3.1 Governing equations and boundary conditions

We assume that the concentration of the surfactants adsorbed on the drop interface is small so that it does not deviate significantly from the equilibrium concentration. Thus the surface tension $(\widetilde{\unicode[STIX]{x1D6FE}})$ can be related to the local surfactant concentration by the relation (Stone Reference Stone1990; Leal Reference Leal2007)

(3.1) $$\begin{eqnarray}\widetilde{\unicode[STIX]{x1D6FE}}(\widetilde{\unicode[STIX]{x1D6E4}})=\widetilde{\unicode[STIX]{x1D6FE}}_{cl}-R_{g}T\,\widetilde{\unicode[STIX]{x1D6E4}},\end{eqnarray}$$

where $\widetilde{\unicode[STIX]{x1D6FE}}_{cl}$ is the surface tension of a clean (surfactant-free) drop, $T$ stands for the absolute temperature and $R_{g}$ denotes the ideal gas constant. Here onwards we denote the variables in their dimensionless forms and drop the tilde symbol from the notations. Based on the present considerations of a steady-state process and bulk insoluble surfactant molecules, the surfactant transport is governed by the following convection–diffusion equation (Leal Reference Leal2007):

(3.2) $$\begin{eqnarray}Pe_{S}\unicode[STIX]{x1D735}_{S}\boldsymbol{\cdot }(\boldsymbol{u}_{S}\unicode[STIX]{x1D6E4})=\unicode[STIX]{x1D6FB}_{S}^{2}\unicode[STIX]{x1D6E4},\end{eqnarray}$$

where $Pe_{S}=a^{2}\,\widetilde{G}/D_{S}$ is the surface Péclet number which represents the relative importance of convection and diffusion at the drop interface. It is to be noted that although the above surfactant transport equation has the same mathematical form as the one reported previously in the absence of electrical effects (Das, Mandal & Chakraborty Reference Das, Mandal and Chakraborty2017a ; Das et al. Reference Das, Mandal, Som and Chakraborty2017b ), the electrical effects implicitly affect the surface transport through the surface velocity $(\boldsymbol{u}_{S})$ , and thus the surfactant distribution is expected to become altered due to the electrical effects.

In order to model the electrical effects, we consider the situation when both the drop and continuous phases are of a leaky dielectric nature and adopt the Taylor–Melcher leaky dielectric model (Melcher & Taylor Reference Melcher and Taylor1969; Saville Reference Saville1997; Vlahovska Reference Vlahovska2019). According to this model, the electric potentials in the drop and continuous phase $(\unicode[STIX]{x1D713}_{d},\unicode[STIX]{x1D713}_{c})$ satisfy

(3.3) $$\begin{eqnarray}\unicode[STIX]{x1D6FB}^{2}\unicode[STIX]{x1D713}_{d,c}=0.\end{eqnarray}$$

Apart from the usual boundedness of potential in and around the drop, the matching condition at the drop interface and the condition of uniformly imposed electric field at the far stream (i.e. $r\rightarrow \infty$ , $\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}_{c}=-\boldsymbol{E}_{\infty }$ ), the interfacial charge balance turns out to be an important condition for the present problem. In the quasi-steady state this reads (Xu & Homsy Reference Xu and Homsy2006)

(3.4a,b ) $$\begin{eqnarray}\text{at}~r=r_{S}(\unicode[STIX]{x1D703},\unicode[STIX]{x1D719}),\quad \boldsymbol{n}\boldsymbol{\cdot }(R\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}_{d}-\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}_{c})=-Re_{E}\unicode[STIX]{x1D735}_{S}\boldsymbol{\cdot }(q_{S}\boldsymbol{u}_{S}),\end{eqnarray}$$

where $\boldsymbol{n}$ is the unit vector in the normal direction at the drop surface and $\unicode[STIX]{x1D735}_{S}$ is the surface gradient operator $(\unicode[STIX]{x1D735}_{S}=(\boldsymbol{I}-\boldsymbol{n}\boldsymbol{n})\boldsymbol{\cdot }\unicode[STIX]{x1D735})$ . Here $Re_{E}=\unicode[STIX]{x1D716}_{c}\widetilde{G}/\unicode[STIX]{x1D70E}_{c}$ is the electrical Reynolds number quantifying the relative importance of the charge relaxation time scale and charge convection time scale. Charge convection (3.4a,b ) implies that the charge distribution and in effect the electric potential are coupled with the velocity field. Since the velocity field is affected by the surfactant non-uniformity, the electrical potential is also influenced by the surfactants, following (3.4a,b ).

Neglecting the inertial effect in the creeping flow limit (where hydrodynamic Reynolds number $Re=\unicode[STIX]{x1D70C}_{c}\widetilde{G}a^{2}/\unicode[STIX]{x1D707}_{c}\ll 1$ ), the hydrodynamics can be described by the following equations (Happel & Brenner Reference Happel and Brenner1981):

(3.5a,b ) $$\begin{eqnarray}-\unicode[STIX]{x1D735}p_{d}+\unicode[STIX]{x1D706}\unicode[STIX]{x1D6FB}^{2}\boldsymbol{u}_{d}=0,\quad \unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}_{d}=0\quad \text{and}\quad -\unicode[STIX]{x1D735}p_{c}+\unicode[STIX]{x1D6FB}^{2}\boldsymbol{u}_{c}=0,\quad \unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}_{c}=0.\end{eqnarray}$$

Similar to the electric potential, the pressure and velocity both in the drop as well as in the outer region must be bounded and they are continuous at the drop interface. Also the fluid velocity at the far stream matches with the applied background flow velocity, i.e. $r\rightarrow \infty ,\boldsymbol{u}_{c}=\boldsymbol{V}_{\infty }$ . Again in the quasi-steady state the normal velocity at the interface becomes zero, i.e. at $r=r_{S}(\unicode[STIX]{x1D703},\unicode[STIX]{x1D719})~\boldsymbol{u}_{d}\boldsymbol{\cdot }\boldsymbol{n}=\boldsymbol{u}_{c}\boldsymbol{\cdot }\boldsymbol{n}=0$ . Finally the interface stress balance can be expressed as

(3.6a,b ) $$\begin{eqnarray}\text{at}~r=r_{S}(\unicode[STIX]{x1D703},\unicode[STIX]{x1D719}),\quad \unicode[STIX]{x27E6}\boldsymbol{T}^{H}\unicode[STIX]{x27E7}+M\,\unicode[STIX]{x27E6}\boldsymbol{T}^{E}\unicode[STIX]{x27E7}=\frac{1}{Ca}(\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{n})\boldsymbol{n}+\underbrace{Ma\,[(1-\unicode[STIX]{x1D6E4})(\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{n})\boldsymbol{n}+\unicode[STIX]{x1D735}_{S}\unicode[STIX]{x1D6E4}]}_{Marangoni\,stress},\end{eqnarray}$$

