2. Configuration, governing equations and numerical method

We consider a LIS texture consisting of a periodic array of rectangular transverse grooves subjected to steady laminar flow. Due to the texture periodicity, it is sufficient to consider one interface unit cell with a single groove, sketched in figure 1(a). The groove depth was $k = 50\,\mathrm {\mu }{\rm m}$, the width $w = 2k$, and the pitch $p = 3k$ (groove centre-to-centre distance).

Figure 1. (a) Interface unit cell, illustrating the balance of stresses at the interface. The velocity profile of the external flow is also shown (red). (b) The physical processes transporting surfactants in the bulk and on the interface and controlling their exchange (red, black and blue arrows, respectively). The external liquid is blue, the infusing liquid green and the solid brown.

The external and infusing fluids have viscosities $\mu _\infty$ and $\mu _i$, respectively ($\mu _\infty = 1.0$ mPa s for water), forming interfaces aligned with the ridges between the grooves. Flat, non-deformable interfaces are assumed in this study so that the sole effect of the surfactants is the Marangoni force. An increased deformation due to a reduction in surface tension is expected to be of secondary importance (Landel et al. Reference Landel, Peaudecerf, Temprano-Coleto, Gibou, Goldstein and Luzzatto-Fegiz2020).

The flow velocities and texture dimensions are small. Therefore, we use the Stokes equations for a steady incompressible flow,

(2.1a,b)\begin{equation} 0 ={-}\boldsymbol{\nabla} P + \boldsymbol{\nabla} \boldsymbol{\cdot} \mu\left(\boldsymbol{\nabla} \boldsymbol{u} + (\boldsymbol{\nabla} \boldsymbol{u})^\mathrm{T}\right) \quad \text{and} \quad \boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{u} = 0, \end{equation}

where $P$ is the pressure, $\mu$ is the fluid viscosity and $\boldsymbol {u} = (u,v)$ is the fluid velocity with streamwise and wall-normal components $u$ and $v$, respectively. The streamwise coordinate is $x$ with $x = 0$ in the centre of the considered fluid–fluid interface, and $y$ is the wall-normal coordinate with $y = 0$ at the interface. Equations (2.1a,b) are valid for both the external and infusing fluids. No-slip and impermeability conditions ($\boldsymbol {u} = 0$) were used at solid boundaries. At the interface, the wall-normal velocity was $v = 0$ and the streamwise velocity $u$ was assumed to be continuous. The balance of shear stress on the interface is (Leal Reference Leal2007)

(2.2)\begin{equation} \mu_\infty\left.\frac{\partial u}{\partial y}\right|_{y = 0^+} = \mu_i\left.\frac{\partial u}{\partial y}\right|_{y = 0^-} - \frac{\mathrm{d} \gamma}{\mathrm{d}\kern0.06em x}, \end{equation}

where $\gamma$ is the surface tension. The velocity gradients in (2.2) have been evaluated precisely above and below the interface ($y = 0^+$ and $0^-$, respectively). The imposed shear stress that drives the flow is assumed to be $\tau _\infty$ at $y \to \infty$.

In order to simplify subsequent notation, we introduce

(2.3a,b)\begin{equation} \tau_{LIS} = \mu_i\left.\frac{\partial u}{\partial y}\right|_{y = 0^-} \quad \text{and} \quad \tau_{{Ma}} ={-}\frac{\mathrm{d} \gamma}{\mathrm{d}\kern0.06em x}. \end{equation}

The balance of stresses is shown in figure 1(a). For no surfactants ($\tau _{{ {Ma}}} = 0$), (2.2) relaxes to the classical interface condition, and for $\tau _{LIS} = 0$, ideal (gas infused) SHS are regained.

The surfactants are assumed to be soluble in the water phase and can adsorb at the interfaces (see the sketch in figure 2). Surfactant transport mechanisms are illustrated in figure 1(b). An advection–diffusion equation governs the interfacial surfactant concentration $\varGamma$,

(2.4)\begin{equation} \frac{\mathrm{d}}{\mathrm{d}\kern0.06em x}(u_s \varGamma) = D_s\frac{\mathrm{d}^2 \varGamma}{\mathrm{d}\kern0.06em x^2} + S, \end{equation}

where $u_s$ is the velocity of the interface plane ($y = 0$), $D_s$ is the interface diffusivity and $S$ is a source term describing adsorption and desorption. The source term would be zero for insoluble surfactants. No-flux conditions apply at interface edges ($\mathrm {d} \varGamma /\mathrm {d}\kern 0.06em x = 0$), and (2.4) is valid for $-w/2 < x < w/2$. The surface coverage of the interfacial surfactants is $\theta = \varGamma /\varGamma _m = \omega \varGamma$, where $\varGamma _m$ is the maximum possible interface concentration and $\omega$ is the molar area. To increase brevity of the expressions throughout the paper, we use the non-dimensional surfactant concentration, $\theta$, but the theory is otherwise presented using dimensional quantities.

Figure 2. Schematic illustration of surfactants dissolved in water and adsorbed at a water–alkane interface. The interface is shown pinned to a corner of the solid texture. SDS and C$_{12}$TAB molecules consist of hydrocarbon tails with 12 carbon atoms and hydrophilic head groups (red). Alkane molecules are also hydrocarbon chains. Dodecane and hexane molecules have $12$ and $6$ carbon atoms, respectively, here assumed to be arranged in straight chains. The different phases have the same colours as in figure 1.

The equation governing bulk surfactant concentration $c$ is

(2.5)\begin{equation} \boldsymbol{\nabla} \boldsymbol{\cdot} (\boldsymbol{u}c) = D\nabla^2 c, \end{equation}

where $D$ is the (bulk) diffusivity. The adsorption and desorption of surfactants at an interface balance the diffusive flux:

(2.6)\begin{equation} D\left.\frac{\partial c}{\partial y}\right|_{y = 0} = S. \end{equation}

At solid boundaries, $\partial c/\partial y = 0$. Sufficiently far above the interface, we assume a constant surfactant concentration $c_0$. The diffusivities were set to $D = D_s = 7.0 \times 10^{-10}\,{\rm m}^2\,{\rm s}^{-1}$, which also was used by Peaudecerf et al. (Reference Peaudecerf, Landel, Goldstein and Luzzatto-Fegiz2017).

The source term and the Marangoni stresses are described in the following sections. We adopt surface tension and adsorption/desorption models for the two extensively studied surfactants SDS and C$_{12}$TAB on water–air (Peaudecerf et al. Reference Peaudecerf, Landel, Goldstein and Luzzatto-Fegiz2017) and water–alkane interfaces (Fainerman et al. Reference Fainerman, Aksenenko, Makievski, Nikolenko, Javadi, Schneck and Miller2019). SDS can be found in personal care products, and C$_{12}$TAB can, for example, be used to stabilise foam (Carey & Stubenrauch Reference Carey and Stubenrauch2009; Rosen & Kunjappu Reference Rosen and Kunjappu2012). These surfactants are illustrated schematically in figure 2. They have hydrophobic hydrocarbon tails with 12 carbon atoms but different hydrophilic (head) groups.

3. Analytical model of a viscous infusing liquid

To construct an analytical model of the flow over LIS with surfactants, we assume that the stresses $\tau _{LIS}$ and $\tau _{{ {Ma}}}$ are constant. Essentially, we combine the models presented by Schönecker, Baier & Hardt (Reference Schönecker, Baier and Hardt2014) and Landel et al. (Reference Landel, Peaudecerf, Temprano-Coleto, Gibou, Goldstein and Luzzatto-Fegiz2020). The former considered the flow over LIS without surfactants and assumed $\tau _{LIS}$ was constant. The latter work considered flow over SHS and modelled the Marangoni stresses $\tau _{{ {Ma}}}$ as constant. In this section, we present the analytical model of our system and refer the reader to Appendix B for derivations.

