \unicode[STIX]{x1D60D}_{inner}=\frac{4\sqrt{2}\unicode[STIX]{x03C0}}{c}\mathop{\sum }_{n=0}^{\infty }(E_{n}+F_{n})=0.\end{eqnarray}$$

By enforcing (4.15) in (4.16), we can deduce two independent linear equations for $U_{2}$ and $U_{3}$ of the following form:

(4.17a,b ) $$\begin{eqnarray}\unicode[STIX]{x1D614}_{21}U_{2}+\unicode[STIX]{x1D614}_{31}U_{3}+\unicode[STIX]{x1D615}_{1}=0,\quad \unicode[STIX]{x1D614}_{22}U_{2}+\unicode[STIX]{x1D614}_{32}U_{3}+\unicode[STIX]{x1D615}_{2}=0.\end{eqnarray}$$

In (4.17), the coefficients are given by $\unicode[STIX]{x1D614}_{21}=\sum _{n=1}^{N}(\unicode[STIX]{x1D610}_{2}^{(n)}+\unicode[STIX]{x1D611}_{2}^{(n)})$ , $\unicode[STIX]{x1D614}_{22}=\sum _{n=1}^{N}(\unicode[STIX]{x1D612}_{2}^{(n)}+\unicode[STIX]{x1D613}_{2}^{(n)})$ , while $\unicode[STIX]{x1D615}_{1}=\sum _{n=1}^{N}(\unicode[STIX]{x1D610}_{4}^{(n)}+\unicode[STIX]{x1D611}_{4}^{(n)})$ , and so on. Equation (4.17) can be easily solved to determine the droplet velocities. It should be noted that, although we have not taken gravity into account, it can be incorporated into the present framework (i.e. in (4.17)) without much difficulty. The MATLAB files containing the detailed forms of the matrices involved can be found in the supplementary information.

We end our analysis of eccentric drop motion with the determination of drop velocities. One can easily infer that the solutions thus obtained can be employed to find the deformation of the drops in a conceptually straightforward manner. However, the calculations for evaluating the pressure and normal stresses are rather cumbersome and obscure due to a lack of the presence of any proper analytical solutions. We further note that it is rather conventional to evaluate the effective viscosity in linear flows, as already done in § 3.5, since the suspensions behave like Newtonian fluids (Faroughi & Huber Reference Faroughi and Huber2015), at least theoretically, when subjected to linear flows. In other kinds of flows (for example quadratic flows), the effective viscosity does not follow the framework of § 3.5. Therefore, to keep the analysis focused, we restrain ourselves from evaluating the deformation and effective viscosity for eccentric drops and concentrate only on the drop velocities as the main indicator of the flow pattern.

4.3 Representative example – Poiseuille flow

As a representative example to obtain quantitative results, we focus on axisymmetric Poiseuille far-field flow, wherein the velocity is expressed as

(4.18) $$\begin{eqnarray}\boldsymbol{u}_{1,\infty }=\left[\left(1-\frac{\unicode[STIX]{x1D70C}^{2}}{R_{0}^{2}}\right)-U_{2}\right]\hat{\boldsymbol{e}}_{z}.\end{eqnarray}$$

It should be noted that the velocity in (4.18) has been expressed with the frame of reference attached to the outer drop. Noting that (Brenner Reference Brenner1961) $u_{z}=-(1/\unicode[STIX]{x1D70C})(\unicode[STIX]{x2202}S/\unicode[STIX]{x2202}\unicode[STIX]{x1D70C})$ ( $S$ is the stream function) and using the relations in (4.2), the stream function can be expressed in terms of the bipolar coordinates as follows:

(4.19) $$\begin{eqnarray}S_{1,\infty }=(\cosh \unicode[STIX]{x1D709}-\unicode[STIX]{x1D701})^{-3/2}\left[-\frac{(1-U_{2})}{2}\frac{c^{2}(1-\unicode[STIX]{x1D701}^{2})}{(\cosh \unicode[STIX]{x1D709}-\unicode[STIX]{x1D701})^{2}}+\frac{1}{4R_{0}^{2}}\frac{c^{4}(1-\unicode[STIX]{x1D701}^{2})^{2}}{(\cosh \unicode[STIX]{x1D709}-\unicode[STIX]{x1D701})^{4}}\right],\quad \text{as}~\unicode[STIX]{x1D709},\unicode[STIX]{x1D702}\rightarrow 0.\end{eqnarray}$$

We can use the following identity to express the above stream function in terms of a series of Gegenbauer polynomials (Brenner Reference Brenner1961):

(4.20a,b ) $$\begin{eqnarray}(r_{1}^{2}-2r_{1}\unicode[STIX]{x1D701}+1)^{-1/2}=\mathop{\sum }_{n=0}^{\infty }r_{1}^{n}P_{n}(\unicode[STIX]{x1D701}),\quad |r_{1}|<1.\end{eqnarray}$$

In (4.20), $P_{n}$ is the Legendre polynomial of degree $n$ . With repeated differentiation with respect to $\unicode[STIX]{x1D701}$ and using the recursion relations for Legendre and Gegenbauer polynomials, we can deduce (using $r_{1}=e^{-\unicode[STIX]{x1D709}}$ )

