4.1. Validation

We first compare our results with those of Hanna & Vlahovska (Reference Hanna and Vlahovska2010), who investigated the impact of bulk-insoluble ( $C = M = 0$ ) and non-diffusing ( $Pe_s \rightarrow \infty$ ) surfactants on droplet migration in an unbounded plane Poiseuille flow. The far-field velocity in their study was taken as $\boldsymbol{u}_{\infty } = (-2 x_d X - X^2)\hat {\boldsymbol{e}}_z - \boldsymbol{U}$ , where the velocity scale is $u_c \sim \alpha 'a'^2$ and $2\alpha 'x^{\prime}_d$ is the local shear rate. With this $\boldsymbol{u}_{\infty }$ , our results show excellent agreement with theirs when the presence of bulk-soluble surfactants is neglected by setting $Bi = 0$ (and also, $C = M = 0$ ), and interfacial diffusion is rendered negligible by setting $Pe_s = 10^4 \gg 1$ . Figure 3(a) compares our findings (lines) with those of Hanna & Vlahovska (Reference Hanna and Vlahovska2010) (symbols) for the cross-stream position of the drop ( $x_d$ ) as a function of time ( $t$ ), when $\lambda = 5$ . Values of all other relevant parameters are mentioned in the caption. One observes that the presence of surfactants propels a spherical drop toward the channel centreline.

Figure 3.

Comparison between our results (lines) and those of Hanna & Vlahovska (Reference Hanna and Vlahovska2010) (symbols) for (a) the drop’s cross-stream location ( $x_d$ ) versus $t$ for $\lambda = 5$ , and (b) rescaled cross-stream migration velocity ( $U_x\lambda$ ) versus rescaled cross-stream distance from the centreline ( $x_d\lambda /Ma$ ) for various values of $\lambda \:(= 10,\;50)$ . Other relevant parameters are $Ma = 10, Pe_s = 10^4, Bi=0$ (insoluble surfactant). (c) Comparison of $\varGamma -1$ (in the $XZ$ plane) versus $\psi$ (angle sweeping the $XZ$ plane), between our results (lines) and those of Pak et al. (Reference Pak, Feng and Stone2014) (symbols), for droplet positions $b = 0$ (at centreline), $0.75$ and $1.5$ . Other relevant parameters are $Bi = 0$ (insoluble surfactants), $Pe_s = 0.2, Ma = 10, \lambda = 5$ and $R_0 = 5$ .

Figure 3(b) shows the comparison of the variations in the rescaled cross-stream migration velocity ( $U_x\lambda$ ) with the rescaled distance from the centreline ( $x_d\lambda /Ma$ ) for various choices of viscosity ratio ( $\lambda$ ), between our study (lines) and that of Hanna & Vlahovska (Reference Hanna and Vlahovska2010) (symbols). Again excellent agreement between the two is noted. This figure establishes that the rate of cross-stream migration becomes maximum at an intermediate distance from the centreline and eventually vanishes as the drop approaches the centreline.

In figure 3(c) we compare our results (lines) for the interfacial concentration ( $\varGamma$ ) of an insoluble surfactant (again, $Bi = C = M = 0$ ), with those of Pak et al. (Reference Pak, Feng and Stone2014) (symbols), who analysed droplet migration in an unbounded Hagen–Poiseuille flow. In their study, the interfacial transport of surfactants was assumed to be diffusion dominated, which entails a small surface Péclet number ( $Pe_s \ll 1$ ). Pak et al. derived perturbative (in terms of $Pe_s$ ) analytical solutions for the droplet migration velocity ( $\boldsymbol{U}$ ) and the interfacial surfactant concentration ( $\varGamma$ ) up to second-order in $Pe_s$ . For the purposes of comparison, we have set $Pe_s = 0.2 \ll 1$ , while $\boldsymbol{u}_{\infty }$ remains the same as defined in (3.8). In figure 3(c), $\varGamma -1$ is plotted in the $XZ$ plane against the angle $\psi \in [0,360]$ , which represents an angle that sweeps the plane, as shown in the inset. The comparison is made at various cross-stream distances of the drop’s centre, given by $b = 0,\; 0.75$ and $1.5$ , with $R_0 = 5$ ; values of all other parameters are mentioned in the caption. Again, excellent agreement is observed between the two sets of results. As expected, $\varGamma$ is symmetric when $b=0$ ; this variation results from the ensuing flow sweeping the surfactants from the front of the drop ( $\psi \lt 90$ or $\psi \gt 270$ , where $\varGamma \lt 1$ ), towards the rear end ( $90 \lt \psi \lt 270$ , where $\varGamma \gt 1$ ). The distribution however becomes asymmetric when $b\neq 0$ – we shed further light on this in the subsections ahead.

Our simulations reveal that $U_y = 0$ due to symmetry, which agrees with the results of Pak et al. (Reference Pak, Feng and Stone2014). Our results for $U_x$ and $U_z$ also show excellent agreement with those of Pak et al., although we have not included them for brevity. We have further compared our results with those of Das et al. (Reference Das, Mandal, Som and Chakraborty2017, Reference Das, Mandal and Chakraborty2018), who examined the migration of a surfactant-laden droplet (bulk insoluble, $Bi = 0$ ) in Poiseuille flow in the presence of a temperature gradient. This comparison is presented in § S3 of the supplementary material.

