2 The dilute limit

Crowdy (Reference Crowdy2010) has previously derived the formula (1.2a−c ) for the longitudinal slip length relevant in the dilute limit $c/l\rightarrow 0$ ; it is natural to think of fixing $c=1$ , say, and taking $l\rightarrow \infty$ . Those prior results can be summarized as follows. With $z=x+\text{i}y$ we consider a complex potential $h_{s}(z)$ , analytic in the fluid region, such that

(2.1) $$\begin{eqnarray}w(x,y)=\dot{{\it\gamma}}\,\text{Im}[h_{s}(z)].\end{eqnarray}$$

The imposed far-field condition of simple shear requires that $w(x,y)\sim \dot{{\it\gamma}}y$ , hence

(2.2) $$\begin{eqnarray}h_{s}(z)\sim z+O(1/z),\quad \text{as}~z\rightarrow \infty .\end{eqnarray}$$

To satisfy the no-slip condition on the wall we need

(2.3) $$\begin{eqnarray}\text{Im}[h_{s}(z)]=0,\quad \text{on}~y=0,x\notin [-c,c],\end{eqnarray}$$

while the no-stress condition on the meniscus requires that

(2.4) $$\begin{eqnarray}\text{Re}[h_{s}(z)]=0,\quad \text{on the meniscus},\end{eqnarray}$$

which follows from the condition $\partial w/\partial n=0$ on use of the Cauchy–Riemann equations.

To solve this problem Crowdy (Reference Crowdy2010) employed conformal mapping techniques. With ${\it\beta}={\rm\pi}-{\it\theta}$ the conformal mapping transplanting the upper half of the unit disc in a parametric complex ${\it\zeta}$ -plane to the fluid region, and its inverse mapping function, are

(2.5a,b ) $$\begin{eqnarray}z({\it\zeta})=c\left[\frac{(1-{\it\zeta})^{2{\it\beta}/{\rm\pi}}+(1+{\it\zeta})^{2{\it\beta}/{\rm\pi}}}{(1-{\it\zeta})^{2{\it\beta}/{\rm\pi}}-(1+{\it\zeta})^{2{\it\beta}/{\rm\pi}}}\right],\quad {\it\zeta}(z)=\frac{(z/c-1)^{{\rm\pi}/2{\it\beta}}-(z/c+1)^{{\rm\pi}/2{\it\beta}}}{(z/c-1)^{{\rm\pi}/2{\it\beta}}+(z/c+1)^{{\rm\pi}/2{\it\beta}}}.\end{eqnarray}$$

Having derived these, Crowdy (Reference Crowdy2010) uses them to establish that

(2.6) $$\begin{eqnarray}h_{s}(z)=-\frac{2{\rm\pi}c}{{\it\beta}}\left[\frac{((z/c)^{2}-1)^{{\rm\pi}/2{\it\beta}}}{(z/c-1)^{{\rm\pi}/{\it\beta}}-(z/c+1)^{{\rm\pi}/{\it\beta}}}\right]\sim z+\frac{{\it\Delta}_{1}}{z}+\frac{{\it\Delta}_{3}}{z^{3}}+\cdots ,\quad \text{as}~|z|\rightarrow \infty ,\end{eqnarray}$$

where

(2.7a−c ) $$\begin{eqnarray}a=\frac{{\rm\pi}}{2({\rm\pi}-{\it\theta})},\quad {\it\Delta}_{1}=-\frac{c^{2}}{3}(2a^{2}+1),\quad {\it\Delta}_{3}=-\frac{2c^{4}}{45}(-7a^{4}+5a^{2}+2).\end{eqnarray}$$

Actually, only ${\it\Delta}_{1}$ is needed to derive (1.2a−c ) based on a simple superposition argument.

