4.1.1. Drag reduction

Figure 4(a) shows how ${DR}_0$ varies with the bulk diffusion ($\alpha$) and surfactant strength ($\gamma$), for $\beta = 1$. In § 4.1, we set $\alpha = \delta$ for simplicity. However, general asymptotic solutions are derived for any $\delta$. The governing equations simplify in different regions of the parameter space, where subsets of the terms in (3.30) are dominant. These distinct physical balances are reflected by limits or transitions of ${DR}_0$, as well as variations in the concentration profiles. The three primary areas of the parameter space, regions $M$, $D$ and $A$, are separated by black lines in figure 4(a). They are analysed asymptotically in Appendix C. Another more general region ($G$) exists where cross-channel gradients in concentration can be comparable to the cross-channel average (shaded in figure 4a). As region $G$ lies beyond the model predictions, its boundaries delimit the domain of validity of the model (Appendix B.2). To contextualise the changes in ${DR}_0$, we examine how $c_0$ and $\varGamma _0$ vary across one period in figure 4(b–e). Recall that $\mathcal {D}_1$ and $\mathcal {D}_2$ represent the subdomains over the plastron and solid ridge, respectively ($\mathcal {D}_1$ and $\mathcal {D}_2$ are separated by the vertical dotted line at $x = \phi _x=0.5$), and $c_0 = \varGamma _0$ in $\mathcal {D}_1$.

Figure 4. The leading-order drag reduction (${DR}_0$) and surfactant distribution ($c_0=\varGamma _0$) in the strong-exchange problem, for $\beta = 1$, $\phi _x = 0.5$, $\epsilon = 0.1$, $\phi _z = 0.5$, ${Da} = 1$ and $P_z = 0.5$, computed using (3.30). (a) Contours of ${DR}_0$, where ${DR}_0 = 0$ exhibits a no-slip SHS and ${DR}_0 = 1$ exhibits a shear-free plastron. The Marangoni ($M$), advection ($A$) and diffusion-dominated ($D$) regions are separated by black lines and ${DR}_0$ is approximated by (4.1), (4.7) and (4.5) for the $M$, $A$ and $D$ regions, respectively. The generalised ($G$) region shaded in brown is discussed in Appendix B.2. The dashed magenta lines describing when ${DR}_0 \approx 0.5$ are given by (4.3) and (4.4). (b–e) Plots of $c_0$ for varying surfactant strength ($\gamma$) and bulk diffusion ($\alpha$), where the asymptotic curves are: (b) $\alpha = \delta = 1$, $A$: – – (4.8), $M$: ${\cdot }{\cdot }{\cdot }$ (4.2) with $\gamma = 100$, (c) $\gamma = 0.1$, $A$: – – (4.8) with $\alpha = \delta = 0.01$, $D$: ${\cdot }{\cdot }{\cdot }$ (4.6) with $\alpha = \delta = 10$, (d) $\alpha = \delta = 0.01$, $A$: – – (4.8), $M$: ${\cdot }{\cdot }{\cdot }$ (4.2) with $\gamma = 100$, and (e) $\gamma = 100$, $M$: – – (4.2) with $\alpha = \delta = 1$, $D$: ${\cdot }{\cdot }{\cdot }$ (4.6) with $\alpha = \delta = 1000$. The star identifies the point of the $(\alpha, \gamma )$-plane where we examine the flow field in § 4.1.2.

To start our journey around the parameter space in figure 4(a), we address region $M$, in which Marangoni effects are sufficiently strong to render the interface almost immobile. That is, the surfactant is transported along the liquid–gas interface by the flow and accumulates at stagnation points, where the adverse Marangoni stress generated is large enough to reduce the streamwise velocity at the liquid–gas interface to negligible values along the whole length of the plastron. As shown in Appendix C.1, ${DR}_0$ is close to zero in region $M$ and takes the value

(4.1)\begin{equation} {DR}_0 \approx \frac{1}{\gamma}\left(\alpha + \delta + \frac{\epsilon^2 s}{\alpha} + \frac{\phi_x(E+1)}{\left(E-1\right)} \right)\quad \text{for}\ \gamma \gg \max\left(1,\alpha,\beta,\delta,\frac{\epsilon^2}{\alpha}\right), \end{equation}

where $E \equiv \exp (2\alpha (1-\phi _x)/(\alpha ^2 + \epsilon ^2 s_4))$ and $s \equiv s_1 + s_2 + s_3$ (note that $s > 0$ for all $\phi _z$, $P_z$ and ${Da}$ examined in Appendix B.1). The leading-order drag reduction can be simplified within two sub-regions: ${M}_{D}$, in which Marangoni effects start to compete with bulk diffusion, and ${M}_{G}$, in which Marangoni effects start to compete with shear dispersion. In the ${M}_{D}$ case, (4.1) reduces to ${DR}_0 \approx \alpha / (\gamma (1 - \phi _x)) + \delta / \gamma$ for $\gamma \gg (\alpha,\delta ) \gg \max (1,\beta,\epsilon ^2/\alpha )$, demonstrating how bulk diffusion (noting that $\alpha /\gamma \propto \hat {\mu } \hat {D}\hat {P}_z/(\hat {A}\hat {G} \hat {H}\hat {L}_d)$) and surface diffusion (noting that $\delta /\gamma \propto \hat {\mu } \hat {D}_I \phi _z \hat {P}_z / (\hat {A}\hat {G}\hat {H}^2)$) weaken the immobilising effects of surfactant. In the ${M}_{G}$ case, (4.1) reduces to ${DR}_0 \approx \epsilon ^2 (s_4 \phi _x + s (1 - \phi _x)) / (\gamma \alpha (1 - \phi _x))$ for $\gamma \gg \epsilon ^2 / \alpha \gg \max (1,\alpha,\beta,\delta )$, demonstrating how shear dispersion starts to mobilise the interface (noting that $\epsilon ^2/(\alpha \gamma ) \propto \hat {\mu } \hat {U}^2\hat {H}^5/(\hat {D}\hat {A}\hat {G}\hat {L}_d\hat {P}_z\hat {P}_x^2)$ shows a quadratic dependence on the bulk flow speed).

