Slip of submerged two-dimensional liquid-infused surfaces in the presence of surfactants

Abstract

Using numerical simulations, we investigate the effects of Marangoni stresses induced by surfactants on the effective slip length of liquid-infused surfaces (LIS) with transverse grooves. The surfactants are assumed soluble in the external liquid shearing the surface and can adsorb onto the interfaces. Two different adsorption models are used: a classical Frumkin model and a more advanced model that better describes the decrease of surface tension for minuscule concentrations. The simulations show that LIS may face even more severe effects of surfactants than previously investigated superhydrophobic surfaces. Constructing an analytical model for the effective slip length, we can predict the critical surfactant concentration for which the slip length decreases significantly. This analytical model describes both adsorption models of LIS on a unified framework if properly adjusted. We also advance the understanding of when surfactant advection gives rise to highly skewed interfacial concentrations: the so-called partial stagnant cap regime. To a good approximation, this regime can only exist below a specific surfactant concentration given by the Marangoni number and the strength of the surfactants.

1. Introduction

Liquid-infused surfaces (LIS) are promising candidates for reducing drag, resisting biofouling and increasing heat transfer in liquid flows (Epstein et al. 2012; Solomon, Khalil & Varanasi 2014; Rosenberg et al. 2016; Sundin et al. 2022). These surfaces consist of a solid surface texture with a lubricating liquid that is immiscible with the external fluid. The fluid–fluid interfaces and mobility of the lubricant give rise to a slipping effect of the external flow. LIS can self-repair and are not sensitive to hydrostatic pressure, thereby being more robust than superhydrophobic surfaces (SHS) if designed properly (Wong et al. 2011; Wexler, Jacobi & Stone 2015; Sundin, Zaleski & Bagheri 2021).

The functionality of LIS has been mostly characterised assuming perfectly clean external liquid flows. However, both in applications and in laboratory set-ups (Jacobi, Wexler & Stone 2015; Peaudecerf et al. 2017), LIS are exposed to trace amounts of surfactants. These are substances that can adsorb onto interfaces and alter the surface (or interfacial) tension. Surfactants influence phenomena such as foaming, wetting, dispersion and emulsification, appearing in a large variety of products, e.g. cleaning agents, paints, cosmetics, pharmaceuticals and motor oils (Rosen & Kunjappu 2012). It has recently been acknowledged that traces of surfactants may induce Marangoni stresses that counteract the slip of SHS (Peaudecerf et al. 2017). As we demonstrate in this paper, the presence of surfactants in the system also modifies the slip of LIS. In applications where the mobility of the infusing liquid is crucial, it is necessary to understand the influence of surfactants on the performance, particularly when measured slip lengths deviate significantly from their expected values.

Surfactants adsorbed onto interfaces accumulate at stagnation points when subjected to flow, building up concentration gradients and corresponding Marangoni stresses. The Marangoni stress counteracts the shear stress from the overlying flow, reducing the slip length. Using numerical simulations, a significant slip length reduction of SHS has been observed at bulk concentrations $c_0 \approx 10^{-3}\,{\rm mol}\,{\rm m}^{-3}$ using properties of the surfactant sodium dodecyl sulphate (SDS) (Peaudecerf et al. 2017). Most recent experimental studies of SHS, which propose that surfactant gradients lead to slip degradation, have not added surfactants artificially (Kim & Hidrovo 2012; Bolognesi, Cottin-Bizonne & Pirat 2014; Peaudecerf et al. 2017; Song et al. 2018; Temprano-Coleto et al. 2021). Instead, the fluid systems have likely been contaminated by the surrounding environment. Indeed, it is generally accepted that surfactants appear as ‘hidden variables’ because their concentrations are unknown (Manikantan & Squires 2020). Analytical models that relate the slip length to surfactant concentration are therefore crucial to interpreting experimental measurements.

Landel et al. (2020) developed an analytical theory to predict the effective slip length of SHS from relevant non-dimensional numbers in a two-dimensional channel flow with surfactants. The model assumed low concentrations and uniform interfacial concentration gradients. A high shear rate can result in the upstream part of an interface having almost no surfactant gradients, rendering the model inaccurate. Inspired by the analysis of buoyantly rising bubbles (Palaparthi, Papageorgiou & Maldarelli 2006), Landel et al. (2020) found that the surface could enter this regime when interfacial surfactant advection overcomes interfacial diffusion and bulk exchange rates. However, they were unable to find a quantitative condition for the transition. The analytical model compared favourably with numerical simulations for low shear rates, containing four fitted parameters. Baier & Hardt (2021) developed an analytical model for slip length degradation by insoluble surfactants. They considered a flow driven by an imposed shear stress and could therefore use the analytical flow solution of Philip (1972a).