where $\unicode[STIX]{x27E6}\unicode[STIX]{x1D74C}\unicode[STIX]{x27E7}$ denotes the jump of a variable $\unicode[STIX]{x1D74C}$ at the interface (e.g.  $\unicode[STIX]{x27E6}\unicode[STIX]{x1D74C}\unicode[STIX]{x27E7}=\unicode[STIX]{x1D74C}_{c}-\unicode[STIX]{x1D74C}_{d}$ ); $\boldsymbol{T}^{H,E}=\boldsymbol{n}\boldsymbol{\cdot }\unicode[STIX]{x1D749}^{H,E}$ are the viscous (superscript ‘ $H$ ’) and electrical (superscript ‘ $E$ ’) traction vectors at the interface. The Marangoni number $(Ma)$ , defined as $Ma=R_{g}T\,\widetilde{\unicode[STIX]{x1D6E4}}_{eq}/\unicode[STIX]{x1D707}_{c}\,a\,\widetilde{G}$ , as appearing in (3.6a,b ), represents the relative importance of the characteristic Marangoni stress due to surfactants and the characteristic viscous stress. Also, $Ca$ is the capillary number, denoting the ratio of the relative magnitudes of viscous and capillary stresses at the interface. Here $Ma$ can be related to $Ca$ by the relation $Ma=\unicode[STIX]{x1D6FD}/(Ca(1-\unicode[STIX]{x1D6FD}))$ , where elasticity number $\unicode[STIX]{x1D6FD}$ is defined as $\unicode[STIX]{x1D6FD}=-d(\widetilde{\unicode[STIX]{x1D6FE}}/\widetilde{\unicode[STIX]{x1D6FE}}_{c})/d(\widetilde{\unicode[STIX]{x1D6E4}}/\widetilde{\unicode[STIX]{x1D6E4}}_{eq})=R_{g}T\,\widetilde{\unicode[STIX]{x1D6E4}}_{eq}/\widetilde{\unicode[STIX]{x1D6FE}}_{c}$ . The parameter $\unicode[STIX]{x1D6FD}$ signifies the sensitivity of the surface tension to the concentration of surfactant molecules at the drop interface. For most practical purposes, $\unicode[STIX]{x1D6FD}$ varies between 0 and 0.8 (Stone Reference Stone1990; Li & Pozrikidis Reference Li and Pozrikidis1997).

3.2 Asymptotic solution

The governing equations depicting the present physical problem are coupled in such a manner that an exact solution for arbitrary values of the dimensionless parameters is not possible. To circumvent this problem, we consider a physically realistic condition for which the charge convection effect is weak in comparison to the ohmic conduction (i.e. $Re_{E}\ll 1$ ) and the drop deviates only slightly from an initial spherical shape (i.e. $Ca\ll 1$ ) (Xu & Homsy Reference Xu and Homsy2006; Mandal et al. Reference Mandal, Bandopadhyay and Chakraborty2016b ). In addition we consider that the surface Péclet number $(Pe_{S})$ is low (i.e. $Pe_{S}\ll 1$ ) (Ha & Yang Reference Ha and Yang1995; Pak, Feng & Stone Reference Pak, Feng and Stone2014; Mandal, Das & Chakraborty Reference Mandal, Das and Chakraborty2017b ), depicting the condition when the interfacial transport is mainly governed by the diffusion mechanism rather than convection. Thereafter, a dimensionless physicochemical constant, $k$ , can be defined as (Stone Reference Stone1990; Mandal, Bandopadhyay & Chakraborty Reference Mandal, Bandopadhyay and Chakraborty2016a )

(3.7) $$\begin{eqnarray}k=\frac{Pe_{S}}{Ca}=\frac{\widetilde{\unicode[STIX]{x1D6FE}}_{eq}\,a}{\unicode[STIX]{x1D707}_{c}\,D_{S}}.\end{eqnarray}$$

In this context, the physical condition of $k=0$ stands for a uniformly contaminated drop (Stone Reference Stone1990). Now for the assumed situation of $Ca\ll 1,Pe_{S}\ll 1$ , the parameter $k$ turns out to be ${\sim}O(1)$ . These considerations let us adopt a regular perturbation scheme which for any generic field variable $\unicode[STIX]{x1D712}$ reads

(3.8) $$\begin{eqnarray}\unicode[STIX]{x1D712}=\unicode[STIX]{x1D712}^{(0)}+Ca\,\unicode[STIX]{x1D712}^{(Ca)}+Re_{E}\,\unicode[STIX]{x1D712}^{(Re_{E})}+O(Ca^{2},CaRe_{E},Re_{E}^{2}).\end{eqnarray}$$

Further details regarding the perturbation scheme and the general expressions of the velocity field, electric potential, surfactant concentration and surface deformation can be found in earlier works (Bandopadhyay et al. Reference Bandopadhyay, Mandal, Kishore and Chakraborty2016; Mandal & Chakraborty Reference Mandal and Chakraborty2017b ; Poddar et al. Reference Poddar, Mandal, Bandopadhyay and Chakraborty2018) and hence are not repeated here.

3.3 Emulsion rheology

In the present physical situation, the simultaneous action of the imposed shear flow and a uniform electric field significantly alters the flow field around the drop and also modifies the drop shape. The presence of surfactant-induced Marangoni stress further complicates the scenario. Thus the effective rheology of the emulsion containing the drops is expected to deviate from the condition when the surrounding fluid is considered alone. The contribution of microlevel flow physics in the dispersed phase to the effective macroscale rheology of a dilute emulsion can be calculated by determining a volume-averaged stress of the emulsion (Schowalter et al. Reference Schowalter, Chaffey and Brenner1968; Batchelor Reference Batchelor1970; Guazzelli & Morris Reference Guazzelli and Morris2011):

(3.9) $$\begin{eqnarray}\langle \unicode[STIX]{x1D749}\rangle =-\langle p\rangle \boldsymbol{I}+2\boldsymbol{D}_{\infty }+\frac{\unicode[STIX]{x1D708}}{V_{d}}\boldsymbol{S}.\end{eqnarray}$$

Here $\boldsymbol{D}_{\infty }=(\unicode[STIX]{x1D735}\boldsymbol{V}_{\infty }+(\unicode[STIX]{x1D735}\boldsymbol{V}_{\infty })^{\text{T}})/2$ is the straining component of the externally imposed linear shear flow and $\boldsymbol{S}$ is the stresslet defined as

(3.10) $$\begin{eqnarray}\displaystyle \boldsymbol{S} & = & \displaystyle \int _{\unicode[STIX]{x1D719}=0}^{2\unicode[STIX]{x03C0}}\int _{\unicode[STIX]{x1D703}=0}^{\unicode[STIX]{x03C0}}\left[\frac{1}{2}\left((\unicode[STIX]{x1D749}_{c}^{H}\boldsymbol{\cdot }\boldsymbol{n})\boldsymbol{r}+((\unicode[STIX]{x1D749}_{c}^{H}\boldsymbol{\cdot }\boldsymbol{n})\boldsymbol{r})^{\text{T}}-\frac{2}{3}\boldsymbol{I}(\unicode[STIX]{x1D749}_{c}^{H}\boldsymbol{\cdot }\boldsymbol{n})\boldsymbol{r}\right)\right.\nonumber\\ \displaystyle & & \displaystyle -\left.(\boldsymbol{u}_{\boldsymbol{c}}\boldsymbol{n}+(\boldsymbol{u}_{\boldsymbol{c}}\boldsymbol{n})^{\text{T}})\right]r_{S}^{2}\sin (\unicode[STIX]{x1D703})\,\text{d}\unicode[STIX]{x1D703}\,\text{d}\unicode[STIX]{x1D719}.\end{eqnarray}$$

The stresslet represents the change in the total stress brought about by the velocity and stress modifications in the continuous phase by the presence of the drop phase.

