2. A boundary value problem of mixed type

In a Cartesian $(x,y,Z)$ plane, let $\boldsymbol {u}=(0,0,w(x,y))$ be the steady, fully developed, pressure-driven flow in the axial $Z$ direction of a fluid, specifically a liquid rather than a gas, of viscosity $\mu$ , through a circular pipe of radius $R_0$ having a cross-section in the $(x,y)$ plane, denoted by $D_z$ . The pipe is superhydrophobic and is assumed to have $M \geqslant 1$ no-shear menisci separated by solid wall portions; later, superhydrophobic pipes with an $N$ -fold rotationally symmetric array of menisci with $M$ menisci per $2\pi /N$ sector will also be considered. It is assumed that all menisci match the curvature of the pipe wall. However, depending on external influences, there may be additional interface deflection, in which case the present interface shape should be understood as a first approximation; this matter is discussed again in § 8. For the analysis that follows, no potential influence of surfactant depositions at the interface is considered. The axial, or longitudinal, velocity $w(x,y)$ in such a pipe is governed by

(2.1) \begin{equation} \nabla ^2 w = - \frac {s R_0^2}{\mu } = -1, \quad \nabla ^2 = \frac {\partial }{\partial x^2} + \frac {\partial }{\partial y^2}, \end{equation}

where the flow is driven by a constant pressure gradient $s$ along the $Z$ -axis in the domain $0 \leqslant |(x,y)| \lt 1$ . Lengths are normalised with respect to the outer-wall radius $R_0$ , and the velocity with respect to $s R_0^2 /\mu$ . At the pipe boundary, the Poisson problem (2.1) has a mixture of two types of boundary conditions: either a no-slip condition on any portion of solid wall, or a no-shear condition on any meniscus. These can be expressed as

(2.2) \begin{eqnarray} w = 0 \quad \mathrm {on} \ |(x,y)| = 1 \quad (\mathrm {no \ slip \ parts}), \end{eqnarray}

(2.3) \begin{eqnarray} \frac {\partial w}{\partial {n}} = 0 \quad \mathrm {on} \ |(x,y)| = 1 \quad (\mathrm {no \ shear \ parts}), \end{eqnarray}

where $\partial /\partial n$ denotes the derivative in the normal direction, pointing into the viscous fluid, at the circular pipe boundary.

It is expedient to express this mixed boundary value problem in terms of the complex variable $z=x+\textrm {i}y$ . A flow governed by the linear partial differential equation (2.1) subject to (2.2) and (2.3) can be written as the sum of a particular solution $w_{ps}$ and a harmonic part $w_{h}$ , i.e.

(2.4) \begin{equation} w = w_{ps} + w_{h}, \end{equation}

with

(2.5) \begin{equation} w_{{ps}} = \frac {1-|z|^2}{4},\quad w_h= \mathrm {Re} \left [ h(z) \right ], \end{equation}

where non-dimensionalised Poiseuille flow is chosen as the particular solution $w_{ps}$ , and $\textrm {Re}[\cdot]$ denotes the real part of the quantity in square brackets. The harmonic part $w_{h}$ , which is the real part of an analytic function $h(z)$ , a complex potential, serves as a correction to the Poiseuille flow that takes into account the influence of the shear-free menisci on the velocity field. In this way, the problem is reduced to finding a complex potential $h(z)$ , analytic in $D_z$ , whose real part is the required $w_h$ .

The challenge then is to determine $h(z)$ . To do so, it is necessary to express the mixed-type boundary conditions (2.2) and (2.3) in terms of it. By the choice of $w_{ps}$ , the no-slip condition (2.2) gives

(2.6) \begin{equation} \mathrm {Re} \left [ h(z) \right ] = 0 \quad \mathrm {on} \ |z| = 1 \quad (\mathrm {no \ slip \ parts}). \end{equation}

The no-shear condition (2.3) on $|z|=1$ can be written as

(2.7) \begin{equation} {\pmb \nabla} w \boldsymbol {\cdot } \boldsymbol {n} = \mathrm {Re} \left [ 2 \frac {\partial w}{\partial z} (-z) \right ] = 0, \end{equation}

where use has been made of the fact that if the complex analogues of two vectors $\textbf{a}=(a_x,a_y)$ and $\textbf{b}=(b_x,b_y)$ are $a=a_x+\textrm {i} a_y$ and $b=b_x+\textrm {i} b_y$ , then the analogue of the dot product $\textbf{a}\boldsymbol{\cdot}\textbf{b}$ is $\textrm {Re}[\overline {a} b]$ , where $\bar {a}$ is the complex conjugate of $a$ . Note that the complex form of ${\pmb \nabla} w$ is $2\,\partial w / \partial \overline {z}$ , and the complex form of the normal vector, pointing into the pipe, is $-z$ . On use of (2.4), condition (2.7) is equivalent to

(2.8) \begin{equation} \mathrm {Re} \left [ z h'(z) \right ] = \frac {1}{2} \quad \mathrm {on} \ |z| = 1 \quad (\mathrm {no \ shear \ parts}). \end{equation}

An important observation is that by the chain rule, and using the parametrisation $z=\textrm{e}^{\textrm {i} \theta }$ for $|z|=1$ ,

(2.9) \begin{equation} {\partial \over \partial \theta } = \textrm {i} z\, {\partial \over \partial z}, \end{equation}

so that (2.8) can alternatively be written as

(2.10) \begin{equation} \mathrm {Re} \left [ - \textrm {i}\, {\partial \over \partial \theta } h(z) \right ] = \textrm {Im} \left [ {\partial \over \partial \theta } h(z) \right ] = \frac {1}{2} \quad \mathrm {on} \ |z| = 1 \quad (\mathrm {no \ shear \ parts}). \end{equation}

This can be integrated with respect to $\theta$ on the $j$ th no-shear portion to give

(2.11) \begin{equation} \textrm {Im}[h(z)] = {\theta \over 2} + c_j, \end{equation}

where $c_j$ is a constant of integration. Without loss of generality, one of these constants can be set equal to zero.

In summary, the mathematical problem in the case of $M$ no-shear menisci is to find a function $h(z)$ analytic in $|z| \lt 1$ satisfying the mixed boundary conditions

(2.12) \begin{equation} \left\{\begin{array}{ll} \textrm {Re}[h(z)] = 0 & \textrm {on solid portions of}\ |z|=1, \\[2pt] \textrm {Im}[h(z)] = {\theta /2} + c_j & \textrm {on meniscus}\ j\ \textrm {of}\ |z|=1, \end{array}\right. \end{equation}

where $z=\textrm{e}^{\textrm {i}\theta }$ for some set of real constants $\lbrace c_j : j=1, \dots , M \rbrace$ .