Representative concentration profiles in region $M$ are shown in figure 4(b,d,e). Here, the leading-order concentration solution in $\mathcal {D}_1$ (Appendix C.1) is linear with a shallow gradient,

(4.2)\begin{equation} c_{0} \approx 1 + \frac{\beta}{\gamma}\left(x - \frac{\phi_x (E + 1)}{E - 1}\right) \quad \text{for}\ \gamma \gg \max\left(1,\alpha,\beta,\delta,\frac{\epsilon^2}{\alpha}\right). \end{equation}

The gradient of $c_0$ is controlled by both advection and Marangoni effects (noting that $\beta /\gamma \propto 1/Ma= \hat {\mu }\hat {U}/(\hat {A}\hat {G})$, which shows that an advective flow is needed to set up a Marangoni gradient), providing a surface shear stress sufficient to immobilise the liquid–gas interface. From (4.2), the magnitude of the concentration in $\mathcal {D}_1$ is weakly regulated by fluxes driven by bulk diffusion or shear dispersion in domain $\mathcal {D}_2$, via the factor $\alpha +s_4 \epsilon ^2/\alpha$ in $E$ that appears due to continuity of concentration across $\mathcal {D}_1$ and $\mathcal {D}_2$.

The $M$ region transitions into the $G$ region across the $GM$ boundary, where the concentration field starts to develop appreciable cross-channel gradients. In Appendix B.2, we show that a 3D perturbation ($c_0'$) to the cross-channel-averaged surfactant field ($\langle c_0 \rangle$) becomes comparable to $\langle c_0 \rangle$ when Marangoni effects compete with shear dispersion, with $\gamma = O(\epsilon ^2/\alpha )$ for $\beta =O(1)$. Similar to streamwise diffusion, shear dispersion increases drag reduction when approaching the $GM$ boundary, as shown in figure 4(a). We conjecture that at 50 % drag reduction, shear dispersion becomes large enough to induce an appreciable cross-channel gradient, defining a possible boundary between regions $M$ and $G$. However, 3D numerical simulations would be required to test this hypothesis and determine precisely the $GM$ boundary, beyond which our asymptotic results (4.1) and (4.2) in region $M$ are no longer valid. Therefore, our asymptotic approximation below to determine the $GM$ boundary based on estimating ${DR}_0 = 0.5$ should be used qualitatively rather than quantitatively. At the $GM$ boundary, we find (Appendix C.2)

(4.3)\begin{equation} {DR}_0 \approx 0.5 \quad \text{when}\ \gamma = \frac{\epsilon^2(2 s_{1}(1 - \phi_x) + s_4 \phi_x (2 + \beta))}{2 \alpha \left(1- \phi_x\right)} \quad \text{for}\ \alpha \ll 1, \end{equation}

which gives the leftmost dashed magenta line in figure 4(a). This approximation agrees with the numerical solution of the 1D strong-exchange problem in the limit $\alpha \rightarrow 0$. As the immobilising effect of surfactant decreases across the $GM$ boundary, larger streamwise velocities in the cross-plane imply that $c_0$ decreases from $c_0\approx 1$ to smaller values along the channel (see figure 4d). This is due to the velocity and surfactant flux being fixed.

The $M$ region transitions into the $D$ region through the $DM$ boundary for $\gamma =O(\alpha )$ or $\gamma =O(\delta )$, which can be defined (see Appendix C.3) by

(4.4)\begin{equation} {DR}_0 \approx 0.5 \quad \text{when}\ \gamma = \delta + \frac{\alpha(2 + \beta \phi_x)}{2(1 - \phi_x)} \quad \text{for}\ \min(\alpha,\delta) \gg 1, \end{equation}

which gives the rightmost dashed magenta line in figure 4(a). The asymptotic approximation (4.4) agrees with the numerical solution of (3.30) (thick black line in figure 4a) for $\min (\alpha,\delta ) \gg 1$. Similar to the $GM$ transition, the immobilising effect of surfactant decreases as we move from $M$ to $D$ by increasing the strength of bulk diffusion. By increasing the strength of bulk diffusion, bulk and interfacial surfactant gradients are attenuated because bulk–surface exchange is strong; this allows the streamwise velocity and drag reduction to increase. As the velocity and surfactant flux are fixed, the increase in velocity leads to a decrease in $c_0$, as shown in figure 4(e).