The heterogeneity in the emulsion is quantified by defining an effective shear viscosity of the emulsion, the dimensionless form of which is given as (Schowalter et al. Reference Schowalter, Chaffey and Brenner1968; Bird et al. Reference Bird, Armstrong and Hassager1987)

(3.11) $$\begin{eqnarray}\unicode[STIX]{x1D702}_{eff}=\frac{\unicode[STIX]{x1D707}_{eff}}{\unicode[STIX]{x1D707}_{c}}=\langle \unicode[STIX]{x1D749}\rangle _{xy}.\end{eqnarray}$$

The isotropy of normal stress is disturbed in a sheared emulsion (Guazzelli & Morris Reference Guazzelli and Morris2011). This anisotropy is quantified by the first and second normal stress differences which are calculated as

(3.12) $$\begin{eqnarray}\text{first normal stress difference:}\quad N_{1}=\frac{\langle \widetilde{\unicode[STIX]{x1D749}}\rangle _{xx}-\langle \widetilde{\unicode[STIX]{x1D749}}\rangle _{yy}}{\unicode[STIX]{x1D707}_{c}\,\widetilde{G}}=\langle \unicode[STIX]{x1D749}\rangle _{xx}-\langle \unicode[STIX]{x1D749}\rangle _{yy}\end{eqnarray}$$

and

(3.13) $$\begin{eqnarray}\text{second normal stress difference:}\quad N_{2}=\frac{\langle \widetilde{\unicode[STIX]{x1D749}}\rangle _{yy}-\langle \widetilde{\unicode[STIX]{x1D749}}\rangle _{zz}}{\unicode[STIX]{x1D707}_{c}\,\widetilde{G}}=\langle \unicode[STIX]{x1D749}\rangle _{yy}-\langle \unicode[STIX]{x1D749}\rangle _{zz}.\end{eqnarray}$$

We proceed to calculating the electric potential, flow field and surfactant distribution in the different orders of perturbation, namely the leading order $O(1)$ , $O(Re_{E})$ and $O(Ca)$ . Next the effective emulsion stress and the normal stress difference expressions of orders $O(\unicode[STIX]{x1D708}),O(\unicode[STIX]{x1D708}Re_{E})$ and $O(\unicode[STIX]{x1D708}Ca)$ are determined by considering a dilute emulsion $(\unicode[STIX]{x1D708}\ll 1)$ where drop–drop interactions are not significant. We refer the reader to Bandopadhyay et al. (Reference Bandopadhyay, Mandal, Kishore and Chakraborty2016) and Mandal et al. (Reference Mandal, Bandopadhyay and Chakraborty2016b ) for a detailed solution strategy at different perturbation orders. The different arbitrary constants used in the definition of bulk rheology parameters are related to the growing and decaying harmonics of the velocity field as detailed in § S-1 of Supplementary_2 of the supplementary material available at https://doi.org/10.1017/jfm.2019.745.

3.4 Description of bulk rheology

The leading-order effective viscosity, obtained from the solution of velocity field, turns out to be of the form

(3.14) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D702}_{eff}^{(0)} & = & \displaystyle \underbrace{1+\unicode[STIX]{x1D708}\frac{(5\,\unicode[STIX]{x1D706}+2)}{(2\,\unicode[STIX]{x1D706}+2)}}_{\unicode[STIX]{x1D702}_{h,clean}^{(0)}}+\underbrace{\frac{3}{2}\frac{\unicode[STIX]{x1D708}k\,\unicode[STIX]{x1D6FD}}{((k-5\,\unicode[STIX]{x1D706}-5)\unicode[STIX]{x1D6FD}+5\,\unicode[STIX]{x1D706}+5)(1+\unicode[STIX]{x1D706})}}_{surfactant\,contribution\,(\unicode[STIX]{x1D702}_{h,surf}^{(0)})}\nonumber\\ \displaystyle & & \displaystyle \underbrace{-\,\frac{27M\unicode[STIX]{x1D708}\sin (2\unicode[STIX]{x1D6F7}_{t})(R-S)}{10(\unicode[STIX]{x1D706}+1)(R+2)^{2}}}_{\unicode[STIX]{x1D702}_{e,clean}^{(0)}}\times \underbrace{\frac{(1-\unicode[STIX]{x1D6FD})(1+\unicode[STIX]{x1D706})}{(k-5\unicode[STIX]{x1D706}-5)\unicode[STIX]{x1D6FD}+5\unicode[STIX]{x1D706}+5}}_{surfactant\,contribution\,(\unicode[STIX]{x1D702}_{e,surf}^{(0)})}.\end{eqnarray}$$

It is noteworthy to observe that surfactant-induced Marangoni stress affects both the hydrodynamic $(\unicode[STIX]{x1D702}_{h,surf}^{(0)})$ and electrical $(\unicode[STIX]{x1D702}_{e,surf}^{(0)})$ contributions to the leading-order effective viscosity. Moreover, the complicated mathematical expression for the influence of the electrical perturbation obviates the applicability of a trivial linear superposition of the influencing effects involved.

In a similar manner, the first and second normal stress differences are obtained as

(3.15a ) $$\begin{eqnarray}N_{1}^{(0)}=-6\,\unicode[STIX]{x1D708}A_{-3,2}^{(0)}=-\frac{27\unicode[STIX]{x1D708}M\cos (2\unicode[STIX]{x1D6F7}_{t})(R-S)}{(\unicode[STIX]{x1D706}+1)(R+2)^{2}}\times \underbrace{\frac{(1-\unicode[STIX]{x1D6FD})(1+\unicode[STIX]{x1D706})}{(k-5\unicode[STIX]{x1D706}-5)\unicode[STIX]{x1D6FD}+5(\unicode[STIX]{x1D706}+1)}}_{{\mathcal{N}}_{1,surf}^{(0)}=surfactant\,contribution}\end{eqnarray}$$

and

(3.15b ) $$\begin{eqnarray}N_{2}^{(0)}=\unicode[STIX]{x1D708}\left(3\,A_{-3,2}^{(0)}+\frac{3}{2}A_{-3,0}^{(0)}\right)=-\frac{27\unicode[STIX]{x1D708}M\sin ^{2}(\unicode[STIX]{x1D6F7}_{t})(R-S)}{(\unicode[STIX]{x1D706}+1)(R+2)^{2}}\times \underbrace{\frac{(1-\unicode[STIX]{x1D6FD})(1+\unicode[STIX]{x1D706})}{(k-5\unicode[STIX]{x1D706}-5)\unicode[STIX]{x1D6FD}+5(\unicode[STIX]{x1D706}+1)}}_{{\mathcal{N}}_{2,surf}^{(0)}=surfactant\,contribution},\end{eqnarray}$$

where both expressions contain a common surfactant contribution term:

(3.16) $$\begin{eqnarray}{\mathcal{N}}_{1,surf}^{(0)}={\mathcal{N}}_{2,surf}^{(0)}={\mathcal{N}}_{surf}^{(0)}=\frac{(1-\unicode[STIX]{x1D6FD})(1+\unicode[STIX]{x1D706})}{(k-5\unicode[STIX]{x1D706}-5)\unicode[STIX]{x1D6FD}+5(\unicode[STIX]{x1D706}+1)}.\end{eqnarray}$$

To obtain the $O(Re_{E})$ description, we consider an undeformed drop but take into account the charge convection at the drop surface. In contrast to the leading order, the interfacial charge distribution of $O(Re_{E})$ gets affected by the flow field and the surfactant effects and subsequently we get $q_{S}^{Re_{E}}=q_{S}^{Re_{E}}(\unicode[STIX]{x1D703},\unicode[STIX]{x1D719};R,S,\unicode[STIX]{x1D706},M,\unicode[STIX]{x1D6F7}_{t},\unicode[STIX]{x1D6FD},k)$ (see Supplementary_2 for the full expression of $q_{S}^{(Re_{E})}$ ).

