2.1. Model

We consider a linear shear flow of an outer liquid of viscosity $\mu ^{*}$ over a shallow rectangular groove of width $\delta$ and depth $e\ll \delta$ (see figure 1a), filled with a lubricant of viscosity $\mu ^{l*}$ (hereafter the asterisk denotes dimensional variables). The $y^{*}$-axis is aligned with the shear direction, while the $z^{*}$-axis is defined normal to the wall, and the dimensionless coordinates are introduced as $y=y^{*}/\delta$ and $z=z^{*}/\delta$. We assume that the groove aspect ratio is small, $\varepsilon =e/\delta \ll 1$, and that, at rest, the static liquid/lubricant interface is flat and located at $z=0$, i.e. there is no pressure difference across the meniscus that is pinned at the edges of the groove. When the meniscus is perturbed by an external flow, its slightly deformed shape sketched in figure 1(b) can be described by a function $\varepsilon \eta (y)$, where $\eta =O(1)$. It is convenient to introduce the stationary angle $\theta \geq {\rm \pi}/2$ defined (relative to the vertical direction) at the point where the concave or flat meniscus meets the groove edge (here, the front one). Thanks to the lubricant volume conservation, the meniscus deformation should be rather close to antisymmetric, so that the stationary angle at the opposite grove edge (here the rear one) is approximately ${\rm \pi} - \theta$. The observed angle $\theta$ cannot exceed a limiting value, known as the advancing angle, beyond which the contact line depins from the groove edge and moves. Likewise, when ${\rm \pi} - \theta$ decreases down to a limiting value of the receding angle, the contact line should suddenly shift laterally.

Figure 1.

Sketches of ($a$) an outer shear flow past a shallow lubricant-infused groove of width $\delta$ and depth $e$, and of ($b$) a liquid/lubricant interface $\varepsilon \eta (y)$ in dimensionless coordinates. The concave liquid/lubricant interface meets the front groove edge with an angle $\theta$ defined relative to the vertical.

If we consider a chemically homogeneous (ideal) surface, the bounds of attainable values of $\theta$ are determined unambiguously by a liquid (Young) contact angle $\varTheta$ (above ${\rm \pi} /2$ to provide a lubricant-infused state) on a planar horizontal surface (see figure 2). Its value can, of course, always be adjusted by a suitable modification of the solid surface (Jung & Bhushan Reference Jung and Bhushan2009; Grate et al. Reference Grate, Dehoff, Warner, Pittman, Wietsma, Zhang and Oostrom2012; Dubov et al. Reference Dubov, Mourran, Möller and Vinogradova2015), but note that, on most solids, $\varTheta$ never exceeds $2 {\rm \pi}/3$ or $120^{\circ }$ (Yakubov, Vinogradova & Butt Reference Yakubov, Vinogradova and Butt2000; Jung & Bhushan Reference Jung and Bhushan2009; Wexler et al. Reference Wexler, Jacobi and Stone2015). Following Quere (Reference Quere2008), Herminghaus, Brinkmann & Seemann (Reference Herminghaus, Brinkmann and Seemann2008) and Dubov et al. (Reference Dubov, Nizkaya, Asmolov and Vinogradova2018) one can argue that the contact angle of the liquid at the groove edges can be anywhere between $\varTheta$ (receding) and $\varTheta + {\rm \pi}/2$ (advancing), as illustrated in figure 2 by the coloured regions that are symmetric relative to a midplane of the groove. The attainable contact angles redefined relative to $z$-direction are then confined between $\varTheta - {\rm \pi}/2$ (receding) and $\varTheta$ (advancing). We remark and stress, however, that the bounds of attainable angles are asymmetric relative to $z=0$, except the case of $\varTheta = 3 {\rm \pi}/4$ (or $135^{\circ }$). Of the two possible unstable angles, one is normally attained faster and overshadows the other. Since for our surfaces $\varTheta$ is smaller than $3 {\rm \pi}/4$, one can argue that the depinning will occur on that edge of the groove, where the meniscus is concave, and where $\theta$ will reach the value of $\varTheta$. For the meniscus shape sketched in figures 1 and 2 this would be the front (left) edge, but of course such a shape is by no means obvious, and will be justified below by solving a hydrodynamic problem. As a side note we mention that the depinning would occur simultaneously at the front and rear (right) edges if $\varTheta = 3 {\rm \pi}/4$, and for larger $\varTheta$ – on a rear edge, when $\theta \simeq \varTheta - {\rm \pi}/2$. Clearly, these estimates hold only for ideal surfaces and relatively low speed. They would become approximate when the surface is chemically heterogeneous. However, they provide us with some guidance.

Figure 2.