(4.21) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle \frac{(1-\unicode[STIX]{x1D701}^{2})}{(\cosh \unicode[STIX]{x1D709}-\unicode[STIX]{x1D701})^{1/2}}=\mathop{\sum }_{n=0}^{\infty }\frac{n(n+1)}{\sqrt{2}}\left[\frac{\exp \{-(n-{\textstyle \frac{1}{2}})\unicode[STIX]{x1D709}\}}{n-{\textstyle \frac{1}{2}}}-\frac{\exp \{-(n+{\textstyle \frac{3}{2}})\unicode[STIX]{x1D709}\}}{n+{\textstyle \frac{3}{2}}}\right]\unicode[STIX]{x1D60A}_{n+1}^{-1/2}(\unicode[STIX]{x1D701}),\\ \displaystyle \frac{(1-\unicode[STIX]{x1D701}^{2})^{2}}{(\cosh \unicode[STIX]{x1D709}-\unicode[STIX]{x1D701})^{5/2}}=\frac{1}{3}\mathop{\sum }_{n=0}^{\infty }\frac{n(n+1)}{4\sqrt{2}}\left[\begin{array}{@{}c@{}}\displaystyle \frac{(n+2)(n+3)}{2n+3}\exp \left\{-\left(n+\frac{3}{2}\right)\unicode[STIX]{x1D709}\right\}\\ \displaystyle \frac{(n-2)(n-1)}{2n-1}\exp \left\{-\left(n-\frac{1}{2}\right)\unicode[STIX]{x1D709}\right\}\end{array}\right]\unicode[STIX]{x1D60A}_{n+1}^{-1/2}(\unicode[STIX]{x1D701}).\end{array}\right\}\end{eqnarray}$$

Comparing (4.21) and (4.7), we can easily deduce

(4.22) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle \tilde{c}_{n}=-\frac{n(n+1)c^{2}}{2\sqrt{2}}\left[\frac{1-U_{2}}{n-1/2}+\frac{c^{2}}{24R_{0}^{2}}\frac{(n-2)(n-1)}{2n-1}\right],\\ \displaystyle \tilde{d}_{n}=\frac{n(n+1)c^{2}}{2\sqrt{2}}\left[\frac{1-U_{2}}{n+3/2}+\frac{c^{2}}{24R_{0}^{2}}\frac{(n+2)(n+3)}{2n+3}\right].\end{array}\right\}\end{eqnarray}$$

Once we know $\tilde{c}_{n}$ and $\tilde{d}_{n}$ , we can follow the procedure outlined in §§ 4.1 and 4.2 to evaluate the flow field, surfactant concentration and the drop migration velocities. The detailed numerical results are given in § 5.4.

5.1 Validation

Previously, Stone & Leal (Reference Stone and Leal1990a ) derived the deformation of a double emulsion in a uniaxial extensional flow, without any surface active agents being present on the droplet interfaces. We test the correctness of our predictions, derived using Lamb’s solution, by comparing them with those of Stone & Leal (Reference Stone and Leal1990a ). It should be noted that, from (3.14) and (3.15), we can recover the deformation in the absence of surfactants (i.e. that of Stone & Leal (Reference Stone and Leal1990a )) by nullifying their effects. This can be achieved by setting $\unicode[STIX]{x1D714}_{2}=0$ in (3.14) and (3.15), which then read $f_{12,2,0}^{(\mathit{ext.})}=\unicode[STIX]{x1D6EC}_{0}/\unicode[STIX]{x1D6F9}_{0}$ and $f_{23,2,0}^{(\mathit{ext.})}=\unicode[STIX]{x1D6EC}_{3}/\unicode[STIX]{x1D714}_{1}\unicode[STIX]{x1D6F9}_{3}$ .

In figure 3, we compare our results with those of Stone & Leal (Reference Stone and Leal1990a ) and show that exact agreement is observed between the two solutions. We compare the deformations for both of the drops, for different viscosity ratios (details of the parameters are given in the figure caption). We therefore conclude that the analytical solutions derived in the present study can successfully reproduce the results for cases with no surfactants being present on the droplet interfaces.

Figure 3.

Deformation of outer $(\mathscr{D}_{O})$ and inner $(\mathscr{D}_{I})$ drops as a function of capillary number $(Ca_{s})$ in a uniaxial extensional flow disregarding the effect of surfactant on the drop interface for two different values of $\unicode[STIX]{x1D706}_{3}=0.1$ and 10. Here, we compare our result in the limiting case of $\unicode[STIX]{x1D6FD}=0$ (represented by solid and dashed lines) with the analytical result of Stone & Leal (Reference Stone and Leal1990a ) (represented by filled markers). Other parameters have the following values: $\unicode[STIX]{x1D706}_{2}=1$ and $R=0.5$ .

Figure 4.