Since his focus was to find the slip length Crowdy (Reference Crowdy2010) did not report the corresponding dilute-limit complex potential, but it will be useful for what follows. It turns out to be

(2.8a,b ) $$\begin{eqnarray}h_{0}(z)=h_{s}(z)-\frac{{\it\Delta}_{1}}{z}-{\it\lambda}_{0}\cot ({\it\delta}z),\quad -\frac{{\it\lambda}_{0}}{{\it\delta}}={\it\Delta}_{1}.\end{eqnarray}$$

We now explain this result because it helps to understand our derivations of the higher-order approximations. If $h(z)$ denotes the exact solution to the periodic problem then $h(z)$ must be analytic in the period window $D$ shown in figure 1 and satisfy the quasi-periodicity condition

(2.9) $$\begin{eqnarray}h(z+2l)=h(z)+2l\end{eqnarray}$$

in order that the velocity $w=\dot{{\it\gamma}}\text{Im}[h(z)]$ is periodic; the quasi-periodicity here is induced by the far-field $z$ behaviour. Similarly, $h_{0}(z)$ is required to be analytic in $D$ . This is true of $h_{0}(z)$ given in (2.8) since $h_{s}(z)$ is analytic there – it is analytic, by construction, everywhere in the upper half-plane and above the meniscus – while the apparent singularity of $h_{0}(z)$ at $z=0$ , which is inside the period window if ${\it\theta}<0$ , is in any case removable by the choice of ${\it\lambda}_{0}$ . Moreover, the cotangent function has a periodic array of singularities along the real axis but the two closest to the one at $z=0$ are in the two period windows neighbouring $D$ and not inside $D$ itself.

The first term added to $h_{s}(z)$ in (2.8) serves the purpose of removing the $O(1/z)$ behaviour of $h_{s}(z)$ as $z\rightarrow \infty$ so that, as $z\rightarrow \infty$ ,

(2.10) $$\begin{eqnarray}h_{0}(z)\sim z+\text{i}{\it\lambda}_{0}+O(1/z^{3}),\end{eqnarray}$$

where we have used the fact that $\cot ({\it\delta}z)\rightarrow -\text{i}$ as $y\rightarrow +\infty$ and the known far-field asymptotics (2.6) of $h_{s}(z)$ . This means that, on the edges of the period window where $|z|\geqslant l$ , the function $h_{0}(z)$ satisfies

(2.11) $$\begin{eqnarray}h_{0}(z+2l)=h_{0}(z)+2l+O({\it\delta}^{3}),\end{eqnarray}$$

which is a good approximation to the quasi-periodicity condition (2.9) if ${\it\delta}$ is small. It is easily checked that $h_{0}(z)$ is real when $z$ is real since this is true of $h_{s}(z)$ , implying that $h_{0}(z)$ satisfies the no-slip condition on the wall. Also, provided ${\it\delta}$ is small, then on the meniscus

(2.12) $$\begin{eqnarray}\displaystyle \text{Re}[h_{0}(z)] & = & \displaystyle \text{Re}\left[h_{s}(z)-\frac{{\it\Delta}_{1}}{z}-{\it\lambda}_{0}\cot ({\it\delta}z)\right]\nonumber\\ \displaystyle & {\sim} & \displaystyle \text{Re}\left[-\frac{{\it\Delta}_{1}}{z}-{\it\lambda}_{0}\left\{\frac{1}{{\it\delta}z}-\frac{{\it\delta}z}{3}+\cdots \right\}\right]=\text{Re}\left[\frac{{\it\lambda}_{0}{\it\delta}z}{3}+\cdots \right]\nonumber\\ \displaystyle & = & \displaystyle O({\it\lambda}_{0}c{\it\delta})=O(c^{2}{\it\delta}^{2}).\end{eqnarray}$$

Thus $h_{0}(z)$ also satisfies the meniscus boundary condition correct to $O(c^{2}{\it\delta}^{2})$ . In this way we have verified that $h_{0}(z)$ is the approximation to the required complex potential provided that ${\it\delta}$ is small.