3. A new derivation of the Philip (Reference Philip1972) single slot solution

It is useful to first retrieve Philip’s solution (1.1)–(1.2) with $M=1$ in a novel manner that differs from the original derivation in Philip (Reference Philip1972). It is then shown in §§ 5 and 6 how the analysis of this section generalises, in a straightforward way, to the more general case of $M \gt 1$ no-shear slots in general position on the pipe wall; this generalisation makes use of the prime function and conformal slit mapping machinery developed in Crowdy et al. (Reference Crowdy, Kropf, Green and Nasser2016)and Crowdy (Reference Crowdy2020).

Figure 1.

Conformal slit mappings $Z(\zeta )$ and $\mathcal {R}(\zeta )$ from an upper half unit disc $\lbrace D_\zeta ^+: |\zeta |\lt 1,\ \textrm {Im}[\zeta ] \geqslant 0 \rbrace$ to a bounded circular slit domain (in a complex $z$ plane) and a bounded radial slit domain (in a complex $\chi$ plane). Corresponding images under the respective maps are colour-coded. The point $\zeta =\alpha$ is the pre-image of the origin in both cases.

The problem to be solved is the situation where the pipe wall has a single shear-free meniscus occupying the sector

(3.1) \begin{equation} -\Phi \lt \textrm {arg}[z] \lt \Phi \end{equation}

for some $\Phi \in [0, \pi )$ . The first step is to introduce a conformal mapping from the upper half unit $\zeta$ disc, denoted by $D_\zeta ^+ = \lbrace \zeta \ : |\zeta | \lt 1,\ \textrm {Im}[\zeta ] \gt 0 \rbrace$ to the fluid inside the circular pipe; figure 1 shows a schematic. It is given by

(3.2) \begin{equation} z = {Z}(\zeta ) \equiv - {\omega (\zeta ,\alpha )\, \omega (\zeta , 1/\alpha ) \over \omega (\zeta ,\overline {\alpha })\, \omega (\zeta , 1/\overline {\alpha })} , \quad \omega (\zeta ,\alpha ) = \zeta -\alpha , \quad \alpha = \textrm {i} r, \end{equation}

for some $0 \leqslant r \lt 1$ . Using this definition, it is easy to check that the upper half unit semicircle, denoted by $C_0^+$ , is transplanted to a portion of the circle $|z|=1$ , while the real diameter $-1 \lt \zeta \lt 1$ is transplanted to the remainder of this circle. It is also easily verified that

(3.3) \begin{equation} {Z}(0) = -1, \quad {Z}(1) = \left ( {1-\alpha \over 1 +\alpha } \right )^2, \quad {Z}(\textrm {i}) = 1. \end{equation}

The semicircle $C_0^+$ can be identified with the pre-image of the meniscus, and the real diameter $-1 \lt \zeta \lt 1$ is the pre-image of the no-slip portion of the pipe wall. The centre of the meniscus at $z=1$ is $\zeta = \textrm {i}$ . The pre-image of the edge of the meniscus in the lower half-plane, i.e. at $z=\textrm{e}^{-\textrm {i} \Phi }$ , is $\zeta =1$ , meaning that

(3.4) \begin{equation} {Z}(1) = \left ( {1-\alpha \over 1 +\alpha } \right )^2 = \left ( {1-\textrm {i}r \over 1 +\textrm {i} r} \right )^2 = \textrm{e}^{-\textrm {i} \Phi }, \quad \textrm {or} \quad {1-\textrm {i}r \over 1 +\textrm {i} r} = \textrm{e}^{-\textrm {i} \Phi /2}. \end{equation}

Simple manipulation of this relation and use of familiar trigonometric identities yields

(3.5) \begin{equation} r = \tan (\Phi /4). \end{equation}

The simple monomial $\omega (\zeta ,\alpha ) = \zeta -\alpha$ can be identified as the prime function for the unit disc, while the form of the mapping (3.2) is an example of a bounded circular slit mapping. These concepts and their generalisations are discussed in detail by Crowdy (Reference Crowdy2020).

It is useful now to write down another conformal slit mapping to a complex $\chi$ plane, called a radial slit mapping (Crowdy Reference Crowdy2020), having a form very similar to (3.2) but defined by

(3.6) \begin{equation} \chi = {\mathcal {R}}(\zeta ) \equiv - {\omega (\zeta ,\alpha )\, \omega (\zeta , 1/\overline {\alpha }) \over \omega (\zeta ,\overline {\alpha })\, \omega (\zeta , 1/\alpha )}. \end{equation}

Notice that while the first terms in the numerators and denominators of (3.2) and (3.6) are the same, the second terms have been swapped. The image of $D_\zeta ^+$ under this mapping is the interior of the circular disc $|\chi | \lt 1$ , with $C_0^+$ being transplanted to a radial slit on the positive real $\chi$ -axis, and the real diameter $-1 \lt \zeta \lt 1$ being transplanted to the entire circle $|\chi |=1$ . It is easy to check, using (3.6), that

(3.7) \begin{equation} {\mathcal {R}}(1) = 1, \quad {\mathcal {R}}(-1) = 1. \end{equation}

Given explicit knowledge of the conformal mapping $Z(\zeta )$ , the composed function

(3.8) \begin{equation} H(\zeta ) \equiv h({Z}(\zeta )) \end{equation}

can be introduced, which, by virtue of the respective analyticity properties of $h(z)$ and ${ Z}(\zeta )$ , is analytic in $D_\zeta ^+$ . In view of (2.12), on the boundary of $D_\zeta ^+$ it must satisfy

(3.9) \begin{equation} \left\{\begin{array}{ll} \textrm {Re}[H(\zeta )] = 0 & \textrm {on}\ {-1} \lt \zeta \lt 1, \\[2pt] \textrm {Im}[H(\zeta )] = {\theta /2} & \textrm {on}\ C_0^+, \end{array}\right. \end{equation}

where $z={Z}(\zeta ) = \textrm{e}^{\textrm {i} \theta }$ , and where the single constant of integration in this $M=1$ problem has been set to zero. This can also be written as

(3.10) \begin{equation} \left\{\begin{array}{ll} \textrm {Re}[H(\zeta )] = 0 & \textrm {on}\ {-1} \lt \zeta \lt 1, \\ \textrm {Im}[H(\zeta )] = - \displaystyle {\textrm {i} \log {Z}(\zeta ) \over 2} & \textrm {on}\ C_0^+. \end{array}\right. \end{equation}

It will now be argued that the expression

(3.11) \begin{equation} H(\zeta ) = {1 \over 2} \left ( \log Z(\zeta ) - \log {\mathcal {R}}(\zeta ) \right ), \end{equation}

involving both the bounded circular slit mapping ${Z}(\zeta )$ and the radial slit mapping ${\mathcal {R}}(\zeta )$ , is the required solution for $H(\zeta )$ . Both mappings ${Z}(\zeta )$ and ${\mathcal {R}}(\zeta )$ have simple zeros at $\zeta =\alpha$ in $D_\zeta ^+$ , but the resulting logarithmic singularities cancel out in (3.11), meaning that the function $H(\zeta )$ in (3.11) is analytic in $D_\zeta ^+$ , as required. Since the real diameter $-1 \lt \zeta \lt 1$ is transplanted to a unit circle (in the respective target plane) by both slit mappings ${Z}(\zeta )$ and ${\mathcal {R}}(\zeta )$ , it is easy to check that $\textrm {Re}[H(\zeta )]=0$ on this real diameter; thus (3.11) satisfies the first of the boundary conditions in (3.10). And since $C_0^+$ is transplanted by ${\mathcal {R}}(\zeta )$ to a radial slit on the positive real $\chi$ -axis, so that $\textrm {arg}[{\mathcal {R}}(\zeta )] = 0$ for $\zeta \in C_0^+$ , it is just as easy to verify that (3.11) also satisfies the second of the boundary conditions in (3.10).