(a) Deformation of outer $(\mathscr{D}_{O})$ inner $(\mathscr{D}_{I})$ drops as a function of capillary number $(Ca_{s})$ for three different values of $R=0.3$ , 0.5 and 0.7, while $\unicode[STIX]{x1D706}_{2}=\unicode[STIX]{x1D706}_{3}=1$ , $k=5$ , $\unicode[STIX]{x1D6FD}=0.7$ and $\unicode[STIX]{x1D6FE}=\unicode[STIX]{x1D6FC}=d=1$ . (b) Deformation of outer $(\mathscr{D}_{O})$ and inner $(\mathscr{D}_{I})$ drop as a function of capillary number $(Ca_{s})$ for three different values of $\unicode[STIX]{x1D6FD}=0$ , 0.7 and 0.9, while $\unicode[STIX]{x1D706}_{2}=\unicode[STIX]{x1D706}_{3}=1$ , $R=0.7$ , $k=5$ and $\unicode[STIX]{x1D6FE}=\unicode[STIX]{x1D6FC}=d=1$ . (c) Deformation of outer $(\mathscr{D}_{O})$ and inner $(\mathscr{D}_{I})$ drops as a function of capillary number $(Ca_{s})$ for three different values of $\unicode[STIX]{x1D706}_{2}=0.1$ , 1 and 10. (d) Shapes of the inner and outer droplets for $\unicode[STIX]{x1D706}_{2}=10$ . Other parameters are $\unicode[STIX]{x1D706}_{3}=1$ , $R=0.5$ , $k=5$ , $\unicode[STIX]{x1D6FD}=0.7$ and $\unicode[STIX]{x1D6FE}=\unicode[STIX]{x1D6FC}=d=1$ .

5.2 Deformation characteristics

In this section, we will only concentrate on deformation characteristics in linear flows. To this end, we note that equations (3.14), (3.15) and (3.20) in §§ 3.4.1 and 3.4.2 show that the deformations for shear flows and uniaxial extensional flows have the exact same form and only differ by a multiplicative constant, namely $1/3$ . Therefore, it will suffice to study the deformation characteristics for any one of these flows, since the deformation for the other one will follow the exact same pattern. Accordingly, here we will only seek the deformation for uniaxial extensional flows.

We begin our discussion with figure 4(a–d), where the deformations for both outer and inner drops have been plotted as a function of the capillary number $(Ca_{S})$ for different values of $R$ in figure 4(a), different values of $\unicode[STIX]{x1D6FD}$ in figure 4(b) and different values of $\unicode[STIX]{x1D706}_{2}$ in figure 4(c). In figure 4(d) we depict the deformed shapes of the inner and the outer drops for three different values of $Ca_{S}$ . The values of the other relevant parameters are given in the caption. We first focus on figure 4(a) and note that the deformations of the inner and the outer interfaces are actually in different phases: the outer drop deforms to prolate shape $(\mathscr{D}_{O}>0)$ while the inner drop deforms to oblate shape $(\mathscr{D}_{I}<0)$ . This is natural in a compound droplet system, as previously demonstrated in a number of studies (Stone & Leal Reference Stone and Leal1990a ; Kan et al. Reference Kan, Udaykumar, Shyy and Tran-Son-Tay1998; Qu & Wang Reference Qu and Wang2012; Chen et al. Reference Chen, Liu and Shi2013; Chen, Liu & Zhao Reference Chen, Liu and Zhao2015b ). The reason is that the directions of flow around the two interfaces are in opposite directions, as required by the viscous stress balance equation. The outer drop encounters imposed uniaxial extensional flow, while the tangential stress balance at the outer drop interface creates a recirculating biaxial extensional flow in the annular region. This biaxial flow leads to oblate deformation of the inner drop, whereas the deformation of the outer drop remains prolate. We further note that an increase in the size of the inner drop increases (i.e. as $R$ increases) the deformation of both of the drops. This variation can be explained based on the fact that, as $R$ increases, the thickness of the annular region decreases. Therefore, the flow in this annular region becomes more squeezed, which leads to higher viscous stresses on both of the interfaces. The increased viscous stresses finally lead to higher deformation.