4.2. Variations in the surfactant concentration

Figure 4 illustrates the evolution of the interfacial surfactant concentration, $\varGamma$ , at various cross-stream distances ( $b = 1.5,\;1.4,\;0.2$ and 0.01) relative to the channel centreline, as shown schematically at the top of the figure. Panels (a–d) display the contours of $\varGamma$ along with vector plots representing the interfacial velocity field, $\boldsymbol{u}_s = u_{\theta }\hat {\boldsymbol{e}}_{\theta } + u_{\phi }\hat {\boldsymbol{e}}_{\phi }$ , at those four locations mentioned above. Panels (e–h) show the corresponding Mercator projections of $\varGamma$ and $\boldsymbol{u}_s$ , while panels (i–l) represent the contours of net adsorption flux at the interface, given by $J_b = Bi(C-\varGamma )$ , as Mercator projections at the same four positions of the droplet centre. The black and red dashed lines respectively represent the equator ( $\theta = 90^{\circ }$ ) and the prime meridian ( $\phi =0^{\circ }$ ). The caption furnishes further details on other relevant parameters.

Figure 4.

The schematic at the top depicts the cross-stream positions of the drop where the subsequent variations in the surfactant concentrations have been shown. The first row depicts the surface contours of $\varGamma$ along with the vector plots of $\boldsymbol{u}_s$ (white arrows) at $b =$ (a) $1.5$ , (b) $1.4$ , (c) $0.2$ and (d) $0.01$ . (e)–(h) Mercator projections of the $\varGamma$ and $\boldsymbol{u}_s$ , on the $\theta -\phi$ space at the same positions as considered in the first row. (i)–(l) Mercator projections of the contours of the net adsorption flux onto the interface ( $J_b = Bi(C-\varGamma )$ ) for the same droplet positions as in panels (a)–(d). The dashed black and red lines are respectively the equator ( $\theta = 90^{\circ }$ ) and the prime meridian ( $\phi =0^{\circ }$ ). The coordinate system attached to the drop is shown in panel (a) and in the schematic on top. The north–south–east–west directions are also shown in panel (a). Other relevant parameters are $Pe_s = 100,\;Pe_b = 20,\;Pe_m = 40,\;Bi = 0.1,\;\omega = 10, C_t = 3,\;C_{\infty } = 1.07,\;M_{\infty } = 1.93,\;N = 10,\;K_m = 0.1$ and $Ma = 0.1$ .

Figure 5 exhibits the associated variations in the bulk monomer (panels a–d) and micelle (panels e–f) concentrations (for $C_t = 3$ ) in the $XZ$ plane at the same positions of the drop during the course of its motion as considered in figure 4. The $XZ$ plane is divided into two half-planes: $\phi =0^{\circ }$ on the right and $\phi =180^{\circ }$ on the left of the red dotted line (same as the $Z$ axis) in panel (a). The angle $\psi$ in the $XZ$ plane is then defined as $\psi = \theta$ in the right half-plane ( $\phi =0^{\circ }$ ) and $\psi = 360^{\circ }-\theta$ in the left half-plane ( $\phi = 180^{\circ }$ ) of the drop’s surface. The caption provides the values of other relevant parameters.

Figure 5.

Contours of the bulk monomer concentration ( $C$ ) in the $XZ$ plane at (a) $b = 1.5$ , (b) $b = 1.4$ , (c) $b = 0.2$ and (d) $b=0.01$ . (e)–(h) Contours of the bulk micelle concentration ( $M$ ) in the $XZ$ plane at those same positions of the drop (also the same as the schematic at the top of figure 4). Panel (a) shows the $XZ$ axes attached to the drop along with the half-plane $\phi =0^{\circ }$ where $\psi = \theta$ , and the half-plane $\phi =180^{\circ }$ where $\psi = 360^{\circ } - \theta$ . Other relevant parameters are the same as in figure 4.

At $b= 1.5$ , in figures 4(a) and 4(e) the polar component of the velocity ( $u_{\theta }\gt 0$ ) near the prime meridian ( $\phi =0^{\circ }$ ) sweeps the interfacial surfactants from the northern region ( $\theta \lt 90^{\circ }$ ) to the southern one ( $\theta \gt 90^{\circ }$ ), leading to monomer accumulation near the south pole ( $\varGamma \gt C_{\infty }$ ) and depletion in the north ( $\varGamma \lt C_{\infty }$ ), and alters the initially uniform interfacial concentration $\varGamma = C_{\infty }$ . Conversely, the opposite dynamics takes place near $\phi =180^{\circ }$ , as evident from the Mercator projection in figure 4(e). On the other hand, the azimuthal velocity component $u_{\phi }$ acts to reduce the variability in $\varGamma$ by transporting surfactants from regions of accumulation to regions of depletion. However, since the polar velocity component ( $u_{\theta }$ ) is significantly stronger than the azimuthal flows ( $u_{\phi }$ ), the accumulation and the depletion zones persist prominently. Notably, the extent of these zones is smaller near $\phi =180^{\circ }$ compared with $\phi =0^{\circ }$ , due to weaker flow strength in the former region. A keen examination of panels (a) and (e) further reveals that the aggregate sizes of the depletion and accumulation zones differ slightly between the two hemispheres, which also creates a mismatch in the surface tension gradients in these two regions. This results in a net pulling force on the interface in the positive $x$ direction, and hence, the drop as a whole migrates in the opposite direction toward the flow centreline.

