2.1 Governing equations

We consider hydrodynamically and thermally fully developed flow in a parallel-plate channel, with uniformly spaced parallel ridges aligned with the flow direction, as in figure 1. The top and bottom walls are structured identically, so that the flow is symmetric about the centre plane of the channel. Surface tension maintains the liquid in the Cassie state, i.e. the liquid contacts the solid substrate only on the tops of the ridges. The grooves between ridges are filled with vapour (and perhaps non-condensible gas). The pressure difference between the liquid and the gas causes the menisci to protrude into (or out of) the cavities. The protrusion is considered to be small enough that the liquid–solid contact angle is small enough for the contact line to stay pinned at the corners of the ridges.

Figure 1.

Schematic of the flow configuration, showing the liquid–solid contact and groove dimensions. Domain $A$ is the cross-section used for the analysis.

The period of the ridges is denoted by $2d^{\ast }$ , the width of each groove by $2a^{\ast }$ and the distance from the top of the ridges to the channel centre line by $h^{\ast }$ , as shown in figure 1. The use of an asterisk indicates a dimensional quantity. The meniscus protrudes at an angle $\unicode[STIX]{x1D703}$ relative to the horizontal at the triple (liquid–solid–gas) contact lines. Positive $\unicode[STIX]{x1D703}$ corresponds to protrusion out of the groove. The $x^{\ast }$ , $y^{\ast }$ and $z^{\ast }$ directions are taken to be the spanwise, transverse (wall-normal) and streamwise directions respectively, with $x^{\ast }$ and $y^{\ast }$ measured from the midpoint between the top of two adjacent ridges. Assuming the number of ridges to be sufficiently large, we can consider the flow to be periodic in the spanwise ( $x^{\ast }$ ) direction, with the same period as the ridges, $2d^{\ast }$ . Hence, we only need to consider a single period window, from the centre of one ridge to the next, say $|x^{\ast }|\leqslant d^{\ast }$ . In addition, symmetry about the line $x^{\ast }=0$ means that we can further restrict attention to just half of a period, $0\leqslant x^{\ast }\leqslant d^{\ast }$ . The appropriate flow domain to consider for analysis, $A$ , is shown in figure 1, with its bottom boundary consisting of half of one liquid–solid interface and half of one curved meniscus, $S$ . Lengths are non-dimensionalised with respect to the width of this domain, $d^{\ast }$ , and we introduce the solid fraction, $\unicode[STIX]{x1D719}=1-a^{\ast }/d^{\ast }$ , and the complementary cavity fraction, $\unicode[STIX]{x1D6FF}=a^{\ast }/d^{\ast }$ . The dimensionless (half) channel height is defined as $h=h^{\ast }/d^{\ast }$ . Neglecting thermocapillary stress-induced convection and evaporation/condensation at the meniscus, there will be no transverse or spanwise flow in fully developed conditions; only the streamwise velocity component $w^{\ast }$ will be non-zero. We non-dimensionalise $w^{\ast }$ by $(d^{\ast 2}/\unicode[STIX]{x1D707})(-\text{d}p^{\ast }/\text{d}z^{\ast })$ , a velocity scale based on the prescribed streamwise pressure gradient $-\text{d}p^{\ast }/\text{d}z^{\ast }$ , giving

where $\unicode[STIX]{x1D707}$ is the dynamic viscosity of the liquid. Then, throughout the domain $w$ satisfies

where $\unicode[STIX]{x1D6FB}^{2}=\unicode[STIX]{x2202}^{2}/\unicode[STIX]{x2202}x^{2}+\unicode[STIX]{x2202}^{2}/\unicode[STIX]{x2202}y^{2}$ is the two-dimensional Laplacian. Symmetry across the left ( $x=0$ ), right $(x=1)$ and top ( $y=h$ ) boundaries of the domain $A$ translates to the conditions

On the solid ridge we apply the no-slip condition,

while on the meniscus $S$ we assume no tangential stress,

where $\boldsymbol{n}$ is the unit (outward) normal to the meniscus. The domain, equations and boundary conditions for the $w$ (and $T$ ) problem are summarised in figure 2.

Figure 2.

Domains, governing equations and boundary conditions for the $w$ and $T$ problems. Subscripts denote partial differentiation.