Pinning of a contact line on a rectangular groove edge. The liquid meets the solid with a contact angle $\varTheta$. Hence, the contact angle at the groove edges can take any value (if the horizontal direction is considered as the reference one) between $\varTheta$ and $\varTheta + {\rm \pi}/2$, as illustrated by coloured region confined between the dashed lines. The attainable contact angles at the edge, redefined relative to the $z$-direction, are then confined between $\varTheta - {\rm \pi}/2$ and $\varTheta$.

The dimensionless velocity and pressure are defined as ${\boldsymbol {u}}=\boldsymbol {u}^{*}/(G\delta )$ and $p=p^{*}/(G\mu ^{*})$, where $G$ is an undisturbed shear rate. We stress, that near the groove the outer flow is modified due to a slippage at the liquid/lubricant interface, and that the lubricant flow, induced by a reverse pressure gradient, has zero flow rate in any cross-section (see figure 3a). The flows in a lubricant and an outer liquid are stationary and satisfy Stokes equations

(2.1)\begin{gather} \boldsymbol{\nabla }\boldsymbol{\cdot} \boldsymbol{u}=0,\quad {\rm \Delta} \boldsymbol{u}-\boldsymbol{\nabla }p= \mathbf{0}, \end{gather}

(2.2)\begin{gather}\boldsymbol{\nabla }\boldsymbol{\cdot} \boldsymbol{u}^{l}=0,\quad \mu {\rm \Delta} \boldsymbol{u}^{l} -\boldsymbol{\nabla}p^{l}=\mathbf{0}, \end{gather}

where $\boldsymbol {u}=(0,v,w)$ and $\boldsymbol {u}^{l}=(0,v^{l},w^{l})$ are velocity fields in liquid and in lubricant, $p, p^{l}$ are corresponding pressure distributions and $\mu =\mu ^{l*}/\mu ^{*}$ is the lubricant/liquid viscosity ratio. Far from the lubricant-induced surface the liquid flow represents a linear shear flow, $\boldsymbol {u}|_{z\to \infty }=z{\boldsymbol {e}}_y$, where ${\boldsymbol {e}}_y$ is a unit vector along the $y$-axis. We apply a no-slip condition at solid boundaries, and at the liquid/lubricant interface, $z=\varepsilon \eta(y)$, we use the conditions of impermeability

(2.3)\begin{equation} \boldsymbol{u}\boldsymbol{\cdot} \boldsymbol{n}={\boldsymbol{u}}^{l}\boldsymbol{\cdot} {\boldsymbol{n}}=0, \end{equation}

and of continuity of tangential velocity and tangential stress,

(2.4)\begin{gather} \boldsymbol{u}\boldsymbol{\cdot} \boldsymbol{\tau}= \boldsymbol{u}^{l}\boldsymbol{\cdot} \boldsymbol{\tau}, \end{gather}

(2.5)\begin{gather}\boldsymbol{\tau} \boldsymbol{\cdot} \boldsymbol{\sigma} \boldsymbol{\cdot} \boldsymbol{n}=\boldsymbol{\tau} \boldsymbol{\cdot} \boldsymbol{\sigma}^{l}\boldsymbol{\cdot} \boldsymbol{n}, \end{gather}

where ${\boldsymbol {\tau }}$ and ${\boldsymbol {n}}$ are the unit tangent and outward normal (to the meniscus) vectors, and the stress tensors are $\boldsymbol {\sigma } = \boldsymbol {\nabla } \boldsymbol {u}+( \boldsymbol {\nabla } \boldsymbol {u})^\textrm {T} {-}p{\boldsymbol{\mathsf{I}}}$ and ${\boldsymbol {\sigma }}^{l} =\mu ({\boldsymbol {\nabla } \boldsymbol {u}}^{l}+ ({\boldsymbol {\nabla } \boldsymbol {u}}^{l})^\textrm {T}){-}p^{l} {\boldsymbol{\mathsf{I}}}$, where I is the unit tensor.

Figure 3.

($a$) Recirculation flow in a shallow groove. ($b$) Shear flow and a local slip length in outer fluid.

The condition for normal stresses at the interface can be derived using the Laplace equation,

(2.6)\begin{equation} \boldsymbol{n}\boldsymbol{\cdot} (\boldsymbol{\sigma}^{\boldsymbol{l}}-\boldsymbol{\sigma})\boldsymbol{\cdot} \boldsymbol{n}=\frac{\kappa}{{Ca}}, \end{equation}

where $Ca=\mu ^{\ast }G\delta /\gamma$ is the capillary number defined using an outer liquid viscosity and surface tension of the interface γ, and $\kappa \simeq \varepsilon \eta ''$ is the interface curvature (negative for the meniscus protruding into the outer liquid).