In figures 4(b) and 4(f), at $b=1.4$ , the variation in $\varGamma$ is qualitatively similar to that in figure 4(a). The simulations suggest that at this position, the variability in $\varGamma$ across the interface reaches its maximum, with the most pronounced accumulation and depletion regions. At $b = 0.2$ (figures 4 c and 4 g), as the drop approaches the channel centreline, the overall strength of $\boldsymbol{u}_s$ decreases and consequently so does the variability in $\varGamma$ . The flow is now nearly axisymmetric, and hence, the polar (north–south) component of $\boldsymbol{u}_s$ dominates, with only minor contributions from the azimuthal (east–west) component. As a result, the accumulation and the depletion zones near the prime meridian ( $\phi =0^{\circ }$ , red dotted line) extend eastward and westward, covering a larger portion of the droplet’s surface. At the centreline ( $b=0.01$ ), as shown in figures 4(d) and 4(h), the azimuthal velocity ( $u_{\phi }$ ) vanishes entirely because of axisymmetry. Hence, for all $\phi$ , the flow transports the interfacial surfactants from north to south ( $u_{\theta }\gt 0$ – see the white arrows), creating a large accumulation zone in the southern hemisphere and a similar depletion zone in the northern hemisphere.

Accumulation of surfactants at any region of the interface leads to desorption into the bulk ( $J_b \lt 0$ ), while depleted regions promote net adsorption of monomers onto the interface ( $J_b \gt 0$ ). These features are clearly visible in figure 4(i–l). Akin to $\varGamma$ , the maximum variation in the net adsorption flux $J_b$ , also occurs at $b=1.4$ (see panel j), and subsequently decreases as the drop approaches the centreline.

The associated variations in the bulk monomer and micelle concentrations as depicted in figure 5 may be explained based on trends for $\varGamma$ observed in figure 4. Recall that the bulk monomer concentration ( $C$ ) is influenced by several mechanisms: adsorption and desorption at the interface, described by the flux $\omega (C-\varGamma )\sim J_b$ , advection–diffusion in the bulk, and the consumption and release of monomers due to micelle formation and breakup, represented by $J_m$ . Figure 5(a–d) clearly shows that the bulk monomer concentration ( $C$ ) exhibits accumulation ( $C \gt C_{\infty }$ ) and depletion zones ( $C \lt C_{\infty }$ ), close to the drop, similar to the patterns observed for $\varGamma$ . When the drop is away from the channel centreline ( $b = 1.5$ or $1.4$ ), there are multiple distinct accumulation and depletion regions. However, as the drop approaches the centreline ( $b = 0.2$ and $0.01$ ), these accumulation and depletion zones become axisymmetric, with a single accumulation region in the southern hemisphere ( $\theta \gt 90^{\circ }$ ) and a single depletion region in the northern hemisphere ( $\theta \lt 90^{\circ }$ ). It is clear from panel (b) that the maximum variability in $C$ is observed at $b=1.4$ – again exactly similar to the trend of $\varGamma$ .

The accumulation and depletion of bulk monomers exactly mirror the same phenomena occurring for the interfacial surfactant concentration ( $\varGamma$ ), which drives adsorption–desorption between the bulk and the interface. For instance, the accumulation of interfacial surfactants occurs in the southern region around $\phi =0^{\circ }$ ( $90^{\circ }\lt \psi \lt 180^{\circ }$ – see figure 4 b,f). Therefore, monomers get dumped in the bulk because of desorption ( $J_b \lt 0$ ), which leads to local accumulation in the bulk and a spike in $C$ . Conversely, depletion of $\varGamma$ is observed in the northern region near $\phi =0^{\circ }$ and in the southern region near $\phi =180^{\circ }$ (see panel f). Therefore, adsorption from the bulk ( $J_b \gt 0$ ) takes place in these regions, leading to a depletion of monomers therein.

Since the total bulk concentration $C_t = 3 \gt$ CMC in figure 5, concentrations of monomers ( $C$ ) and micelles ( $M$ ) in the bulk remain strongly coupled. Figure 5(e–h) show that the micelles also exhibit distinct accumulation ( $M \gt M_{\infty }$ ) and depletion zones ( $M \lt M_{\infty }$ ), and indeed these regions mirror those of $C$ , driven by the self-aggregation rate $J_m$ . In regions of monomer accumulation in the bulk ( $C \gt C_{\infty }$ ), the excess monomers drive local micelle formation ( $J_m \gt 0$ ), and this leads to a spike in the micelle concentration ( $M \gt M_{\infty }$ ), whilst capping the maximum monomer concentration. Conversely, in monomer depletion zones in the bulk ( $C \lt C_{\infty }$ ), the local equilibrium is disturbed, which leads to spontaneous dissociation of micelles ( $J_m \lt 0$ ) into monomers to replenish their deficit. The extent of accumulation and depletion regions for both monomers and micelles strongly depend on the respective Péclet numbers. Lastly, akin to $\varGamma$ and $C$ , the maximum variation in the micelle concentration, $M$ , also occurs at $b=1.4$ , gradually decreasing afterwards as the drop approaches the flow centreline.