In the thermal problem for the temperature field $T^{\ast }$ , we assume that the only source of heating of the liquid is through its contact with the solid ridges, where it experiences a constant heat flux per unit area, $q_{sl}^{\prime \prime }$ . Specifically, on the solid–liquid interface we have

The numerical results of Maynes et al. (Reference Maynes, Webb and Davies2008) show that, for a flat meniscus, the rate of heat conducting into the liquid through the gas cavities is negligible in comparison to the rate entering through the solid ridges for cavity depths greater than approximately 25 % of the cavity width. We will assume this here, and take the gas phase to be adiabatic, applying a zero-heat-flux condition on the meniscus:

Viscous dissipation, Marangoni induced convection, and evaporation and condensation at the meniscus are neglected. The dimensional thermal energy equation is then

where $\unicode[STIX]{x1D6FC}=k/(\unicode[STIX]{x1D70C}c_{p})$ is the thermal diffusivity of the liquid, $k$ is the thermal conductivity, $\unicode[STIX]{x1D70C}$ is the density and $c_{p}$ is the specific heat, all assumed to be constant. The thermally fully developed assumption, i.e. that we are sufficiently far from the channel inlet so as to neglect entrance effects, is characterised by the self-similarity property (Bergman et al. Reference Bergman, Lavine, Incropera and DeWitt2011):

where $T_{sl}^{\ast }$ is the temperature on the solid–liquid interface and $T_{m}^{\ast }$ is the mixed-mean liquid temperature,

For the constant-heat-flux boundary condition (2.6), the property (2.9) can be shown, conveniently, to imply $\unicode[STIX]{x2202}T^{\ast }/\unicode[STIX]{x2202}z^{\ast }=\text{d}T_{m}^{\ast }/\text{d}z^{\ast }=\text{const.}$ As a consequence, the axial conduction term, $\unicode[STIX]{x2202}^{2}T^{\ast }/\unicode[STIX]{x2202}z^{\ast 2}$ , is exactly zero and can be dropped from (2.8). The constant can be found by considering an energy balance on a control volume of cross-section $A$ and depth $\text{d}z^{\ast }$ , yielding

where $Q^{\ast }$ is the total volume flow rate through the cross-section $A$ .

We non-dimensionalise temperature by subtracting $T_{m}^{\ast }$ and then scaling by the characteristic temperature scale, $\unicode[STIX]{x0394}T^{\ast }=q_{sl}^{\prime \prime }d^{\ast }/k$ , from the heat flux condition (2.6), to give

Then, the dimensionless form of the thermal energy equation (2.8) is

where it is convenient to introduce

with $Q$ the non-dimensional flow rate. Thus, the forcing $Pw$ is determined completely by the hydrodynamic problem.

Just as with the velocity, the left ( $x=0$ ), right ( $x=1$ ) and top ( $y=h$ ) boundaries of the domain are symmetry boundaries of $T$ , and so

The constant-heat-flux condition (2.6) on the ridge becomes

and the adiabatic condition on the meniscus (2.7) is unchanged. Just as for the $w$ problem, see figure 2 for a summary of the problem for $T$ . The $w$ problem can be solved in isolation, and then the solution $w$ appears as a forcing in the Poisson problem for $T$ . It should be noted that this is possible for a meniscus of any shape or curvature.

The quantity that characterises the heat transfer problem is the Nusselt number, $Nu$ , which captures the influence of the structured surface (including apparent slip and periodic heating) on the overall thermal transport due to convection. We follow Maynes & Crockett (Reference Maynes and Crockett2014) and define a local Nusselt number

where $D$ is the hydraulic diameter, $D=4h^{\ast }$ , and $h_{T}(x)$ is the local heat transfer coefficient (not to be confused with $h$ , the non-dimensional half-channel thickness), defined as $h_{T}(x)=q_{sl}^{\prime \prime }/(T_{sl}^{\ast }-T_{m}^{\ast })$ on the solid–liquid interface (where the temperature is $T_{sl}^{\ast }$ ) and zero on the meniscus. Hence,

It should be noted that $T_{sl}(x)$ is not a function of $z$ , which can be shown from the non-dimensionalisation (2.12) and fully developed assumption (2.9). Averaging $Nu(x)$ over the composite interface (solid–liquid interface and meniscus) gives the average Nusselt number characterising the structured surface,

where $\unicode[STIX]{x1D719}+|S|$ is the total length of the composite interface. This length factor will, however, have no effect on the results in this paper (see (3.25)). We remark that there are other ways of defining a Nusselt number to represent the structured surface, such as using only the temperature at the centre of the ridge, where it will be highest.