The (effective) slip length $b$ is defined by

(3.1)\begin{equation} U_s = b\left.\left\langle \frac{\partial u}{\partial y}\right\rangle\right|_{y = 0^+} = b\frac{\tau_\infty}{\mu_\infty}, \end{equation}

where $\left \langle \right \rangle$ gives the average in the streamwise direction ($-p/2 < x \le p/2$), and $U_s = \left \langle u \right \rangle$ at $y = 0$ is the slip velocity. The resulting analytical expression of the effective slip length is (B24)

(3.2)\begin{equation} b = b_{SHS}\beta_{LIS} \left(1 - \frac{\tau_{{Ma}}}{\tau_\infty}\right), \end{equation}

where we have introduced the ideal SHS slip length ($\mu _i/\mu _\infty = \tau _{{ {Ma}}}/\tau _\infty = 0$)

(3.3)\begin{equation} b_{SHS} ={-}\frac{p}{2{\rm \pi}}\ln(\cos \alpha), \end{equation}

with $\alpha = ({\rm \pi} /2)(w/p)$. The factor $\beta _{LIS} = C_t/(1 + C_t)$ describes the effects due to the viscous infusing liquid, where $C_t$ depends on the groove geometry and the viscosity ratio,

(3.4)\begin{equation} C_t = \frac{8\alpha D_t\mu_\infty/\mu_i}{\ln\left(\dfrac{1 + \sin(\alpha)}{1 - \sin(\alpha)}\right)}. \end{equation}

The parameter $D_t$ is a normalised maximum local slip length (B22). Equation (3.2) describes the slip length as a linear function of the Marangoni stress. Similarly, the velocity at the centre of the interface is (B25)

(3.5)\begin{equation} u_{s}^0 = u_{s,{SHS}}^0\beta_{LIS}\left(1 - \frac{\tau_{{Ma}}}{\tau_\infty}\right), \end{equation}

where

(3.6)\begin{equation} u_{s,{SHS}}^0 = \frac{\tau_\infty w}{4\mu_\infty\alpha}\mathrm{arccosh}\left( \frac{1}{\cos\alpha}\right). \end{equation}

What remains is to find an explicit expression for $\tau _{{ {Ma}}}$. However, we can compare the expression for $b$ in (3.2) to simulations of (2.1a,b) and (2.2) with an artificially applied $\tau _{{ {Ma}}}$. A comparison is shown in figure 3, with a convincing agreement. The two examined viscosity ratios were $\mu _i/\mu _\infty = 1.4$ and $0.33$, corresponding to water–dodecane and water–hexane interfaces, respectively (table 1). For both viscosity ratios, $b$ decreases from $b_{SHS}\beta _{LIS}$ to $0$ when $\tau _{{ {Ma}}}/\tau _\infty$ increases from $0$ to $1$, as predicted by (3.2). The resulting slip length is higher for water–hexane interfaces (for the same $\tau _{{ {Ma}}}/\tau _\infty$) because of the lower viscosity ratio.

Figure 3. Comparison of (3.2) to simulation results where $\tau _{{ {Ma}}}$ was applied artificially. The viscosity ratio was set to $\mu _i/\mu _\infty = 1.4$ or $0.33$ (corresponding to water–dodecane or water–hexane, respectively).

Table 1. Parameters used for SDS at water–air and water–dodecane interfaces and C$_{12}$TAB at water–hexane interfaces. Reg. Frumkin refers to the model expressed by (4.1) and (4.2) and Adv. Frumkin to (6.1), (6.2) and (6.4). For all set-ups, we assume $D = D_s = 7.0 \times 10^{-10}\,{\rm m}^2\,{\rm s}^{-1}$.

4. Adsorption and desorption with regular Frumkin kinetics

The interfacial adsorption and desorption rates determine the source term $S$. These rates have in recent studies of SHS been modelled by Frumkin kinetics (Peaudecerf et al. Reference Peaudecerf, Landel, Goldstein and Luzzatto-Fegiz2017; Landel et al. Reference Landel, Peaudecerf, Temprano-Coleto, Gibou, Goldstein and Luzzatto-Fegiz2020), which are consistent with the Frumkin isotherm (Chang & Franses Reference Chang and Franses1995). Therefore, we also adopt Frumkin kinetics for this investigation. The source term becomes

(4.1)\begin{equation} S = \frac{1}{\omega}\left(\kappa_{a} c_s\left(1 - \theta\right) - \kappa_{d}\theta e^{{-}2a\theta}\right), \end{equation}

where $c_s$ is the value of $c$ at the interfaces (dependent on $x$), $a$ is a constant, $\kappa _{a}$ is the adsorption coefficient and $\kappa _{d}$ is the desorption coefficient. If the surfactant molecules are mutually attractive, $a$ is positive, and if they are repulsive, $a$ is negative (Manikantan & Squires Reference Manikantan and Squires2020). In equilibrium, adsorption and desorption fluxes balance (i.e. $S = 0$), and it is sufficient to state $\kappa _{a}/\kappa _{d}$ instead of the individual values of $\kappa _{a}$ and $\kappa _{d}$.

The expression for the surface tension (equation of state) corresponding to the Frumkin isotherm is

(4.2)\begin{equation} \gamma = \gamma_c + \frac{nRT}{\omega}\left[\ln(1 - \theta) + a{\theta}^2\right], \end{equation}

where $\gamma _c$ is the surface tension for $\theta = 0$, $R = 8.314\,\mathrm {J}\,(\mathrm {K}\,\mathrm {mol})^{-1}$ is the universal gas constant, $T = 293\,\mathrm {K}$ (at $20\,^{\circ }{\rm C}$) is the absolute temperature and $n$ is a constant. The factor $n$ can account for the adsorption of counter-ions (Chang & Franses Reference Chang and Franses1995; Fainerman et al. Reference Fainerman, Aksenenko, Makievski, Nikolenko, Javadi, Schneck and Miller2019). Without a supporting electrolyte, SDS has $n = 2$, which is the value used here. We assume that the data used for C$_{12}$TAB were obtained using a solution with a sufficiently high concentration of supporting electrolyte so that $n = 1$. It is generally accepted that the equilibrium model (4.2) can be used in non-equilibrium conditions (Chang & Franses Reference Chang and Franses1995).

We performed simulations of SDS on water–dodecane interfaces with regular Frumkin kinetics ((4.1) and (4.2)). We increased the concentration $c_0$ in steps starting from $10^{-8}\,{\rm mol}\,{\rm m}^{-3}$. Equilibrium parameters were adapted from Fainerman et al. (Reference Fainerman, Aksenenko, Makievski, Nikolenko, Javadi, Schneck and Miller2019), compensated for not explicitly considering interactions of surfactant–alkane molecules (§ 6). These parameters are summarised in table 1 and result in an equilibrium-state surface tension shown in figure 4. The figure also contains experimental data from Fainerman et al. (Reference Fainerman, Aksenenko, Makievski, Nikolenko, Javadi, Schneck and Miller2019). Regular Frumkin kinetics can describe the measurements but not the correct asymptotic behaviour at minuscule concentrations.

Figure 4. Equilibrium surface tension ($S = 0$) of SDS on a water–dodecane interface, comparing experimental data of Fainerman et al. (Reference Fainerman, Aksenenko, Makievski, Nikolenko, Javadi, Schneck and Miller2019) with the regular Frumkin model and a Frumkin model taking into account alkane–surfactant interaction. The surface tension is $\gamma _c = 52.87\,{\rm mN}\,{\rm m}^{-1}$ for water–dodecane without surfactants at $20\,^{\circ }{\rm C}$ (Zeppieri, Rodríguez & López de Ramos Reference Zeppieri, Rodríguez and López de Ramos2001). In contrast to water–air interfaces, the regular Frumkin model does not exhibit the correct asymptotic behaviour when $c_s \to 0$ at the water–oil interfaces.