Figure 6.

Micelle formation flux ( $J_m= K_m(C^N-M)$ ) in the $XZ$ plane versus $\overline {r}\;\;(= 1/r)$ along (a) $\psi =135^{\circ }$ and (b) $\psi =60^{\circ }$ , for $b=1.5,\;1.4,\;0.2$ and $0.01$ – see figures 3(c) and 5(a) for the definitions of $\psi$ . Other relevant parameters are the same as in figure 4.

Figure 6 displays the variations in the micelle formation rate ( $J_m$ ) with radial distance ( $\overline {r}\:(=1/r)$ ) at two distinct angular positions, namely at $\psi = 135^{\circ }$ (panel a) where the formation rate is close to its maximum, and $\psi =60^{\circ }$ (panel b) where this rate is close to its minimum. The same droplet positions as in figure 5 have been considered here, while the values of all other relevant parameters are mentioned in the caption. As expected, panel (a) depicts a positive micelle formation rate ( $J_m \gt 0$ ) for being located in the southern region ( $90^{\circ }\lt \psi \lt 180^{\circ }$ ) near $\phi =0^{\circ }$ , where monomer accumulation takes place, which then rapidly self-aggregate to form excess micelles. In contrast, panel (b) shows that $J_m\lt 0$ leads to the dissociation of micelles within the monomer depletion zones, located in the northern region ( $0^{\circ }\lt \psi \lt 90^{\circ }$ ) near the $\phi =0^{\circ }$ half-plane. Indeed, the difference between $J_{m,{max}}$ and $J_{m,{min}}$ is at its maximum when $b=1.4$ (when the variability in $C$ , $M$ and $\varGamma$ are also at their respective peaks), beyond which both $J_{m,{max}}$ and $J_{m,{min}}$ decrease as the drop approaches the channel centreline, diminishing the intensity of the respective accumulation and depletion regions, as corroborated by figures 4 and 5.

4.3. Impact of micelles on the cross-stream migration of the drop: parametric study

Figure 7 demonstrates the influence of total surfactant concentration in the bulk ( $C_t$ ) on the cross-stream migration velocity ( $|U_x|$ ) for various choices of the Marangoni number, $Ma = 0.1,\;0.5$ and $1$ , when the drop is located at $b=1.5$ . Panel (a) depicts the variations in $|U_x|$ with $C_t$ (over the same range as in figure 2 c); the inset shows the variations in the coefficient $|\varGamma ^1_1|$ , which dictates the migration velocity, as a function of $C_t$ . Panels (b) and (c) display the variations in the interfacial concentration, $\varGamma$ with $\psi$ (see figure 5 a for the definition of $\psi$ ) for $Ma = 0.5$ and $Ma = 0.1$ , respectively. The insets in these panels illustrate the corresponding variations in the net adsorption flux onto the interface ( $J_b$ ). Panels (d) and (e) showcase the variations in the bulk monomer concentration near the interface ( $C(r=1)$ ) with $\psi$ for $Ma = 0.5$ and $Ma = 0.1$ , respectively. The two insets in each of these panels portray the corresponding variations in the bulk micelle concentration ( $M(r=1)$ ) and the micelle formation rate near the interface ( $J_m(r=1)$ ). In panels (b–e) we have chosen $C_t = 0.1,0.8,3$ and the designated entities are plotted in the $XZ$ plane. The caption outlines the values of all other relevant parameters.

Figure 7.

(a) Plot of the cross-stream migration velocity ( $|U_x|$ ) versus $C_t$ for $Ma = 0.1,\;0.5$ and $1$ , when the drop is located at $b = 1.5$ . The inset shows $|\varGamma ^1_1|$ vs $C_t$ for the same parameters. Panels (b) and (d) respectively show plots of $\varGamma$ vs $\psi$ and $C(r=1)$ vs $\psi$ in the $XZ$ plane for $C_t = 0.1,\;0.8$ and $3$ when $Ma = 0.5$ . Panels (c) and (e) show those same plots when $Ma = 0.1$ . The insets in (b) and (c) depict the net adsorption flux onto the interface ( $J_b$ ) versus $\psi$ . The insets in (d) and (e) exhibit $M(r=1)$ vs $\psi$ (the first inset) and the aggregation rate at the interface ( $J_m\:(r=1)$ ) versus $\psi$ (the second inset). Other relevant parameters remain identical to those in figure 4.