The effects of surfactants on LIS have not been investigated thoroughly, although its importance has been acknowledged. Certain interfacial observations of LIS, that cannot be explained fully, have been attributed to the presence of surfactants. One example is the absence of interface deformations in the vicinity of stagnation points (Jacobi et al. 2015). The influence of surfactants on LIS drag reduction was also highlighted as a future challenge in a recent review (Hardt & McHale 2022).

There are indications that LIS are more sensitive to surfactants compared with SHS. Dodecane, hexane and other alkanes are promising infusing liquids for drag reduction applications (Van Buren & Smits 2017). However, the interfaces of aqueous surfactant solutions and saturated hydrocarbons (e.g. alkanes) generally face a more significant decrease in surface tension than the interfaces of corresponding water–air systems (assuming only minor or no solubility of the surfactant in the hydrocarbon) (Rosen & Kunjappu 2012). The relatively early publication by Gillap, Weiner & Gibaldi (1968) reported this effect for sodium decyl sulphate and SDS, and various water–alkane interfaces (e.g. water–hexane with SDS).

Measurements have also shown that the surface tension of water–alkane interfaces can experience an initial decrease of several ${\rm mN}\,{\rm m}^{-1}$ at minuscule surfactant concentrations. Fainerman et al. (2019) illustrated this effect for water–hexane interfaces with dodecyl and tridecyl dimethyl phosphine oxide (C$_{12}$DMPO and C$_{13}$DMPO, lowest concentration $c_0 = 10^{-6}\,{\rm mol}\,{\rm m}^{-3}$). This effect is also present for several other surfactants such as SDS and trimethyl ammonium bromides (C$_{n}$TAB). It is only recently that adsorption models for water–oil interfaces have been developed to account for such phenomena (Fainerman et al. 2020).

Based on the current knowledge about surfactants at water–oil interfaces, we have investigated the dependency of the slip length of LIS on surfactant concentration. The coupled system of equations for the flow and the surfactants has been solved numerically for transverse grooves in laminar shear flow. In contrast to SHS, it is necessary to resolve the flow both inside and outside the textures of LIS. We also developed an analytical model for the slip length in the presence of surfactants. The model can be used to predict the reduction of slip lengths if the surfactant type and concentration in bulk liquid are known. In settings where surfactants are hidden in the system, the model may be used to estimate the concentration of surfactants given measurements of the slip length.

The flow configuration, governing equations and numerical methods are described in the next section. Section 3 introduces the analytical model for an imposed Marangoni stress. Interfacial surfactant adsorption and desorption of surfactants using regular Frumkin kinetics are described and used for simulations in § 4, followed by the corresponding analytical model (§ 5). A more advanced model giving consistent surface tensions at minuscule concentrations is introduced in § 6. Higher applied shear stresses, resulting in highly skewed interfacial surfactant concentrations (the partial stagnant cap (SC) regime), are treated in § 7. Final remarks and conclusions are presented in §§ 8 and 9, respectively.

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. 2020).

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 2007)

(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. (2017).

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. 2017) and water–alkane interfaces (Fainerman et al. 2019). SDS can be found in personal care products, and C$_{12}$TAB can, for example, be used to stabilise foam (Carey & Stubenrauch 2009; Rosen & Kunjappu 2012). These surfactants are illustrated schematically in figure 2. They have hydrophobic hydrocarbon tails with 12 carbon atoms but different hydrophilic (head) groups.

2.1. Numerical method

The numerical simulations were performed using the finite-element solver FreeFem++ (Hecht 2012; Lācis et al. 2020). Velocities and surfactant concentrations were discretised using quadratic ($P_2$) finite elements while linear ($P_1$) elements were used for the pressure. The system of (2.1a,b), (2.4) and (2.5) was solved iteratively. The domain consisted of one interface unit cell (figure 1). The external flow domain had a height of $3k$. At the upper boundary, we imposed constant shear stress $\tau _\infty$, zero wall-normal stress and bulk surfactant concentration $c_0$. Periodic boundary conditions were applied in the streamwise direction.