Expression (3.11) is a convenient parametric form of Philip’s single slot solution (Philip Reference Philip1972). It is shown in Appendix A how to retrieve (1.1)–(1.2) for $N=1$ from (3.11). A significant advantage of the parametric form (3.11) is that the square-root branch points in $h(z)$ at the edge of the no-shear slots do not appear explicitly, and are accounted for implicitly by use of the parametric variable $\zeta$ .

4. Effective slip length

It is now possible to calculate the so-called effective slip length $\lambda _{\textit{eff}}$ representing the averaged influence of the no-shear boundary segments on the axial velocity field, providing a tool to quantify the degree to which given SHS diverge from being a no-slip wall. It involves equating the total volume flux $Q$ caused by a given pressure gradient $s$ to that of a suitable comparison flow $(0,0,w_{ns}(x,y))$ driven by the same pressure gradient, containing a purely no-slip wall at $|z| = 1$ . The effective slip length is the value of $\lambda _{\textit{eff}}$ for which

(4.1) \begin{equation} Q_{ns} = Q(\lambda _{\textit{eff}}). \end{equation}

To evaluate this quantity, consider first a general axisymmetric pipe flow solution of

(4.2) \begin{equation} {\pmb \nabla} ^2 w^* = -1 \end{equation}

subject to

(4.3) \begin{equation} w^* = \lambda _{\textit {eff}}\, {\partial w^* \over \partial n}. \end{equation}

This problem has the solution

(4.4) \begin{equation} w^* = \frac {1 - |z|^2}{4} + \frac {\lambda _{\textit{eff}}}{2} , \end{equation}

describing an axisymmetric pressure-driven pipe flow with constant slip $\lambda _{\textit{eff}}$ at the outer wall, which can be interpreted as the effective slip length. By integrating (4.4) over the pipe cross-section, the associated volume flux is found to be

(4.5) \begin{equation} Q = \frac {\pi }{8} + \frac {\pi }{2} \lambda _{\textit{eff}}. \end{equation}

An explicit expression for the effective slip length is then found by rearranging the ratio $Q / Q_{ns}$ such that

(4.6) \begin{equation} \lambda _{\textit{eff}} = \frac {1}{4} \left ( \frac {Q}{Q_{ns}} - 1 \right ), \end{equation}

where $Q_{ns}$ denotes the reference no-slip flow value. It is natural to take the comparison flow to be Poiseuille flow through the pipe for which $Q_{ns} = \pi / 8$ .

It remains to find the mass flux $Q$ associated with the solution of the mixed boundary value problems solved earlier. In view of the decomposition (2.4), it is natural to write

(4.7) \begin{equation} Q = Q_{ns} + Q_{h}, \end{equation}

where

(4.8) \begin{equation} Q_{h} = \iint _{D_z} w_{h} \, \mathrm {d}A = \iint _{D_z} \mathrm {Re} \left [ h(z) \right ] \, \mathrm {d}A \end{equation}

so that the effective slip length is given by

(4.9) \begin{equation} \lambda _{\textit{eff}} = \frac {1}{4} \left ( \frac {Q_{ns} + Q_{h}}{Q_{ns}} -1 \right ) = \frac {1}{4} \frac {Q_{h}}{Q_{ns}} = {2 \over \pi } Q_{h}. \end{equation}

Let $\partial D_z$ denote the pipe boundary. The complex form of the Stokes theorem implies

(4.10) \begin{equation} Q_{h} =\textrm {Re} \left [ \int _{D_z} h(z)\, \textrm{d}A \right ] =\textrm {Re} \left [ {1 \over 2\textrm {i}} \oint _{\partial D_z} h(z)\, \overline {z}\, \textrm{d}z \right ] =\textrm {Re} \left [ {1 \over 2\textrm {i}} \oint _{\partial D_z} h(z)\, {\textrm{d}z \over z} \right ], \end{equation}

where the fact that $\overline {z} =1/z$ on $\partial D_z$ has been used. But $h(z)$ is analytic in $D_z$ , so by the residue theorem,

(4.11) \begin{equation} Q_{h} =\textrm {Re} \left [ {2 \pi \textrm {i}} \times {h(0) \over 2 \textrm {i}} \right ] = \pi\, \textrm {Re}[h(0)]. \end{equation}

It follows that

(4.12) \begin{equation} Q_{h} = \pi \log \left | {\omega (\alpha ,1/\alpha ) \over \omega (\alpha ,1/\overline {\alpha })} \right |, \end{equation}

where (3.11), (3.2) and (3.6) have been used. It follows from (4.9) and (4.12) that

(4.13) \begin{equation} \lambda _{\textit {eff}} = {2 \over \pi } Q_h = 2 \log \left | {\omega (\alpha ,1/\alpha ) \over \omega (\alpha ,1/\overline {\alpha })} \right |. \end{equation}

This expression for the effective slip length in terms of the prime function of the pre-image domain, in this case the unit $\zeta$ disc, is an important new result of this paper. Exactly the same formula will be found in the following sections to give the effective slip length for a wide range of superhydrophobic pipes with more general no-shear patterns, the only modification being that the appropriate prime function $\omega (\cdot,\cdot)$ relevant to each pipe must be used in (4.13).

On use of the definition (3.2) of the prime function for the unit disc, it follows from (4.13) that

(4.14) \begin{equation} \lambda _{\textit {eff}} = 2 \log \left | {\alpha -1/\alpha \over \alpha -1/\overline {\alpha }} \right | = 2 \log \left | {1+r^2 \over 1-r^2} \right | = 2 \log \left ( \sec \left ( \frac {\Phi }{2}\right ) \right ), \end{equation}

where (3.5) and some trigonometric double-angle formulae have been used. In this way, Philip’s slip length result (1.3) for a single slot is retrieved. As will be seen shortly, other choices of prime function used in (4.13) will give different results.