Next, we move on to figure 4(b), where the deformations have been plotted for different values of the parameter $\unicode[STIX]{x1D6FD}$ . The most interesting fact to note from this figure is that the primary effect of the surfactant is to reduce the deformations in both of the drops. This effect is far more pronounced for the inner drop, as is clearly noted from figure 4(b). To explain the pronounced effect of the surfactant distribution in the inner drop, we recall that the flow around the inner interface, which dictates its deformation (i.e. flow in fluids 2 and 3), is influenced by the surfactants on both of the interfaces. On the other hand, the flow in fluid 1, which contributes to the deformation of the outer interface, is mostly affected by surfactants on the outer interface. Therefore, we can expect that the deformation of the inner interface will be more sensitive to the changes in surfactant distribution compared with that of the outer interface. This reduction in deformation is in sharp contrast to the case of a single drop, where the presence of surfactants in uniaxial extensional flow always leads to an increase in the deformation, as shown by Stone & Leal (Reference Stone and Leal1990b ). The decrease in deformation of the compound drop can be explained by considering the concentration distribution of the surfactant on the interfaces. We have explicitly depicted the surfactant and velocity distributions as a function of the polar angles on the inner and outer drop surfaces in figure 5, for different values of $\unicode[STIX]{x1D6FD}=0$ , 0.7 and 0.9, while values of the other relevant parameters are given in the caption. We note from figure 5(a) that the surfactants are driven by the fluid velocities at the outer drop interface $(S_{12})$ towards the droplet poles (i.e.  $\unicode[STIX]{x1D703}=0,\unicode[STIX]{x03C0}$ ) due to the imposed uniaxial extensional flow. This leads to an accumulation of surfactants near the poles of the outer drop (refer to figure 5 a), which in turn reduces the interfacial tension near the poles as compared with the interfacial tension near the equator (i.e.  $\unicode[STIX]{x1D703}=\unicode[STIX]{x03C0}/2$ ) of the interface $S_{12}$ . Thus, the uniaxial extensional flow creates a gradient in the interfacial tension from poles towards equator, which in turn drives a Marangoni flow in the same direction. Hence, the gradient in interfacial tension actually opposes the flow at the interface $S_{12}$ . This results in a net reduction of flow strength around the interface $S_{12}$ , which finally leads to lower viscous stresses and eventually lower deformation. Similar reduction in flow strength and subsequent reduction in deformation is also present for the inner drop interface $(S_{23})$ . This effect of reduced flow strength is evident directly from figure 5(b,d), where the surface velocity is clearly observed to decrease as $\unicode[STIX]{x1D6FD}$ is augmented. Since increase in $\unicode[STIX]{x1D6FD}$ arrests fluid motion, it also leads to smaller variations in surfactant concentration along the interface, as seen in figure 5(a,c). An important point to note is that the surfactants, in essence, play a twofold role to influence the deformation. On one hand, the concentration gradient opposes the externally imposed flow, which tends to reduce the viscous stresses and hence the total deformation. On the other hand, the presence of surfactants also reduces the total surface tension, which allows larger deformation to take place. We naturally conclude that the net influence of surfactant on the total deformation will depend upon whichever of these two effects is the most dominant. For example, later we show that, depending on the radius ratio $R$ , surfactants can also increase the deformation.

Figure 5.

(a) Variation of surfactant concentration $(\unicode[STIX]{x1D6E4}_{12})$ at the outer drop interface. (b) Variation of tangential velocity $(u_{s_{12}})$ at the outer drop interface. (c) Variation of surfactant concentration $(\unicode[STIX]{x1D6E4}_{23})$ at the inner drop interface. (d) Variation of tangential velocity $(u_{s_{23}})$ at the inner drop interface. Other parameters are taken as $\unicode[STIX]{x1D706}_{2}=\unicode[STIX]{x1D706}_{3}=1$ , $\unicode[STIX]{x1D6FC}=d=\unicode[STIX]{x1D6FE}=1$ , $k=5$ , $R=0.7$ and $Ca_{S}=0.1$ . As the distribution of surfactant concentration and velocity is symmetric about the $z$ axis, we have shown the results for $\unicode[STIX]{x1D703}\in [0,\unicode[STIX]{x03C0}]$ and any value of the azimuthal angle  $(\unicode[STIX]{x1D719})$ .

Figure 4(c) demonstrates that increase in the viscosity of the outer drop leads to increase in the deformation of both of the drops. In fact, in general, an increase in the viscosity ratio of any of the drops results in increased deformation, as previously pointed out by Stone & Leal (Reference Stone and Leal1990a ) (see figure 7a in their paper). The reason for such behaviour is quite simple. Increased viscosity leads to increased viscous stresses (given that the suspending fluid velocity does not change), which finally results in increased deformation. Figure 4(d) shows the explicit shapes of the inner and outer drops for three different values of $Ca_{S}=0$ , 0.05 and 0.1. This figure verifies our earlier claim that the inner drop deforms to an oblate shape, whereas the outer drop takes a prolate shape.

We next concentrate on figure 6, where the relative deformation of the outer drop is plotted as a function of $\unicode[STIX]{x1D6FD}$ , for different values of $R$ . The relative deformation here is defined as $\mathscr{D}/\mathscr{D}_{\unicode[STIX]{x1D6FD}=0}$ , i.e. the ratio between the deformation at a given value of $\unicode[STIX]{x1D6FD}$ and the deformation without the presence of surfactant. In the inset, the deformation of the inner drop is shown as a function of $\unicode[STIX]{x1D6FD}$ , while the values of the other relevant parameters are given in the caption. The most interesting point to note from figure 6 is that surfactant influences the outer drop deformation in two separate ways, depending on the value of $R$ . We note that, for lower values of $R$ , i.e. for a very small inner drop, the presence of surfactant increases the deformation, whereas for higher values of $R$ , the deformation is decreased in the presence of surfactants. However, the presence of surfactants always decreases the deformation of the inner drop, as is clearly evident from the inset. The change in deformation of the outer drop can be attributed to the quadratic terms involving $\unicode[STIX]{x1D714}_{2}$ present in the expression (3.14), which impart a non-monotonic behaviour to the deformation of the outer drop. In the case of the inner drop, however, the presence of surfactant always opposes the imposed flow (as already explained in the discussion for the previous figure), which ultimately leads to a decrease in the viscous stresses and the resulting deformation. Recalling our discussion related to figure 4(b), we conclude that, in figure 6, for $R<0.5$ , the effect of surfactant on the total surface tension becomes more dominant, which leads to an increase in deformation. On the other hand, for $R>0.5$ , the effect of surfactant on the velocity field by imposing a surface tension gradient remains dominant, which ultimately leads to a decrement in the total deformation.