2.2 Meniscus curvature

We will take a perturbative approach to investigate the effect of meniscus curvature on the heat transfer problem, based on the assumption that the deflection of the interface relative to a flat one is small. As in Sbragaglia & Prosperetti (Reference Sbragaglia and Prosperetti2007), we represent the position of the meniscus in the form

where $\unicode[STIX]{x1D702}(x)$ is some shape function and $\unicode[STIX]{x1D716}$ is the parameter that is taken to be small compared with 1. To relate $\unicode[STIX]{x1D716}$ to dimensional quantities and find an expression for $\unicode[STIX]{x1D702}(x)$ , we can briefly return to dimensional variables and invoke the Young–Laplace equation relating the capillary pressure difference across a static interface to its radius of curvature. Denoting the pressure on the liquid side of the interface by $p_{l}^{\ast }$ , that on the gas side by $p_{g}^{\ast }$ and the local radius of curvature by $R^{\ast }$ ,

where $\unicode[STIX]{x1D70E}$ is the surface tension along the meniscus. As there is no flow in the transverse directions, $p_{l}^{\ast }$ is constant throughout any given flow cross-section, and we have neglected the gas phase so will assume that $p_{g}^{\ast }$ is constant throughout the length of the cavity. Hence, for a given value of $z^{\ast }$ , the difference $p_{l}^{\ast }-p_{g}^{\ast }$ is constant along the meniscus, giving a constant radius of curvature. Assuming a small streamwise capillary number, as we move down the channel, the pressure $p_{l}^{\ast }$ will fall and $R^{\ast }$ will increase, but the change will be slow enough not to invalidate the assumption of locally unidirectional flow (Ou et al. Reference Ou, Perot and Rothstein2004; Ou & Rothstein Reference Ou and Rothstein2005).

If $p_{l}^{\ast }>p_{g}^{\ast }$ , then $R^{\ast }>0$ , and the meniscus protrudes into the cavity (negative protrusion angle $\unicode[STIX]{x1D703}$ ), and if $p_{l}^{\ast }<p_{g}^{\ast }$ , then $R^{\ast }<0$ , which we can interpret as the meniscus protruding upwards into the liquid (positive protrusion angle $\unicode[STIX]{x1D703}$ ). In general, both signs of $\unicode[STIX]{x1D703}$ are possible since the Young equilibrium angle can lie anywhere between the advancing and receding contact angles.

For protrusion into the cavity, a circle of radius $R^{\ast }$ passing through the corners of neighbouring ridges, i.e. $x^{\ast }=\pm a^{\ast }$ , is defined by

Defining $R=R^{\ast }/d^{\ast }$ , in non-dimensional form this is

which when expanded for $R\gg 1$ gives

Thus, we define our small parameter as $\unicode[STIX]{x1D716}=1/(2R)$ and our shape function as $\unicode[STIX]{x1D702}(x)=\unicode[STIX]{x1D6FF}^{2}-x^{2}$ . Then, for $\unicode[STIX]{x1D716}\ll 1$ , the meniscus is prescribed by

As we will not calculate up to third order in $\unicode[STIX]{x1D716}$ in our perturbative expansion, only first order, the leading-order approximation $y=-\unicode[STIX]{x1D716}\unicode[STIX]{x1D702}$ will be sufficient. The case of protrusion into the liquid corresponds to negative $\unicode[STIX]{x1D716}$ . The relation between $\unicode[STIX]{x1D716}$ and the protrusion angle $\unicode[STIX]{x1D703}$ is found from geometry,

Water on low-surface-energy fluoropolymer has a maximum advancing contact angle of approximately $110^{\circ }$ (measured from vertical), corresponding to a maximum (negative) protrusion angle of $\unicode[STIX]{x1D703}=-20^{\circ }$ . For low solid fractions, e.g. $\unicode[STIX]{x1D719}=1-\unicode[STIX]{x1D6FF}=0.01$ , this gives a maximum value of $\unicode[STIX]{x1D716}=0.17$ .