Next, we consider the non-equilibrium system by imposing an external shear stress. As a reference to SDS on water–dodecane interfaces, we also simulated SDS on water–air interfaces, starting at $c_0 = 10^{-7}\,{\rm mol}\,{\rm m}^{-3}$. Air–water parameters for equilibrium were taken from Prosser & Franses (Reference Prosser and Franses2001). The relatively low imposed shear stress was ${\tau _\infty = 0.33}$ mPa; the effects of increased shear stress are described in § 7. The non-equilibrium parameter $\kappa _d = 500\,{\rm s}^{-1}$ of the air–water interfaces corresponds to the value reported by Chang & Franses (Reference Chang and Franses1995). This value was determined using empirically modified Langmuir–Hinshelwood kinetics. This model allows $\kappa _d$ to vary with concentration and includes an exponential factor in the adsorption flux similar to the desorption flux (cf. (4.1)). The value used here corresponds to the lowest bulk concentration ($1.7\,{\rm mol}\,{\rm m}^{-3}$). For the water–dodecane system, we used both $\kappa _d = 500$ and $5\,{\rm s}^{-1}$, with minor differences. The latter implies a lower desorption rate. For all other simulations, we used $\kappa _d = 5\,{\rm s}^{-1}$.

Figure 5(a) shows the resulting slip lengths as a function of $c_0$, and figure 5(b) provides an enlarged view. The water–air and water–dodecane results are shown in blue and red, respectively. Results for the advanced adsorption model are also included in the figure (yellow markers, see § 6). The slip length is approximately five times higher for the SHS without surfactants ($c_0 = 0\,{\rm mol}\,{\rm m}^{-3}$). As $c_0$ increases, the slip length of both systems decreases. The bulk concentration giving a significantly decreased slip length ($b/(b_{SHS}\beta _{LIS}) \approx 0.5$) is around one order of magnitude smaller for the water–dodecane than the water–air system, marked by vertical dash-dotted lines in figure 5. For the water–air system and the water–dodecane systems ($\kappa _d = 500\,{\rm s}^{-1}$), these concentrations are $c_0 = 4\times 10^{-4}$ and $5\times 10^{-5}\,{\rm mol}\,{\rm m}^{-3}$, respectively. This difference reflects the equilibrium surface tension behaviour. In order to explain the results in more detail, we continue to develop the analytical model of § 3 in § 5. This analytical model is also included in figure 5 (dashed lines).

Figure 5. Parametric study of slip length at $\tau _\infty = 0.33$ mPa for different bulk concentrations of SDS with a water–air (blue) and a water–dodecane interface (red regular model, yellow advanced model), together with the analytical model (dashed lines). SHS and LIS results are shown in (a), and an enlarged view of the LIS results is shown in (b). Symbols refer to $\kappa _d = 5$ ($\circ$) and $500$ ($+$) s$^{-1}$. The vertical dash-dotted lines mark $\alpha _{diff} + \alpha _S = 1$, corresponding to $b/(b_{SHS}\beta _{LIS}) \approx 1/2$ (5.16).

The analytical model predicts that the slip length becomes independent of $\kappa _d$ when this parameter is sufficiently large. At these desorption rates, diffusion of bulk surfactants becomes the limiting process (Damköhler number ${ {Da}}_\delta \gg 1$). Temprano-Coleto et al. (Reference Temprano-Coleto, Smith, Peaudecerf, Landel, Gibou and Luzzatto- Fegiz2021) recently discussed this independence, using $\kappa _d = 0.75\,{\rm s}^{-1}$ with good agreement with the experimental results of SHS with surfactants naturally occurring in their experimental setting.

5. Analytical model of surfactant transport

The analytical model of surfactant transport discussed here uses similar core assumptions to previous works (Landel et al. Reference Landel, Peaudecerf, Temprano-Coleto, Gibou, Goldstein and Luzzatto-Fegiz2020). It is assumed that interfacial concentrations are low ($\theta \ll 1$) so that the governing equations (4.1) and (4.2) can be linearised. A second assumption is that bulk and interfacial surfactant concentrations vary linearly over the interfaces (figure 6). Such distributions imply approximately constant surface tension gradients, i.e. the uniformly retarded regime.

Figure 6. Illustration of the distributions of bulk and interfacial surfactant concentrations ($c$ and $\theta$, respectively) of the analytical model. It is assumed that $c$ and $\theta$ vary linearly over the interfaces. The infusing liquid and the solid colours are the same as in figure 1.

5.1. Modelling bulk exchange

For a steady flow, there is a balance between adsorption and desorption processes. The integral of (2.6) over the interfaces must then be zero,

(5.1)\begin{equation} D\int_{{-}w/2}^{w/2}\left.\frac{\partial c}{\partial y}\right|_{y = 0}\, \mathrm{d}\kern0.06em x = \int_{{-}w/2}^{w/2}S \,\mathrm{d}\kern0.06em x = 0. \end{equation}

This condition implies the existence of a point $x_0$ on an interface where $D\partial c/\partial y |_{y = 0} = S = 0$. As the wall-normal derivative of $c$ is zero at this location, we have $c_s \approx c_0$. The linearised source term (4.1) is

(5.2)\begin{equation} S \approx \frac{\kappa_{d}}{\omega}\left(c_{k} \frac{c_s}{c_0} - \theta\right), \end{equation}

where we have introduced the non-dimensional bulk surfactant concentration ${c_k = \kappa _ac_0/\kappa _d}$. From the linearised source term,

(5.3)\begin{equation} \theta \approx c_k, \quad \text{at} \ x = x_0. \end{equation}

Due to advection, $\theta$ decreases upstream and increases downstream of $x_0$. It is assumed that $x_0 \approx 0$, that $\theta$ varies around $c_{k}$ by $\pm \Delta \theta$, and in the same way $c_s$ around $c_0$ by $\pm \Delta c_s$. These assumed concentration distributions imply (using (2.6) and (5.2)),

(5.4)\begin{equation} \left.\frac{\partial c}{\partial y}\right|_{y = 0} \approx \frac{\kappa_d c_k}{D \omega}\left(\frac{\Delta \theta}{c_k} - \frac{\Delta c_s}{c_0}\right)\frac{-x}{w/2} = \frac{c_0}{w}{{Da}}\left(\frac{\Delta \theta}{c_k} - \frac{\Delta c_s}{c_0}\right)\frac{-x}{w/2}, \end{equation}

where ${ {Da}} = \kappa _a w/(\omega D)$ is the Damköhler number (Temprano-Coleto et al. Reference Temprano-Coleto, Smith, Peaudecerf, Landel, Gibou and Luzzatto- Fegiz2021).

From (5.2), a characteristic adsorption flux is given by

(5.5)\begin{equation} j_a = \kappa_d c_k c_s/(\omega c_0) = \kappa_a c_s/\omega \sim \kappa_a c_0/\omega, \end{equation}

in ${\rm mol}\,({\rm s}\,{\rm m}^2)^{-1}$. Equation (5.3) entails that the characteristic desorption flux is the same: $\kappa _d \theta /\omega \sim \kappa _d c_k/\omega = \kappa _a c_0/\omega$. A corresponding scale for the diffusive flux of bulk surfactants is $D c_0/w$ (2.6). Hence, ${ {Da}}$ expresses characteristic adsorption/desorption flux to diffusive flux of bulk surfactants. Corresponding rates are found by multiplication by $\omega$. The actual values of $\Delta \theta$ and $\Delta c_s$ are neglected, but we do include the characteristic sizes of $\theta$ and $c_s$. This interpretation of ${ {Da}}$ is also seen in (5.4); large diffusion flux and low adsorption result in smaller wall-normal derivative of $c$.