Figure 6.

Normalized deformation of the outer drop $(\mathscr{D}_{O}/\mathscr{D}_{O,\unicode[STIX]{x1D6FD}=0})$ as a function of $\unicode[STIX]{x1D6FD}$ for four different values of $R=0.3$ , 0.5 and 0.7, while $\unicode[STIX]{x1D706}_{2}=\unicode[STIX]{x1D706}_{3}=1$ , $Ca_{s}=0.1$ , $\unicode[STIX]{x1D6FE}=\unicode[STIX]{x1D6FC}=d=1$ and $k=5$ . Here, we have normalized the deformation of the outer drop $(\mathscr{D}_{O})$ by the deformation of the outer drop in the absence of surfactant (represented by $\mathscr{D}_{O,\unicode[STIX]{x1D6FD}=0}$ ). The inset shows the variation of the inner drop deformation $(\mathscr{D}_{I})$ as a function of $\unicode[STIX]{x1D6FD}$ for the same parameters as for the outer drop.

5.3 Effective viscosity

We now seek the variation in the effective viscosity of a dilute suspension of double emulsions, subject to linear flows, in the presence of surface active agents. To this end, we study the relative viscosity, as obtained by numerically integrating (3.32).

5.3.1 Comparison with experiments

Following Einstein’s prediction (Leal Reference Leal2007) of the effective viscosity of a dilute suspension of rigid particles, a number of experimental studies (Faroughi & Huber Reference Faroughi and Huber2015) have been executed, which investigate the effective viscosity of suspensions of particles (Chan & Powell Reference Chan and Powell1984), bubbles (Rust & Manga Reference Rust and Manga2002) and single as well as compound drops (Pal Reference Pal1999, Reference Pal2007). These studies have revealed that, depending on the suspension concentration as well as the fluid properties, a wide range of constitutive behaviours are exhibited by the suspensions, which include Newtonian behaviour (Pal Reference Pal2007) as well as shear thinning behaviour (Chan & Powell Reference Chan and Powell1984; Pal Reference Pal2007). Pal (Reference Pal2007), among many others, has extensively studied the viscosities of double and single emulsions in shear flows. We test the accuracy of our predicted results (based on (3.32)) for effective viscosities, by comparing them with the experimental results of Pal (Reference Pal2007).

Pal (Reference Pal2007) used two sets of double emulsions for his experiments, both of the type oil-in-water-in-oil (O/W/O) emulsions. Set 1 was prepared by mixing Corena-32 and water, while set 2 was prepared by mixing petroleum oil with water. The experimental results clearly indicated that the suspension prepared from the first set exhibited shear thinning behaviour, even for low shear rates and low droplet concentration. However, the suspension prepared from set 2 exhibited Newtonian behaviour for low to moderate droplet concentration for a very wide range of imposed shear rates (see figure 6 in Pal (Reference Pal2007)’s work). Since our effective viscosity model is strictly valid for Newtonian behaviour, we will compare our results with the experimental data obtained from set 2. Here, fluid 1 is petroleum oil with $\unicode[STIX]{x1D707}_{1}=5.54~\text{mPa}~\text{s}$ , fluid 2 is water with $\unicode[STIX]{x1D707}_{2}=1~\text{mPa}~\text{s}$ and fluid 3 is again petroleum oil with $\unicode[STIX]{x1D707}_{3}=\unicode[STIX]{x1D707}_{1}$ . Moreover, Pal used surfactants of type Triton X-100 (see table 1 in Pal’s (Reference Pal2007) work) to form the emulsion. Therefore, we can expect the presence of surfactant on the droplet interfaces. However, it is rather difficult to have any accurate measure of the surface diffusivity ( $k$ , in non-dimensional form) and $\unicode[STIX]{x1D6FD}$ for this particular surfactant and fluid combination. Hence, we will treat $k$ and $\unicode[STIX]{x1D6FD}$ as fitting parameters for the comparisons. As a matter of fact, here we have kept $k$ fixed at $k=3$ and only varied $\unicode[STIX]{x1D6FD}$ . It should be recalled that $k$ is defined as follows: $k=\unicode[STIX]{x1D70E}_{12}R_{O}/D_{12}\unicode[STIX]{x1D707}_{1}$ . It should be noted that in Pal’s (Reference Pal2007) work values of $D_{12}$ , $\unicode[STIX]{x1D70E}_{12}$ and $R_{O}$ were not mentioned explicitly. However, in a separate study, Pal (Reference Pal2011) experimented with drops of size 5– $18~\unicode[STIX]{x03BC}\text{m}$ . Therefore, in the present study, we observe that by choosing $R_{O}=10~\unicode[STIX]{x03BC}\text{m}$ , $\unicode[STIX]{x1D70E}_{12}^{0}=0.5~\text{mN}~\text{m}^{-1}$ , $\unicode[STIX]{x1D6FD}=0.8$ and $\unicode[STIX]{x1D707}_{1}=5.54~\text{mPa}~\text{s}$ , for $k=3$ , we must have $D_{12}=6\times 10^{-8}~\text{m}^{2}~\text{s}^{-1}$ , which is close to usual surface diffusivity values (Valkovska & Danov Reference Valkovska and Danov2000).