4.1 Zeroth-order solution, $w^{(0)}$

The problem for $w^{(0)}$ is

along with vanishing normal derivatives on the remaining boundaries,

We will solve for $w^{(0)}$ in the form of an eigenfunction expansion, and the solution is provided by, e.g., Teo & Khoo (Reference Teo and Khoo2009). We write the solution as the one-dimensional flow profile (satisfying Poisson’s equation and no-slip on $y=0$ ) and a perturbation part:

Then, $\tilde{w}^{(0)}$ satisfies Laplace’s equation, with a constant shear stress on the flat meniscus,

along with the symmetry conditions (4.4). Using separation of variables, or noting that $\tilde{w}^{(0)}$ is even in $x$ with period $2$ , the solution can be written as a Fourier cosine series in $x$ ,

where $k_{n}=n\unicode[STIX]{x03C0}$ . It should be noted that this satisfies the requirement that $\unicode[STIX]{x2202}_{y}\tilde{w}^{(0)}=0$ at $y=h$ . The coefficients, $a_{n}^{(0)}$ , are then determined from the conditions on the composite interface, equations (4.7) and (4.8), which lead to the dual series equations

In the limit of a channel of infinite height, $h\rightarrow \infty$ , these can be solved analytically (see § 6), but for a finite channel they must be solved numerically. The numerical method used is the same as that of previous authors (Lauga & Stone Reference Lauga and Stone2003; Sbragaglia & Prosperetti Reference Sbragaglia and Prosperetti2007; Teo & Khoo Reference Teo and Khoo2009), and is outlined in an appendix of Sbragaglia & Prosperetti (Reference Sbragaglia and Prosperetti2007). Briefly, the method is to truncate each series at $n=N$ , and then project each equation, in the region where it holds, onto the eigenfunctions $\{\cos (k_{n}x):0\leqslant n\leqslant N\}$ to form an ( $N+1$ )-dimensional linear algebra problem for $\boldsymbol{a}^{(0)}=(a_{0}^{(0)},a_{1}^{(0)},\ldots ,a_{N}^{(0)})^{T}\in \mathbb{R}^{N+1}$ , written as $\unicode[STIX]{x1D648}\boldsymbol{a}=\boldsymbol{f}$ . The series structure (left-hand sides) and the forcing (right-hand sides) of (4.10)–(4.11) determine $\unicode[STIX]{x1D648}\in \mathbb{R}^{N+1}\times \mathbb{R}^{N+1}$ and $\boldsymbol{f}\in \mathbb{R}^{N+1}$ respectively.

From this solution, the leading-order flow rate $Q^{(0)}$ is found by integrating (4.5) over the rectangular domain, giving

and hence

Lastly, from differentiating the series solution (4.9) term by term,

Hence, $\unicode[STIX]{x1D6FB}^{2}\tilde{w}^{(0)}=0$ and (via (4.5)) $\unicode[STIX]{x1D6FB}^{2}w^{(0)}=-1$ are satisfied on the meniscus, $y=0$ , as well as in the interior of the domain. This justifies simplifying the meniscus boundary condition (3.9) for $w^{(1)}$ in the manner of Sbragaglia & Prosperetti (Reference Sbragaglia and Prosperetti2007) to give condition (4.16) below.

4.2 First-order solution, $w^{(1)}$

The problem for the first-order perturbation $w^{(1)}$ is

along with vanishing normal derivatives on the remaining boundaries,

We write the solution in a similar form to $w^{(0)}$ , but a one-dimensional base profile is unnecessary, i.e. $w^{(1)}=h\tilde{w}^{(1)}$ . Then, $\tilde{w}^{(1)}$ can also be written as a Fourier series,

The no-slip condition (4.17) and meniscus condition (4.16) give another set of dual series equations,

This system for $a_{n}^{(1)}$ takes the same form as for $a_{n}^{(0)}$ , with the only difference being the forcing on the right-hand side of the second equation (4.21). Again, this can be solved analytically in the limit $h\rightarrow \infty$ (see § 6), but for finite $h$ we must solve this numerically.