These equations should be supplemented by the condition of volume conservation in the lubricant phase and by the pinning conditions at the edges of the groove,

(2.7a,b)\begin{equation} \int\limits_{0}^{1}\eta(y)\,\textrm{d}y=0, \quad \eta(0)=\eta(1)=0. \end{equation}

Equations (2.1)–(2.7a,b) represent a closed system governing liquid and lubricant flows. They involve three dimensionless parameters, i.e. the shallow groove aspect ratio $\varepsilon$, the lubricant/liquid viscosity ratio $\mu$ and the capillary number $Ca$. The two latter parameters could be any positive value, but $\varepsilon$ is small, so that we could use it to construct asymptotic solutions for the meniscus shape, velocity fields and pressure.

Since $\varepsilon$ is small, ${\boldsymbol {\tau } }\simeq ( 0,1,\varepsilon \partial _{y}\eta )$ and ${\boldsymbol {n}}\simeq ( 0,-\varepsilon \partial _{y}\eta ,1)$ and the boundary conditions (2.3)–(2.5) at a curved interface can be simplified to

(2.8a–c)\begin{equation} w=w^{l}=0, \quad v=v^{l}=v_{\eta}(y), \quad\dfrac{\partial v}{\partial z}=\mu\dfrac{\partial v^{l}}{\partial z}, \end{equation}

which defines the coupling between the liquid and the lubricant. Here, $v_{\eta }(y)$ is the tangential velocity of both the liquid and the lubricant at the interface. From (2.6) we then obtain

(2.9)\begin{equation} \sigma^{l}_{zz}=\sigma_{zz}+\dfrac{\varepsilon\eta''}{{{Ca}}},\end{equation}

which determines the meniscus shape, with normal stresses given by

(2.10a,b)\begin{equation} \sigma_{zz}=2 \partial _{z}w-p,\quad \sigma^{l}_{zz}=2\mu \partial _{z}w^{l}-p^{l}. \end{equation}

Note that the normal stresses include not only the pressure but also the gradients of the normal velocities since, unlike the no-slip case, the latter do not vanish at slippery surfaces.

2.2. Inner flow

The lubricant flow in the groove is generated by the interface velocity $v_{\eta }(y)$. For this inner problem the boundary condition, (2.8a–c), should be imposed at the curved meniscus $z=\eta (y)$, since its local deviation from the flat one, $z=0$, is comparable to the depth of the groove, $\varepsilon$. Since the local slopes of the meniscus are small, we apply the lubrication theory and consider a locally parabolic velocity profile of zero flow rate

(2.11a,b)\begin{equation} v^{l}\simeq v_{\eta }( y) \zeta\left(3\zeta-2\right) ,\quad w^{l}\simeq 0, \end{equation}

where $\zeta =(z+\varepsilon )/[\varepsilon (1+\eta )]$ varies from $0$ at the bottom wall to $1$ at the interface. Equations (2.11a,b), which are equivalent to those derived by Nizkaya et al. (Reference Nizkaya, Asmolov and Vinogradova2013) for a flat meniscus, but varying local thickness of the thin lubricant film, allow one to immediately calculate both the lubricant shear rate at the interface

(2.12)\begin{equation} \partial_z v^{l}=\dfrac{4 v_{\eta}}{\varepsilon(1+\eta)}, \end{equation}

and the transverse local slip length

(2.13)\begin{equation} b(y)=\frac{v_{\eta}}{\mu \partial_z v^{l}} \simeq \frac{\varepsilon}{4 \mu}(1+\eta), \end{equation}

where $\varepsilon /\mu =O(1)$. Equation (2.13) implies that $b(y)$ is proportional to a local thickness of the lubricant layer and inversely proportional to $\mu$, similarly to infinite systems (Miksis & Davis Reference Miksis and Davis1994; Vinogradova Reference Vinogradova1995). We should also note that ${\varepsilon }/{4\mu }$ may be interpreted as a local transverse slip length, $b_0$, on a flat (undisturbed) liquid/lubricant interface (Nizkaya et al. Reference Nizkaya, Asmolov and Vinogradova2013, Reference Nizkaya, Asmolov and Vinogradova2014).

For SH grooves, pressure induced by flow changes in the inner gas is usually neglected since $\mu \ll 1$. However, $\mu$ takes on finite values for lubricant-infused surfaces, and the gradient of pressure in closed shallow grooves could become very large, even when the lubricant viscosity is small, $\mu \sim \varepsilon$. Using simple scaling arguments one can show that $\partial _{y}p^{l} \simeq \mu ({\partial ^{2}v^{l}}/{\partial y^{2}})\sim \mu \varepsilon ^{-2}\gg 1$. Indeed, the corresponding to (2.11a,b) pressure profile satisfies

(2.14a,b)\begin{equation} \partial _{y}p^{l}\simeq \dfrac{6\mu v_{\eta }( y) }{ \varepsilon^{2}\left( 1+\eta \right) ^{2}},\quad \partial _{z}p^{l}\simeq 0. \end{equation}

Finally, using (2.14a,b) and (2.2) one can estimate that $2\mu \partial _z w^{l} \simeq -2\mu \partial _y v^{l}\sim \mu \ll |p^{l}|$. From (2.10a,b) it then follows that $\sigma ^{l}_{zz} \simeq -p^{l}$, which may be determined by integrating (2.14a,b).