The computational mesh used for the flow (bulk and cavity) and the bulk surfactants ($y \ge 0$) was generated using a built-in tool. The cells were triangular, and the number of cells was prescribed at the different domain boundaries with a spacing of $w/N$, where $N = 64$. The grid was refined close to the interface. For the simulations with a low flow speed (§§ 4 and 6), two refinements were made, reducing the sides of the cells by a factor of four in total, i.e. $N = 256$. The simulations with artificially applied Marangoni stress (§ 3) also had $N = 256$ at the interface. For simulations with high flow speed, we used four refinements ($N = 1024$, § 7). We have also performed grid refinement studies, see Appendix A.

The interfacial surfactant transport equation (2.4) was solved on a one-cell-high mesh with equal cell spacing to the flow and bulk surfactant mesh in the streamwise direction. We imposed periodic boundary conditions in the $y$-direction. The interface velocity and the bulk surfactant concentration appearing in the source term ($u_s$ and $c_s$, respectively) were taken at $y = 0$. There were no variations in the $y$-direction in the solution of the interfacial surfactant concentration; it exactly corresponds to the one-dimensional solution.

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 (2014) and Landel et al. (2020). 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. 2017; Landel et al. 2020), which are consistent with the Frumkin isotherm (Chang & Franses 1995). 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 2020). 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 1995; Fainerman et al. 2019). 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 1995).

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. (2019), 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. (2019). 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. (2019) 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 2001). 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 (2001). 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 (1995). 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. (2021) 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. 2020). 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. 2021).

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. (2020), (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. 2006). 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. 2019). 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. 2020; Müller et al. 2021). In this paper, we adopt a model consistent with the equilibrium model of Fainerman et al. (2019). 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. (2010). 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. 2001). (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. 2020). 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. 2006). 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 (2021) 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. 2019).

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. (2020), 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. 2020; Temprano-Coleto et al. 2021). 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. 2015; Wexler et al. 2015; Fu et al. 2019).

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. 2017).

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 1966, cf. figure 4). C$_{12}$TAB holds a similar CMC (Klevens 1948, 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 2020).

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. (2020), 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 2020). 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. 2014). 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. 2020). 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.

9. Conclusions

If surfactants are present in a flow over SHS or LIS, they might severely decrease the slip lengths of these surfaces. The surfactants adsorb at the fluid interfaces and are advected towards the downstream stagnation points, resulting in surface tension gradients that oppose the flow. Surface tension gradients are known as Marangoni stresses. Using numerical simulations of laminar flow over LIS with transverse grooves, we have explored how the effective slip length changes when the flow has a non-zero concentration of the commonly used surfactants SDS and C$_{12}$TAB. The external fluid was water, and the infusing liquid an alkane. The surfactants have been assumed to be soluble in the water but insoluble in the alkane.

For low applied shear stresses ($\tau _\infty = 0.33$ mPa), the distribution of surfactants on an interface is approximately linear. It is then possible to construct an analytical theory for the LIS slip length, similar to what has been done for SHS (Landel et al. 2020). In the numerical simulations, we used classical Frumkin kinetics and a more advanced Frumkin model that considers interactions between the surfactants and the alkane molecules. Both models can be described by the (properly adjusted) analytical theory. The predicted slip length is given by (5.16), with parameters $\alpha _{diff}$ and $\alpha _S$ specified by (5.17) or (6.9a,b), depending on the model.

The interfacial surfactants might be swept towards the downstream stagnation point for large applied shear stresses ($\tau _\infty = 0.33$ Pa). The interfaces are then in the partial SC regime. Some slip length is regained, as a large part of an interface does not have a significant surface tension gradient. However, this regime can only exist below a particular bulk concentration. Above this concentration, the surfactant distribution is linear, and the analytical model becomes accurate. The concentration is given by (7.8), with $1 - C_{SC}$ typically having a value close to unity.

Surfactants in fluid systems are difficult to detect directly yet affect flows significantly; they act like hidden variables (Manikantan & Squires 2020). The concentrations for which the slip is profoundly reduced for LIS and SHS can be assumed to correspond to those naturally occurring in experimental set-ups (Peaudecerf et al. 2017). Temprano-Coleto et al. (2021) estimated such naturally occurring surfactant concentrations to be $c_0 = 3\times 10^{-4}\,{\rm mol}\,{\rm m}^{-3}$ by comparing experimental results of SHS with polydimethylsiloxane (PDMS) textures to theoretical predictions (cf. figures 5 and 8b). It is the hope that the models presented here can be used to interpret experimental results of LIS both with and without artificially added surfactants. They can also aid the design of grooves to avoid severe performance degradation. For example, LIS can be designed to remain in the partial SC regime at similar concentrations. We also hope that by including infusing liquid viscosity in this model, we have taken a step towards three-dimensional models of LIS with textures more similar to those typically used in experimental set-ups.