5. Exact solution for $M=2$ unequal menisci

Consider a superhydrophobic pipe, with cross-section again denoted by $D_z$ , now having $M=2$ generally unequal menisci separated by solid portions with total solid fraction equal to $\phi$ , and each having equal circumferential length. The latter restriction is not necessary, and is easily lifted, but it makes presentation of the results clearer. According to the analysis in § 2, the problem reduces to finding a function $h(z)$ , analytic inside $D_z$ and satisfying the mixed boundary conditions

(5.1) \begin{equation} \left\{\begin{array}{ll} \textrm {Re}[h(z)] = 0 & \textrm {on solid portions}\ \textrm {of}\ |z|=1, \\[2pt] \textrm {Im}[h(z)] = {\theta /2} & \textrm {on meniscus}\ 1\ \textrm {of}\ |z|=1, \\[2pt] \textrm {Im}[h(z)] = {\theta /2} + c_2 & \textrm {on meniscus}\ 2\ \textrm {of}\ |z|=1, \end{array}\right. \end{equation}

for some real constant $c_2$ .

In view of the new derivation of Philip’s single meniscus solution in § 3, and motivated by the use of conformal slit mappings in solving mixed boundary value problems of this kind as originally proposed by Crowdy (Reference Crowdy2011), it is now posed that a parametric solution for $h(z)$ satisfying the boundary conditions (2.12) is given by

(5.2) \begin{align} z &= Z(\zeta ), \quad h = H(\zeta ) = \tfrac{1}{2} \left ( \log Z(\zeta ) - \log {\mathcal {R}}(\zeta ) \right ),\end{align}

where the parametric complex variable $\zeta$ takes values in the upper half-annulus $\lbrace D_\zeta ^+: \rho \lt |\zeta |\lt 1,\ \textrm {Im}[\zeta ] \geqslant 0 \rbrace$ , and

(5.3) \begin{equation} Z(\zeta ) \equiv -{\omega (\zeta ,\alpha )\, \omega (\zeta , 1/\alpha ) \over \omega (\zeta ,\overline {\alpha })\, \omega (\zeta , 1/\overline {\alpha })}, \quad {\mathcal {R}}(\zeta ) \equiv -{\omega (\zeta ,\alpha )\, \omega (\zeta , 1/\overline {\alpha }) \over \omega (\zeta ,\overline {\alpha })\, \omega (\zeta , 1/{\alpha })}, \end{equation}

for

(5.4) \begin{equation} \omega (\zeta ,\alpha ) = -{\alpha \over C^2}\, P(\zeta /\alpha ,\rho ), \quad C= \prod _{n=1}^\infty \left(1-\rho ^{2n}\right), \end{equation}

with $\alpha = \textrm {i} r$ for $\rho \lt r\lt 1$ , and where

(5.5) \begin{equation} P(\zeta ,\rho ) = (1-\zeta )\,\hat P(\zeta ,\rho ), \quad \hat P(\zeta ,\rho ) \equiv \prod _{n=1}^\infty \left(1-\rho ^{2n}\zeta \right) \left(1-\rho ^{2n}/\zeta \right). \end{equation}

In (5.2), the principal branch of the complex logarithm is assumed with $\log X = \log |X| + \textrm {i} \textrm { arg}[X]$ , where $-\pi \lt \textrm {arg}[X] \leqslant \pi$ so that the logarithm of a negative real quantity has imaginary part $\textrm {i} \pi$ .

Functionally, these formulae look identical to those in § 3, but there is a small but significant difference: the function $\omega (\zeta ,\alpha )$ is no longer defined by $\omega (\zeta ,\alpha ) =\zeta -\alpha$ , which is the prime function associated with the unit $\zeta$ disc. Instead, $\omega (\zeta ,\alpha)$ now denotes the prime function for the concentric annulus $\lbrace D_\zeta : \rho \lt |\zeta |\lt 1 \rbrace$ (see chapters 5 and 14 of Crowdy Reference Crowdy2020), although its dependence on the parameter $\rho$ is suppressed in the notation. An important feature is that the prime function $\omega (\zeta ,\alpha )$ has a simple zero when $\zeta =\alpha$ . On substitution of (5.4) into (5.3), it is also possible to write

(5.6) \begin{equation} Z(\zeta ) \equiv -{P(\zeta /{\alpha }, \rho )\, P(\zeta {\alpha }, \rho ) \over P(\zeta /\overline {\alpha }, \rho )\, P(\zeta \overline {\alpha }, \rho ) }, \quad {\mathcal {R}}(\zeta ) \equiv -{P(\zeta /{\alpha }, \rho )\, P(\zeta \overline {\alpha }, \rho ) \over P(\zeta /\overline {\alpha }, \rho )\, P(\zeta {\alpha }, \rho ) }, \end{equation}

which highlights the dependence of $Z(\zeta )$ and ${\mathcal {R}}(\zeta )$ on $\rho$ . The formulae (5.6) will be used in what follows. It is a straightforward exercise to show, directly from the infinite product representation (5.5), that $P(\zeta ,\rho )$ satisfies the two functional relations

(5.7) \begin{equation} P(1/\zeta ,\rho ) = - \zeta ^{-1}\,P(\zeta ,\rho ), \quad P(\rho ^2 \zeta ,\rho ) = - \zeta ^{-1}\,P(\zeta ,\rho ). \end{equation}

Figure 2.

Conformal slit mappings $Z(\zeta )$ and ${\mathcal {R}}(\zeta )$ from an upper half-annulus $\lbrace D_\zeta ^+: \rho \lt |\zeta |\lt 1,\ \textrm {Im}[\zeta ] \geqslant 0 \rbrace$ to a bounded circular slit domain (in a complex $z$ plane) and a bounded radial slit domain (in a complex $\chi$ plane). Corresponding images under the respective maps are colour-coded. The point $\zeta =\alpha$ is the pre-image of the origin in both the $z$ and $\chi$ planes.

The two functions (5.6) built using the prime function are, respectively, further examples of a circular slit map and a radial slit map as described in Crowdy & Marshall (Reference Crowdy and Marshall2006) and Crowdy (Reference Crowdy2011, Reference Crowdy2020). The difference compared with those already presented in § 3 is that there are now $M=2$ pre-image circles $C_0^+$ and $C_1^+$ , and consequently, two circular arc slit images and two radial slit images; in § 3, $C_1^+$ was clearly absent. The properties (5.7) can be used to confirm that the colour-coded images of the boundary portions of the upper half-annulus $D_\zeta ^+$ under these two mappings are as shown in figure 2. In particular, the upper half unit circle, denoted by $C_0^+$ , is taken to be the pre-image of the blue meniscus centred at $z=1$ , and the upper half-circle $|\zeta | =\rho$ , denoted by $C_1^+$ , is the pre-image of the red meniscus centred at $z=-1$ . The two intervals on the real axis, $-1 \lt \zeta \lt -\rho$ and $\rho \lt \zeta \lt 1$ , are taken to be the pre-images of the two no-slip boundary portions of the pipe. From these geometrical properties, it can be verified, using properties of the complex logarithm, that $h(z)$ as given in (5.2) satisfies all the boundary conditions (2.12).