Once $a_{n}^{(0)}$ and $a_{n}^{(1)}$ are computed, the changes to the flow rate, $Q_{1}^{(1)}$ and $Q_{2}^{(1)}$ , can be calculated by inserting the series (4.19) and (4.9) into (4.22) and (4.23) respectively. Then,

where each integral in the last sum can be calculated using integration by parts and $\unicode[STIX]{x1D702}(x)=\unicode[STIX]{x1D6FF}^{2}-x^{2}$ to give

We can define an apparent slip length by equating the flow rate $Q$ with what it would be, $Q_{\mathit{slip}}$ , if the ridges were replaced with a flat surface subjected to a Navier slip condition, $w_{\mathit{slip}}(x,0)=\unicode[STIX]{x1D706}\unicode[STIX]{x2202}_{y}w_{\mathit{slip}}(x,0)$ , from which we find

Here, $\unicode[STIX]{x1D706}$ is the slip length normalised by half the ridge period, i.e. $\unicode[STIX]{x1D706}=\unicode[STIX]{x1D706}^{\ast }/d^{\ast }$ . We expand $\unicode[STIX]{x1D706}$ similarly to $Q$ ,

where $\unicode[STIX]{x1D706}^{(0)}$ , $\unicode[STIX]{x1D706}_{1}^{(1)}$ and $\unicode[STIX]{x1D706}_{2}^{(1)}$ are the contributions from $Q^{(0)},Q_{1}^{(1)}$ and $Q_{2}^{(1)}$ respectively. Thus, using (4.25), (4.12), (4.22) and (4.23), we find

5.1 Zeroth-order solution, $T^{(0)}$

The problem for $T^{(0)}$ , the temperature for the case of a flat meniscus, is given by (see (2.13))

along with vanishing normal derivatives on the remaining boundaries,

It should be recalled that $P^{(0)}=\unicode[STIX]{x1D719}/Q^{(0)}=\unicode[STIX]{x1D719}/(1/3h^{3}+h^{2}a_{0}^{(0)})$ , and $w^{(0)}$ is given by (4.5) and (4.9). Substitution of $w^{(0)}$ into (5.1) gives

and we see that the forcing consists of a quadratic in $y$ plus an infinite series, each term in the series depending on both $x$ and $y$ . This can be handled by setting $T^{(0)}=T_{p}^{(0)}+\widetilde{T}^{(0)}$ , i.e. splitting it into a particular solution, $T_{p}^{(0)}$ , that satisfies the forcing and a remaining homogeneous (harmonic) part, $\widetilde{T}^{(0)}$ , that enforces the boundary conditions on the composite interface, $y=0$ . In addition, if $T_{p}^{(0)}$ satisfies the symmetry properties across $x=0$ , $x=1$ and $y=h$ , then $\widetilde{T}^{(0)}$ will also. To find such a $T_{p}^{(0)}$ , satisfying $\unicode[STIX]{x1D6FB}^{2}T_{p}^{(0)}=P^{(0)}w^{(0)}$ and the symmetry conditions, we can consider the two parts of the forcing $P^{(0)}w^{(0)}$ separately and then linearly superimpose their solutions. The polynomial part of this forcing, $P^{(0)}(-1/2y^{2}+hy+ha_{0}^{(0)}),$ is satisfied by the quartic solution

which is even about the channel centreline, $y=h$ , and $x=0,1$ . In the infinite series part of the forcing, each term takes a form proportional to $\cosh [k_{n}(y-h)]\cos (k_{n}x),$ and so the particular solution in this case is

which is also even about $y=h$ and $x=0,1$ . It should be noted that each series term in the forcing is harmonic, so each corresponding particular solution $f_{n}$ is biharmonic ( $\unicode[STIX]{x1D6FB}^{4}f_{n}=0$ ). Thus, $f_{n}$ can also be found from the biharmonic general solution in Cartesian coordinates, and enforcing $\unicode[STIX]{x1D6FB}^{2}f_{n}=\cosh (k_{n}(y-h))\cos (k_{n}x)$ and symmetry. Finally, combining all of these components, we find

which satisfies (5.1). Recalling that $T^{(0)}=T_{p}^{(0)}+\widetilde{T}^{(0)}$ , the remaining part, $\widetilde{T}^{(0)}$ , then satisfies the homogeneous equation

but with a modification to the condition on $y=0$ , which can be written as $\unicode[STIX]{x2202}_{y}\widetilde{T}^{(0)}(x,0)=-H(x-\unicode[STIX]{x1D6FF})-\unicode[STIX]{x2202}_{y}T_{p}^{(0)}(x,0)$ , or

where $H(x)$ is the Heaviside step function. Lastly, $\widetilde{T}^{(0)}$ satisfies the symmetry conditions (5.4).