The effective viscosity is obtained as a correction to the leading-order term, i.e. $\unicode[STIX]{x1D702}_{eff}=\unicode[STIX]{x1D702}_{eff}^{(0)}+Re_{E}\,\unicode[STIX]{x1D702}_{eff}^{(Re_{E})}$ . Unlike the leading order, the surfactant contribution parameters of order $Re_{E}$ are not equal, i.e. ${\mathcal{N}}_{1}^{(Re_{E})}\neq {\mathcal{N}}_{2}^{(Re_{E})}$ . In addition to being functions of the surfactant parameters $\unicode[STIX]{x1D6FD}$ and $k$ , they are also strongly dependent on electrical parameters, namely the tilt angle ( $\unicode[STIX]{x1D6F7}_{t}$ ), the Mason number ( $M$ ) and the electrical property ratios ( $R,S$ ).

In the $O(Ca)$ perturbation, we consider the deformable nature of the drop and disregard the charge convection effect. This is mathematically represented with the help of deformed drop radius, $r_{S}(\unicode[STIX]{x1D703},\unicode[STIX]{x1D719})=1+Caf^{(Ca)}$ , where the detailed expression of $f^{(Ca)}$ is provided in appendix B. In each of the surface deformation harmonics equations (B 2) the second term corresponds to the modification in the drop shape brought about by the non-uniform surfactant distribution. Interestingly, for the deformation modes $L_{2,0}^{(Ca)},L_{2,2}^{(Ca)}$ the surfactant has its role only if there is an applied electric field. On the other hand, the form of $\hat{L}_{2,2}^{(Ca)}$ suggests that the deformation arising from the hydrodynamic origin is itself influenced by the Marangoni effect. The final forms of the $O(Ca)$ bulk rheological parameters $(\unicode[STIX]{x1D702}_{eff}^{(Ca)},N_{1}^{(Ca)},N_{2}^{(Ca)})$ (see the supplementary material for full expressions) again suggest a strong coupling between the Marangoni stress, hydrodynamic stress and electrical Maxwell stress under the condition of a deformed drop surface.

4.1 Non-uniform surfactant distribution

Figure 2.

Effect of electric field on the surfactant concentration distribution on the drop surface $\unicode[STIX]{x1D6E4}=\unicode[STIX]{x1D6E4}^{(0)}+Ca\,\unicode[STIX]{x1D6E4}^{(Ca)}$ . (a) Only the effect of background shear flow is considered. (b–e) The electrical effects are shown by varying the tilt angle of the applied electric field as $\unicode[STIX]{x1D6F7}_{t}=0,\unicode[STIX]{x03C0}/4,\unicode[STIX]{x03C0}/2$ and $3\unicode[STIX]{x03C0}/4$ , respectively. Here the electrohydrodynamic parameters are chosen as per system-A. The other parameters are: capillary number, $Ca=0.2$ ; Mason number, $M=5$ ; elasticity parameter, $\unicode[STIX]{x1D6FD}=0.8$ ; and physicochemical constant, $k=1$ .

Figure 3.

Phase diagram for the location of the peak surfactant concentration $(\unicode[STIX]{x1D6E4}_{max})$ on the $\unicode[STIX]{x1D6F7}_{t}$ – $M$ plane. Here parameters are same as in figure 2. The points (b)–(e) on the phase diagram correspond to the specific parametric states of figures 2(b) to 2(e).

In the absence of any electric field, when only a linear shear flow is imposed, the surfactant molecules obtain a non-uniform but antisymmetric distribution $(\unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D703},\unicode[STIX]{x1D719}))$ as shown in figure 2(a). Distribution $\unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D703},\unicode[STIX]{x1D719})$ follows the straining axis of the flow which is inclined at an angle $\unicode[STIX]{x1D719}=\unicode[STIX]{x03C0}/4$ to the background flow direction (Bentley & Leal Reference Bentley and Leal1986). In that case, the surfactant molecules are observed to accumulate more towards the polar regions of this axis and a dominant depletion takes place orthogonal to these locations. However, this antisymmetric pattern changes when an electric field is applied (shown in figure 2 b–e). With the application of an external field the surfactant distribution does not follow the axisymmetric deposition along $\unicode[STIX]{x1D719}=\unicode[STIX]{x03C0}/4$ . While for the tilt angles $\unicode[STIX]{x1D6F7}_{t}=0,\unicode[STIX]{x03C0}/2$ and $3\unicode[STIX]{x03C0}/4$ this common trend is followed, a completely different characteristic of surfactant distribution emerges for $\unicode[STIX]{x1D6F7}_{t}=\unicode[STIX]{x03C0}/4$ . In this case the maximum surfactant concentration points are no longer in the plane of shear ( $x$ – $y$ plane in this case) and the maximum points shift to the $y$ – $z$ plane which is orthogonal to the plane of shear. A comprehensive picture of the critical tilt angle of the applied electric field for the maxima shift of surfactant distribution can be obtained from the phase map shown in figure 3. It shows that along with the tilt angle, the Mason number ( $M$ ), indicating the relative importance of the electrical stress in comparison to the hydrodynamic stress, also strongly influences the surfactant distribution along the interface, thereby modulating the location of $\unicode[STIX]{x1D6E4}_{max}$ . Until a critical Mason number of $M\approx 1.56$ (for this specific set of parameters) is reached the maximum lies in the $x$ – $y$ plane for any value of the tilt angle. However, beyond this critical point, there exists an intermediate regime of tilt angles, for which the maximum of surfactant concentration shifts to the $y$ – $z$ plane, while for the rest of the tilt angles the maximum appears in the plane of shear ( $x$ – $y$ plane). The range of this band of intermediate tilt angles widens gradually with increasing Mason number. It has been found that surfactant characterization parameters $\unicode[STIX]{x1D6FD}$ and $k$ have negligible effect on the maxima shift. This can be further confirmed by observing figure S-1(a,c), which shows the dependence of the maxima value on both $\unicode[STIX]{x1D6FD}$ and $k$ , although the angular positions of the peaks remain unaltered.

Undulations in the local surfactant concentration trigger corresponding changes in the interfacial tension $\unicode[STIX]{x1D6FE}(\unicode[STIX]{x1D703},\unicode[STIX]{x1D719})$ and gradients in surface tension $(|\unicode[STIX]{x1D6FE}_{max}-\unicode[STIX]{x1D6FE}_{min}|)$ along the drop surface. This influence of Mason number $(M)$ on the surface tension is described in figure S-2 for the configuration of a vertically applied electric field. It shows that the symmetric deviation of the maximum and minimum surface tension about the equilibrium condition, in the case of simple shear flow, is not maintained when an electric field is applied. Also, the gradients in surface tension $(|\unicode[STIX]{x1D6FE}_{max}-\unicode[STIX]{x1D6FE}_{min}|)$ become progressively higher as the influence of electrical stresses increases relative to their hydrodynamic counterparts.