3 An improved slip length formula

To produce an improved formula for the slip length, accurate at even larger no-shear fractions, it is natural to seek a higher-order approximation in ${\it\delta}c$ . Consider the modified complex potential

(3.1) $$\begin{eqnarray}h_{1}(z)=h_{s}(z)+\frac{{\it\lambda}_{1}}{{\it\delta}}\frac{1}{z}-{\it\lambda}_{1}\cot ({\it\delta}z)+\frac{{\it\lambda}_{1}{\it\delta}}{3}[h_{s}(z)-z],\end{eqnarray}$$

with ${\it\lambda}_{1}$ now chosen to satisfy

(3.2) $$\begin{eqnarray}{\it\Delta}_{1}+\frac{{\it\lambda}_{1}}{{\it\delta}}+\frac{{\it\lambda}_{1}{\it\delta}{\it\Delta}_{1}}{3}=0.\end{eqnarray}$$

With this choice the $1/z$ term in the far-field expansion of the three non-cot terms

(3.3a−c ) $$\begin{eqnarray}h_{s}(z),\quad +\frac{{\it\lambda}_{1}}{{\it\delta}}\frac{1}{z},\quad +\frac{{\it\lambda}_{1}{\it\delta}}{3}[h_{s}(z)-z]\end{eqnarray}$$

of $h_{1}(z)$ vanishes. This ensures that, as $z\rightarrow \infty$ ,

(3.4) $$\begin{eqnarray}h_{1}(z)\sim z+\text{i}{\it\lambda}_{1}+O(1/z^{3}),\end{eqnarray}$$

where we have again used the fact that $\cot ({\it\delta}z)\rightarrow -\text{i}$ as $y\rightarrow +\infty$ and the known far-field asymptotics (2.6) of $h_{s}(z)$ . Hence $h_{1}(z)$ has the correct far-field behaviour with slip length ${\it\lambda}_{1}$ . On the edges of the period window where $|z|\geqslant l$ , we still have

(3.5) $$\begin{eqnarray}h_{1}(z+2l)=h_{1}(z)+2l+O({\it\delta}^{3}),\end{eqnarray}$$

which is a good approximation to the required quasi-periodicity (2.9) if ${\it\delta}$ is small. The apparent singularity of the function (3.1) at $z=0$ is again removable while the nearest other singularities of the cotangent function are in the neighbouring period windows as before. It is therefore confirmed that $h_{1}(z)$ is analytic in $D$ . When $z$ is real, which is true on the no-slip wall, so is $h_{1}(z)$ , confirming that the no-slip condition is satisfied there. Finally, on expansion of the cotangent for small ${\it\delta}$ , notice that on the meniscus,

(3.6) $$\begin{eqnarray}\displaystyle \text{Re}[h_{1}(z)] & = & \displaystyle \text{Re}\left[h_{s}(z)+\frac{{\it\lambda}_{1}}{{\it\delta}}\frac{1}{z}-{\it\lambda}_{1}\cot ({\it\delta}z)+\frac{{\it\lambda}_{1}{\it\delta}}{3}[h_{s}(z)-z]\right]\nonumber\\ \displaystyle & {\sim} & \displaystyle \text{Re}\left[+\frac{{\it\lambda}_{1}}{{\it\delta}}\frac{1}{z}-{\it\lambda}_{1}\left\{\frac{1}{{\it\delta}z}-\frac{{\it\delta}z}{3}+O(c^{3}{\it\delta}^{3})\right\}-\frac{{\it\lambda}_{1}{\it\delta}z}{3}\right]\nonumber\\ \displaystyle & = & \displaystyle O({\it\lambda}_{1}c^{3}{\it\delta}^{3})=O(c^{4}{\it\delta}^{4}),\end{eqnarray}$$

where we have used (2.4) twice as well as the Laurent expansion of $\cot ({\it\delta}z)$ about $z=0$ . Hence, (3.1) satisfies the boundary conditions on the meniscus, the no-slip wall and the edges of the period window correct to $O(c^{3}{\it\delta}^{3})$ . The quantity ${\it\lambda}_{1}$ as given by (3.2) is now the required slip length at this order of approximation and produces the result (1.3). Teo & Khoo (Reference Teo and Khoo2010) report the slip lengths ${\it\lambda}_{TK}$ , say, with the renormalizations

(3.7) $$\begin{eqnarray}{\it\lambda}_{TK}=\frac{{\it\lambda}_{1}}{2/l}=\frac{{\it\lambda}_{1}}{4{\it\delta}/{\rm\pi}},\end{eqnarray}$$

where we have taken $c=1$ . Figure 2 shows ${\it\lambda}_{TK}$ , as a function of protrusion angle ${\it\theta}$ , for $c/l=0.1$ , 0.25, 0.5, 0.75 and 0.9 together with corresponding data points from Teo & Khoo (Reference Teo and Khoo2010), which serve as our benchmark solution. The maximum relative error of this approximation for $c/l=0.75$ is between 6 and 7 % and for $c/l=0.9$ is around 25 %.