Just as for the previous Laplace problems (4.6) and (4.15), $\widetilde{T}^{(0)}$ has a series solution,

and the coefficients $b_{n}^{(0)}$ can be determined from the condition (5.10) at $y=0$ . The problems for $\tilde{w}^{(0)}$ and $\tilde{w}^{(1)}$ had mixed Dirichlet and Neumann conditions at $y=0$ , and resulted in dual series equations. The condition (5.10) for $\widetilde{T}^{(0)}$ is not mixed, only discontinuous, and so can be handled in the usual way by substituting (5.11) into condition (5.10), then multiplying by $\cos (k_{m}x)$ and integrating over $x\in [0,1]$ . The result is

It should be noted that since the boundary condition is in terms of $\unicode[STIX]{x2202}_{y}\widetilde{T}^{(0)}$ , it cannot determine the first coefficient, $b_{0}^{(0)}$ . Instead, it is fixed by enforcing the zero-mixed-mean condition, $T_{m}=0$ , which to leading order is

This holds because we used the mixed mean in the definition of $T$ as per (2.12). Substituting the series expressions for $w^{(0)}$ and $T^{(0)}$ , and then using orthogonality and integration by parts gives

Therefore, given the Fourier coefficients $a_{n}^{(0)}$ of the velocity field $w^{(0)}$ , the temperature field $T^{(0)}$ can be decomposed into a particular solution $T_{p}^{(0)}$ , given explicitly by (5.8), and a Fourier series $\widetilde{T}^{(0)}$ , with coefficients given explicitly by (5.12) and (5.14).

Setting $y=0$ in expressions (5.8) and (5.11) then gives the temperature $T_{s}^{(0)}(x)$ on the composite interface,

The local and average Nusselt numbers are given by $Nu^{(0)}(x)=4h/T_{s}^{(0)}(x)$ and $\overline{Nu}^{(0)}=\int _{\unicode[STIX]{x1D6FF}}^{1}4h/T_{s}^{(0)}\,\text{d}x$ , with the integration performed numerically.

5.2 First-order solution, $T^{(1)}$

In this section, we develop the first-order correction, $T^{(1)}$ , to the temperature field due to the curvature of the meniscus. From § 3, the problem for $T^{(1)}$ is

along with vanishing normal derivatives on the remaining boundaries,

Here, $P^{(0)}$ and $P^{(1)}$ are constants given by (3.22), in terms of the flow rates $Q^{(0)}$ , $Q_{1}^{(1)}$ and $Q_{2}^{(1)}$ of the velocity problem. It should be noted that the boundary condition (5.17) for $T^{(1)}$ on the meniscus comes from expanding the normal derivative condition $\boldsymbol{n}\boldsymbol{\cdot }\unicode[STIX]{x1D735}T=0$ , and so takes the same form as that for $w^{(1)}$ , (3.9). The condition (5.17) can be simplified in a similar fashion to (3.9) by noting from the series solution that $\unicode[STIX]{x1D6FB}^{2}T^{(0)}=P^{(0)}w^{(0)}$ applies not only in the domain interior but also on $y=0$ , $0\leqslant x<\unicode[STIX]{x1D6FF}$ (excluding the triple contact line, i.e. $y=0$ , $x=\unicode[STIX]{x1D6FF}$ ). Thus, (5.17) can be written