In summary, the factors promoting drag reduction from a state of interfacial immobilisation (in region $M$) are bulk diffusion (moving into region $D$), shear dispersion (moving into region $G$) or reduced surfactant strength (moving into region $G$, $D$ or $A$, of which more details are given below). As we move into region $D$ (or $G$), the surfactant in the bulk of the channel spreads out via diffusion (or shear dispersion), transporting surfactant from areas of high concentration to low concentration. Then, because exchange between the bulk and interfacial surfactant is strong, the concentration fields rapidly equilibrate; the interfacial surfactant gradient is also attenuated by these diffusive processes and the drag reduction $DR_0$ increases.

We next turn our attention to region $D$, where diffusion is sufficiently strong for surfactant to be distributed almost uniformly along the bulk and interface. This implies that there are almost no Marangoni stresses to increase the drag and the liquid–gas interface is almost shear-free. As shown in Appendix C.4, ${DR}_0$ is close to unity in region $D$ and is given by

(4.5)\begin{equation} {DR}_0 \approx 1 - \frac{(1 - \phi_x)\gamma}{(1 +\phi_x \beta) \alpha + (1 - \phi_x)\delta} \quad \text{for}\ \min(\alpha,\delta) \gg \max(1,\gamma), \end{equation}

for $\beta =O(1)$. Equation (4.5) shows how drag-promoting Marangoni effects are weakened by strong diffusion in the bulk ($\alpha$) or at the interface ($\delta$). Figure 4(c,e) shows the surfactant profiles in region $D$ in the limit of strong diffusion, where $\alpha$ or $\delta$ are large compared with bulk advection and Marangoni effects. The leading-order solution in $\mathcal {D}_1$ is uniform along $x$, such that

(4.6)\begin{equation} c_{0} \approx \frac{\alpha + \delta(1-\phi_x)}{\alpha(\beta\phi_x + 1) + \delta(1-\phi_x)} \quad \text{for}\ \min(\alpha,\delta) \gg \max(1,\gamma), \end{equation}

demonstrating how diffusion eliminates gradients of $c_0$ throughout $\mathcal {D}_1$ and $\mathcal {D}_2$ (the weak gradients contributing to ${DR}_0$ in (4.5) appear at higher order and are detailed in Appendix C.4). Therefore, drag reduction in region $D$ is impeded by decreasing the bulk and surface diffusion (if $\gamma \gg 1$) or increasing the surfactant strength (i.e. moving into region $M$).

We finally consider region $A$, in which surfactants can adsorb onto the interface but generate weak Marangoni stresses, allowing the streamwise flow to advect the interfacial surfactant towards the downstream stagnation point. In this advection-dominated region, we show in Appendix C.5 that ${DR}_0$ is once again close to the shear-free value

(4.7)\begin{equation} {DR}_0 \approx 1 -\frac{\gamma}{2\phi_x\left(\beta +1\right)} \quad \text{for}\ \epsilon^2 \ll \alpha \ll 1, \quad \gamma \ll \min(1,\beta). \end{equation}

Region $A$ can also be divided into two sub-regions: ${A}_{G}$ which balances advection and shear dispersion, and ${A}_{D}$ which balances advection and diffusion; these balances give the same ${DR}_0$ as in (4.7). The corresponding concentration profiles in region $A$ are shown in figure 4(b–d). The leading-order solution in $\mathcal {D}_1$ exhibits a surfactant gradient that increases monotonically towards the downstream end of the plastron

(4.8)\begin{align} c_0 \approx \frac{1}{\beta +1} + \frac{\beta}{(\beta+1)}\exp\left(\frac{(1+\beta)(x-\phi_x)}{\alpha +\delta + \epsilon^2 s_1 / \alpha }\right) \quad \text{for}\ \epsilon^2 \ll \alpha \ll 1,\enspace \gamma \ll \min(1,\,\beta). \end{align}

The downstream boundary layer in (4.8) can be short compared with the plastron length, such that the surface shear stress is negligible almost everywhere on the interface and the bulk flow experiences a largely shear-free boundary.

Region $A$ transitions into region $D$ across the $AD$ boundary, where advection and diffusion balance. The concentration and drag reduction at the $AD$ boundary can be obtained in closed form following the method shown in Appendix C.5. However, in figure 4(a), we see that crossing the $AD$ boundary (shown by the vertical black line at $\alpha =1$ and $\gamma \ll 1$) does not affect ${DR}_0$ to leading order. When $\gamma \ll \min (1,\beta )$ and the surfactant strength is weak, either the streamwise velocity at the surface is large enough to advect most of the interfacial surfactant to the downstream stagnation point ($\alpha \ll 1$) or diffusion is strong enough to attenuate any interfacial surfactant gradient that forms ($\alpha \gg 1$). As the interface is mostly shear-free in both $A$ and $D$, we do not pursue this limit. Bulk diffusion dominates over bulk advection for $\alpha \gg O(1)$, so we employ $\alpha = 1$ to illustrate the $AD$ boundary in figure 4(a) (via the thick vertical black line on the right of the $A$ region). When bulk diffusion is weak, shear dispersion dominates over bulk advection for $\epsilon ^2 / \alpha \gg 1$ (Appendix B.2), so we use $\alpha = \epsilon ^2$ to illustrate the $AG$ boundary (the thick vertical black line on the left of the $A$ region in figure 4a). Region $A$ transitions into region $M$ across the $AM$ boundary, such that Marangoni effects dominate advection for $\gamma \gg \max (1,\beta )$.