Figure 2.

Normalized slip length (3.7) as a function of protrusion angle ${\it\theta}$ for $c/l=0.1$ (a), 0.25 (b), 0.5 (c), 0.75 (d) and 0.9 (e). Crosses show numerical data from Teo & Khoo (Reference Teo and Khoo2010).

4 Higher-order analysis

The analysis of the previous section is readily extended to even higher order. We introduce $\tilde{h}_{s}(z)$ as the complex potential for a single protruding bubble with all the same boundary conditions as for $h_{s}(z)$ but now satisfying the modified far-field condition

(4.1) $$\begin{eqnarray}\tilde{h}_{s}(z)\sim z^{3}+O(1/z),\quad \text{as}~z\rightarrow \infty .\end{eqnarray}$$

This far-field flow is no longer a simple shear. By the same conformal mapping arguments (Crowdy Reference Crowdy2010) used to find $h_{s}(z)$ it can be shown that

(4.2) $$\begin{eqnarray}\tilde{h}_{s}(z)=-a^{3}c^{3}\left[\frac{1}{{\it\zeta}^{3}}-{\it\zeta}^{3}\right]+c^{2}(1-a^{2})\left[\frac{1}{{\it\zeta}}-{\it\zeta}\right]\sim z^{3}+\frac{\tilde{{\it\Delta}}_{1}}{z}+\frac{\tilde{{\it\Delta}}_{3}}{z^{3}}+\cdots ,\quad \text{as}~|z|\rightarrow \infty ,\end{eqnarray}$$

where we have used (2.5) and

(4.3) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle \tilde{{\it\Delta}}_{1}=-\frac{2c^{4}}{15}(-7a^{4}+5a^{2}+2),\\ \displaystyle \tilde{{\it\Delta}}_{3}=-\frac{c^{6}}{3780}(4855a^{6}+945a^{5}-5817a^{4}+6615a^{3}-6090a^{2}+3780a-508).\end{array}\right\}\end{eqnarray}$$

Now consider the new complex potential, constructed using $h_{s}(z)$ and $\tilde{h}_{s}(z)$ , given by

(4.4) $$\begin{eqnarray}\displaystyle h_{2}(z) & = & \displaystyle h_{s}(z)+\frac{{\it\lambda}_{2}}{{\it\delta}}\frac{1}{z}+\frac{{\it\mu}_{2}}{{\it\delta}^{3}}\frac{1}{z^{3}}-{\it\lambda}_{2}\cot ({\it\delta}z)-{\it\mu}_{2}\cot ({\it\delta}z)\text{cosec}^{2}({\it\delta}z)\nonumber\\ \displaystyle & & \displaystyle +\left[\frac{1}{3}{\it\lambda}_{2}{\it\delta}+\frac{1}{15}{\it\mu}_{2}{\it\delta}\right](h_{s}(z)-z)+\left[\frac{1}{45}{\it\lambda}_{2}{\it\delta}^{3}+\frac{20}{945}{\it\mu}_{2}{\it\delta}^{3}\right](\tilde{h}_{s}(z)-z^{3}),\qquad\end{eqnarray}$$

where the real parameters ${\it\lambda}_{2}$ and ${\it\mu}_{2}$ are chosen to satisfy