As pointed out by Landel et al. (Reference Landel, Peaudecerf, Temprano-Coleto, Gibou, Goldstein and Luzzatto-Fegiz2020), (5.4) can be used to estimate the boundary layer thickness $\delta$ of $c$. Approximating $\delta$ by

(5.6)\begin{equation} \left.\frac{\partial c}{\partial y}\right|_{y = 0, x ={-}w/2} \approx \frac{\Delta c_s}{\delta} \implies \frac{\Delta c_s}{c_0} \approx \frac{\Delta \theta}{c_k}\frac{{{Da}} \dfrac{\delta}{w}}{1 + {{Da}} \dfrac{\delta}{w}}. \end{equation}

The modified Damköhler number ${ {Da}}_\delta = { {Da}} \delta /w$ considers the diffusion length scale of bulk surfactants to be $\delta$, which is more appropriate than $w$ (Palaparthi et al. Reference Palaparthi, Papageorgiou and Maldarelli2006). We see that if the adsorption/desorption rate is much larger than the diffusion rate (${ {Da}}_\delta \gg 1$), $\Delta c_s/c_0 \approx \Delta \theta /c_k$. Otherwise, if ${ {Da}}_\delta \ll 1$, $\Delta c_s/c_0 \approx { {Da}}_\delta \Delta \theta /c_k$.

The bulk surfactant transport equation (2.5) implies that $\delta$ depends on $w$ and Péclet number ${ {Pe}} = Uw/D$, where $U = w\tau _\infty /\mu _\infty$ is the characteristic velocity at $y = w$. Diffusion between boundary layers of adjacent grooves also introduces a dependency on $p$. An analytical estimate (Appendix C) resulted in

(5.7)\begin{equation} \frac{\delta}{w} = \dfrac{\dfrac{1}{2}\sqrt{1 - \dfrac{w}{p}}}{\left(1 + \dfrac{2}{3}{{Pe}}\left(\dfrac{1}{2}\sqrt{1 - \dfrac{w}{p}}\right)^3\right)^{1/3}}. \end{equation}

The left relation of (5.6) was also used to compute $\delta$ explicitly. It was found that $\delta /w = 0.3$ was a relatively good approximation for all tested configurations for the current geometry and ${ {Pe}}$ (Appendix C and figure 12a,b), in agreement with (5.7). Deviations were around ${\pm }0.1$.

5.2. Interfacial surfactant transport balance

The transport equation for the interfacial surfactant concentration (2.4) expresses a balance between advection, diffusion and adsorption/desorption (compare with figure 1b). We integrate this equation from $x = -w/2$ to $x = x_0$ and use the boundary condition (2.6),

(5.8)\begin{equation} \left.\left( u_s\theta\right)\right|_{x=x_0} = D_s\left.\frac{\mathrm{d} \theta}{\mathrm{d}\kern0.06em x}\right|_{x=x_0} + D\omega\int_{{-}w/2}^{x_0} \left.\frac{\partial c}{\partial y}\right|_{y = 0} \,\mathrm{d}\kern0.06em x. \end{equation}

Using $x_0 \approx 0$ and (5.6) with the streamwise dependency (5.4), we have

(5.9)\begin{equation} \int_{{-}w/2}^{x_0} \left.\frac{\partial c}{\partial y}\right|_{y = 0} \,\mathrm{d}\kern0.06em x \approx \frac{c_0}{w}\int_{{-}w/2}^{0} \frac{\Delta \theta}{c_k}\frac{{{Da}}}{1 + {{Da}}\dfrac{\delta}{w}}\frac{-x}{w/2} \,\mathrm{d}\kern0.06em x = c_0\frac{1}{4}\frac{\Delta \theta}{c_k}\frac{{{Da}}}{1 + {{Da}}_\delta}. \end{equation}

With (5.3) and $\mathrm {d} \theta /\mathrm {d}\kern 0.06em x|_{x=x_0} \approx 2\Delta \theta /w$, we obtain an expression for the interface centre velocity,

(5.10)\begin{equation} u_s^0 \approx U\left(c_1 \frac{1}{{{Pe}}_s} + c_2\frac{{{Bi}}}{1 + {{Da}}_\delta}\right)\frac{\Delta \theta}{c_k} \end{equation}

with coefficients $c_1 \approx 2$ and $c_2 \approx 1/4$ resulting from the linear distributions. The interface Péclet number is defined by ${ {Pe}}_s = Uw/D_s$ and the Biot number is defined by ${ {Bi}} = c_0D \omega { {Da}} /(U c_k) = \kappa _d w/U$.

The Péclet number ${ {Pe}}_s$ is a measure of characteristic advection to diffusion rates of the interfacial surfactants, which with (5.3) can be written as

(5.11a,b)\begin{equation} r_{adv} = Uc_k/w \quad \text{and} \quad r_{diff} = D_sc_k/w^2, \end{equation}

respectively. The characteristic adsorption/desorption fluxes (5.5) correspond to a rate $\kappa _d c_k$. Hence, the Biot number expresses the adsorption/desorption rate to the advection rate of interfacial surfactants.

For a specific value of $\Delta \theta /c_k$, (5.10) implies that for $u_s^0/U$ to be close to zero, the characteristic advection of interfacial surfactants must dominate (i) diffusion of interfacial surfactants (${ {Pe}}_s \gg \Delta \theta /c_k$) and (ii) bulk exchange (cf. figure 1b). The bulk exchange may be limited by either adsorption/desorption rate (${ {Da}}_\delta \ll 1$) or diffusion of bulk surfactants (${ {Da}}_\delta \gg 1$). If ${ {Da}}_\delta \ll 1$, we must have ${ {Bi}} \ll \Delta \theta /c_k$, i.e. advection of interfacial surfactants must dominate over the adsorption/desorption rate. If ${ {Da}}_\delta \gg 1$, then we must have ${ {Bi}}/{ {Da}}_\delta \ll \Delta \theta /c_k$: advection of interfacial surfactants must dominate over the diffusion of bulk surfactants.

The Marangoni stress also dictates the velocity on the interface (3.5). This relationship and (5.10) give a condition for $\Delta \theta$, derived in the next section.

5.3. Corresponding slip length

Linearisation of the Marangoni stresses with surface tension (4.2) implies

(5.12)\begin{equation} \tau_{{Ma}} ={-}\frac{\mathrm{d} \gamma}{\mathrm{d}\kern0.06em x} ={-}\frac{\mathrm{d} \gamma}{\mathrm{d} \theta}\frac{\mathrm{d} \theta}{\mathrm{d}\kern0.06em x} \approx \frac{nRT}{\omega}\frac{\mathrm{d} \theta}{\mathrm{d}\kern0.06em x} = \mu_\infty U{Ma}\frac{\mathrm{d} \theta}{\mathrm{d}\kern0.06em x}, \end{equation}

where we have introduced the Marangoni number ${ {Ma}} = nRT/(\omega \mu _\infty U)$. We have chosen to neglect thermal effects ($T$ is a constant). The gradient of the surface coverage $\mathrm {d} \theta /\mathrm {d}\kern 0.06em x$ is generally positive (figure 6), meaning that $\mathrm {d} \gamma /\mathrm {d}\kern 0.06em x$ is negative, and the Marangoni stresses act in the negative streamwise direction, as shown by figure 1(a). By assuming linear interfacial concentrations, $\mathrm {d} \theta /\mathrm {d}\kern 0.06em x$ can be estimated as $2\Delta \theta /w$, resulting in

(5.13)\begin{equation} \tau_{{Ma}} \approx 2\mu_\infty U{Ma}\frac{\Delta \theta}{w}. \end{equation}

It follows from (3.5) that

(5.14)\begin{equation} u_{s}^0 \approx u_{s,{SHS}}^0\beta_{LIS}\left(1 - 2{Ma}\Delta \theta \right). \end{equation}

Equation (5.13) implies a scaling $nRT/(\omega w)$ of the Marangoni stress, neglecting $\Delta \theta$. As $\tau _\infty = \mu _\infty U/w$, the Marangoni number expresses the ratio of characteristic Marangoni to imposed shear stresses. If the characteristic size of $\theta$ is considered, the Marangoni number transforms to ${ {Ma}} c_k$.