Figure 7(a) establishes that $|U_x|$ shows a non-monotonic variation with $C_t$ depending on Marangoni number $Ma$ , and the inset reveals that this variation mirrors the trend of $|\varGamma ^1_1|$ vs $C_t$ , simply because $|U_x|\propto Ma |\varGamma ^1_1|$ (see (3.12)). We recall that in the absence of surfactants ( $C_t = 0$ ), a clean spherical droplet will not undergo cross-stream migration in the given situation. However, even a small amount of surfactant ( $C_t = 0.1 \ll 1$ ) induces migration due to Marangoni stresses caused by the variations in interfacial surfactant concentration ( $\varGamma$ ). As the total bulk concentration $C_t$ increases, so do the variations in $\varGamma$ , which enables Marangoni stresses to intensify (see figures 7 b and 7 c), enhancing the cross-stream migration (i.e. $|U_x|$ ) in the process. However, this process ceases to be effective beyond a threshold value of $C_t$ , especially for relatively higher values of the Marangoni number ( $Ma=0.5,\;1$ ). One possible explanation for this stems from the fact that, at large enough $C_t$ , the Marangoni stresses become sufficiently strong and immobilises the droplet interface itself. Specifically, an increase in $Ma$ weakens the polar flows ( $u_{\theta }$ ) and strengthens the azimuthal flows ( $u_{\phi }$ ). As discussed in figure 4, polar flows are responsible for creating gradients in $\varGamma$ , while azimuthal flows act to diminish these variations and make $\varGamma$ uniform. Indeed panels (b) and (c) show that $\varGamma$ remains relatively closer to a uniform value throughout the interface when $Ma$ increases from $0.1$ to $0.5$ . Consequently, $|\varGamma ^1_1|$ decreases with $Ma$ (see the inset in panel a), and this decrement dominates over any change in $Ma$ as far as $|U_x|$ is concerned (recall that $|U_x|\propto Ma |\varGamma ^1_1|$ ). Therefore, the intensity of cross-stream migration also sees a dip beyond a critical $C_t$ . Understandably, this turn around in $|U_x|$ when $C_t$ is increased is expected for relatively larger $Ma$ values, as is indeed observed in panel (a). In fact, the precise value of $C_t$ where the maximum $|U_x|$ is realised depends strongly on the Marangoni number ( $Ma$ ). For instance, for $Ma = 0.5$ , the peak value of $|U_x|$ is observed around $C_t \sim 0.4$ ; this peak shifts to lower $C_t$ values as $Ma$ increases. However, at very low concentrations ( $C_t=0.1$ ), $|\varGamma ^1_1|$ remains unaffected by $Ma$ , and thus, in such cases, $|U_x|$ increases with $Ma$ . The above results align well with the findings of Pak et al. (Reference Pak, Feng and Stone2014), who observed that $|U_x|\propto Ma$ for bulk-insoluble surfactants at low surface Péclet number ( $Pe_s$ ). When non-ideal surfactant behaviour is incorporated using the Frumkin isotherm, it is found that the finite size of the surfactant molecules tend to reduce the extent of cross-stream migration by nearly saturating the interface beyond a given bulk concentration – see figure 14 and the related discussion in Appendix C.3.

Beyond the CMC ( $C_t \gt 1$ ), rapid self-aggregation leads to micelle formation and, hence, limits the availability of the monomers. This puts a cap on the magnitude of the interfacial concentration $\varGamma$ and also restricts its variations across the interface to a large extent. As a result, both $|\varGamma _1^1|$ and thereby $|U_x|$ essentially plateau beyond $C_t = 1$ . In fact, figures 7(b) and 7(c) do indicate that $\varGamma$ changes by a relatively small amount, despite a nearly four-fold increase in $C_t$ from $0.8$ to $3$ , regardless of $Ma$ . This behaviour is consistent with earlier observations in the literature (Stebe et al. Reference Stebe, Lin and Maldarelli1991; Langevin Reference Langevin2014; Manikantan & Squires Reference Manikantan and Squires2020), which report that above the CMC, the interface becomes remobilised, as the diminished Marangoni stresses have a limited impact on droplet motion.

At sufficiently low surfactant concentrations, such as for $C_t = 0.1 \ll 1$ , only monomers are expected to be present in the continuous phase ( $M = 0$ everywhere), as is indeed observed in the insets of figures 7(d) and 7(e). Just below the CMC however (specifically at $C_t = 0.8$ ), something very interesting takes place. Because of desorption from the interface, the local bulk monomer concentration may exceed the CMC ( $C_t \gt 1$ ) close to the droplet, triggering micelle formation in these areas (see the insets in panels d and e). As a consequence, despite the far-field concentration ( $C_t$ ) being lower than the CMC, a small amount of micelles do exist in the continuous phase, purely because of the coupling between the bulk and the interfacial transport. A somewhat similar mechanism for micelle formation had earlier been proposed by Stebe & Maldarelli (Reference Stebe and Maldarelli1994) in the context of settling drops, and our simulations suggest that such pathways may indeed be feasible.

Figure 8 portrays the influence of the Biot number ( $Bi$ ) and the bulk interaction parameter ( $\omega$ ) on the cross-stream migration velocity of the drop positioned at $b=1.5$ . Panel (a) depicts the variations in $|U_x|$ with $Bi$ for $\omega = 10$ , while panel (d) plots $|U_x|$ vs $\omega$ for $Bi=0.1$ , both for $C_t = 0.1,\;0.8$ and $3$ . The insets in these two panels show the variations in the coefficient $|\varGamma ^1_1|$ as a function of $Bi$ and $\omega$ , respectively. Panels (b) and (e) respectively represent the variations in $\varGamma$ with $\psi$ for $Bi = 0.1,0.9,5$ and $\omega = 0.1,3.5,10$ , when $C_t=3$ . The insets in these panels illustrate the corresponding variations in the net adsorption flux at the interface ( $J_b = Bi(C-\varGamma )$ ). Panels (c) and (f) display the variations in the bulk monomer concentration near the interface ( $C(r=1)$ ) with $\psi$ , for the same choices of $Bi$ , $\omega$ and $C_t$ as in panels (b) and (e). The insets in these panels depict the corresponding variations in the bulk micelle concentration ( $M(r=1)$ ) and the micelle formation rate near the interface ( $J_m\:(r=1)$ ). All variables are plotted in the $XZ$ plane. Values of other relevant parameters are given in the caption.