We next investigate the dependence of the drag reduction on the partition coefficient ($\beta$) by evaluating ${DR}_0$ for $\beta =10$ in figure 5(a) and $\beta = 100$ in figure 5(b). As $\beta$ grows, the advective flux of surfactant at the liquid–gas interface increases, sweeping more of the interfacial surfactant towards the downstream stagnation point. This means that a larger portion of the upstream part of the interface is shear-free, however, the surfactant concentration remains unchanging at the downstream end of the interface. Overall, the net (nonlinear) effect with increasing $\beta$ is a reduction in drag. The limit $\beta \gg 1$ also corresponds to the near-insoluble surfactant limit, with nearly all surfactant being transported along the interface in region $\mathcal {D}_1$. A number of the asymptotic approximations given in (4.1)–(4.8) simplify for $\beta \gg 1$: (4.3) reduces to ${DR}_0\approx 0.5$ when $\gamma = \epsilon ^2 s_4 \beta \phi _x/(2\alpha (1-\phi _x))$ for $\alpha \ll 1 \ll \beta$, such that shear dispersion in $\mathcal {D}_2$ determines ${DR}_0$; (4.4) reduces to ${DR}_0\approx 0.5$ when $\gamma = \alpha \beta \phi _x/(2(1-\phi _x))$ for $\alpha \gg \beta \gg 1$ and $\alpha = O(\delta )$, with bulk diffusion determining ${DR}_0$.

Figure 5. The leading-order drag reduction (${DR}_0$) and surfactant distribution ($c_0=\varGamma _0$) in the strong-exchange problem, for $\phi _x = 0.5$, $\epsilon = 0.1$, $\phi _z = 0.5$, ${Da}=1$ and $P_z = 0.5$, computed using (3.30). Contours of ${DR}_0$ for (a) $\beta = 10$ and (b) $\beta = 100$, where ${DR}_0 = 0$ exhibits a no-slip SHS and ${DR}_0 = 1$ exhibits a shear-free plastron at the SHS. The Marangoni ($M$), advection ($A$) and diffusion-dominated ($D$) regions are separated by black lines and ${DR}_0$ is approximated by (4.1), (4.7) and (4.5) for the $M$, $A$ and $D$ regions, respectively. The dashed magenta lines describing when ${DR}_0 \approx 0.5$ are given by (4.9) and (4.4) for the $AM$ and $DM$ boundaries, respectively. Plots of $c_0$ for varying surfactant strength ($\gamma$) and $\alpha = \delta = 1$, where (c) $\beta = 10$, $A$: – – (4.8), $M$: ${\cdot }{\cdot }{\cdot }$ (4.2) with $\gamma = 1000$, and (d) $\beta = 100$, $A$: – – (4.8), $M$: ${\cdot }{\cdot }{\cdot }$ (4.2) with $\gamma = 10\,000$.

Another feature that emerges from figure 5(a,b) is the flattening of the central contours of ${DR}_0$ at the $AM$ boundary as $\beta$ increases for $\beta \gg 1$, due to advection at the interface becoming stronger. To extract this feature asymptotically, in Appendix C.6 we assume $\beta \gg 1 \gg \max (\alpha,\delta,\epsilon ^2/\alpha )$ and $\gamma = O(\beta )$ at the $AM$ boundary, showing that ${DR}_0 \approx 1 - \gamma /(2 \beta \phi _x)$ for $\gamma / \beta \leq 2\phi _x$ and $\epsilon ^2\ll \alpha \ll 1$ (the case where $\gamma / \beta > 2\phi _x$ is also considered in Appendix C.6 and gives $DR_0\approx -(1/\beta +\gamma \ln (1-2\beta \phi _x/\gamma )/(2\phi _x\beta ^2))$ provided $1-2\phi _x \beta / \gamma \gg \exp {(-\beta )}$). This demonstrates how a large surface advective flux weakens the immobilising effects of surfactant (noting that $\gamma /\beta \propto \hat {A}\hat {G}/\hat {\mu }\hat {U}$). Using this expression for ${DR}_0$ at the $AM$ boundary, we predict that

(4.9)\begin{equation} {DR}_0 \approx 0.5 \quad \text{when}\ \gamma = \phi_x \beta \quad \text{for}\ \beta \gg 1 \gg \max\left(\alpha, \delta, \frac{\epsilon^2}{\alpha}\right), \end{equation}

which gives the central dashed magenta lines in figure 5(a,b) (agreement with the 1D numerical solution of the strong-exchange problem improves for increasing $\beta$). At the $AM$ boundary, the surfactant concentration at the interface exhibits a stagnant cap (He et al. Reference He, Maldarelli and Dagan1991), or a piecewise-linear distribution, if $\gamma /\beta \leq 2\phi _x$, where the upstream interface is almost shear-free and the downstream interface is effectively no-slip. In Appendix C.6, we show that