The mapping function $Z(\zeta )$ satisfies the functional identity

(5.8) \begin{equation} {Z}(-\overline {\zeta }) = -{P(-\overline {\zeta }/{\alpha }, \rho )\, P(-\overline {\zeta } {\alpha }, \rho ) \over P(-\overline {\zeta }/\overline {\alpha }, \rho )\, P(-\overline {\zeta } \overline {\alpha }, \rho ) } = -{P(\overline {\zeta }/\overline {\alpha }, \rho )\, P(\overline {\zeta } \overline {\alpha }, \rho ) \over P(\overline {\zeta }/{\alpha }, \rho )\, P(\overline {\zeta } {\alpha }, \rho ) } = \overline {{Z}(\zeta )}, \end{equation}

which reveals that the image of any point $\zeta$ and that of $-\overline {\zeta }$ (the reflection of $\zeta$ in the imaginary $\zeta$ -axis) are complex conjugates of each other in the $z$ plane. This symmetry has two consequences. First, since $C_0^+$ and $C_1^+$ are taken to be the pre-images of the two menisci, the pre-images of their centres are at $\zeta = \textrm {i}, \textrm {i} \rho$ . Indeed, it can be verified that

(5.9) \begin{equation} Z(\textrm {i}) = 1, \quad Z(\textrm {i} \rho ) = -1, \end{equation}

and the two menisci are reflectionally symmetric about the $x$ -axis. The symmetry (5.8) also confirms that the lengths of the images of the two segments of the real axis $-1 \lt \zeta \lt -\rho$ and $\rho \lt \zeta \lt 1$ , which are taken to be the pre-images of the two solid portions, must have the same circumferential lengths since the image of one is the reflection about the $x$ -axis of the other.

It is useful to remark here that to generalise the analysis to no-slip boundary portions of different lengths, it is simply necessary to allow the parameter $\alpha$ to move off the imaginary $\zeta$ -axis but to remain in the upper half-annulus.

The conformal mapping in (5.6) depends on two unknown real parameters: $r$ and $\rho$ . For given values of the angles $\Phi$ and $\Psi$ indicated in figure 2, the values of these two parameters are found by solving the equations

(5.10) \begin{equation} Z(-1) = \textrm{e}^{\textrm {i} \Phi }, \quad Z(-\rho ) = \textrm{e}^{\textrm {i} \Psi }, \quad \Psi = \Phi + \pi \phi , \end{equation}

where $\phi$ is the solid fraction, so that $\pi \phi$ is the circumferential length of one of the solid portions. This is readily done using a nonlinear solver such as Newton’s method.

Concerning the radial slit mapping, the two functional relations (5.7) can be used to show that the images of both $C_0^+$ and $C_1^+$ have constant argument. Indeed, it can be verified using (5.6) that

(5.11) \begin{equation} {R}(\textrm {i}) = \left ({P(+r,\rho ) \over P(-r, \rho )} \right )^2, \quad { R}(\textrm {i}\rho ) = -\left ({P(\rho r,\rho ) \over P(-\rho r, \rho )} \right )^2, \end{equation}

revealing that the images under the radial slit mapping, in a complex $\chi$ plane say, of the points $\zeta = \textrm {i}, \textrm {i} \rho$ (corresponding to the centres of menisci 1 and 2), are, respectively, on the positive and negative real $\chi$ -axis; indeed, these correspond to the ends, inside the unit $\chi$ disc, of the blue and red radial slits shown in figure 2.

With these geometrical properties of the two conformal slit mappings $Z(\zeta )$ and ${\mathcal {R}}(\zeta )$ established, it is easy to verify, using the properties of the principal branch of the complex logarithm, that they satisfy

(5.12) \begin{equation} \left\{\begin{array}{ll} \textrm {Im}[ \log Z(\zeta )] = \theta & \textrm {on meniscus 1}, \\[2pt] \textrm {Re}[\log Z(\zeta )] = 0 & \textrm {on solid portions}, \\[2pt] \textrm {Im}[ \log Z(\zeta )] = \theta & \textrm {on meniscus 2}, \end{array}\right. \end{equation}

and

(5.13) \begin{equation} \left\{\begin{array}{ll} \textrm {Im}[- \log {\mathcal {R}}(\zeta )] = 0 & \textrm {on meniscus 1}, \\[2pt] \textrm {Re}[- \log {\mathcal {R}}(\zeta )] = 0 & \textrm {on solid portions}, \\[2pt] \textrm {Im}[- \log {\mathcal {R}}(\zeta )] =- \pi & \textrm {on meniscus 2}. \end{array}\right. \end{equation}

From these observations, the form of the required solution (5.2) satisfying (5.1) is now clear. Arguments the same as in § 4 lead to the deduction that formula (4.13) continues to give the relevant effective slip length, but now with $\omega$ representing the appropriate prime function. In this case, it can be rewritten as

(5.14) \begin{equation} \lambda _{\textit {eff}} = 2 \log \left | {P(-r^2,\rho ) \over P(+r^2, \rho )} \right |. \end{equation}

It is worth mentioning that this closely resembles a slip length formula for shear flow over SHS with periodic arrays of no-shear slots as found by Crowdy (Reference Crowdy2011).

The problem of a single meniscus, the case $M=1$ , corresponds to the $\rho \to 0$ limit of the above analysis. The pre-image $\zeta$ annulus reduces in this limit to the unit $\zeta$ disc. And from (5.4), as $\rho \to 0$ , the prime function for the concentric annulus $\rho \lt |\zeta | \lt 1$ reduces to the prime function for the unit disc (Crowdy Reference Crowdy2020), namely,

(5.15) \begin{equation} \omega (\zeta ,\alpha ) = \zeta -\alpha, \end{equation}

while $P(\zeta ,\rho ) \to (1-\zeta )$ so that (5.14) retrieves (4.14).

5.1. Illustrative calculations

Suppose that the solid fraction is chosen to be $\phi =1/2$ so that the circumferential length of each solid portion is $2 \pi \phi /2 = \pi /2$ ; figure 3 shows graphs of $\lambda _{\textit {eff}}$ , calculated using (5.14) as a function of $\Phi$ . Thus $\Phi$ is in the range $0 \leqslant \Phi \leqslant \pi /2$ : as $\Phi \to 0$ , meniscus 1 (shown in blue in figure 2) disappears, while as $\Phi \to \pi /2$ , meniscus 2 (shown in red in figure 2) disappears. These graphs are computed by first solving (5.10) for $r$ and $\rho$ given a value of $\Phi$ (and $\phi =1/2$ ), and using these values to find $\lambda _{\textit {eff}}$ from (5.14).