Figure 8.

(a) Plot of $|U_x|$ vs $Bi$ for $\omega = 10$ and (d) $|U_x|$ vs $\omega$ for $Bi = 0.1$ and $Ma = 0.5$ . The insets in (a) and (d) respectively show $|\varGamma ^1_1|$ vs $Bi$ and $|\varGamma ^1_1|$ vs $\omega$ , for $C_t = 0.1,\;0.8$ and $3$ . Panels (b) and (c) respectively depict $\varGamma$ vs $\psi$ and $C(r=1)$ vs $\psi$ in the $XZ$ plane for $Bi = 0.1,0.9$ and $5$ and a fixed $\omega = 10$ . Panels (e) and (f) respectively exhibit those same plots for $\omega = 0.1,\,3.5$ and $10$ , and a fixed $Bi = 0.1$ . The insets in (b) and (e) show the plot of $J_b(r=1)$ vs $\psi$ . The insets in panels (c) and (f) plot the bulk micelle concentration ( $M(r=1)$ ) and micelle formation rate near the interface ( $J_m\:(r=1)$ ) versus $\psi$ , all in the $XZ$ plane. We have taken $C_t = 3$ in panels (b), (c), (e) and (f); all other relevant parameters are identical to figure 7.

Figure 8(a) reveals that $|U_x|$ decreases with $Bi$ (which represents the rate of monomer adsorption at the interface) for all choices of the total bulk concentration $C_t$ , and $|\varGamma ^1_1|$ exhibits an exactly similar trend (see inset). As $Bi$ increases, the exchange of monomers between the bulk and the interface becomes more pronounced, and this also gets reflected as a commensurate rise in $J_b$ shown in the inset of panel (b). The relatively larger $J_b$ tends to diminish the existing non-uniformities in $\varGamma$ , as evident from panel (b), resulting in a reduction in $|\varGamma ^1_1|$ and, therefore, the cross-stream migration velocity. In fact, examining panels (b) and (c), it becomes evident that the largest variability in $\varGamma$ as well as $C(r=1)$ occurs at a relatively low Biot number ( $Bi = 0.1$ ), which also results in greater local variations in the micelle concentration ( $M(r=1)$ ) because of a larger micelle formation rate ( $J_m(r=1)$ ). The respective variations in $\varGamma ,C, J_m$ and $M$ progressively diminish with increasing $Bi$ , making the aforementioned quantities closer to a uniform value.

Figure 8(d) shows that the bulk interaction parameter ( $\omega$ ) has an opposite impact to that of $Bi$ on the cross-stream migration, since both $|U_x|$ and $|\varGamma ^1_1|$ increase with $\omega$ for all values of the total bulk concentration $C_t$ . Recall that a relatively lower $\omega$ indicates that adsorption–desorption between the interface and the bulk has relatively little impact on the bulk monomer concentration and, hence, it tends to stay uniform due to fast diffusion; see panel (f). On the contrary, a large $\omega$ tends to slow down the movement of the monomers and, hence, delays the replenishment or removal of monomers from near the drop interface. This feature becomes clear from the increased variations in $C(r=1)$ with $\omega$ in panel (f). In other words, a relatively larger $\omega$ tends to make the accumulation and the depletion zones more pronounced, which obviously impedes any further adsorption–desorption of monomers at the interface (see the variations in $J_b(r=1)$ in the inset of panel e). In a way, this counteracts the impact of $Bi$ and results in increasingly non-uniform surface concentration ( $\varGamma$ ), as evident from panel (e). As a consequence, $|\varGamma ^1_1|$ increases and so does the cross-stream migration velocity. In fact, panel (f) shows that any increment in $\omega$ also causes the variations in $C$ , $M$ as well as $J_m$ to increase, at least close to the droplet interface.

Figure 9 illustrates the influence of micelle-specific parameters, in particular, the micelle formation rate ( $K_m$ , panel a), the micelle Péclet number ( $Pe_m$ , panelb) and the aggregation number ( $N$ , panel c), on the cross-stream migration velocity. Corresponding variations in $|\varGamma ^{1}_1|$ have been shown in the respective insets, while values of all other relevant parameters are mentioned in the caption.

Figure 9.

Cross-stream migration velocity ( $|U_x|$ ) versus $C_t$ for (a) various $K_m\;(\: = 0.1,\;1,\;10)$ with $Pe_m=40$ and $N = 10$ , (b) various $Pe_m\;(\: = 20,\;40,\;60,200)$ with $K_m=10$ and $N = 10$ , and (c) various $N\;(\: = 10,30,50)$ with $K_m=0.1$ and $Pe_m = 40$ . The insets plot $|\varGamma ^1_1|$ vs $C_t$ for the same parameters as in the main panels. Other relevant parameters are identical to those in figure 7.