(4.10)\begin{align} c_0 \approx \begin{cases} 0 & \text{for} \ -\phi_x \leq x \leq x_0, \\ \dfrac{\beta}{\gamma}(x - \phi_x) +1 & \text{for} \ x_0 \leq x \leq \phi_x, \end{cases}\quad \text{for} \min(\beta, \gamma) \gg 1 \gg \max\left(\alpha,\delta, \frac{\epsilon^2}{\alpha}\right), \end{align}

where $x_0 = \phi _x - \gamma /\beta$ (see, e.g., the red curve for $\gamma =100$ in figure 5d). If $\gamma /\beta > 2\phi _x$, then the interfacial concentration distribution at the $AM$ boundary is linear, $c_0\approx \beta /\gamma (x-\phi _x)+1$ at leading order. When the surface advection overcomes Marangoni effects, for $\beta \gg \gamma$, surfactants can accumulate at the downstream stagnation point. The blue curve for $\gamma = 10$ in figure 5(d) demonstrates that $c_0 \approx 0$ throughout most of $\mathcal {D}_1$: the large surfactant advective flux at the interface contracts the downstream boundary layer, making the interface almost shear-free. In summary, stronger nonlinearity in the form of stagnant cap profiles in the interfacial concentration can appear with increasing $\beta$ near the transition from region $A$ to $M$. This results in a sharp decline in ${DR}_0$ towards zero, which can be seen as the $AM$ transition becomes less smooth, i.e. the vertical distance between ${DR}_0=0.5$ and ${DR}_0 = 0.05$ contours decreases from figure 5(a) to (b) as the surface advection increases from $\beta =10$ to $\beta =100$. However, we note that the location of the transition is captured by the simple expression (4.9) when $\gamma /\beta \leq 2\phi _x$.

4.1.2. Flow field

Whenever there is partial drag reduction, $0 \leq {DR}_0 < 1$, the surfactant has a non-uniform concentration in $\mathcal {D}_1$, reducing the streamwise velocity at the plastron and generating a secondary flow in the bulk. To illustrate, videos of the leading-order streamwise velocity $u_0$, wall-normal velocity $v_1$, transverse velocity $w_1$ and the streamwise gradient of the surfactant distribution $c_{0x}$ are given in supplementary movie 1 available at https://doi.org/10.1017/jfm.2023.264, evaluated at a point in the parameter space where ${DR}_0 \approx 0.5$, shown by the star in figure 4(a). The corresponding flow field is shown in figure 6. Figure 6(a) shows $u_0$ at the centre of domain $\mathcal {D}_1$, where $x=0$. The streamwise velocity, built from the components shown in figure 2, attains a maximum value at the channel centre, decaying towards either SHS ($y = 0,2$) to satisfy no-slip at the solid wall ($-P_z \leq z \leq \phi _z$ and $\phi _z\leq z \leq P_z$), whilst allowing for slip over the plastron ($-\phi _z \leq z \leq \phi _z$). The slip velocity at $\mathcal {I}$ reduces as one progresses through $\mathcal {D}_1$, in accordance with the rise in $c_{0x}$ for increasing $x$; see figure 6(c) and supplementary movie 1. The associated secondary cross-channel velocity field $(v_1,w_1)$ shown in figure 6(b) (recall that the leading-order components are $(v_0,w_0)=(0,0)$) advects particles from $\mathcal {I}$ towards the centre of the channel ($y=1$ and $z=\pm P_z$). The absolute maximum of $v_1$ occurs above the centre of the plastron ($z=0$), whereas the absolute maximum of $w_1$ occurs above the transverse contact lines ($z=\pm \phi _z$). Supplementary movie 1 and figure 6(d) show how $v_1$ and $w_1$ grow in magnitude as one progresses through $\mathcal {D}_1$ along the positive $x$-direction as the interface is immobilised. As mentioned in § 3.2.1, the present long-wave model does not capture rapid adjustments of the flow field near the contact lines at $x= \pm \phi _x$, where domains $\mathcal {D}_1$ and $\mathcal {D}_2$ meet.

Figure 6. Contour maps of the flow field in the strong-exchange problem, for $\alpha = 0.1$, $\beta = 1$, $\gamma = 1.3$, $\delta = 0.1$, $\phi _x = 0.5$, $\phi _z = 0.5$ and $P_z = 0.5$, corresponding to 50 % normalised drag reduction given by the star in figure 4(a). (a) Leading-order streamwise velocity $u_0$ and (b) leading-order wall-normal velocity $v_1$ with $(v_1,w_1)$ streamlines at the centre of the plastron, $x=0$. (c) Leading-order streamwise velocity $u_0$ and (d) leading-order transverse velocity $w_1$ with $(u_0, w_1)$ streamlines at the interfaces, $y=0$ or $2$. The thick black lines in (a–d) represent the solid regions of the SHS.

4.2.1. Drag reduction

Figure 7(a) shows how ${DR}_0$ varies with surface exchange strength ($\nu$) and surfactant strength ($\gamma$), for $\alpha = 1$, $\beta = 1$ and $\delta =1$. Similar to the strong-exchange problem in § 4.1.1, the moderate-exchange governing equations (3.1)–(3.2) simplify in different regions of the parameter space, representing distinct physical balances. The analysis in the moderate-exchange problem is simplified by the fact that for $\nu /\epsilon ^2 \gg \max (1,\alpha,\beta,\delta )$ the moderate-exchange problem transitions to the strong-exchange problem, which is discussed in detail in § 4.1.1. In the strong-exchange limit, $\nu /\epsilon ^2 \gg \max (1,\alpha,\beta,\delta )$, we reach the $AD$ boundary and region ${M}$ identified in figure 4(a). Accordingly, the regimes $AD$ and ${M}$ are also identified in figure 7(a) where bulk–surface exchange is strong. New regimes ${M}_{E}$ and ${AD}_{E}$ appear for weak exchange, $\nu /\epsilon ^2 \ll \max (1,\alpha,\beta,\delta )$, and are analysed asymptotically in Appendix D. The four primary areas of the parameter space, namely ${M}_{E}$, ${AD}_E$, $M$ and $AD$, are separated by black lines in figure 7(a). Corresponding surface concentration distributions $\varGamma _0(x)$ (where, unlike the strong-exchange problem, $c_0 \neq \varGamma _0$ in $\mathcal {D}_1$) are illustrated in figure 7(b–e). The primary feature of figure 7(a) is that the drag-reduction transition from large $DR_0$ to small $DR_0$ shifts only modestly, despite the exchange strength $\nu$ varying across many orders of magnitude, for reasons that we explain in the following paragraph.