The velocities expressed by (5.10) and (5.14) must be the same. This equality results in the expression

(5.15a,b)\begin{equation} \left.\Delta \theta \approx \frac{ u_{s,{SHS}}^0\beta_{LIS}}{U}c_k\right/\left(c_1 \frac{1}{{{Pe}}_s} + c_2\frac{{{Bi}}}{1 + {{Da}}_\delta} + 2\frac{u_{s,{SHS}}^0\beta_{LIS}}{U}{Ma} c_k\right), \end{equation}

which by (5.13) gives $\tau _{{ {Ma}}}$. Equation (3.2) then gives the slip length

(5.16)\begin{equation} b \approx b_{SHS}\beta_{LIS}\left(1 - \frac{1}{\alpha_{diff}+ \alpha_S + 1}\right). \end{equation}

We have introduced

(5.17)\begin{equation} \alpha_{diff} = c_1\frac{1}{{{Pe}}_s'}\frac{1}{2{Ma} c_k} \quad \text{and} \quad \alpha_S = c_2\frac{{{Bi}}' }{1 + {{Da}}_\delta}\frac{1}{2{Ma} c_k}, \end{equation}

where ${ {Pe}}_s' = wu_{s,{SHS}}^0\beta _{LIS}/D_s$ and ${ {Bi}}' = \kappa _d w /(u_{s,{SHS}}^0\beta _{LIS}$) are the Péclet and Biot numbers, respectively, with more appropriate velocity scales. The two expressions of (5.17) represent the effects of interfacial surfactant diffusion and bulk exchange on the slip length, respectively. In order to reduce $b$, they must both be small, $\alpha _{diff} \lesssim 1$ and $\alpha _S \lesssim 1$. Based on the interpretations of the non-dimensional numbers,

(5.18)$$\begin{gather} \alpha_{diff} \sim \frac{\text{diffusion rate of interfacial surfactant}}{\text{advection rate of interfacial surfactant}}\frac{\text{imposed shear stress}}{\text{Marangoni stress scale}}, \end{gather}$$

(5.19)$$\begin{gather}\alpha_{S} \sim \dfrac{\dfrac{\text{adsorption/desorption rate}}{\text{advection rate of interfacial surfactant}}}{1 + \dfrac{\text{adsorption/desorption rate}}{\text{diffusion rate of bulk surfactant}}}\dfrac{\text{imposed shear stress}}{\text{Marangoni stress scale}}. \end{gather}$$

Returning to figure 5(a), we can plot the results also from the analytical model, showing a satisfactory agreement with the simulation results. Some central non-dimensional numbers are summarised in table 2.

Table 2. Non-dimensional numbers corresponding to the results presented in §§ 4, 6 and 7.

If $\alpha _S \gg \alpha _{diff}$, the diffusion of interfacial surfactants is insignificant compared with the bulk exchange. In the opposite situation, $\alpha _S \ll \alpha _{diff}$, the interfacial diffusion dominates (insoluble limit). For the $\kappa _d = 500\,{\rm s}^{-1}$ water–dodecane and water–air systems, the bulk exchange is much faster than interfacial diffusion ($\alpha _S \gg \alpha _{diff}$). The parameter $\alpha _S$ thereby governs the decrease of slip length for these systems. The values of ${ {Da}}_\delta$ are $97$ and $14$, respectively. As ${ {Da}}_\delta \gg 1$, the bulk exchange rate is limited by the diffusion of bulk surfactants, implying that $\alpha _S$ depends on $\kappa _d/\kappa _a$ (${ {Bi}}'/{ {Da}}_\delta$) instead of $\kappa _d$ (${ {Bi}}'$). For the water–dodecane system, $\kappa _d = 5\,{\rm s}^{-1}$ corresponds to ${ {Da}}_\delta = 0.97$, below which the condition (5.16) becomes more strict as then $\alpha _S$ becomes approximately proportional to $\kappa _d$.

Figure 7 illustrates how $\alpha _{diff}$ and $\alpha _S$ can be used to predict whether there will be a considerable decrease in slip length. These two parameters span a two-dimensional space. If

(5.20)\begin{equation} \alpha_{diff} + \alpha_S = 1, \end{equation}

the slip length has halved compared with surfactant-free interfaces ($b/(b_{SHS}\beta _{LIS}) \approx 0.5$). In the region bounded by (5.20), the slip length decrease is larger and outside it is smaller. We use this limit to denote a significant slip length reduction, but other threshold values could also be used. As we only varied $c_0$, $\alpha _S/\alpha _{diff}$ is constant, describing a straight line in the $(\alpha _{diff},\alpha _S)$ space, shown in figure 7 for the water–air and the $\kappa _d = 5\,{\rm s}^{-1}$ water–dodecane systems (cf. figure 5). The water–dodecane system has $\alpha _S$ and $\alpha _{diff}$ of similar magnitude ($\alpha _S/\alpha _{diff} = 4.5$). Bulk exchange is more prominent than interfacial diffusion, but both are considerable. For the air–water system, bulk exchange dominates and the points are shifted towards the upper left corner of the figure.

Figure 7. Normalised slip length $b/(b_{SHS}\beta _{LIS})$ of the SDS water–air and the SDS $\kappa _d = 5\,{\rm s}^{-1}$ water–dodecane systems ($\alpha _S/\alpha _{diff} = 59$ and $4.5$, respectively), plotted in the space spanned by $\alpha _{diff}$ and $\alpha _S$. The dash-dotted line shows $\alpha _{diff} + \alpha _S = 1$.

6. Adsorption and desorption taking into account alkane–surfactant interaction

The bulk exchange model presented in § 4 resulted in a more considerable surface tension decrease for the LIS than the SHS for the same SDS concentration. However, it cannot capture the initial decrease in surface tension at low concentrations appearing in water–alkane systems. To be able to describe this phenomenon, more advanced models are needed.

The water–alkane interface abnormalities must be caused by interactions between adsorbed surfactants and alkane molecules adjacent to the interface. The level of interaction between adsorbed surfactant and alkane molecules at the interface can be modelled by an effective alkane interface concentration $\varGamma ^a$ (Fainerman et al. Reference Fainerman, Aksenenko, Makievski, Nikolenko, Javadi, Schneck and Miller2019). It is associated with a molar area $\omega ^a \sim \omega$ and surface coverage $\theta ^a = \omega ^a\varGamma ^a$. This interaction has recently been investigated on molecular levels (Kartashynska et al. Reference Kartashynska, Lylyk, Aksenenko, Makievski, Vysotsky, Fainerman and Miller2020; Müller et al. Reference Müller, Bonthuis, Miller and Schneck2021). In this paper, we adopt a model consistent with the equilibrium model of Fainerman et al. (Reference Fainerman, Aksenenko, Makievski, Nikolenko, Javadi, Schneck and Miller2019). They assumed that the molar area $\omega$ decreased with surface coverage. However, this decrease is only notable for higher concentrations ($\theta \sim 1$), and, therefore, we neglect this correction.