Several checks are available in this case. When $\Phi = 0, \pi /2$ , the two no-slip zones merge, and one of the menisci disappears, leaving a pipe with a single meniscus and no-slip fraction equal to $\phi =1/2$ . The results of (5.14) should therefore coincide with (1.3) for $N=1$ and $\phi =1/2$ , as corroborated by open circles in figure 3. When $\Phi =\pi /4$ , the two no-slip zones have the same size as the two menisci, and the patterning on the pipe has symmetries about both the $x$ - and $y$ -axes (and a four fold rotational symmetry). The value of the effective slip length should then coincide with that given by (1.3) when $N=2$ and $\phi =1/2$ , as corroborated by a cross in figure 3. The new analytical solution (5.2), for two non-equal menisci generally devoid of any rotational symmetry, can therefore be viewed as providing a continuous interpolating set between the rotationally symmetric $N=1$ and $N=2$ solutions, both with solid fraction $\phi =1/2$ .

Figure 4 shows contour lines of constant axial velocity for three different geometric configurations of superhydrophobic pipes, with $M=2$ unequal shear-free menisci, for $\Phi = 0.5,1, 1.3$ . The slip lengths associated with these SHS pipes are indicated in figure 3 by a triangle, a filled circle and an open square, respectively.

Figure 3.

Graph of $\lambda _{\textit {eff}}$ as a function of $\Phi$ for $M=2$ no-shear menisci and two equal no-slip zones with solid fraction $\phi =1/2$ . The two open circles, at $\Phi =0, \pi /2$ , show Philip’s result (1.3) for $N=1$ and $\phi =1/2$ since the configuration reduces to Philip’s single meniscus solution retrieved in § 3. The cross, at $\Phi =\pi /4$ , shows Philip’s result (1.3) for $N=2$ and $\phi =1/2$ since the no-shear pattern has two fold rotational symmetry in this case. The new solutions interpolate between these known special cases. The triangle, filled circle and open square correspond to the contour plots in figure 4.

Figure 4.

Constant axial velocity contours for superhydrophobic pipes with $M=2$ unequal shear-free menisci, assuming $\phi = 1/2$ : (a) $\Phi = 1/2$ , (b) $\Phi = 1$ , (c) $\Phi = 1.3$ . The parameters $r,\rho$ are found by solving (5.10).

Figure 5.

Conformal slit mappings $Z(\zeta )$ and ${\mathcal {R}}(\zeta )$ as given by (6.3) to a bounded circular slit domain (in a complex $z$ plane) and a bounded radial slit domain (in a complex $\chi$ plane). The image in the $z$ plane is the interior of a circular superhydrophobic pipe with $M=3$ menisci punctuated by three solid portions of equal circumferential length. Corresponding images under the respective maps are colour-coded. The point $\zeta =\alpha$ is the pre-image of the origin in both the $z$ and $\chi$ planes.

6. Superhydrophobic pipe with $M \gt 2$ menisci

In the $M=2$ analysis in § 5, the central role played by the prime function defined in (5.4), and associated with the concentric annulus relevant to that example, is apparent. The concentric annulus is an example of a doubly connected circular domain, a domain whose boundaries are a union of circles. Prime functions can naturally be associated with any multiply connected circular domain, a mathematical fact whose theory has been expounded in detail in a recent monograph (Crowdy Reference Crowdy2020). Equipped with this mathematical machinery, the generalisation of the $M=2$ solution to any value of $M \gt 2$ is straightforward, a circumstance underscoring the theoretical importance of prime functions in applied mathematics (Crowdy Reference Crowdy2020).

Suppose now that a superhydrophobic pipe, with cross-section denoted again by $D_z$ , has $M=3$ menisci in general positions around the boundary, punctuated by $M$ no-slip zones. Figure 5 shows such a superhydrophobic pipe with total solid fraction equal to $\phi$ , and each having equal circumferential length; again, the latter restriction is not necessary and can be lifted. The solution construction for this $M=3$ case will now be set out, after which generalisation to any $M \geqslant 1$ should be obvious.

According to the analysis in § 2, the problem reduces to finding a function $h(z)$ , analytic inside $D_z$ and satisfying the mixed boundary conditions

(6.1) \begin{equation} \left\{\begin{array}{ll} \textrm {Re}[h(z)] = 0 & \textrm {on solid portions}\ \textrm {of}\ |z|=1, \\[2pt] \textrm {Im}[h(z)] = {\theta /2} & \textrm {on meniscus}\ 1\ \textrm {of}\ |z|=1, \\[2pt] \textrm {Im}[h(z)] = {\theta /2} + c_2 & \textrm {on meniscus}\ 2\ \textrm {of}\ |z|=1, \\[2pt] \textrm {Im}[h(z)] = {\theta /2} + c_3 & \textrm {on meniscus}\ 3\ \textrm {of}\ |z|=1. \end{array}\right. \end{equation}

The parametric solution for $h(z)$ using generalised conformal slit mappings is immediate given the prime function framework (Crowdy Reference Crowdy2020). It is

(6.2) \begin{align} z &= Z(\zeta ), \quad h(z) = H(\zeta ) = \tfrac{1}{2} \left ( \log Z(\zeta ) - \log {\mathcal {R}}(\zeta ) \right ), \end{align}

where the parametric complex variable $\zeta$ takes values in the upper half circular domain $D_\zeta ^+$ also shown in figure 5, and

(6.3) \begin{equation} Z(\zeta ) \equiv -{\omega (\zeta ,\alpha )\, \omega (\zeta , 1/\alpha ) \over \omega (\zeta ,\overline {\alpha })\, \omega (\zeta , 1/\overline {\alpha })}, \quad {\mathcal {R}}(\zeta ) \equiv -{\omega (\zeta ,\alpha )\, \omega (\zeta , 1/\overline {\alpha }) \over \omega (\zeta ,\overline {\alpha })\, \omega (\zeta , 1/{\alpha })}, \end{equation}

with $\alpha = \textrm {i} r$ for $0 \lt r\lt 1$ . The formulae (6.2) and (6.3) are exactly the same as in §§ 3 and 5, with the only difference being that the function $\omega (\cdot,\cdot)$ must now be taken to be the prime function associated with a triply connected circular domain $D_\zeta$ comprising the unit disc $|\zeta |\leqslant 1$ with two circular discs $|\zeta +\delta | \lt q$ and $|\zeta -\delta | \lt q$ excised; the unit circle will be denoted by $C_0$ , and the two smaller circular boundaries $|\zeta +\delta | = q$ and $|\zeta -\delta | = q$ by $C_1$ and $C_2$ , respectively. The domain $D_\zeta ^+$ shown in figure 5 is the upper half of this circular domain $D_\zeta$ . The general theory of prime functions associated with circular domains such as $D_\zeta$ can be found in Crowdy (Reference Crowdy2020), and Appendix B surveys methods by which these special functions can be evaluated. An important feature of every prime function $\omega (\zeta ,\alpha )$ is that it has a simple zero when $\zeta =\alpha$ . It has other mathematical properties that ensure that (6.3) continues to provide the requisite conformal slit mappings with the geometrical effect shown in figure 5 (Crowdy Reference Crowdy2020).