Figure 7. The leading-order drag reduction (${DR}_0$), bulk surfactant ($c_0$) and interfacial surfactant distribution ($\varGamma _0$) in the moderate-exchange problem, for $\alpha = 1$, $\beta = 1$, $\delta = 1$, $\phi _x = 0.5$, $\epsilon = 0.1$, $\phi _z = 0.5$ and $P_z = 0.5$, computed using (3.1)–(3.3a–d). (a) Contours of ${DR}_0$, where ${DR}_0 = 0$ exhibits a no-slip SHS and ${DR}_0 = 1$ exhibits a shear-free plastron. The Marangoni ($M$), advection–diffusion ($AD$), Marangoni-exchange (${M}_{E}$) and advection–diffusion-exchange (${AD}_{E}$) regions are separated by black lines and ${DR}_0$ is approximated by (4.1), (4.7), (4.11a,b) and (4.15) in the $M$, $AD$, ${M}_{E}$ and ${AD}_{E}$ regions, respectively. The dashed magenta lines describing when ${DR}_0\approx 0.5$ are given by (4.4) and (4.13). Plots of $\varGamma _0$ and $c_0$ for varying surfactant strength ($\gamma$) and exchange strength ($\nu$), where the asymptotic curves are: (b) $\nu = 10$, $AD$: – – (4.2) with $\gamma = 100$, $M$: ${\cdot }{\cdot }{\cdot }$ (4.8), (c) $\gamma = 0.03$, ${AD}_{E}$: – – (4.16), $AD$: ${\cdot }{\cdot }{\cdot }$ (4.8), (d) $\nu = 0.001$, ${AD}_{E}$: – – (4.16), ${M}_{E}$: ${\cdot }{\cdot }{\cdot }$ (4.12a,b) with $\gamma = 10$, and (e) $\gamma = 100$, ${M}_{E}$: – – (4.12a,b), $M$: ${\cdot }{\cdot }{\cdot }$ (4.2).

We begin our exploration of the parameter space in the moderate-exchange problem (figure 7) with region ${M}_{E}$, where Marangoni effects are strong and surfactant exchange between the bulk and interface is weak. In Appendix D.1 we show that ${DR}_0$ remains close to the immobilised value

(4.11a,b)\begin{equation} {DR}_0 \approx \frac{\delta}{\gamma} \quad \text{for}\ \gamma \gg \max(1,\alpha,\beta,\delta),\quad \frac{\nu}{\epsilon^2} \ll \min(1,\alpha,\beta,\delta). \end{equation}

Here, the immobilising effects of surfactant are weakened by strong interfacial diffusion ($\delta /\gamma \propto \hat {\mu } \hat {D}_I \phi _z \hat {P}_z / (\hat {A}\hat {G}\hat {H}^2)$). The corresponding concentration fields $c_0$ and $\varGamma _0$ are shown in figure 7(d,e). The leading-order solution on $\mathcal {I}$ in ${M}_{E}$ is given by

(4.12a,b)\begin{equation} \varGamma_{0} \approx 1 + \frac{\beta x}{\gamma} \quad \text{for}\ \gamma \gg \max(1,\alpha,\beta,\delta), \quad \frac{\nu}{\epsilon^2} \ll \min(1,\alpha,\beta,\delta). \end{equation}

The gradient of $\varGamma _0$ in region ${M}_{E}$ is controlled by the relative strength of surface advection and Marangoni effects, $\beta /\gamma$, which is sufficiently small to immobilise the liquid–gas interface. Furthermore, there is very little adsorption and desorption at the interface and the bulk and interfacial surfactant concentrations are close to their background values at leading order (i.e. $c_0 \approx 1$ and $\varGamma _0 \approx 1$). In contrast to (4.2) in $M$ and the strong-exchange problem of § 4.1.1, where $c_0=\varGamma _0$ and $\varGamma _0 < 1$ for all $x$, in ${M}_{E}$ and the weak-exchange problem, $c_0 \approx 1$ and $\varGamma _0 > 1$ for $x > 0$ (see, e.g., the dashed and blue curves in figure 7e), in order to satisfy the net flux condition (2.26) at leading order. Furthermore, $c_0$ has turning points for $-\phi _x < x < \phi _x$ where the adsorption–desorption fluxes attain a local maximum and then decrease towards the contact lines, as shown by the green and orange curves in the inset in figure 7(d).