(4.5) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle {\it\Delta}_{1}+\frac{{\it\lambda}_{2}}{{\it\delta}}+\left[\frac{1}{3}{\it\lambda}_{2}{\it\delta}+\frac{1}{15}{\it\mu}_{2}{\it\delta}\right]{\it\Delta}_{1}+\left[\frac{1}{45}{\it\lambda}_{2}{\it\delta}^{3}+\frac{20}{945}{\it\mu}_{2}{\it\delta}^{3}\right]\tilde{{\it\Delta}}_{1}=0,\\ \displaystyle {\it\Delta}_{3}+\frac{{\it\mu}_{2}}{{\it\delta}^{3}}+\left[\frac{1}{3}{\it\lambda}_{2}{\it\delta}+\frac{1}{15}{\it\mu}_{2}{\it\delta}\right]{\it\Delta}_{3}+\left[\frac{1}{45}{\it\lambda}_{2}{\it\delta}^{3}+\frac{20}{945}{\it\mu}_{2}{\it\delta}^{3}\right]\tilde{{\it\Delta}}_{3}=0.\end{array}\right\}\end{eqnarray}$$

We claim that $h_{2}(z)$ is the required higher-order solution. To see this, the expansions

(4.6a,b ) $$\begin{eqnarray}\cot z=\frac{1}{z}-\frac{z}{3}-\frac{z^{3}}{45}-\frac{2}{945}z^{5}+\cdots ,\quad \cot (z)\text{cosec}^{2}(z)=\frac{1}{z^{3}}-\frac{z}{15}-\frac{20}{945}z^{3}+\cdots ,\end{eqnarray}$$

reveal that the third-order singularity of $h_{2}(z)$ at $z=0$ is removable. Furthermore,

(4.7) $$\begin{eqnarray}h_{2}(z)\rightarrow z+\text{i}{\it\lambda}_{2}+O(1/z^{5}),\end{eqnarray}$$

where we have used the facts that $\cot ({\it\delta}z)\rightarrow -\text{i}$ and $\cot ({\it\delta}z)\text{cosec}^{2}({\it\delta}z)$ decays exponentially as $y\rightarrow +\infty$ , as well as the far-field forms (2.6) and (4.2) of $h_{s}(z)$ and $\tilde{h}_{s}(z)$ . Then on the edges of the period window where $|z|\geqslant l$ ,

(4.8) $$\begin{eqnarray}h_{2}(z+2l)=h_{2}(z)+2l+O({\it\delta}^{5})\end{eqnarray}$$

so that $h_{2}(z)$ satisfies the required quasi-periodicity condition there (now to higher order than before). ${\it\lambda}_{2}$ is the required slip length. For $z$ on the no-slip wall $h_{2}(z)$ is real, implying that the no-slip condition is satisfied there. On expanding (4.4) for small ${\it\delta}$ , and on use of (4.5),

(4.9) $$\begin{eqnarray}\displaystyle \text{Re}[h_{2}(z)] & = & \displaystyle \text{Re}\left[\frac{{\it\lambda}_{2}}{{\it\delta}}\frac{1}{z}+\frac{{\it\mu}_{2}}{{\it\delta}^{3}}\frac{1}{z^{3}}-{\it\lambda}_{2}\left[\frac{1}{{\it\delta}z}-\frac{{\it\delta}z}{3}-\frac{{\it\delta}^{3}z^{3}}{45}+O({\it\delta}^{5}c^{5})\right]\right.\nonumber\\ \displaystyle & & \displaystyle -\,{\it\mu}_{2}\left[\frac{1}{{\it\delta}^{3}z^{3}}-\frac{{\it\delta}z}{15}-\frac{20}{945}{\it\delta}^{3}z^{3}+O({\it\delta}^{5}c^{5})\right]-\left[\frac{1}{3}{\it\lambda}_{2}{\it\delta}+\frac{1}{15}{\it\mu}_{2}{\it\delta}\right]z\nonumber\\ \displaystyle & & \displaystyle -\left.\left[\frac{1}{45}{\it\lambda}_{2}{\it\delta}^{3}+\frac{20}{945}{\it\mu}_{2}{\it\delta}^{3}\right]z^{3}\right]=O({\it\lambda}_{1}c^{5}{\it\delta}^{5}),\qquad\end{eqnarray}$$

where we have used the Laurent expansions (4.6) and the fact that

(4.10) $$\begin{eqnarray}\text{Re}[h_{s}(z)]=\text{Re}[\tilde{h}_{s}(z)]=0,\quad \text{on the meniscus}.\end{eqnarray}$$