The upper half of the unit circle is denoted again by $C_0^+$ , while the upper halves of the circles $|\zeta +\delta | = q$ and $|\zeta -\delta | =q$ will be denoted by $C_1^+$ and $C_2^+$ , respectively. For the superhydrophobic pipe problem with three no-shear menisci, the images of the upper half semicircles $C_0^+, C_1^+, C_2^+$ under the mapping $Z(\zeta )$ are taken to be the three no-shear menisci in the $z$ plane, with the correspondences colour-coded in figure 5; the no-slip zones are the images of the three portions of the real $\zeta$ -axis, shown in black in figure 5. The semicircles $C_1^+$ and $C_2^+$ are taken to have the same radius, with centres symmetrically disposed about the $\zeta$ origin, and $\alpha$ has been placed on the imaginary axis inside $D_\zeta ^+$ because these are necessary (but not sufficient) conditions for the three solid portions to have the same circumferential length.

In (6.2), the principal branch of the complex logarithm is assumed, and it is easy to show that the constants $c_2$ and $c_3$ in (6.1) are related to the arguments of the radial slits shown schematically in figure 5. Formulae, in terms of the prime function, for these constants can be derived, but they are not important for determining the flow or the effective slip length, so they are omitted here.

The mapping $Z(\zeta )$ now depends on three real parameters, $q$ , $\delta$ and $r$ . Suppose that the solid fraction is chosen to be $\phi =1/2$ so that the circumferential length of each no-slip region is $2 \pi \phi /3 = \pi /3$ . Then $q$ , $\delta$ and $r$ must be chosen to satisfy the three conditions

(6.4) \begin{equation} Z(-1) = \textrm{e}^{\textrm {i} \Phi }, \quad Z(-\delta -q) = \textrm{e}^{\textrm {i} (\Phi + \pi /3)}, \quad Z(-\delta +q) = \textrm{e}^{5 \pi \textrm { i}/6}, \end{equation}

where the angle $\Phi$ is shown in figure 5. A nonlinear solver such as Newton’s method applied to (6.4) leads easily to the requisite values of $q, \delta$ $q$ , $\delta$ and $r$ . It is important to emphasise that although the same function name $\omega (\cdot,\cdot)$ as in § 5 has been used, it now refers to a different function that can be readily calculated following the discussion in Appendix B. More details on how to compute the prime function can be found in Chapter 14 of Crowdy (Reference Crowdy2020).

The same arguments as used in § 4 lead to the deduction that (4.13) continues to give the relevant effective slip length, but now with $\omega$ representing the appropriate prime function.

6.1. Illustrative calculations

Figure 6 shows graphs of $\lambda _{\textit {eff}}$ as functions of $\Phi$ for three no-slip zones with solid fraction $\phi =1/2$ so that the circumferential length of each is $\pi /3$ . This means that $\Phi$ takes values in the interval $0 \leqslant \Phi \leqslant \pi /2$ : when $\Phi \to 0$ , meniscus 1 (shown in blue in figure 5) disappears, while menisci 2 and 3 have the same circumferential length equal to $\pi /2$ ; when $\Phi =\pi /6$ , all three menisci and the three solid portions have equal length $\pi /3$ , and the pipe has three fold rotational symmetry; when $\Phi \to \pi /2$ , menisci 2 and 3 have disappeared, and the problem reduces to the $M=1$ solution with solid fraction $\phi =1/2$ . For each $\Phi$ , the relevant values of $q$ , $\delta$ and are found by solving (6.4) with those values, then used in (4.13) to compute the effective slip length. As a check on the calculation, the open circle in figure 6 shows the result (1.3) with and when; in this case, the superhydrophobic pipe has three fold rotational symmetry, and (1.3) applies.

Figure 7 illustrates the axial velocity contour lines for three distinct SHS pipes, each featuring $M=3$ unequal shear-free menisci at $|z|=1$ , for $\Phi = 0.24, 0.9, 1.3$ . The associated slip lengths for these SHS pipes are indicated in figure 6 by a triangle, a filled circle and an open square, respectively.

Figure 6.

Graph of $\lambda _{\textit {eff}}$ as a function of $\Phi$ for $M=3$ no-shear menisci and three no-slip zones of fixed length $\pi /3$ so that the solid fraction is $\phi =1/2$ . The cross shows Philip’s result (1.3) with $N=3$ and $\phi =1/2$ when $\Phi =\pi /6$ , since at this value, the no-shear pattern has three fold rotational symmetry. The open circle shows the result (1.3) with $N=1$ and $\phi =1/2$ when $\Phi =\pi /2$ , since at this value, the configuration corresponds to Philip’s single meniscus case studied in § 3. The new solutions interpolate between these known special cases. The triangle, filled circle and open square correspond to the contour plots in figure 7.

Figure 7.

Axial velocity field contour lines for SHS pipes with $M=3$ unequal shear-free menisci at $|z|=1$ with $\phi = 1/2$ : (a) $\Phi = 2.4$ , (b) $\Phi = 0.9$ , (c) $\Phi = 1.3$ . The parameters $r,q, \delta$ are found by solving the three equations in (6.4).

7. The $N$ -fold rotationally symmetric pipes with $M$ menisci per sector

Suppose now that there is an $N$ -fold rotationally symmetric array of menisci with a single meniscus in each sector of opening angle $2\pi /N$ . By the assumed rotational symmetry, it is enough to consider the problem in the sector

(7.1) \begin{equation} -{\pi \over N} \lt \textrm {arg}[z] \lt {\pi \over N}, \end{equation}

where the meniscus is taken to occupy

(7.2) \begin{equation} -\Phi \lt \textrm {arg}[z] \lt \Phi . \end{equation}

Under the conformal mapping

(7.3) \begin{equation} {\mathcal {Z}} = z^N, \end{equation}

this principal sector in the $z$ plane is transplanted to the entire unit disc in a complex $\mathcal {Z}$ plane with the meniscus now corresponding to the sector

(7.4) \begin{equation} -\tilde \Phi \lt \textrm {arg}[z] \lt \tilde \Phi , \quad \tilde \Phi = N \Phi . \end{equation}

Figure 8 shows a schematic in the case $N=4$ where there is just a single meniscus in the principal sector.

Figure 8.

Conformal mapping (7.3) from a $2\pi /N$ sector in the $z$ plane to a full circle in a complex $\mathcal {Z}$ plane. The case $N=4$ is shown with $M=1$ meniscus per $2\pi /N$ sector.

The original boundary value problem for $w(x,y)$ is a Poisson problem with mixed boundary conditions and is not therefore conformally invariant. However, the solution for the $N$ -fold symmetric generalisation can still be written as (2.4), but with $h(z)$ now sought as a function of $\mathcal {Z}$ , i.e.