Figure 7 shows the comparison between the experimental data of Pal (Reference Pal2007) and theoretical predictions based on (3.32), for $\unicode[STIX]{x1D706}_{eff}$ versus $\unicode[STIX]{x1D711}$ data, for three different values of $\unicode[STIX]{x1D6FD}=0$ , 0.5 and 0.8, while the values of all other relevant parameters are given in the caption. The experimental data were picked from figure 6 of Pal’s (Reference Pal2007) work, for imposed 0.06 Pa shear rate. We have deliberately chosen experimental data up to $\unicode[STIX]{x1D711}=0.489$ , since for higher droplet concentrations, the suspension started to exhibit shear thinning behaviour. We note from figure 7 that very good agreement is observed between the experimental data and the $\unicode[STIX]{x1D6FD}=0.8$ curve. We further note that for all values of $\unicode[STIX]{x1D6FD}$ the qualitative trend is correctly captured by the theoretical predictions, while the predicted value gets closer to the experimental data as $\unicode[STIX]{x1D6FD}$ is increased.

Figure 7.

Effective emulsion viscosity as a function of volume fraction $(\unicode[STIX]{x1D711})$ for three different values of $\unicode[STIX]{x1D6FD}=0$ , 0.5 and 0.8, while $\unicode[STIX]{x1D711}_{m}=0.584$ , $\unicode[STIX]{x1D707}_{1}=5.54\times 10^{-3}~\text{Pa}~\text{s}$ , $\unicode[STIX]{x1D707}_{2}=10^{-3}~\text{Pa}~\text{s}$ , $\unicode[STIX]{x1D707}_{3}=\unicode[STIX]{x1D707}_{1}$ , $R=0.905$ , $\unicode[STIX]{x1D6FE}=\unicode[STIX]{x1D6FC}=d=1$ and $k=3$ . Here, we compare our result with the experimental result of Pal (Reference Pal2007) (represented by filled markers).

Figure 8.

(a) Effective emulsion viscosity as a function of viscosity ratio $(\unicode[STIX]{x1D706}_{2})$ for three different values of $\unicode[STIX]{x1D6FD}=0$ , 0.5 and 0.8, while $\unicode[STIX]{x1D711}=0.5$ , $\unicode[STIX]{x1D711}_{m}=0.637$ , $\unicode[STIX]{x1D707}_{3}/\unicode[STIX]{x1D707}_{1}=0.1$ , $R=0.9$ , $\unicode[STIX]{x1D6FE}=\unicode[STIX]{x1D6FC}=d=1$ and $k=3$ . Here, we compare our result in the limiting case of $\unicode[STIX]{x1D6FD}=0$ (represented by solid line) with the analytical result of Pal (Reference Pal2007) (represented by filled markers). (b) Effective emulsion viscosity as a function of $\unicode[STIX]{x1D714}_{2}(=\unicode[STIX]{x1D6FD}k/(1-\unicode[STIX]{x1D6FD}))$ for four different values of $R=0$ , 0.2, 0.5 and 0.8, while $\unicode[STIX]{x1D711}=0.2$ , $\unicode[STIX]{x1D706}_{2}=\unicode[STIX]{x1D706}_{3}=1$ , $k=5$ and $\unicode[STIX]{x1D6FE}=\unicode[STIX]{x1D6FC}=d=1$ .

5.4 Motion and deformation in Poiseuille flow

5.4.1 Motion of eccentric drops

This section is dedicated largely to the motion of eccentric drops. Here, we present the results from the specific example of eccentric drops in Poiseuille flow, as given in § 4.3. We begin our discussion with figures 9(a) and 9(b), where variations in the normalized outer (a) and inner (b) drop velocities (normalized as $U/U_{k=0}$ , where $U_{k=0}$ represents the drop velocity in the absence of surfactants) are depicted as a function of $\unicode[STIX]{x1D6FD}$ for different values of $e$ , while the other parameters are given in the caption. We recall that the velocities for eccentric drops can be deduced by solving (4.17). For comparison, we have also plotted the drop velocities as obtained from concentric drop solutions (§ 3.4.3, (3.23) and (3.24)). We first note that good agreement is observed between the eccentric drop solutions and the concentric solutions for low values of $e(=0.1)$ . This essentially verifies the correctness of the eccentric drop solutions. There are several other interesting points to note from these figures. First, the presence of surfactant tends to decrease the migration velocities of both of the drops. This decrement is more pronounced for the inner drop, as is clearly evident from figure 9(b). The second point to note is that the primary effect of eccentricity is to suppress the influence of surfactants, since increasing $e$ increases the (normalized) drop velocities. Again, the influence of eccentricity is far more prominent on the inner drop as compared with the outer drop; noticeable change due to eccentricity occurs only when the inner and outer drop interfaces come very close to each other. This is a general trend that can be observed throughout this study. For example, we recall that the presence of surfactants alters the deformation of the inner drop (see figure 4) more prominently.

Figure 9.