To proceed, we take the same approach that we took for $T^{(0)}$ , and set $T^{(1)}=T_{p}^{(1)}+\widetilde{T}^{(1)}$ , where $T_{p}^{(1)}$ is the particular solution that satisfies the forcing in (5.16), and $\widetilde{T}^{(1)}$ is a remaining homogeneous (harmonic) part that enforces the conditions at $y=0$ . In (5.16), the first forcing term, $P^{(1)}w^{(0)}$ , is the same as at the previous order, (5.1), with $P^{(0)}$ replaced with $P^{(1)}$ . The second term, $P^{(0)}w^{(1)}$ , is similar, but with $w^{(0)}$ replaced with $w^{(1)}$ , which has the same Fourier series structure. Thus, we can construct $T_{p}^{(1)}$ from two copies of $T_{p}^{(0)}$ , with the above changes made. It can be verified that

satisfies $\unicode[STIX]{x1D6FB}^{2}T_{p}^{(1)}=P^{(1)}w^{(0)}+P^{(0)}w^{(1)}$ , and the symmetry conditions (5.19). The remaining part, $\widetilde{T}^{(1)}$ , in the decomposition then satisfies the homogeneous problem

but with the following modification to the conditions on $y=0$ :

where the last term is from differentiating (5.21). Similarly to the previous Laplace problems ( $\tilde{w}^{(0)}$ , $\tilde{w}^{(1)}$ , and $\widetilde{T}^{(0)}$ ), $\widetilde{T}^{(1)}$ has a series solution

and the $b_{n}^{(1)}$ are found from condition (5.23) by substituting (5.21), then multiplying by $\cos (k_{m}x)$ (for $m\geqslant 1$ ) and integrating $x\in [0,1]$ , giving

where $\unicode[STIX]{x1D613}_{mn}=\int _{0}^{\unicode[STIX]{x1D6FF}}\unicode[STIX]{x1D702}(x)\cos (k_{m}x)\cos (k_{n}x)\,\text{d}x$ and $\tilde{\unicode[STIX]{x1D613}}_{mn}=\int _{0}^{\unicode[STIX]{x1D6FF}}\unicode[STIX]{x1D702}(x)\sin (k_{m}x)\sin (k_{n}x)\,\text{d}x$ are known matrices, given by (A 1) and (A 2) respectively (see appendix A). As with $\widetilde{T}^{(0)}$ , the coefficient $b_{0}^{(1)}$ is fixed by the zero-mixed-mean condition. Expanding this condition for $0<\unicode[STIX]{x1D716}\ll 1$ ,

At $O(\unicode[STIX]{x1D716}^{0})$ this recovers (5.13), and at $O(\unicode[STIX]{x1D716}^{1})$ we have

Substitution of our series solutions for $w^{(0)}$ , $w^{(1)}$ , $T^{(0)}$ and $T^{(1)}$ gives an expression for $b_{0}^{(1)}$ . The computation is straightforward but long, and the expression for $b_{0}^{(1)}$ is unwieldy, so the details can be found in appendix B. Despite this, the expression (B 7) for $b_{0}^{(1)}$ is explicit and is given exactly in terms of the velocity coefficients, $a_{n}^{(0)}$ and $a_{n}^{(1)}$ , and the geometric parameters $h$ and $\unicode[STIX]{x1D719}$ (or $\unicode[STIX]{x1D6FF}=1-\unicode[STIX]{x1D719}$ ).

Defining $T_{s}^{(1)}(x):=T^{(1)}(x,0)$ , the corrections to the local and average Nusselt numbers are found numerically from $Nu^{(1)}=-4hT_{s}^{(1)}/(T_{s}^{(0)})^{2}$ and $\overline{Nu}^{(1)}=-\int _{\unicode[STIX]{x1D6FF}}^{1}4hT_{s}^{(1)}/(T_{s}^{(0)})^{2}\,\text{d}x$ respectively.

6.1 Zeroth-order velocity

First, consider the dual series equations (4.10)–(4.11) for $a_{n}^{(0)}$ . The part of (4.11) preventing an analytical solution is the $\tanh (n\unicode[STIX]{x03C0}h)$ factor appearing in the sum. However, $\tanh (n\unicode[STIX]{x03C0}h)\sim 1+\text{e.s.t.}$ as $h\rightarrow \infty$ , where ‘e.s.t.’ denotes ‘exponentially small terms’. Therefore, by replacing $\tanh (n\unicode[STIX]{x03C0}h)$ with 1, the equations are those for a semi-infinite channel (coefficients denoted $a_{n}^{(0,\infty )}$ ) solved by Sbragaglia & Prosperetti (Reference Sbragaglia and Prosperetti2007). Thus, $a_{n}^{(0)}\sim a_{n}^{(0,\infty )}+\text{e.s.t.}$ For $y=O(1)$ (the inner region), taking $h\rightarrow \infty$ in the solution (4.5) for $w^{(0)}$ ,

where

We note that $\tilde{w}^{(0,\infty )}$ is the perturbation to the shear-driven flow in a semi-infinite domain, with slip length $\unicode[STIX]{x1D706}^{(0,\infty )}=a_{0}^{(0,\infty )}$ , given by Philip (Reference Philip1972),