The interfacial and bulk surfactant concentrations are assumed to follow the same transport equations as the previous model ((2.4) and (2.5), respectively). However, the source term is (cf. (4.1))

(6.1)\begin{equation} S = \frac{1}{\omega}\left(\kappa_{a} c_s\left(1 - \theta^t\right) - \kappa_{d}\theta e^{{-}2a\theta - 2a^{as}\theta^a}\right), \end{equation}

where we have introduced the additional interaction constant $a^{as}$ and the total surface coverage $\theta ^t = \theta + \theta ^a$. The adsorption term contains the total surface coverage, but the desorption term maintains its $\theta$-dependency (together with an additional exponential factor).

We assume that the alkane molecules are in local equilibrium with the adsorbed surfactants, resulting in the corresponding expression

(6.2)\begin{equation} \kappa_a^a c_0^a\left(1 - \theta^t\right) - \kappa_d^a\theta^a e^{{-}2a^a\theta^a - 2a^{as}\theta} = 0, \end{equation}

where $\kappa _a^a$ and $\kappa _d^a$ are the adsorption and desorption coefficients of the alkane phase, respectively, $a^a$ is a constant and $c_0^a$ is the (constant) bulk alkane concentration. The current model assumes that the alkane adsorption coefficient depends on $\theta$ by

(6.3)\begin{equation} \kappa_a^a = \min(\kappa_{a,0}^{a}\theta, \kappa_{a,{max}}^a), \end{equation}

where $\kappa _{a,0}^{a}$ and $\kappa _{a,{max}}^a$ are constants. This expression implies that the alkane adsorption rate increases proportionally to $\theta$ for minuscule concentrations. Without surfactants, $\theta ^a = 0$.

The surface tension has been modelled by (cf. (4.2))

(6.4)\begin{equation} \gamma = \gamma_c + \frac{RT}{\omega_0}\left[\ln(1 - \theta^t) + a\theta^2 + a^a{\theta^a}^2 + 2a^{as}\theta \theta^a\right], \end{equation}

where

(6.5)\begin{equation} \omega_0 = \frac{1}{\theta^t}\left(\frac{\omega}{n}\theta + \omega^a\theta^a\right) \end{equation}

is the effective average molar area.

The results plotted in figure 5 show that this improved modelling lowers the critical concentration by an additional order of magnitude. Corresponding results for C$_{12}$TAB and water–hexane LIS are shown in figure 8. We only used the more advanced adsorption/desorption model for C$_{12}$TAB (parameters given in table 1). This system manifests a stronger sensitivity to the surfactants, as $b/p$ decreases significantly at even lower concentrations. The simulations are illustrated in the $(\alpha _{diff},\alpha _S)$ space in figure 8(c). In contrast to SDS (figure 7), C$_{12}$TAB have more prominent interfacial diffusion than bulk exchange ($\alpha _S/\alpha _{diff} = 0.30$). Therefore, the simulations are closer to the bottom right corner of the figure.

Figure 8. (a) Equilibrium surface tension of C$_{12}$TAB at a water–hexane interface with experimental data from Pradines et al. (Reference Pradines, Fainerman, Aksenenko, Krägel, Mucic and Miller2010). Without surfactants, the surface tension is $\gamma _c = 50.8\,{\rm mN}\,{\rm m}^{-1}$ for water–hexane ($20\,^{\circ }{\rm C}$, Zeppieri et al. Reference Zeppieri, Rodríguez and López de Ramos2001). (b) Slip lengths at $\tau _\infty = 0.33$ mPa for different bulk concentrations of C$_{12}$TAB at a water–hexane interface, together with the analytical model (dashed lines). The correspondence for SDS is shown in figure 5(a). (c) Normalised slip length, $b/(b_{SHS}\beta _{LIS})$, plotted in the parameter space of $\alpha _{diff}$ and $\alpha _S$. In both (b,c), the dashed-dotted lines are $\alpha _{diff} + \alpha _S = 1$, corresponding to $b/(b_{SHS}\beta _{LIS}) \approx 1/2$ (5.16).

6.1. Analytical model with alkane–surfactant interaction

Even if the source term has been changed slightly, the analytical model developed in § 5 is essentially the same. The linear approximation of (6.1) is equal to (5.2). With the surfactant transport equations unchanged, the bulk exchange and the interfacial transport predictions do not need to be modified (§§ 5.1 and 5.2, respectively).

By linearising the alkane source term (6.2), an estimation of $\theta ^a$ can be found. We introduce the non-dimensional alkane concentrations $c_{k,0}^a = c_0^a\kappa _0^a/\kappa _{d}^a$ and $c_{k,{max}}^a = c_0^a\kappa _{a,{max}}^a/\kappa _{d}^a$. For large concentrations, $\theta ^a \approx c_{k,{max}}^a$, whereas for small concentrations, $\theta ^a \approx c_{k,0}^a\theta \approx c_{k,0}^a c_k$ (6.3). Capturing both cases,

(6.6)\begin{equation} \theta^a \approx \min(c_{k,0}^a \theta, c_{k,{max}}^a) \approx \min(c_{k,0}^a c_k, c_{k,{max}}^a) \quad \text{at}\ x = x_0. \end{equation}

All simulations have resulted in values in the lower concentration interval ($c_{k,0}^a \theta \le c_{k,{max}}^a$). It follows that $\Delta \theta \approx \Delta \theta ^a/c_{k,0}^a$.

We use the linear approximation of the surface tension (6.4) to estimate the Marangoni stresses. With $c_{k,0}^a > 1$ (table 1), $\theta ^a$ is assumed to be larger than $\theta$ at low concentrations (6.6). With $\omega _0 \approx \omega ^a$,

(6.7)\begin{equation} \frac{\mathrm{d} \gamma}{\mathrm{d}\kern0.06em x} = \frac{\partial \gamma}{\partial \theta}\frac{\mathrm{d} \theta}{\mathrm{d}\kern0.06em x} + \frac{\partial \gamma}{\partial \theta^a}\frac{\mathrm{d} \theta^a}{\mathrm{d}\kern0.06em x} \approx{-}\frac{RT}{\omega^a}\left(\frac{\mathrm{d} \theta}{\mathrm{d}\kern0.06em x} + \frac{\mathrm{d} \theta^a}{\mathrm{d}\kern0.06em x}\right) \approx{-}\mu_\infty U{Ma}^a\frac{\mathrm{d} \theta^a}{\mathrm{d}\kern0.06em x}, \end{equation}

where ${ {Ma}}^a = RT/(\omega ^a \mu _\infty U)$ (cf. (5.12)). Analogous to (5.13), $\mathrm {d} \theta ^a/\mathrm {d}\kern 0.06em x \approx 2\Delta \theta ^a/w \approx 2c_{k,0}^a\Delta \theta /w$, giving

(6.8)\begin{equation} \tau_{{Ma}} \approx 2\mu_\infty U{Ma}^a c_{k,0}^a\frac{\Delta \theta}{w}. \end{equation}

We now adapt the estimation of the slip length (5.16). Equations (3.5), (5.10) and (6.8) give an expression for $\Delta \theta$ corresponding to (5.15). The slip length is found from (3.2) and is identical to (5.16) if ${ {Ma}}$ is replaced by ${ {Ma}}^ac_{k,0}^a$, equivalent to redefining

(6.9a,b)\begin{equation} \alpha_{diff} = c_1\frac{1}{{{Pe}}_s'}\frac{1}{2{Ma}^a c_{k,0}^a c_k} \quad \text{and} \quad \alpha_S = c_2\frac{{{Bi}}'}{1 + {{Da}}_\delta}\frac{1}{2{Ma}^a c_{k,0}^a c_k}. \end{equation}

The results from this analytical model are also shown in figures 5 and 8(b), in good agreement with the simulation results.