It is expected that specific properties of the micelle will only impact $|U_x|$ beyond a critical $C_t$ , i.e. when micelles are abundant in the continuous phase. As per figure 5, the self-aggregation actually begins when $C_t \sim 0.8$ (fuelled by desorption), which is a little below the CMC and, hence, $|U_x|$ should depend on parameters such as $K_m,\;Pe_m$ and $N$ , when $C_t \geqslant 0.8$ . This is indeed what is observed in all panels of figure 9. On a more general note, the micelle properties seem to have relatively little impact on the migration velocity ( $U_x$ ), as opposed to say, $Ma$ , $Bi$ or $\omega$ . One possible reason for this is the absence of any direct interactions between the micelles and the surfactants on the interface; see assumption (iii) in § 2.2. Therefore, although the self-aggregation process itself, which leads to micelle formation, has a strong impact on cross-stream migration (see, for instance, figure 7), the specific details of this process seem to have relatively little influence on the same, especially when the micelles do not directly interact with the droplet interface.

Nevertheless, figure 9 reveals some noteworthy trends. An increase in the micelle formation rate ( $K_m$ ) generally reduces $U_x$ , whereas a relatively higher micelle Péclet ( $Pe_m$ ) and aggregation number ( $N$ ) tend to enhance $U_x$ , especially at high bulk concentrations ( $C_t \geqslant 1$ ). For large $K_m$ , faster monomer–micelle exchange homogenises the bulk monomer concentration, reducing the Marangoni stresses and, thus, $U_x$ in the process. In contrast, a relatively higher $Pe_m$ and $N$ accelerate micelle transport and formation, depleting monomers near the interface and amplifying the interfacial concentration gradients, which boosts $|U_x|$ . Their impact is more pronounced than that of $K_m$ – altering $|U_x|$ by about $8\,\% \unicode{x2013}28\,\%$ . This is because $K_m$ contributes linearly to the micelle formation flux ( $J_m= K_m(C^N - M)$ ), while the same varies exponentially with the aggregation number $N$ , making its influence inherently stronger. The extent to which the micelle formation rate $K_m$ impacts cross-stream migration also depends on the Biot number ( $Bi$ ). In fact, the influence of $K_m$ is expected to diminish when the $Bi$ is relatively small ( $Bi = 0.1$ ), as is the case in panel (a). The variations in $U_x$ driven by $K_m$ at a higher $Bi$ value have been shown in figure 12 in Appendix B.1.

In this context, it may be noted that previous studies such as that of Morozov (Reference Morozov2020) have shown that micelles may directly interact with the interface by solubilising it to form ‘swollen micelles’ (i.e. micelles enclosing nano-droplets). They generally have very low diffusivity and high resistance to breakup, and thus, tend to accumulate near the droplet interface, hindering the natural adsorption–desorption processes between the interface and the bulk. As a result, any non-uniformity in the concentration of these swollen micelles may potentially result in relatively enhanced Marangoni stresses even at concentrations above the CMC, possibly impacting droplet migration. Therefore, unlike regular micelles, the properties of swollen micelles, such as their Péclet number, aggregation number and formation rate, may play a significant role in governing a drop’s trajectory. However, the precise extent of their influence cannot be determined without a dedicated analysis, which lies beyond the scope of the present paper.

4.4. Analysis of droplet trajectories

Figure 10 presents the droplet trajectories for the Marangoni number, $Ma = 0.1$ (panels a,c) and $Ma = 0.5$ (panels b,d), at $C_t = 0.1$ , $0.4$ , $0.8$ and $3$ . Panels (a) and (b) depict the drop’s cross-stream position ( $b = x_d$ ) relative to its streamwise position ( $z_d$ ), while panels (c) and (d) illustrate the variations in the drop’s cross-stream migration velocity ( $U_x$ ) as a function of its cross-stream position ( $x_d$ ). The caption provides the values of all other relevant parameters.

Figure 10.

Plot of the cross-stream position of the drop ( $b = x_d$ ) versus its streamwise position ( $z_d$ ) for (a) $Ma = 0.1$ and (b) $Ma = 0.5$ , both for various $C_t\:(=0.1,\;0.4,\;0.8,\;\text{and}\;3)$ . (c,d) Cross-stream migration velocity ( $U_x$ ) versus $x_d$ for the same choices of parameters respectively as in (a) and (b). Other parameters remain the same as in figure 7.

Figures 10(a) and 10(b) both confirm that the drop consistently migrates toward the flow centreline (indicated by $U_x \lt 0$ ) at all cross-stream positions. This has previously been observed experimentally (Santra et al. Reference Santra, Das, Das and Chakraborty2018; Panigrahi et al. Reference Panigrahi, Santra, Banuprasad, Das and Chakraborty2021), and this behaviour stems from the formation of non-axisymmetric surfactant accumulation and depletion regions near the droplet’s interface, as explained earlier in figure 4. Panels (c) and (d) indicate that the cross-stream migration velocity of the drop reaches a maximum at around $x_d \sim 1.4$ , before gradually diminishing to zero near the centreline (as it must because of axisymmetry) – this again qualitatively agrees with the findings of Hanna & Vlahovska (Reference Hanna and Vlahovska2010). The impact of the total bulk concentration ( $C_t$ ) is however more interesting and in fact depends strongly on the choice of $Ma$ . Recall that in figure 7(a), $U_x$ was shown to decrease with $C_t$ (for $C_t\gt 0.4$ ), when $Ma = 0.5$ (or larger); the opposite is observed when $Ma = 0.1$ . It is also expected that these trends for $U_x$ will remain qualitatively unchanged regardless of the specific location of the drop. Thence, it follows that increasing the total bulk concentration will facilitate faster cross-stream migration when $Ma = 0.1$ , while the reverse should be true for $Ma = 0.5$ . These assertions are clearly found to be satisfied in panels (a) (for $Ma = 0.1$ ) and (b) (for $Ma = 0.5$ ).