(7.5) \begin{equation} h(z) = {\mathcal {H}}({\mathcal {Z}}). \end{equation}

By the rotational symmetry of the original boundary value problem, ${\mathcal {H}}({\mathcal {Z}})$ will be analytic and single-valued in $|{\mathcal {Z}}| \leqslant 1$ . On the solid portions, it is still necessary that

(7.6) \begin{equation} \textrm {Re}[{\mathcal {H}}({\mathcal {Z}})] = 0. \end{equation}

However, on the no-shear portions, the boundary condition differs. This is because, by the chain rule,

(7.7) \begin{equation} z\, {\partial \over \partial z} = N {\mathcal {Z}}\,{\partial \over \partial {\mathcal {Z}}}, \end{equation}

so the no-shear condition (2.7) becomes

(7.8) \begin{equation} \mathrm {Re} \left [ -\frac {z \bar {z}}{2} + N {\mathcal {Z}}\, {\mathcal {H}}'({\mathcal {Z}}) \right ] = 0, \quad \textrm {or} \quad \mathrm {Re} \left [ {\mathcal {Z}}\, {\mathcal {H}}'({\mathcal {Z}}) \right ] = {1 \over 2N}. \end{equation}

Letting ${\mathcal {Z}}= \textrm{e}^{\textrm {i} \Theta }$ , the chain rule implies

(7.9) \begin{equation} {\partial \over \partial \Theta } = \textrm {i} {\mathcal {Z}}\, {\partial \over \partial {\mathcal {Z}}}, \end{equation}

so that (7.8) can be written as

(7.10) \begin{equation} \textrm {Im} \left [ {\partial \over \partial \Theta } {\mathcal {H}}({\mathcal {Z}}) \right ] = \frac {1}{2N} \quad \mathrm {on} \ |{\mathcal {Z}}| = 1 \quad (\mathrm {no \ shear \ part}). \end{equation}

This can be integrated with respect to $\Theta$ :

(7.11) \begin{equation} \textrm {Im} \left [ {\mathcal {H}}({\mathcal {Z}}) \right ] = \frac { \Theta }{2N} \quad \mathrm {on} \ |{\mathcal {Z}}| = 1 \quad (\mathrm {no \ shear \ part}), \end{equation}

where a constant of integration has been set to zero without loss of generality.

This boundary value problem for $\mathcal {H}$ as a function of $\mathcal {Z}$ can be solved using the parametric $\zeta$ plane because it is now formally the same as the problem already solved in § 3. Indeed, a parametric representation of the solution is

(7.12) \begin{equation} {\mathcal {Z}} = Z(\zeta ), \quad {\mathcal {H}} = {1 \over 2N} \left (\log Z(\zeta ) - \log {\mathcal {R}}(\zeta )\right ), \end{equation}

where $Z(\zeta )$ and ${\mathcal {R}}(\zeta )$ are given by (3.2) and (3.6), but now with $\alpha = \textrm {i} r$ , where

(7.13) \begin{equation} r = \tan (\tilde \Phi /4). \end{equation}

The important point is that given any solution for $M \geqslant 1$ no-shear menisci such as those derived in § 3 for $M=1$ , in § 5 for $M=2$ , and for $M=3$ in § 6, it can be composed with a mapping of the form (7.3) for any $N \gt 1$ to generalise that solution to the $N$ -fold rotationally symmetric case. This means that the methodology of this paper provides a solution strategy for a wide class of superhydrophobic patternings.

The formula (4.13) for the effective slip length has to be modified by a simple multiplicative prefactor when $N\gt 1$ . Let $\partial D_z$ denote the pipe boundary of an $N$ -fold rotationally symmetric pattern with $M$ menisci in each $2\pi /N$ sector. Let $\tilde D_z$ denote the principal sector (or pie slice), and let $\partial \tilde D_z$ denote the part of its boundary on $|z|=1$ . The quantity $Q_{\textrm {h}}$ is

(7.14) \begin{equation} Q_{h} =\textrm {Re} \left [ \int _{D_z} h(z)\, \textrm{d}A \right ] = N\, \textrm {Re} \left [ \int _{\tilde D_z} h(z)\, \textrm{d}A \right ], \end{equation}

where the $N$ -fold symmetry has been used. By the complex form of Stokes’ theorem,

(7.15) \begin{equation} Q_{h} = N\, \textrm {Re} \left [ {1 \over 2 \textrm {i}} \int _{\partial \tilde D_z} h(z)\, \overline {z}\, \textrm{d}z \right ], \end{equation}

where the contributions from the two ‘edges’ of the sector can be shown to cancel out by the rotational symmetry. But now

(7.16) \begin{equation} Q_{h} = N\, \textrm {Re} \left [ {1 \over 2 \textrm {i}} \int _{\partial \tilde D_z} h(z)\,{\textrm{d}z \over z} \right ], \end{equation}

where the fact that $\overline {z}=1/z$ on $\partial \tilde D_z$ has been used. Now this contour integral can be transformed to the $\mathcal {Z}$ plane:

(7.17) \begin{equation} Q_{h} = N\, \textrm {Re} \left [ {1 \over 2 \textrm {i}} \int _{\partial \tilde D_{\mathcal {Z}}} {\mathcal {H}}({\mathcal {Z}})\, {\textrm{d}{\mathcal {Z}} \over N {\mathcal {Z}}} \right ] = \textrm {Re} \left [ {1 \over 2 \textrm {i}} \int _{\partial \tilde D_{\mathcal {Z}}} {\mathcal {H}}({\mathcal {Z}})\, {\textrm{d}{\mathcal {Z}} \over {\mathcal {Z}}} \right ], \end{equation}

where $\partial \tilde D_{\mathcal {Z}}$ denotes the image of $\partial \tilde D_z$ under the mapping (7.3). This equation is now identical to (4.10), the only difference being that the solution for ${\mathcal {H}}({\mathcal {Z}})$ as a function of $\mathcal {Z}$ is now a factor of $1/N$ times the solution for $h(z)$ as a function of $z$ ; this is readily seen by comparing (7.12) with (3.11). It follows, on comparing (4.10) and (7.17) and making use of (4.9), that the slip length is also multiplied by a factor of $1/N$ , yielding

(7.18) \begin{equation} \lambda _{\textit {eff}} = {2 \over N} \log \left | {\omega (\alpha ,1/\alpha ) \over \omega (\alpha ,1/\overline {\alpha })} \right |. \end{equation}

This is consistent with (1.3) and (4.14) for the $M=1$ case considered by Philip (Reference Philip1972).

The point is that having found the slip length $\lambda _{\textit {eff}}$ for $M$ menisci in general position around a superhydrophobic pipe, the slip length associated with flow in another pipe where this $M$ -menisci pattern is repeated $N$ times around the pipe in a rotationally symmetric pattern is simply $\lambda _{\textit {eff}}/N$ . This quantity clearly decreases with increasing $N$ , so there is a deterioration in slip that comes from adding higher-order symmetry in the patterning, a feature that has been noticed elsewhere (Crowdy Reference Crowdy2021b ).