Variation of the normalized velocity of (a) the outer $(U_{2}/U_{2,k=0})$ drop and (b) the inner $(U_{3}/U_{3,k=0})$ drop as a function of $\unicode[STIX]{x1D6FD}$ for different values of the eccentricity $(e)$ . Other parameters are taken as $R=0.2$ , $\unicode[STIX]{x1D706}_{2}=0.1$ , $\unicode[STIX]{x1D706}_{3}=5$ , $k=5$ , $\unicode[STIX]{x1D6FE}=\unicode[STIX]{x1D6FC}=d=1$ and $R_{0}=5$ .

Figure 10.

(a) Variation of the eccentricity $(e)$ of the inner drop with time for four different values of $\unicode[STIX]{x1D6FD}=0$ , 0.2, 0.5 and 0.9, while other parameters are taken as $R=0.5$ , $\unicode[STIX]{x1D706}_{2}=\unicode[STIX]{x1D706}_{3}=1$ , $R_{0}=5$ , and $\unicode[STIX]{x1D6FC}=d=\unicode[STIX]{x1D6FE}=1$ . (b) Variation of the eccentricity $(e)$ of the inner drop with time for three different values of the viscosity ratio $\unicode[STIX]{x1D706}_{2}=0.2$ , 1 and 5, while other parameters are taken as $R=0.5$ , $\unicode[STIX]{x1D6FD}=0.3$ , $\unicode[STIX]{x1D706}_{3}=1$ , $k=5$ , $\unicode[STIX]{x1D6FE}=\unicode[STIX]{x1D6FC}=d=1$ and $R_{0}=5$ .

Finally, figure 10(a,b) shows the evolution of eccentricity ( $e(t)$ ) with time for different values of $\unicode[STIX]{x1D6FD}$ in (a) and $\unicode[STIX]{x1D706}_{2}$ in (b), while the values of all other relevant parameters are given in the caption. Since the drops move with different velocities, their relative velocity will lead to an ever-changing eccentricity. Recalling (4.17), we note that the evolution of $e$ with time can be quantified as follows:

(5.1) $$\begin{eqnarray}\frac{\text{d}e}{\text{d}t}=[U_{3}(e(t),R,\unicode[STIX]{x1D706}_{2},\unicode[STIX]{x1D706}_{3},\unicode[STIX]{x1D6FD})-U_{2}(e(t),R,\unicode[STIX]{x1D706}_{2},\unicode[STIX]{x1D706}_{3},\unicode[STIX]{x1D6FD})].\end{eqnarray}$$

We can thus deduce $e$ as a function of time by numerically integrating (5.1) and marching forward in time, subject to the initial condition $e(t=0)=e_{0}$ . We have taken $e_{0}=0.01$ , and a time step of $\unicode[STIX]{x0394}t=0.5$ has been chosen. We have verified time-step independence of the solutions. From the numerical simulations, we can note two distinct regions in the $e$ versus $t$ curve in figure 10(a), based on the relative magnitude of its slope. First, the eccentricity increases monotonically, which signifies that the inner drop is moving towards the outer drop surface. Within this first regime the relative velocity of the inner drop becomes faster in the latter stages, as indicated by an increased slope of the curve, compared with the slopes at the initial times. Finally, when the two interfaces approach each other, the curve becomes flat (see the curve for $\unicode[STIX]{x1D6FD}=0.9$ ), indicating a decrease in the relative velocity. This essentially signifies that, as the inner interface approaches the outer one, the pressure in front of the inner drop increases, which eventually leads to slowing down of the inner drop. It should be noted that the two regimes described here are only distinctly prominent for $\unicode[STIX]{x1D6FD}=0.9$ . For other values of $\unicode[STIX]{x1D6FD}$ , the first regime is prominent, while the final regime, where the inner drop slows down, is not very clear. This is simply because the presence of surfactants usually slows down the motion of the drops. We can easily infer that this influence is strongest when $\unicode[STIX]{x1D6FD}=0.9$ . Therefore, such strong effects of surfactants, combined with the small separation between the interfaces for large values of $e$ , significantly slow down the inner drop, as is evident from the flat portion of the curve (for $\unicode[STIX]{x1D6FD}=0.9$ ). Since the effects of surfactant are not as strong for lower values of $\unicode[STIX]{x1D6FD}$ , the final retardation of the inner drop in these cases only becomes visible for even larger values of $e$ , where the interfaces come very close to each other. Therefore, for the range of $e$ shown in figure 10(a), the final regime is not very prominent for smaller values of $\unicode[STIX]{x1D6FD}$ , although the initiation of this regime can be easily noted. Nevertheless, it can be inferred that the final regime would also be prominent for smaller values of $\unicode[STIX]{x1D6FD}$ if the evolution of $e$ was tracked for longer times. Further, it can be noted that the inner drop always moves faster than the outer drop, as indicated by the positive and monotonically increasing values of $e$ . Previous studies (Sadhal & Oguz Reference Sadhal and Oguz1985) related to motion of eccentric drops have also reported the same trend. The effect of surfactant is consistent with that of figure 10. Increment in $\unicode[STIX]{x1D6FD}$ always slows down both of the drops, which ultimately results in a slower evolution of $e$ . In fact, for large values of $\unicode[STIX]{x1D6FD}$ ( $=0.9$ ), the relative motion of the inner drop is completely arrested after $e$ surpasses 0.4.