Indeed, recent experiments by Santra et al. (Reference Santra, Das, Das and Chakraborty2018) suggest that relatively augmented levels of dissolved surfactants in the bulk may suppress cross-stream migration of drops. A direct quantitative comparison with their results is difficult because of a few critical differences between the two studies. Santra et al. conducted their experiments in a rectangular microchannel, whereas our analysis considers flow inside a circular tube. At the same time, in their experiments, the surfactant was only present inside the aqueous droplet phase, whereas in the present work, they are present in the carrier phase, while no surfactants penetrate the droplet phase. Furthermore, confinements and the deformation of the drop also played a role in migration, but their impact has been ignored in our work. In fact, their results show that droplets undergo cross-stream migration even in the absence of surfactants, primarily due to droplet deformation (as seen in figure 6 in their paper). The presence of surfactants suppressed the deformation, but possibly also caused migration on its own. Further increasing its concentration reduced the extent of migration – a feature that qualitatively agrees with our results. Nevertheless, we can attempt to approximate the Marangoni number based on Santra et al.’s (Reference Santra, Das, Das and Chakraborty2018) results. To this end, we note the following parameter values: the reference temperature: $T' = 298.15$ K; viscosity of the carrier phase (sunflower oil): $\mu ^{\prime}_1 = 0.04914$ Pa s; centreline velocity: $U^{\prime}_0 \approx 8.5 \times 10^{-4}$ m s⁻¹ (from figure 4 in their work); equilibrium surfactant concentration: $\varGamma ^{\prime}_{eq} = 5 \times 10^{-7}$ mol m⁻ $^2$ at a bulk concentration of $100$ ppm, which corresponds to $C_t = 0.25$ (i.e. $25\,\%$ of the CMC $= 400$ ppm). Using these values and substituting $\varGamma ^{\prime}_{c} \approx \varGamma ^{\prime}_{eq}$ , we estimate the Marangoni number as $Ma \approx 29.67$ . Since the interfacial concentration at the CMC ( $\varGamma ^{\prime}_{\textit{cmc}}$ ) is expected to be higher than the equilibrium value at 100 ppm ( $\varGamma ^{\prime}_{\textit{cmc}} \gt \varGamma ^{\prime}_{eq}$ ), the actual Marangoni number defined as based on $\varGamma ^{\prime}_{\textit{cmc}}$ in their experiment would be $Ma \gt 29.67$ . Notwithstanding the major differences between the two studies stated earlier, as per our results, any increment in the bulk surfactant concentration should suppress cross-stream migration at such high Marangoni numbers, as also observed in their experiments.

Figure 11.

Plot of the cross-stream position of the drop ( $b = x_d$ ) versus its streamwise position ( $z_d$ ) for (a) various $Bi\:(=0.1,\;0.6,\;\text{and}\;1.4)$ with $\omega = 10,\;C_t = 0.8$ , and (b) various $\omega \:(=2,\;6,\;\text{and}\;10)$ with $Bi = 0.1$ and $C_t = 0.8$ . (c,d) Cross-stream migration velocity ( $U_x$ ) versus $x_d$ for the same choices of parameters respectively as in (a) and (b). Other parameters are the same as in figure 8.

Figure 11 depicts the influence of the Biot number $Bi$ (panels a,c) and the bulk interaction parameter $\omega$ (panels b,d) on the drop trajectories for $C_t = 0.8$ . Panels (a) and (b) depict the variations in $x_d(t)$ relative to $z_d(t)$ , while panels (c) and (d) illustrate the corresponding variations in $U_x$ as a function of $x_d$ ; values of all other relevant parameters have been mentioned in the caption. Recall from figure 8(a) that a relatively larger $Bi$ diminishes $U_x$ by making the interfacial surfactant concentration close to uniform, while figure 8(d) established that increments in $\omega$ counteract this by hindering adsorption–desorption, and therefore, results in increased $U_x$ . Again, one expects that these trends will remain uncharged regardless of the location of the drop; thus, increments in $Bi$ should force the drop to cover a larger distance before approaching the centreline, while increments in $\omega$ should lead to the opposite effect. This is exactly what is noted in panels (a) and (b). Furthermore, the overall variations in $U_x$ with $x_d$ also remain qualitatively identical to those in figure 10.

The impact of specific micelle properties ( $K_m$ , $Pe_m$ , $N$ ) on the droplet trajectory is presented in figure 13, included in Appendix B.2. One finds their influence on the overall trajectory to be quite limited, as also suggested by figure 9. In essence, the presence of micelles in highly concentrated surfactant solutions thus play a significant role in influencing droplet trajectories by altering the monomer concentrations, even without directly interacting with the droplet. This remains one of the most novel findings of the present study.