The above characteristics of $\unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D703},\unicode[STIX]{x1D719})$ under different conditions become clear by consideration of (A 2). It shows that different modes in $\unicode[STIX]{x1D6E4}^{(Ca)}$ contain the information of various aspects governing the flow physics. While $\unicode[STIX]{x1D6E4}_{2,0}^{(Ca)}$ and $\unicode[STIX]{x1D6E4}_{2,2}^{(Ca)}$ are only present when an external electric field is applied, the harmonic $\hat{\unicode[STIX]{x1D6E4}}_{2,2}^{(Ca)}$ has a non-zero component arising solely from hydrodynamic origin. Again, the surfactant harmonics are strongly dependent on the tilt angle of the applied electric field. Harmonic $\unicode[STIX]{x1D6E4}_{2,0}^{(Ca)}$ appears for any configuration of the electric field, but $\unicode[STIX]{x1D6E4}_{2,2}^{(Ca)}$ vanishes completely when the tilt angle becomes $\unicode[STIX]{x1D6F7}_{t}=\unicode[STIX]{x03C0}/4$ or $3\unicode[STIX]{x03C0}/4$ . Similarly, the electric component of $\hat{\unicode[STIX]{x1D6E4}}_{2,2}^{(Ca)}$ becomes zero when the electric field is applied in either horizontal $(\unicode[STIX]{x1D6F7}_{t}=0)$ or vertical $(\unicode[STIX]{x1D6F7}_{t}=\unicode[STIX]{x03C0}/2)$ configuration. In order to investigate the physics behind the coupled behaviour of imposed linear shear and electric field, we appeal to the flow characteristics around the drop originating from the individual effects, as identified in earlier studies (Vlahovska Reference Vlahovska2011; Mandal & Chakraborty Reference Mandal and Chakraborty2017b ). The linear shear flow can be decomposed as a combination of a pure straining and pure rotational flow. On the other hand, the externally applied tilted electric field can give rise to either a uniaxial (for $R>S$ ) or biaxial (for $R<S$ ) straining flow around the drop. Hence, under the simultaneous action of shear flow and electrically induced flow, the surfactant molecules assume diverse redistribution patterns, the final form of which is determined by a combined interfacial convection and diffusive transport.

4.2 Leading-order bulk rheology

Figure 4.

Leading-order normalized effective shear viscosity $\bar{\unicode[STIX]{x1D702}}_{eff}=(\unicode[STIX]{x1D702}_{eff}^{(0)}-1)/\unicode[STIX]{x1D708}$ versus elasticity parameter for different physicochemical constants, $k$ . The electrohydrodynamic parameters correspond to (a) system-A, (b) system-B and (c) system-C. A Mason number of $M=2$ is chosen.

In figure 4(a–c), the contribution of non-uniform surfactant distribution on the drop surface, in modulating the leading-order normalized effective shear viscosity $(\bar{\unicode[STIX]{x1D702}}_{eff}^{(0)}=(\unicode[STIX]{x1D702}_{eff}^{(0)}-1)/\unicode[STIX]{x1D708})$ , is described. It is observed from figure 4(a,b) that an increase in either of the parameters $\unicode[STIX]{x1D6FD}$ or $k$ causes an enhancement in the effective viscosity. In contrast, figure 4(c) depicts that the surfactant characterization parameters have a completely reverse effect on $\bar{\unicode[STIX]{x1D702}}_{eff}$ . To understand the effects of the surfactant characterization parameters $\unicode[STIX]{x1D6FD}$ and $k$ , we appeal to figure S-1 (see Supplementary_2 of the supplementary material) where the variations in surfactant concentration and surface tension are shown. The surface elasticity number, $\unicode[STIX]{x1D6FD}$ , signifies the sensitivity of the surface tension to the local surfactant concentration. From figure S-1(a,b), we observe that an increase in $\unicode[STIX]{x1D6FD}$ results in an enhanced gradient in surface tension $(|\unicode[STIX]{x1D6FE}_{max}-\unicode[STIX]{x1D6FE}_{min}|)$ , but the deviations in the surfactant concentration from the equilibrium condition $(|\unicode[STIX]{x1D6E4}_{max}-\unicode[STIX]{x1D6E4}_{min}|)$ become suppressed with increasing $\unicode[STIX]{x1D6FD}$ . This behaviour is consistent with the results of Li & Pozrikidis (Reference Li and Pozrikidis1997), where surfactant effects were analysed for the case of a drop in simple shear flow. Greater magnitude of surface tension gradient leads to an enhancement in Marangoni convection which in turn opposes the convective transport of the surfactant. As a result, the distribution of $\unicode[STIX]{x1D6E4}$ approaches equilibrium. An increase in $k$ , on the other hand, signifies a corresponding increase in the convective transport mechanism of the surfactants (see (3.7)). Thus, for a fixed $\unicode[STIX]{x1D6FD}$ , when $k$ is increased, the convective transport of surfactants dominates over the effect due to increased surface tension gradient, resulting in greater undulations in both $\unicode[STIX]{x1D6E4}$ and $\unicode[STIX]{x1D6FE}$ from the equilibrium condition as portrayed in supplementary figures S-1(c) and S-1(d), respectively. The Marangoni effect due to non-uniformity in surfactant distribution modifies the balance between the tangential component of the hydrodynamic and electrical Maxwell stresses at the interface (see (3.6a,b )). In order to visualize the effect of surfactant coating on the drop surface, in figure 5 we demonstrate the flow fields for some representative cases. This figure shows that the orientation of the vorticity poles for system-B (figure 5 a,b) is vertical while it becomes horizontal for system-C (figure 5 c,d). The surfactants create further alterations in both the velocity magnitude and the pattern of the disturbance flow around the drop. This subsequently causes an enhancement or reduction in the viscous dissipation of the bulk mixture and finally a corresponding change in the effective viscosity results.

Figure 5.

Comparison of leading-order flow fields in and around a drop for clean and surfactant-coated cases. The streamlines are shown in the plane of shear and with respect to a body-fitted reference frame. The colourbars show the variation in velocity magnitude, while the green circle in each panel denotes the drop–matrix interface. (a,b) System-B, $M=5$ ; (c,d) system-C, $M=5$ . The parameters are chosen as $\unicode[STIX]{x1D6F7}_{t}=\unicode[STIX]{x03C0}/4,\unicode[STIX]{x1D6FD}=0.8$ and $k=1$ .