Figure 10(b) also exhibits the same trends as in figure 10(a). Quite intuitively, increment in $\unicode[STIX]{x1D706}_{2}$ , i.e. a more viscous outer drop, leads to a decrease in the overall velocities, which finally results in slower evolution of the eccentricity. In this sense, the influences of the viscosity ratio and surfactant are quite similar to each other, since they both tend to slow down the motion of the inner drop. Reviewing figures 9 and 10, we can thus infer that the presence of surfactant has a larger influence on the overall flow dynamics, as compared with the eccentricity, at least for Poiseuille flow. In fact, the relative change in droplet velocities lies well below 10 % for both of the drops when the eccentricity is varied, as is evident from figure 9.

We further note that it is possible to apply lubrication theory (Leal Reference Leal2007) for computing the velocity field and the pressure distribution in fluid 2, lying between the two interfaces, when they almost touch each other. The typical distance between the interfaces then become very small compared with the drop radius, which enables us to apply lubrication theory, using the ratio of these two distances as the gauge function. Naturally, the pressure inside this small region would become very large (as is the norm in lubrication theory (Leal Reference Leal2007)) and would eventually prohibit any further motion of the inner drop. This trend is indeed reflected in figure 10.

5.4.2 Deformation in Poiseuille flow

We note from figure 9 that the concentric drop solutions actually offer quite accurate results for low values of the eccentricity (quantified as $e<0.1$ ). Therefore, we plot the deformation in Poiseuille flow, as evaluated based on concentric drop solutions, keeping in mind that it remains valid for low eccentricity. The corresponding deformations can be evaluated based on (A 14) and (A 15), as given in appendix A. Since the expressions are quite large and algebraically complicated, we do not explicitly give them here. The MAPLE files containing the detailed expressions have been made available as supplementary information. For the sake of completeness, we mention that the only non-zero component of the deformation is $f_{ij,3,0}$ when the compound drop is positioned on the axis of the Poiseuille flow (i.e.  $b=0$ in (3.23) and (3.24)).

Figure 11.

(a) Deformation of the outer $(\mathscr{D}_{O})$ and inner $(\mathscr{D}_{I})$ drops in Poiseuille flow as a function of the capillary number $(Ca_{s})$ for three different values of $\unicode[STIX]{x1D6FD}=0$ , 0.5 and 0.9, while $\unicode[STIX]{x1D706}_{2}=\unicode[STIX]{x1D706}_{3}=1$ , $R=0.5$ , $k=5$ , $R_{0}=5$ and $\unicode[STIX]{x1D6FE}=\unicode[STIX]{x1D6FC}=d=1$ . (b) Deformation of the outer $(\mathscr{D}_{O})$ and inner $(\mathscr{D}_{I})$ drops in Poiseuille flow as a function of the capillary number $(Ca_{s})$ for three different values of $\unicode[STIX]{x1D706}_{2}=0.2$ , 1 and 5, while $\unicode[STIX]{x1D6FD}=0.5$ , $\unicode[STIX]{x1D706}_{3}=1$ , $R=0.5$ , $k=5$ , $R_{0}=5$ and $\unicode[STIX]{x1D6FE}=\unicode[STIX]{x1D6FC}=d=1$ .

Figure 11 depicts the deformation of outer as well as inner drops, as a function of capillary number, for different values of $\unicode[STIX]{x1D6FD}$ in figure 11(a) and different values of $\unicode[STIX]{x1D706}_{2}$ in figure 11(b), while the values of all other relevant parameters are given in the caption. We note from figure 11(a) that the presence of surfactants leads to a decrement in the deformation of the inner drop, while it slightly increases the deformation of the outer drop. We recall that the qualitative trend for the inner drop remains the same as that for linear flows (see figure 4 b), although the trend is altered for the outer drop. Again, we note that the effects of surfactants are more prominent for the inner drop, in keeping with the general trend of the analysis. Figure 11(b), on the other hand, shows the exact same trend as that for the extensional flows (see figure 4 c), i.e. increase in the viscosity ratio leads to an increase in the deformation for both of the drops, the change being far more pronounced for the inner drop. An important point to note from figure 11(a,b) is that the magnitude of deformation is very small, typically of the order of ${\sim}10^{-3}$ , which indicates that in Poiseuille flow, the drops remain nearly spherical for low values of the capillary number. A number of earlier studies (Hetsroni & Haber Reference Hetsroni and Haber1970) have also reached the same conclusion regarding deformation in Poiseuille flows.

To conclude this section, we note that the presence of surface active agents helps to create a surface tension gradient, which opposes the imposed ambient flow. Therefore, the presence of surfactants, in some ways, acts in a very similar way to augmented viscosity. The presence of an inner droplet also leads to increased viscous stresses inside the outer drop, which can again be thought of as an effective augmented viscosity. When the droplets are in motion, they generally show an eccentric configuration, while the effects of eccentricity only become prominent when it surpasses a critical value. The inner drop seems to be the one that is most prominently affected by the change of eccentricity. Moreover, the inner drop always moves faster as compared with the outer drop, which leads to an ever-increasing eccentricity.