The sudden change in the gradient of $\gamma$ in figures 4 and 8(a) ($c_s = 1.1\times 10^{-2}$ and $1.6\times 10^{-4}\,{\rm mol}\,{\rm m}^{-3}$, respectively) coincide with switching the $\theta$-dependency of $\kappa _a^a$ in (6.3). However, the simulations are unaffected because they are performed at lower concentrations. In this interval, $\partial \gamma /\partial \theta ^a \approx -RT/\omega ^a$ (6.7), which is independent of concentration. As $\theta ^a$ depends linearly on $\theta$ (6.6), the total derivative with respect to $\theta$ is constant, and with respect to $c_0$ in equilibrium because $\theta \approx c_k = \kappa _ac_0/\kappa _d$ (5.3). This concentration interval has no experimental data to validate the surface tension curve. A linear decrease in surface tension with concentration (i.e. constant derivative) is a reasonable first assumption.

Another problem of the current adsorption/desorption model is that because $c_{k,0}^a$ usually is large, the ratio between $\varGamma$ and $\varGamma ^a$ becomes unreasonably large at low concentrations, considering the number of alkane molecules that can interact with one surfactant molecule. A more realistic result is achieved by letting the surfactant molecules adsorb in two different states, described by the so-called reorientation model (Kartashynska et al. Reference Kartashynska, Lylyk, Aksenenko, Makievski, Vysotsky, Fainerman and Miller2020). We welcome future studies using more advanced adsorption/desorption models. However, for this investigation, we consider the current model sufficient.

7. SC regime

High shear stresses can result in a portion of the interface with almost no adsorbed surfactants. In such conditions, the interface is no longer in the uniformly retarded regime: an underlying assumption of the analytical model. The downstream part of the interface with surfactants becomes a SC in analogy to bubbles (Palaparthi et al. Reference Palaparthi, Papageorgiou and Maldarelli2006). The interface can be said to be in a partial SC regime. A slip length is partly regained because the flow is not decelerated outside the stagnant part. If the concentration of bulk surfactants cannot be reduced further, a partial SC regime can therefore be desirable to achieve. The partial SC regime was investigated by Baier & Hardt (Reference Baier and Hardt2021) for insoluble surfactants on SHS in the limit of large ${ {Pe}}_s'$. For LIS, the SC can improve the retention of the oil in the groove, but on the other hand, the stagnant part of the interface does not contribute to drag reduction (Fu et al. Reference Fu, Chen, Arnold and Hultmark2019).

This section shows that a SC eventually grows to cover the whole interface if the bulk concentration increases. The interface is then in the so-called full SC regime, again undesirable for drag reduction. There is no apparent difference in the distribution of $\theta$ between the full SC and the uniformly retarded regime, so distinguishing between them is a mere formality.

To form a SC, we increased the imposed shear stress to $\tau _\infty = 0.33$ Pa (factor of 1000) by increasing the applied velocity gradient and thereby $U$. Relevant affected non-dimensional parameters are ${ {Pe}}$, ${ {Pe}}_s'$, ${ {Bi}}'$ and ${ {Ma}}$, of which the first two increase linearly with $U$ and the latter decrease linearly (table 2). The increase in ${ {Pe}}$ makes the concentration boundary layer thinner (5.7). Interestingly, $\alpha _{diff}$ and $\alpha _S$ are independent of $U$ in the Stokes regime, except through $\delta$. The Reynolds number is ${ {Re}} = \rho _\infty Uw/\mu _\infty = 3.3 \sim 1$ (with density $\rho _\infty = 1000\,{\rm kg}\,{\rm m}^{-3}$ for water), so the Stokes equations are still valid to some extent, especially considering the lower velocity close to the interfaces.

A partial SC requires that the advection rate of interfacial surfactants overcomes the interfacial diffusion and bulk exchange rates of the linear surfactant distribution. The non-dimensional numbers comparing these transport mechanisms are reiterated here for convenience, (cf. (5.18) and (5.19))

(7.1)$$\begin{gather} \frac{\text{diffusion rate of interfacial surfactant}}{\text{advection rate of interfacial surfactant}} \sim \frac{1}{{{Pe}}_s'}, \end{gather}$$

(7.2)$$\begin{gather}\frac{\text{diffusion rate of bulk surfactant}}{\text{advection rate of interfacial surfactant}} \sim \frac{{{Bi}}'}{{{Da}}_\delta}, \end{gather}$$

(7.3)$$\begin{gather}\frac{\text{adsorption/desorption rate}}{\text{advection rate of interfacial surfactant}} \sim {{Bi}}'. \end{gather}$$

The quantitative limits of the SC formation can be understood from (5.15). This equation can be reformulated as

(7.4)\begin{equation} 1 - \frac{1}{\alpha_{diff}+ \alpha_S + 1} \approx \left(c_1 \frac{1}{{{Pe}}_s'} + c_2\frac{{{Bi}}'}{1 + {{Da}}_\delta}\right)\frac{\Delta \theta}{c_k}. \end{equation}

A linear interfacial surfactant distribution implies $\Delta \theta /c_k \le 1$ because we cannot have negative values of $\theta$ (figure 6). Therefore, (7.4) cannot hold for large $\alpha _{diff}+ \alpha _S$ if

(7.5)\begin{equation} C_{SC} = c_1 \frac{1}{{{Pe}}_s'} + c_2\frac{{{Bi}}'}{1 + {{Da}}_\delta} < 1. \end{equation}

Instead, the surfactants are advected to the end stagnation point of the interface, where a larger gradient is created. The interface is then in the partial SC regime. The same limitation exists for $\Delta c_s/c_0$. However, from (5.6), we know that $\Delta c_s/c_0 \lesssim \Delta \theta /c_k$, so it does not provide any other peculiarities. If $\alpha _{diff} + \alpha _S \ll 1$, (7.4) would again allow for a linear surfactant profile. Such interface is in the full SC regime. We suggest using (7.5) to distinguish between (i) the uniformly retarded regime ($C_{SC} > 1$) and (ii) the partial and full SC regimes ($C_{SC} < 1$).

Assuming that (7.5) is fulfilled, and considering the maximum value $\Delta \theta /c_k = 1$, we get the limit for $\alpha _{diff}+ \alpha _S$ above which no linear solution can exist as

(7.6)\begin{equation} \alpha_{diff} + \alpha_S \approx \frac{1}{1 - C_{SC}} - 1. \end{equation}

Equation (7.6) is the lower limit for the partial SC regime, below which we enter the full SC regime.

We illustrate the limit (7.6) for SDS (advanced model) at $\tau _\infty = 0.33$ Pa in figure 9. These simulations were performed with a more refined grid (Appendix A). The region where linear solutions cannot exist is shown in grey. For the current parameters (table 2),

(7.7)\begin{equation} \frac{1}{{{Pe}}_s'} = 0.0036, \quad {{Bi}}' = 0.25, \quad \text{and} \quad \frac{{{Bi}}'}{{{Da}}_\delta} = 1.2, \quad \text{giving} \ C_{SC} = 0.059. \end{equation}

In the grey region, the analytical model loses validity as expected. Equation (7.6) also holds for the simulation results of Landel et al. (Reference Landel, Peaudecerf, Temprano-Coleto, Gibou, Goldstein and Luzzatto-Fegiz2020), which are shown in Appendix D. Therefore, the transition between the partial and full SC regime appears consistent over a wide range of geometrical and surfactant parameters.

Figure 9. (a) The analytical model (dashed line) compared with simulations of SDS at $\tau _\infty = 0.33$ Pa. The limit (7.6) forms the boundary of the grey region, in which linear distributions of $\theta$ cannot exist. (b) Three distributions of $\theta /c_k$, going from a nonlinear (blue) to an intermediate (red) and a linear (yellow) for the smallest value of $\alpha _{diff} + \alpha _S$. The inset is identical, apart from a lower maximum vertical axis limit.