The dependence of the normalized effective shear viscosity $(\bar{\unicode[STIX]{x1D702}}_{eff}=(\unicode[STIX]{x1D702}_{eff}^{(0)}-1)/\unicode[STIX]{x1D708})$ on the shear rate is shown in figure 6(a,b) for two different combinations of electrohydrodynamic parameters. First, we define a relative hydrodynamic shear rate with respect to the electrical stress as $\bar{G}=1/M=\unicode[STIX]{x1D707}_{c}\widetilde{G}/\unicode[STIX]{x1D716}_{c}\widetilde{E}_{\infty }^{2}$ . It was previously reported (Mandal & Chakraborty Reference Mandal and Chakraborty2017b ) that in the leading order, the effective viscosity becomes a function of relative shear rate $\bar{G}$ only when the electric field is tilted with respect to the direction of the applied shear flow, while the emulsion shows a Newtonian behaviour when the electric field is applied along either the direction of shear flow or its gradient. A careful inspection of these figures reveals that the non-uniformity surfactant distribution has a potential in altering the non-Newtonian shear rate dependency of the effective shear rate. It is quite interesting to find from figure 6(a) that for a tilt angle of external electric field $\unicode[STIX]{x1D6F7}_{t}=\unicode[STIX]{x03C0}/4$ , there exists a crossover point at $\bar{G}\sim 0.2$ before which an increase in $\unicode[STIX]{x1D6FD}$ causes the effective shear viscosity to decrease in comparison to an uncontaminated drop, while beyond this point a corresponding increase in $\bar{\unicode[STIX]{x1D702}}_{eff}$ results with similar changes in $\unicode[STIX]{x1D6FD}$ . However, for $\unicode[STIX]{x1D6F7}_{t}=3\unicode[STIX]{x03C0}/4$ , $\bar{\unicode[STIX]{x1D702}}_{eff}$ shows a steady enhancement with $\unicode[STIX]{x1D6FD}$ . In this case, for a fixed relative shear rate, the variation in $\unicode[STIX]{x1D6FD}$ can lead to a change in effective shear viscosity to such an extent that it becomes greater than the viscosity of the continuous medium $\bar{\unicode[STIX]{x1D702}}_{eff}>0$ , or $\unicode[STIX]{x1D707}_{eff}>\unicode[STIX]{x1D707}_{c}$ , becoming increased from a condition of $\bar{\unicode[STIX]{x1D702}}_{eff}<0$ , or $\unicode[STIX]{x1D707}_{eff}<\unicode[STIX]{x1D707}_{c}$ , for a clean drop. The role of the electric field tilt angle is reversed for system-B, as shown in figure 6(b).

In the electrohydrodynamics literature, the flow behaviour is often shown to be of opposite nature when the electrical property combinations give rise to a positive ( $R>S$ ) or negative ( $R<S$ ) polarity (Taylor Reference Taylor1966; Tsukada et al. Reference Tsukada, Katayama, Ito and Hozawa1993; Salipante & Vlahovska Reference Salipante and Vlahovska2010; Mandal & Chakraborty Reference Mandal and Chakraborty2017b ). Based on the polarity, the direction of the electrical traction at the interface is altered. However, comparing the present results in figure 4 for different electrohydrodynamic systems, it is observed that the consequence of the Marangoni effect cannot be simply categorized based on the condition of $R>S$ or $R<S$ . Although system-A and system-B have properties $R<S$ and $R>S$ , respectively, they exhibit a similar trend of Marangoni effect, while both system-A and system-C show opposite behaviour despite having the same polarity. Along similar lines, in figure 6(a,b), the Marangoni effect is observed to be dependent on the electric field tilt angle ( $\unicode[STIX]{x1D6F7}_{t}$ ) as well. Inspired by such observations, we analyse the expression of the leading-order effective shear viscosity $\unicode[STIX]{x1D702}_{eff}^{(0)}$ in (3.14). We obtain a unique discriminating factor in terms of the crossover relative shear rate $\bar{G}_{cr}$ given as

(4.1) $$\begin{eqnarray}\bar{G}_{cr}^{(0)}=\frac{9}{5}\,\frac{\sin (2\unicode[STIX]{x1D6F7}_{t})(S-R)}{(R+2)^{2}}.\end{eqnarray}$$

Similar discriminating factors have been previously obtained by others in the context of drop deformation (Taylor Reference Taylor1966) and surface charge convection (Xu & Homsy Reference Xu and Homsy2006). If $\unicode[STIX]{x1D6F7}_{t}<\unicode[STIX]{x03C0}/2$ then $\sin (2\unicode[STIX]{x1D6F7}_{t})$ is a positive quantity. Thus, for a realistic $\bar{G}_{cr}^{(0)}$ to exist, the condition which must be satisfied is $R<S$ . Again, for $\unicode[STIX]{x1D6F7}_{t}>\unicode[STIX]{x03C0}/2$ the property ratios must fulfil the condition $R>S$ . Hence, the Marangoni effect on the electrically induced emulsion viscosity can be suitably controlled by finely tuning the combination of the electric field direction and electrical properties of the drop–surrounding fluid pair. To understand such a behaviour of the Marangoni effect, we plot the surface tension gradient $(|\unicode[STIX]{x1D6FE}_{max}-\unicode[STIX]{x1D6FE}_{min}|)$ with relative shear rate for different values of the tilt angle in figure 6(c) for system-A. It shows that the surface tension gradient vanishes at a relative shear rate which is exactly equal to the crossover point observed in figure 6(a). This indicates that due to competitive effects of the hydrodynamic and electrical stresses on the surface tension variation, for a specific relative shear rate, the Marangoni effect vanishes completely and the drop surface behaves as an uncontaminated one. The corresponding competing effects can be realized by referring back to (3.14) where distinct functional forms of the hydrodynamic and electric surfactant contribution terms $(\unicode[STIX]{x1D702}_{h,surf}^{(0)},\unicode[STIX]{x1D702}_{e,surf}^{(0)})$ appeared in the expression of $\unicode[STIX]{x1D702}_{eff}^{(0)}$ . The lesser extent of variation of $\unicode[STIX]{x1D702}_{eff}^{(0)}$ with $\bar{G}$ , after the crossover point $\bar{G}_{cr}$ is reached (observation from figure 6 a), can also be explained from figure 6(c) where the range of surface tension gradient $(|\unicode[STIX]{x1D6FE}_{max}-\unicode[STIX]{x1D6FE}_{min}|)$ becomes lower after $\bar{G}_{cr}$ than it was before  $\bar{G}_{cr}$ .

The value range of $\bar{G}_{cr}^{(0)}$ is observed to be different in the two sets of electrohydrodynamic parameters. In this context, the parametric mapping of the critical relative shear rate on the $R$ – $S$ plane is shown for two different tilt angles in figure 6(d,e). It is to be noted that a critical normalized shear rate was obtained previously by Mandal & Chakraborty (Reference Mandal and Chakraborty2017b ) in relation to a surfactant-free drop under the combined influence of electric field and shear flow. In that case, the critical parameter had been used to characterize whether the effective shear viscosity is greater (or lesser) than that of the suspending medium. In stark contrast, the presently obtained critical relative shear rate is related to the surfactant-induced Marangoni effect. To the best of our knowledge, for the first time we have obtained a discriminating factor in terms of a relative shear rate for which the Marangoni effect will have no contribution in altering the emulsion rheology.

Figure 6.