Hence the meniscus boundary condition is satisfied correct to $O(c^{6}{\it\delta}^{6})$ . The solution of the $2\times 2$ system (4.5) gives the required slip length ${\it\lambda}_{2}$ . In matrix form it is

(4.11) $$\begin{eqnarray}\left(\begin{array}{@{}cc@{}}a_{11} & a_{12}\\ a_{21} & a_{22}\end{array}\right)\left(\begin{array}{@{}c@{}}{\it\lambda}_{2}\\ {\it\mu}_{2}\end{array}\right)=\left(\begin{array}{@{}c@{}}-{\it\Delta}_{1}{\it\delta}\\ -{\it\Delta}_{3}{\it\delta}^{3}\end{array}\right),\end{eqnarray}$$

where

(4.12) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle a_{11}=1+\frac{{\it\Delta}_{1}{\it\delta}^{2}}{3}+\frac{\tilde{{\it\Delta}}_{1}{\it\delta}^{4}}{45},\quad a_{12}=\frac{{\it\Delta}_{1}{\it\delta}^{2}}{15}+\frac{20\tilde{{\it\Delta}}_{1}{\it\delta}^{4}}{945},\\ \displaystyle a_{21}=\frac{{\it\Delta}_{3}{\it\delta}^{4}}{3}+\frac{\tilde{{\it\Delta}}_{3}{\it\delta}^{6}}{45},\quad a_{22}=1+\frac{{\it\Delta}_{3}{\it\delta}^{4}}{15}+\frac{20\tilde{{\it\Delta}}_{3}{\it\delta}^{6}}{945}.\end{array}\right\}\end{eqnarray}$$

In view of (2.7) and (4.3) all matrix elements are known as explicit functions of $a$ and $c$ , consequently an explicit formula for the slip length ${\it\lambda}_{2}$ is

(4.13) $$\begin{eqnarray}{\it\lambda}_{2}=\frac{{\it\Delta}_{3}{\it\delta}^{3}a_{12}-{\it\Delta}_{1}{\it\delta}a_{22}}{a_{11}a_{22}-a_{12}a_{21}}.\end{eqnarray}$$

The normalized slip length (3.7) – but now with ${\it\lambda}_{1}$ replaced by ${\it\lambda}_{2}$ as given by (4.13) – is plotted in figure 3, as a function of ${\it\theta}$ for $c/l=0.1$ , 0.25, 0.5, 0.75 and 0.9. It gives markedly better agreement with the numerical solution at $c/l=0.9$ than that shown in figure 2. The maximum relative error of this approximation for $c/l=0.75$ is between 1 and 2 %, and for $c/l=0.9$ it is between 8 and 9 %.

Figure 3.

Normalized slip length, now based on the approximation ${\it\lambda}_{2}$ in (4.13), as a function of ${\it\theta}$ for $c/l=0.1$ (a), 0.25 (b), 0.5 (c), 0.75 (d) and 0.9 (e). Crosses show numerical data from Teo & Khoo (Reference Teo and Khoo2010).

Our analysis leads naturally to formula (4.13) but if we use the second equation in (4.5) to approximate

(4.14) $$\begin{eqnarray}{\it\mu}_{2}\approx -{\it\Delta}_{3}{\it\delta}^{3}\end{eqnarray}$$

then, on substitution of this into the first equation in (4.5), we find the approximation

(4.15) $$\begin{eqnarray}{\it\lambda}_{2}\approx -\frac{\displaystyle {\it\delta}{\it\Delta}_{1}\left(1-\frac{{\it\delta}^{4}{\it\Delta}_{3}}{15}\right)}{\displaystyle 1+\frac{{\it\Delta}_{1}{\it\delta}^{2}}{3}+\frac{{\it\delta}^{4}{\it\Delta}_{3}}{15}}.\end{eqnarray}$$

On eliminating $a$ in favour of ${\it\theta}$ , it can be shown that

(4.16) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle {\it\Delta}_{1}=-\frac{c^{2}}{6({\rm\pi}-{\it\theta})^{2}}(2{\it\theta}^{2}-4{\rm\pi}{\it\theta}+3{\rm\pi}^{2}),\\ \displaystyle {\it\Delta}_{3}=-\frac{c^{4}}{360({\rm\pi}-{\it\theta})^{4}}(32{\it\theta}^{4}-128{\rm\pi}{\it\theta}^{3}+212{\rm\pi}^{2}{\it\theta}^{2}-168{\rm\pi}^{3}{\it\theta}+45{\rm\pi}^{4}).\end{array}\right\}\end{eqnarray}$$