Region ${M}_{E}$ gives way to region ${M}$ as the exchange strength increases. It is notable that ${DR}_0 \approx \delta /\gamma$ (see (4.11a,b)) is regulated mainly by surface diffusion in region ${M}_{E}$, whereas both bulk and surface diffusion control $DR_0$ in region ${M}$, where ${DR}_0 \approx \alpha / (\gamma (1 - \phi _x)) + \delta /\gamma$ (see § 4.1.1). This is due to the absence of the strong coupling between $\varGamma _0$ and $c_0$ in region ${M}_{E}$, which imposed $\varGamma _0=c_0$ in region ${M}$. In figure 7(a), we find that $\nu = 10 \epsilon ^2$ best captures the centre of this transition (this can also be tested by varying $\epsilon$). As exchange weakens and the bulk and interfacial surfactant move out of equilibrium, variations in $\varGamma _0$ increase, thus making the interface more susceptible to Marangoni effects that increase the drag.

Depending on the strength of diffusion (via the parameters $\alpha$ and $\delta$, which vary orthogonally to the $(\nu,\gamma )$-plane in figure 7a), the lower half of the $(\nu,\gamma )$-plane is composed of either a sub-region of $A$, $D$, or the transition region $AD$ between $A$ and $D$. In figure 7(a), because $\alpha =\delta =1$, this corresponds to the transition region $AD$. In contrast, in figure 8(a) we plot ${DR}_0$ for $\alpha = \delta =0.1$ and $\alpha = \delta =10$ in figure 8(b). The ${M}_{E}$ region transitions into the diffusion-exchange-dominated (${D}_{E}$) region across the ${M}_{E}{D}_{E}$ boundary for $\min (\alpha, \delta ) \gg O(1)$ (figure 8b), whereas the ${M}_{E}$ region transitions into the advection-exchange-dominated (${A}_{E}$) region across the ${M}_{E}{A}_{E}$ boundary for $\max (\alpha, \delta ) \ll O(1)$ (figure 8a). At the ${M}_{E}{D}_{E}$ boundary, the asymptotic analysis in Appendix D.2 shows that

(4.13)\begin{equation} {DR}_0 \approx 0.5 \quad \text{when}\ \gamma = \delta \quad \text{for} \min(\alpha, \delta) \gg 1,\enspace \frac{\nu}{\epsilon^2} \ll 1, \end{equation}

which gives the leftmost dashed line in figure 8(b). This asymptote agrees with the numerical solution presented in figure 8(b) for $\delta \gg 1$ and $\nu \rightarrow 0$. The range of validity for the asymptote given in (4.13) is extended to $\delta = O(1)$ to include the ${M}_{E}{AD}_{E}$ boundary, for which we give partial justification in Appendix D.3. At the ${M}_{E}{A}_{E}$ boundary, Appendix D.4 demonstrates that

(4.14)\begin{equation} {DR}_0 \approx 0.5 \quad \text{when}\ \gamma = \frac{\beta \phi_x}{4} + \frac{\delta}{2} \quad \text{for} \max(\alpha, \delta) \ll 1, \enspace \frac{\nu}{\epsilon^2} \ll 1, \end{equation}

which gives the leftmost dashed line in figure 8(a). This asymptote approximates the numerical solution presented in figure 8(a) where $\delta = 0.1$ as $\nu \rightarrow 0$. Agreement improves for smaller $\delta$. It is notable that the threshold (4.14) differs from the strong-exchange limit (4.9) by a factor of four.

Figure 8. The leading-order drag reduction (${DR}_0$), bulk surfactant ($c_0$) and interfacial surfactant distribution ($\varGamma _0$) in the moderate-exchange problem, for $\beta = 1$, $\phi _x = 0.5$, $\epsilon = 0.1$, $\phi _z = 0.5$ and $P_z = 0.5$, computed using (3.1)–(3.3a–d). Contours of ${DR}_0$ for (a) $\alpha = \delta = 0.1$ and (b) $\alpha = \delta = 10$, where ${DR}_0 = 0$ exhibits a no-slip SHS and ${DR}_0 = 1$ exhibits a shear-free plastron at the SHS. The Marangoni ($M$), advection ($A$), diffusion ($D$), Marangoni-exchange (${M}_{E}$), advection-exchange (${A}_{E}$), diffusion-exchange (${D}_{E}$) regions are separated by black lines and ${DR}_0$ is approximated by (4.1), (4.7), (4.5), (4.11a,b), (4.15) and (4.15) in M, A, D, ${M}_{E}$, ${A}_{E}$ and ${D}_{E}$, respectively. The dashed magenta lines describing when ${DR}_0 \approx 0.5$ are given by (4.4) in (a,b) (right curves), (4.13) in (b) and (4.14) (a). Plots of $\varGamma _0$ and $c_0$ for varying surfactant strength ($\gamma$), where (c) $\alpha = \delta = 0.1$ and $\nu = 10^{-4}$, ${A}_{E}$: – – (4.16), ${M}_{E}$: ${\cdot }{\cdot }{\cdot }$ (4.12a,b) with $\gamma = 10$, and (d) $\alpha = \delta = 10$ and $\nu = 0.01$, ${D}_{E}$: – – (4.17), ${M}_{E}$: ${\cdot }{\cdot }{\cdot }$ (4.12a,b) with $\gamma = 1000$. The star identifies the point of the $(\nu, \gamma )$-plane where we examine the flow field in § 4.2.2.