Variation of normalized effective shear viscosity $(\bar{\unicode[STIX]{x1D702}}_{eff})$ with relative shear rate $\bar{G}=1/M$ for different values of the elasticity parameter, $\unicode[STIX]{x1D6FD}$ : (a) system-A and (b) system-B. In both panels, $\unicode[STIX]{x1D6F7}_{t}=\unicode[STIX]{x03C0}/4$ is chosen, while the inset of each panel shows the corresponding variations for $\unicode[STIX]{x1D6F7}_{t}=3\unicode[STIX]{x03C0}/4$ . (c) Variation of surface tension gradient $(|\unicode[STIX]{x1D6FE}_{max}-\unicode[STIX]{x1D6FE}_{min}|)$ with relative shear rate for various $\unicode[STIX]{x1D6F7}_{t}$ . Here the demonstration is for system-A. Maps of the critical relative shear rate on the $(R,S)$ plane for electric field tilt angle (d) $\unicode[STIX]{x1D6F7}_{t}=\unicode[STIX]{x03C0}/4$ and (e) $\unicode[STIX]{x1D6F7}_{t}=3\unicode[STIX]{x03C0}/4$ .

Another important aspect of bulk rheology, i.e. the first $(N_{1}^{(0)})$ and second $(N_{2}^{((0))})$ normal stress difference, is significantly affected by the presence of surfactant contamination on the drop surface. Equation (3.15) shows that in the leading order, the normal stress differences in the bulk rheology arise only from electrohydrodynamic origins, even when the surfactant effects are considered. From the functional form of ${\mathcal{N}}_{surf}^{(0)}$ , it is evident that for all the realistic parametric ranges of ( $0\leqslant \unicode[STIX]{x1D6FD}<1,k>0$ and $\unicode[STIX]{x1D706}>0$ ), the surfactant-induced Marangoni effect always acts to reduce the normal stress differences, i.e. ${\mathcal{N}}_{surf}^{(0)}<1$ . In figure 7, a parametric map of the common surfactant contribution factor ${\mathcal{N}}_{surf}^{(0)}$ (defined in (3.16)) is shown on the $\unicode[STIX]{x1D6FD}$ – $k$ plane for different viscosity ratios of the drop–matrix fluid pair. It shows that for lower viscosity of the drop phase as compared to the matrix fluid, the surfactant contribution is more prominently observed. In the limit of $\unicode[STIX]{x1D706}\rightarrow 0$ , when the drop tends behave as a bubble, the surfactant effect is a maximum and we obtain $\lim _{\unicode[STIX]{x1D706}\rightarrow 0}\,{\mathcal{N}}_{surf}^{(0)}=(1-\unicode[STIX]{x1D6FD})/((k-5)\unicode[STIX]{x1D6FD}+5)$ . In the other limit of $\unicode[STIX]{x1D706}\rightarrow \infty$ , the drop behaves like a particle and its surface becomes completely immobilized. In that case, the normal stress differences vanish for a spherical drop.

Figure 7.

Surfactant modification parameter for the normal stress differences of the emulsion $({\mathcal{N}}^{(0)})$ on $\unicode[STIX]{x1D6FD}$ – $k$ plane for different viscosity ratios ( $\unicode[STIX]{x1D706}$ ).

4.3 Order $Re_{E}$ bulk rheology

Figure 8(a) depicts that an increase in the parameter $\unicode[STIX]{x1D6FD}$ suppresses the $O(Re_{E})$ modifications in $\bar{\unicode[STIX]{x1D702}}_{eff}$ . To justify such a phenomenon, we plot the variations in $O(Re_{E})$ surface velocity and the electrical traction at the interface in figure S-4 for different values of $\unicode[STIX]{x1D6FD}$ . The figure shows that the charge-convection-induced modifications in the flow characteristics (exemplified by modulations in $u_{S,\unicode[STIX]{x1D719}}^{(Re_{E})}$ and $[[T_{\unicode[STIX]{x1D719}}^{E}]]^{(Re_{E})}$ ) are severely altered due to the surface elasticity parameter, $\unicode[STIX]{x1D6FD}$ .

The interplay between the charge convection and Marangoni flow (see the discussion in § S-5 of Supplementary_2) at the interface also modifies the shear-rate-dependent response of the emulsion viscosity, as shown in figure 8. The critical relative shear rate corresponding to the crossover point of the Marangoni effect, $(\bar{G}_{cr})$ , is shifted from its leading-order counterpart (shown in figure 8 a). This effect can be substantiated by carefully inspecting the $O(Re_{E})$ effective shear viscosity ( $\unicode[STIX]{x1D702}_{eff}^{(Re_{E})}$ ). Similar to the leading order, there exists a specific range of relative shear rate for which the Marangoni effect has hardly any role in modifying the $O(Re_{E})$ effective viscosity of the medium ( $\unicode[STIX]{x1D702}_{eff}^{(Re_{E})}$ ). However, the critical shear rate in this case is also a function of the surfactant parameters and the viscosity ratio:

(4.2) $$\begin{eqnarray}\bar{G}_{cr,\bar{\unicode[STIX]{x1D702}}}^{(Re_{E})}=\frac{R-S}{(R+2)^{2}}\frac{\sin (2\unicode[STIX]{x1D6F7}_{t})}{\unicode[STIX]{x1D6FF}_{1}\cos (2\unicode[STIX]{x1D6F7}_{t})+\unicode[STIX]{x1D6FF}_{2}},\end{eqnarray}$$

where

(4.3) $$\begin{eqnarray}\unicode[STIX]{x1D6FF}_{1}=\frac{175}{36}\frac{(3R+4)(\unicode[STIX]{x1D706}+1)(R-2S-2)(\unicode[STIX]{x1D6FD}k-5\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D706}-5\unicode[STIX]{x1D6FD}+5\unicode[STIX]{x1D706}+5)}{(69R^{2}-54RS+130R-80S+40)(\unicode[STIX]{x1D6FD}k-10\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D706}-10\unicode[STIX]{x1D6FD}+10\unicode[STIX]{x1D706}+10)}\end{eqnarray}$$

and

(4.4) $$\begin{eqnarray}\unicode[STIX]{x1D6FF}_{2}=-\frac{5}{108}\frac{(701\,R^{2}-506\,RS+1410\,R-760\,S+520)}{(69\,R^{2}-54\,RS+130\,R-80\,S+40)}.\end{eqnarray}$$

In addition, comparing figures 8(a) and 8(b), we observe that the non-Newtonian shear-thickening (shear-thinning) rheology at low shear rates (corresponding to the curve $Re_{E}=0.2,$ clean drop) becomes a shear-thinning (shear-thickening) one in the presence of surfactant (refer to the curve $Re_{E}=0.2,\unicode[STIX]{x1D6FD}=0.8$ ). This reversal in non-Newtonian behaviour is absent when charge convection effects are insignificant.

Figure 8.

(a) Normalized effective shear viscosity ( $\bar{\unicode[STIX]{x1D702}}_{eff}=\bar{\unicode[STIX]{x1D702}}_{eff}^{(0)}+Re_{E}\,\bar{\unicode[STIX]{x1D702}}_{eff}^{(Re_{E})}$ ) variation with elasticity parameter $\unicode[STIX]{x1D6FD}$ for different electric Reynolds numbers. Here $M=2$ and $\unicode[STIX]{x1D6F7}_{t}=\unicode[STIX]{x03C0}/4$ . (b,c) Variation of $\bar{\unicode[STIX]{x1D702}}_{eff}$ with relative shear rate $(\bar{G})$ for different elasticity parameters $\unicode[STIX]{x1D6FD}$ and electric Reynolds number $Re_{E}$ . In the inset of (b), the shift of critical shear rate $\bar{G}_{cr}$ with $Re_{E}$ is highlighted. Here $\unicode[STIX]{x1D6F7}_{t}=\unicode[STIX]{x03C0}/4$ and $\unicode[STIX]{x1D6F7}_{t}=3\unicode[STIX]{x03C0}/4$ in panels (b) and (c), respectively. The other parameters in all panels are chosen following system-C.