Equation (4.15) is now equivalent to (1.4) reported earlier. The maximum relative error of this approximation is not quite as good as for formula (4.13), but is still impressive: at $c/l=0.75$ the maximum relative error is around 2 % (about the same as for (4.13)) and for $c/l=0.9$ it is around 12 % (slightly worse). The surprisingly wide range of no-shear fractions for which these formulae give good accuracy arguably obviates the need for higher-order approximations, but these are easily derivable in principle.

5 Verification against previous results

When formally expanded in powers of the no-shear fraction our formulae cross-check with other known results. First, Philip (Reference Philip1972) has shown that for a flat meniscus with ${\it\theta}=0$ , $c=1$ and at any no-shear fraction,

(5.1) $$\begin{eqnarray}{\it\lambda}=\frac{2l}{{\rm\pi}}\log \sec \left[\frac{{\rm\pi}}{2l}\right]=-\frac{1}{{\it\delta}}\log \cos {\it\delta}\approx {\it\delta}\left[\frac{1}{2}+\frac{{\it\delta}^{2}}{12}+\frac{{\it\delta}^{4}}{45}+\cdots \right],\quad {\it\delta}\equiv \frac{{\rm\pi}}{2l}.\end{eqnarray}$$

But for ${\it\theta}=0$ we find ${\it\alpha}(0)=1/2$ and ${\it\beta}(0)=1/8$ and it can be verified that (1.4) agrees with (5.1) to this order of expansion in ${\it\delta}$ .

By taking their microchannel width to infinity, Sbragaglia & Prosperetti (Reference Sbragaglia and Prosperetti2007) performed an analytical study for small protrusion angles ${\it\theta}$ in a semi-infinite domain. The expansion of (4.15) for small ${\it\delta}$ is

(5.2) $$\begin{eqnarray}\frac{{\it\lambda}_{2}}{c}={\it\delta}c{\it\alpha}({\it\theta})\left[1+\frac{c^{2}{\it\delta}^{2}{\it\alpha}({\it\theta})}{3}+{\it\delta}^{4}c^{4}\left(\frac{2{\it\beta}({\it\theta})}{15}+\frac{{\it\alpha}({\it\theta})^{2}}{9}\right)+\cdots \right].\end{eqnarray}$$

A Taylor expansion of ${\it\alpha}({\it\theta})$ and ${\it\beta}({\it\theta})$ to linear order in the protrusion angle ${\it\theta}$ gives

(5.3a,b ) $$\begin{eqnarray}{\it\alpha}({\it\theta})=\frac{1}{2}+\frac{{\it\theta}}{3{\rm\pi}}+\cdots ,\quad {\it\beta}({\it\theta})=\frac{1}{8}+\frac{{\it\theta}}{30{\rm\pi}}+\cdots .\end{eqnarray}$$

On substitution of (5.3) into (5.2), we find

(5.4) $$\begin{eqnarray}\frac{{\it\lambda}_{2}}{c}={\it\delta}c\left[\frac{1}{2}+\frac{{\it\delta}^{2}c^{2}}{12}+\frac{{\it\delta}^{4}c^{4}}{45}+\cdots \right]+{\it\delta}c\left(\frac{{\it\theta}}{{\rm\pi}}\right)\left[\frac{1}{3}+\frac{c^{2}{\it\delta}^{2}}{9}+\frac{24}{675}c^{4}{\it\delta}^{4}+\cdots \right]+\cdots .\end{eqnarray}$$

The first term agrees with (5.1), valid for ${\it\theta}=0$ , once we set $c=1$ . On setting ${\it\delta}={\rm\pi}{\it\xi}/2c$ the normalized leading-order correction due to the meniscus curvature is