We next analyse the drag reduction in both the ${A}_{E}$ and ${D}_{E}$ regions, where advection or diffusion induce a near-uniform distribution of surfactant at the interface. In Appendix D.5 we show that ${DR}_0$ in regions ${A}_{E}$ and ${D}_{E}$ is close to the shear-free value

(4.15)\begin{equation} {DR}_0 \approx 1 - \frac{\gamma}{\delta} \quad \text{for}\ \gamma \ll \min(1,\alpha,\beta,\delta),\quad \frac{\nu}{\epsilon^2} \ll \min(1,\alpha,\beta,\delta). \end{equation}

Equation (4.15) shows how drag-promoting Marangoni effects are weakened by strong diffusion at the interface, when surfactant exchange between the bulk and interface is weak. Figure 8(c,d) shows the surfactant profiles in ${A}_{E}$ and ${D}_{E}$, the leading-order solution on $\mathcal {I}$ being given by

(4.16)\begin{align} \varGamma_{0} \approx \frac{2 \phi_x \beta \exp\left(\dfrac{\beta \left(\phi_x +x\right)}{\delta }\right)}{\delta \left(\exp\left(\dfrac{2 \beta \phi_x }{\delta }\right)-1\right)} \quad \text{for}\ \gamma \ll \min(1,\alpha,\beta,\delta),\quad \frac{\nu}{\epsilon^2} \ll \min(1,\alpha,\beta,\delta). \end{align}

From (4.16), weak surface diffusion, $\delta \ll O(1)$, decreases (increases) $\varGamma _0$ at the upstream (downstream) end of the interface. A downstream boundary layer forms to satisfy the mass balance condition (2.26), condensing the Marangoni effect to a small region near $x=\phi _x$. The streamwise velocity flows over a almost shear-free boundary at the upstream end of the interface. When surface diffusion is strong, $\delta \gg O(1)$, (4.16) reduces to

(4.17)\begin{equation} \varGamma_0 = 1 + \frac{\beta x}{\delta} \quad \text{for}\ \gamma \ll \min(1,\alpha,\beta,\delta),\quad \frac{\nu}{\epsilon^2} \ll \min(1,\alpha,\beta,\delta). \end{equation}

Noting that $\beta /\delta \propto \hat {H}^2 \hat {U}/(\hat {D}_I \phi _z\hat {P}_z)$, (4.17) demonstrates how the gradient of $\varGamma _0$ is controlled by the ratio of advection to diffusion at the interface, which allows surface diffusion to regulate ${DR}_0$ in (4.15). In figure 8(a,b), the horizontal boundary of the ${M}_{E}$ region (shown with a black solid line) moves downwards when surface diffusion decreases ($\delta =0.1$ in figure 8a) and upwards when it increases ($\delta =10$ in figure 8b). Figure 8(c,d) demonstrates how increasing surface diffusion attenuates gradients of surfactant at the interface, increasing the drag reduction from sub-region ${A}_{E}$ to ${D}_{E}$ in figure 8(a,b).

4.2.2. Flow field

Videos of $u_0$, $v_1$, $w_1$ and $\varGamma _{0x}$ are given in supplementary movie 2. Again, we choose an example in figure 9 for which ${DR}_0=0.5$, shown with a star in figure 8(a). Figure 9(a) shows that $u_0$ has a similar structure to the strong-exchange problem at $x=0$ (see figure 6a). However, when comparing figure 9(c) with figure 6(c), $u_0$ exhibits different behaviour along $\mathcal {I}$. The streamwise velocity decreases slowly at the upstream end of $\mathcal {I}$, $\varGamma _{0x}$ and $\varGamma _{0xx}$ increase slowly with $x$, and there is a uniform distribution of surfactant at the interface which is almost shear-free. The streamwise velocity decreases rapidly at the centre of $\mathcal {I}$, where there is a sudden change of surfactant gradient at the interface. The streamwise velocity then decreases at the downstream end of $\mathcal {I}$, where there is a more linear distribution of surfactant at the interface which is almost no-slip. The cross-channel flow in figure 9(b) closely resembles that observed in figure 6(b). In supplementary movie 2 and figure 9(d), the magnitudes of both $v_1$ and $w_1$ increase until they attain a maximum at the start of the low-slip region, after which, they decrease towards $x=\phi _x$. The streamwise location of the maximum of $v_1$ and $w_1$ is approximately the same as the streamwise location of the maximum of $\varGamma _{0xx}$ on $\mathcal {I}$, the ‘corner’ in the surfactant field.

Figure 9. Contour maps of the flow field in the moderate-exchange problem, for $\alpha = 0.1$, $\beta = 1$, $\gamma = 0.2$, $\delta = 0.1$, $\nu = 10^{-4}$, $\phi _x = 0.5$, $\phi _z = 0.5$ and $P_z = 0.5$, corresponding to 50 % normalised drag reduction given by the star in figure 8(a). (a) Leading-order streamwise velocity $u_0$ and (b) leading-order wall-normal velocity $v_1$ with $(v_1, w_1)$ streamlines at the centre of the plastron, $x=0$. (c) Leading-order streamwise velocity $u_0$ and (d) leading-order transverse velocity $w_1$ with $(u_0, w_1)$ streamlines at the interfaces, $y=0$ or $2$. The thick black lines in (a–d) represent the solid regions of the SHS.