2.3. Outer flow

We now turn to the outer liquid flow (of length scale $\delta$), that practically cannot be affected by small variations in $\varepsilon \eta$, and, therefore, (2.8a–c) can be mapped to the flat interface, $\boldsymbol {u}\vert _{z=\varepsilon \eta }= \boldsymbol {u}\vert _{z=0}+O(\varepsilon ).$ Impermeability condition (2.3) is then reduced to $w(y,0)=0,$ and (2.5) takes the form of a conventional partial slip condition applied at $z=0$,

(2.15)\begin{equation} v = b(y)\dfrac{\partial v}{\partial z}, \end{equation}

with the local slip length described by (2.13).

An outer solution for a flow field with prescribed velocity $v_{\eta }( y)$ and zero normal velocity can be found using $u(y,z)$ for a longitudinal configuration (Asmolov & Vinogradova Reference Asmolov and Vinogradova2012)

(2.16a,b)\begin{gather} v=u+z\frac{\partial u}{\partial z},\quad w=-z\frac{\partial u}{\partial y}, \end{gather}

(2.17)\begin{gather}p=-2\frac{\partial u}{\partial y}. \end{gather}

Here, $u$ satisfies the Laplace equation, ${\rm \Delta} u=0$, with the boundary condition $u( 0,y) =v_{\eta }( y)$. Equations (2.16a,b), (2.17) immediately suggest that $\sigma _{zz}( 0,y)=2 \partial _{z}w-p=0$. In other words, the contributions of the pressure and the gradient of normal velocity to the normal stress cancel out. This implies that the outer flow does not affect the meniscus deformation. Consequently, the meniscus shape depends on the sign of the pressure gradient in the lubricant. When it is positive, the shape will be as shown in figures 1 and 2, i.e. concave at the front groove edge and convex at the rear one.

The equation describing the meniscus shape can be obtained by differentiating (2.9) with respect to $y$. Keeping then only the leading term in $\varepsilon$ and using (2.11a,b) we find that this shape obeys

(2.18)\begin{equation} \eta'''=-\frac{6 \mu {{Ca}}}{\varepsilon^{3}}\dfrac{v_{\eta}(y)}{(1+\eta)^{2}}. \end{equation}

To solve this differential equation, conditions (2.7a,b) should be imposed.

To summarize, the outer asymptotic problem is reduced to (2.1) coupled with the equation for the meniscus shape (2.18) expressed via the local slip length $b(y)$, (2.13), and interface velocity, $v_{\eta }$.

2.4. Limiting cases

In the general case, the inner and outer flows are strongly coupled, and the two-phase problem should be solved numerically. However, in some limits the system can be simplified, thanks to a decoupling of these flows.

In the limit of $\varepsilon /\mu \ll 1$, typical for a very viscous lubricant and/or an extremely thin lubricant layer, (2.15) reduces to a no-slip boundary condition, $v(y,0)\simeq 0$. Consequently, an outer flow remains undisturbed by the inner one, and the shear stress in the liquid is $\partial _{z}v(y,0) \simeq 1$. It follows then from (2.8a–c) that the lubricant shear rate is $\partial _{z}v^{l}(y,0) \simeq \mu ^{-1}$, and the interface velocity, found from (2.15), is $v_{\eta }=b(y)\partial _{z}v(y,0)\simeq \varepsilon (1+\eta )/4 \mu$, i.e. it decreases with $\mu$. From (2.14a,b) it follows then that $\partial _{y}p^{l}$ is finite and does not depend on $\mu$, and (2.18) reduces to

(2.19)\begin{equation} \eta''' =-\dfrac{3 {{Ca}} }{2 \varepsilon^{2} \left( 1+\eta \right) }. \end{equation}

In the opposite limiting case of low lubricant viscosity, $\varepsilon /\mu \gg 1$, the local slip length $b(y)$ is large provided $1+\eta (y)$ remains finite. Thus, we might argue that, at any deformation, a sensible approximation for a local slip would be $b(y) \rightarrow \infty$ that leads to the interface velocity $v_{\eta }^{P}( y)=[1-(2y-1)^{2}]^{1/2}/{4}$ (Philip Reference Philip1972). Substituting this to (2.18) we get

(2.20)\begin{equation} \eta'''=-\frac{6 \mu {{Ca}}}{\varepsilon^{3}}\dfrac{v_{\eta}^{p}(y)}{(1+\eta)^{2}} . \end{equation}