(5.5) $$\begin{eqnarray}\frac{c{\it\theta}{\it\xi}}{2}\left[\frac{1}{3}+\frac{{\it\xi}^{2}{\rm\pi}^{2}}{36}+\frac{{\it\xi}^{4}{\rm\pi}^{4}}{450}\right]=-\frac{c^{2}}{2R}\left[\frac{{\it\xi}}{3}+\frac{{\it\xi}^{3}{\rm\pi}^{2}}{36}+\frac{{\it\xi}^{4}{\rm\pi}^{4}c^{4}}{450}\right],\end{eqnarray}$$

where, for small meniscus deflection downwards into the grooves, ${\it\theta}\approx \sin {\it\theta}=-c/R$ . We thus retrieve the result in (45)–(47) of Sbragaglia & Prosperetti (Reference Sbragaglia and Prosperetti2007), who derived it using quite different techniques. In summary, various series expansions of our formulae (1.3) and (4.13), or (1.4), give results that are consistent with other studies. Note, however, that we continue to present (1.3) and (1.4) in the unexpanded form arising naturally from our analysis.

6 A reciprocity result

In view of their convenient explicit forms we expect that the slip length formulae (1.3) and (1.4), and the associated complex potentials (3.1) and (4.4), will be useful in a variety of studies of superhydrophobic surfaces where, for example, additional physical effects are included (such as heat transfer, Marangoni or thermocapillary effects, or the influence of an enclosed gas phase). We end by showing how to combine the new formulae with a useful reciprocity result based on one of Green’s identities.

Let $w_{2}(z)=\text{Im}[\dot{{\it\gamma}}h_{2}(z)]$ , with associated slip length ${\it\lambda}_{2}$ . Suppose ${\hat{w}}$ is the solution of the same problem of shear flow over the bubble mattress with all the same boundary conditions imposed on $w_{2}(z)$ except on the meniscus, where we now require

(6.1) $$\begin{eqnarray}\frac{\partial {\hat{w}}}{\partial n}=\mathscr{A}(z,\overline{z}),\end{eqnarray}$$

where $\mathscr{A}(z,\overline{z})$ is some specified function (derived, say, from inclusion of additional physical effects). Let the slip length for this modified problem be $\hat{{\it\lambda}}$ . By Green’s identity, the harmonicity of $w_{2}$ and ${\hat{w}}$ , and the divergence theorem, we deduce that

(6.2) $$\begin{eqnarray}0=\int _{D}[w_{2}{\rm\nabla}^{2}{\hat{w}}-{\hat{w}}{\rm\nabla}^{2}w_{2}]\,\text{d}A=\oint _{\partial D}w_{2}\frac{\partial {\hat{w}}}{\partial n}\,\text{d}s-\oint _{\partial D}{\hat{w}}\frac{\partial w_{2}}{\partial n}\,\text{d}s,\end{eqnarray}$$

where $\partial D$ is the boundary of $D$ . On the edges of the period window for which $|z|\geqslant l$ ,

(6.3a,b ) $$\begin{eqnarray}w_{2}=\dot{{\it\gamma}}(y+{\it\lambda}_{2})+O({\it\delta}^{5}),\quad \frac{\partial w_{2}}{\partial n}\,\text{d}s=\pm \frac{\partial w_{2}}{\partial x}\,\text{d}y,\end{eqnarray}$$

where we have used (4.7). Hence the contributions to the right-hand side of (6.2) from the side edges are small (for small  ${\it\delta}$ ) while the no-slip portions of the surface do not contribute at all. We then arrive at the approximation

(6.4) $$\begin{eqnarray}\hat{{\it\lambda}}\approx {\it\lambda}_{2}-\frac{1}{2l}\int _{\mathit{meniscus}}\mathscr{A}(z,\overline{z})\,\text{Im}[h_{2}(z)]\,\text{d}s,\end{eqnarray}$$

where we have integrated around $\partial D$ in an anticlockwise direction and used the far-field conditions. By virtue of the results of this paper the right-hand side of (6.4) is an integral expression for the required slip length $\hat{{\it\lambda}}$ accurate to the same order in the no-shear fraction as the solution $h_{2}(z)$ and ${\it\lambda}_{2}$ used to obtain it.