The limit for the bulk surfactant concentration corresponding to (7.6) is attained by using the definitions of $\alpha _{diff}$ and $\alpha _S$ ((5.17) or (6.9a,b)). The expression for the non-dimensional concentration is

(7.8)\begin{equation} c_k = \frac{1}{2{Ma}}\left(1 - C_{SC}\right) \quad \text{or} \quad c_k^a = \frac{1}{2{Ma}^a}\left(1 - C_{SC}\right), \end{equation}

for the regular and the more advanced Frumkin model, respectively, where $c_k^a = c_{k,0}^a c_k$. No partial SC can exist above these concentrations. As $C_{SC}$ decreases, $c_k$ or $c_k^a$ increases towards a maximum value determined by the Marangoni number. The current parameters (7.7) correspond to $94\,\%$ of the maximum concentration. The only flow-dependent quantity of the maximum concentration is $U$, which is determined by the imposed shear stress and the width of the grooves. Apart from $T$, the other quantities depend only on the liquids and the surfactants.

With the increase of slip length in the partial SC regime, a new condition for significant slip length reduction is needed in place of (5.20). Equation (7.6) can replace (5.20) if smaller than $1$ (this occurs if $C_{SC} < 1/2$) because (5.20) remains valid in the full SC regime. However, it should be noted that (7.6) then gives $b/b_{SHS}\beta _{LIS} < 0.5$ as predicted by the analytical model becoming valid at this $\alpha _{diff} + \alpha _S$. Still, we do think this is an appropriate condition. We summarise these limits by concluding that a significant slip length reduction occurs if

(7.9)\begin{equation} \alpha_{diff} + \alpha_S < \begin{cases} 1, & \text{if } C_{SC} \ge 1/2, \\ \dfrac{1}{1 - C_{SC}} - 1, & \text{if } C_{SC} < 1/2. \end{cases} \end{equation}

The equations provided in this section are intended to serve as convenient tools to improve the understanding and classification of LIS with surfactants subjected to high shear stresses. We have not attempted to model the partial SC regime, which is a potential topic for future studies. However, it can be noted that, even in the partial SC regime, the analytical model can indicate the order of magnitude of the slip length. Uncertainties in parameters and adsorption models might be a larger concern; the difference between the analytical prediction and the simulation results of figure 9(a) can be compared with the deviations in slip length in figure 5. The error of the analytical model can, in particular, be expected to be minor if $C_{SC} \sim 1$, implying that the two conditions for significant slip length reduction ((5.20) and (7.9)) are similar.

8. Remarks

The analytical model has been developed for two-dimensional transverse grooves. However, similar surfactant transport processes are expected to be present for LIS with three-dimensional longitudinal grooves (Landel et al. Reference Landel, Peaudecerf, Temprano-Coleto, Gibou, Goldstein and Luzzatto-Fegiz2020; Temprano-Coleto et al. Reference Temprano-Coleto, Smith, Peaudecerf, Landel, Gibou and Luzzatto- Fegiz2021). Therefore, the analytical model is expected to provide an indication of the effects of surfactants on the slip length also for such configurations, commonly used in experimental LIS studies (Jacobi et al. Reference Jacobi, Wexler and Stone2015; Wexler et al. Reference Wexler, Jacobi and Stone2015; Fu et al. Reference Fu, Chen, Arnold and Hultmark2019).

If the grooves are longer than those used here, the slip length tends to be larger. When disregarding changes in aspect ratio or solid fraction, $b_{SHS}\beta _{LIS}$ and $u_{s,{SHS}}^0\beta _{LIS}$ increase proportional to $w$ (as given by (3.3) and (3.6)). The diffusion rate is suppressed by increasing the groove width. Thereby, ${ {Pe}}_s'$ and ${ {Da}}_\delta$ increase with $w$ (even if the $w$-dependency of $\delta$ in (5.7) is non-trivial), whereas ${ {Bi}}'$ is constant. Correspondingly, an increased groove width decreases $C_{SC}$ (7.5); increasing the groove width may make the interfaces enter the SC regime. As ${ {Ma}}$ and ${ {Ma}}^a$ are inversely proportional to $w$, $\alpha _{diff}$ decreases with $w$, whereas $\alpha _S$ increases ((5.17) or (6.9a,b)). Therefore, with $C_{SC}$ decreasing and $\alpha _S$ eventually dominating, the slip length becomes less sensitive to surfactants with increasing groove width, according to (7.9).

For turbulent flows, the grooves would need to be larger or the imposed shear stress higher to generate a relevant drag reduction, even without surfactants. For $\tau _\infty = 0.33$ mPa and $0.33$ Pa, the viscous length scales correspond to $l_\nu = \mu _\infty /\sqrt {\rho _\infty \tau _\infty } = 1.7$ mm and $55\,\mathrm {\mu }{\rm m}$, respectively. Assuming $b/p \approx 0.02$ also for the turbulent flows (figure 5), $b^+ = b/l_\nu \approx 1.7\times 10^{-3}$ and $5.4\times 10^{-2}$, but these non-dimensional slip lengths would need to be $b^+ \gtrsim 1$ to give meaningful drag reduction (Fu et al. Reference Fu, Arenas, Leonardi and Hultmark2017).

Some surfactants exhibit a critical micelle concentration (CMC), above which they can form micelles (clusters of surfactant molecules) instead of adsorbing at interfaces. Hence, the source term dependencies on $c_s$ ((4.1) and (6.1)) are invalid above the CMC. The CMC is $8.2\,{\rm mol}\,{\rm m}^{-3}$ for SDS, above which the surface tension remains somewhat constant (Elworthy & Mysels Reference Elworthy and Mysels1966, cf. figure 4). C$_{12}$TAB holds a similar CMC (Klevens Reference Klevens1948, cf. figure 8a). The slip length reductions studied here occur at lower concentrations (figures 5a and 8b). Therefore, micelle formation does not affect our conclusions. Micelles can also act as monomer buffers (surfactants not part of micelles), dissociating to reduce monomer concentration gradients. They thereby give rise to a remobilisation effect where interface gradients are reduced (Manikantan & Squires Reference Manikantan and Squires2020).

For $\theta \sim c_k \gg 1$, the analytical model could be inaccurate because nonlinearities in the Frumkin source term then become relevant. As discussed by Landel et al. (Reference Landel, Peaudecerf, Temprano-Coleto, Gibou, Goldstein and Luzzatto-Fegiz2020), such non-dimensional concentrations could appear below the CMC for ‘strong’ surfactants ($\kappa _d/\kappa _a$ much lower than the CMC, cf. table 1). However, they performed some simulations with $c_k \gg 1$, which roughly agreed with the model (Appendix D). Peculiarities that might arise at high concentrations (e.g. for $a \neq 0$) are out of the scope of the current investigation.

We have isolated the effects of Marangoni stresses and viscous stresses imposed by the infusing liquids. However, surfactants might also cause additional rheological stresses at the interface, characterised by the (intrinsic) surface shear and dilatational viscosities (Manikantan & Squires Reference Manikantan and Squires2020). However, these surface viscosities are challenging to measure. Careful measurements have given a reliable upper limit of the surface shear viscosity for surfactants on water–air interfaces (Zell et al. Reference Zell, Nowbahar, Mansard, Leal, Deshmukh, Mecca, Tucker and Squires2014). If it is assumed that this upper limit holds for both surface viscosities, these extra stresses are irrelevant compared with the Marangoni stress at groove dimensions $w \gg 10\,\mathrm {\mu }{\rm m}$ (Landel et al. Reference Landel, Peaudecerf, Temprano-Coleto, Gibou, Goldstein and Luzzatto-Fegiz2020). They can then be neglected for the geometries considered here.

Finally, a possible extension of the current work would be to model surfactant solubility also in the oil phase. Such a model requires a second set of surfactant adsorption and desorption coefficients, describing surfactant exchange in the oil.