In the absence of charge convection, the surfactant contributions to the first and second normal stress differences $(\unicode[STIX]{x0394}\bar{N}_{1}=|\bar{N}_{1}-\bar{N}_{1,clean}|,\unicode[STIX]{x0394}\bar{N}_{2}=|\bar{N}_{2}-\bar{N}_{2,clean}|)$ increase with the tilt angle and are symmetric about $\unicode[STIX]{x1D6F7}_{t}=\unicode[STIX]{x03C0}/2$ , as depicted in figure 9. The value of $\unicode[STIX]{x0394}\bar{N}_{1}$ is a maximum for $\unicode[STIX]{x1D6F7}_{t}=0,\unicode[STIX]{x03C0}/2$ and vanishes for $\unicode[STIX]{x1D6F7}_{t}=\unicode[STIX]{x03C0}/4$ ; while $\unicode[STIX]{x0394}\bar{N}_{1}$ increases from zero to maximum from $\unicode[STIX]{x1D6F7}_{t}=0$ to $\unicode[STIX]{x03C0}/2$ . However, owing to the presence of charge convection, such symmetric nature is broken, although the increasing trends of both $\unicode[STIX]{x0394}\bar{N}_{1}$ and $\unicode[STIX]{x0394}\bar{N}_{2}$ are preserved.

Figure 9.

Variation of normalized (a) first $(\bar{N}_{1}=N_{1}/\unicode[STIX]{x1D708})$ and (b) second $(\bar{N}_{2}=N_{2}/\unicode[STIX]{x1D708})$ normal stress differences with the tilt angle of the applied electric field $\unicode[STIX]{x1D6F7}_{t}$ for different values of elasticity parameter $\unicode[STIX]{x1D6FD}$ and electrical Reynolds number $Re_{E}$ . Here the electrohydrodynamic parameters are chosen according to system-C. Other parameters are $M=2$ and $k=1$ .

4.4 Bulk rheology of deformed drops

Here, we analyse the consequences of the coupled interaction between the deformable nature of the drop and surfactant coating for different electrohydrodynamic systems. From figure 10(a,b), it can be observed that the deformation-induced modification in the Marangoni effect is much pronounced for lower values of the elasticity parameter $\unicode[STIX]{x1D6FD}$ . Also, the effect of capillary number becomes diminished as $\unicode[STIX]{x1D6FD}\rightarrow 0.8.$ Most interestingly, the increase in capillary number causes a decrease in the effective viscosity for system-B, whereas the trend is opposite for system-C.

To understand the physical origin of such behaviour, we investigate the nature of drop deformation in different systems and the consequence of surfactant coating on the drops. For this, we appeal to the stress balance equation (3.6a,b ) where the Marangoni effect is seen to play its role. From a physical perspective, the localized accumulation of surfactant molecules reduces the local surface tension and makes that region more prone to deformation in order to balance the normal stresses. This is known as the ‘tip stretching’ phenomenon in the literature (Pawar & Stebe Reference Pawar and Stebe1996). Again, due to increased surface area of the deformed drop, the overall surfactant concentration gets diluted and the local surface tension tends to decrease. The electrical contribution to the deformation of a surfactant-free drop surface is further modified by surface contamination and the deformation parameter shows a steady increase with $\unicode[STIX]{x1D6FD}$ as shown in figures S-5(a,b). This leads to modifications in the nature as well as the intensity of drop deformation. As a result, the drop poses different kinds of resistances to the imposed shear flow, thereby creating diverse modulations in the effective shear viscosity of the emulsion.

Figure 10.

Normalized effective shear viscosity ( $\bar{\unicode[STIX]{x1D702}}_{eff}=\bar{\unicode[STIX]{x1D702}}_{eff}^{(0)}+Ca\,\bar{\unicode[STIX]{x1D702}}_{eff}^{(Ca)}$ ) versus elasticity parameter $\unicode[STIX]{x1D6FD}$ for different capillary numbers $Ca$ : (a) system-B and (b) system-C. Here $M=2$ . (c) Normalized effective shear viscosity versus relative shear rate for different elasticity parameter $\unicode[STIX]{x1D6FD}$ and capillary number $Ca$ . In the inset, the shift of critical shear rate $\bar{G}_{cr}$ with $Ca$ is highlighted. Here, other parameters are chosen following system-C. In all panels, $\unicode[STIX]{x1D6F7}_{t}=\unicode[STIX]{x03C0}/4$ and $k=1$ .

Figure 10(c) shows that the critical shear rate corresponding to the vanishing Marangoni effect $(\bar{G}_{cr})$ is shifted to a higher value when the capillary number becomes 0.2 from an undeformed state, for the specific parametric choices of system-C. Until $\bar{G}_{cr}$ is reached, the shape deformation causes significant enhancement in the effective shear viscosity. However, with an increase in $\bar{G}$ beyond this critical point, the impact of deformation effects on the effective viscosity becomes vanishingly small.

In figure 11, we demonstrate the variations in the bulk normal stress differences with the tilt angle of the applied electric field for different values of the surface elasticity number and capillary number. It is found that the leading-order symmetry in both $\bar{N}_{1}$ and $\bar{N}_{2}$ about the tilt angle $\unicode[STIX]{x1D6F7}_{t}=\unicode[STIX]{x03C0}/2$ is destroyed, owing to a deformed drop surface. In the deformed condition, the contribution of the surfactants in modifying the first normal stress difference $(\unicode[STIX]{x0394}\bar{N}_{1}=|\bar{N}_{1}-\bar{N}_{1,clean}|)$ is observed to be positive for certain values of the tilt angle, while the reverse happens for other tilt angles (shown in figure 11 a). However, figure 11(b) depicts that under similar circumstances, the surfactant-induced modifications in the second normal stress difference $(\unicode[STIX]{x0394}\bar{N}_{2}=|\bar{N}_{2}-\bar{N}_{2,clean}|)$ show a steady decrease for all the tilt angles between $\unicode[STIX]{x1D6F7}_{t}=0$ and $\unicode[STIX]{x03C0}$ . Variations in the orientation of the applied electric field not only alter the behaviour of the drop deformation but also cause severe changes in the electrohydrodynamic flow pattern around the drop. In turn, the surface tension gradient is redistributed. Finally, an intricate interplay between the physical mechanisms of electrohydrodynamic and Marangoni flow dictates the elastic characteristics of the emulsion, resulting in corresponding changes in the normal stress differences.

Figure 11.

Normalized (a) first $(\bar{N}_{1}=N_{1}/\unicode[STIX]{x1D708})$ and (b) second $(\bar{N}_{2}=N_{2}/\unicode[STIX]{x1D708})$ normal stress differences versus tilt angle of the applied electric field $\unicode[STIX]{x1D6F7}_{t}$ for various elasticity parameter $\unicode[STIX]{x1D6FD}$ and capillary number $Ca$ . Here the electrohydrodynamic parameters are chosen according to system-B. Other parameters are $M=2$ and $k=1$ .