In particular,

For $y=O(h)$ (the outer region), we write $y=hY$ with $Y=O(1)$ (in fact $Y\leqslant 1$ due to the geometry of the problem). In this region, the sum in $w^{(0)}$ is exponentially small, so

which is the Navier slip velocity profile.

6.2 First-order velocity

Using $\tanh (n\unicode[STIX]{x03C0}h)\sim 1+\text{e.s.t.}$ and $\tilde{w}^{(0)}(x,0)\sim \tilde{w}^{(0,\infty )}(x,0)+\text{e.s.t.}$ in the dual series equations (4.20)–(4.21) for $a_{n}^{(1)}$ , we obtain

These can now also be solved analytically, in the manner of Sbragaglia & Prosperetti (Reference Sbragaglia and Prosperetti2007), to give

Sbragaglia & Prosperetti only obtained the first term, $a_{0}^{(1,\infty )}$ , producing an $O(h^{-1})$ error. However, if the $O(h^{-1})$ term is captured, as above, the error is exponentially small – a remarkable improvement. The solution of (6.8)–(6.9) can then be used in asymptotic expressions for $w^{(1)}$ in the inner and outer regions, just as for $w^{(0)}$ .

Lastly, an exponentially accurate expression for $\unicode[STIX]{x1D706}_{2}^{(1)}$ , the slip due to the increase in cross-sectional area, is given by

which is $O(h^{-1})$ and, remarkably, identical to the second term in the expression (6.10) for $\unicode[STIX]{x1D706}_{1}^{(1)}$ . Thus, $\unicode[STIX]{x1D706}^{(1)}=\unicode[STIX]{x1D706}_{1}^{(1)}+\unicode[STIX]{x1D706}_{2}^{(1)}\sim a_{0}^{(1,\infty )}+2\unicode[STIX]{x1D706}_{2}^{(1)}$ .

6.3 Nusselt numbers

Neglecting only exponentially small terms, inner and outer asymptotic solutions as $h\rightarrow \infty$ can be found for the temperature field in a similar manner to the velocity field, but the results are not as useful. For exponential accuracy, or even to go further than $O(h^{-1})$ in the expansion, however, the full solution to the dual series equations (6.8)–(6.9) is needed, which we only know numerically (analytically we only know $a_{0}^{(1)}$ in the limit $h\rightarrow \infty$ ). Using the asymptotic behaviours $a_{0}^{(0)}\sim \unicode[STIX]{x1D706}^{(0,\infty )}+\text{e.s.t}$ , $a_{n}^{(0)}\sim a_{n}^{(0,\infty )}+\text{e.s.t}$ , $P^{(0)}\sim \unicode[STIX]{x1D719}/(h^{3}/3+h^{2}\unicode[STIX]{x1D706}^{(0,\infty )})+\text{e.s.t.}$ from the velocity problem in the series expression (5.15) for $T_{s}^{(0)}(x)$ , we find (after dividing by $h$ )

Here, $b_{n}^{(0,\infty )}$ is the leading order of $b_{n}^{(0)}$ as $h\rightarrow \infty$ , and $b_{n}^{(0)}\sim b_{n}^{(0,\infty )}+O(h^{-1})\sim -2\sin (n\unicode[STIX]{x03C0}\unicode[STIX]{x1D6FF})/(n\unicode[STIX]{x03C0})^{2}+O(h^{-1})$ . Substituting into (3.25), expanding further for $h\rightarrow \infty$ and integrating, we find, correct to $O(h^{-1})$ ,

Moreover, using the behaviour of $a_{0}^{(1)}$ , $a_{n}^{(1)}$ and $b_{n}^{(1)}$ as $h\rightarrow \infty$ in the series expression for $T_{s}^{(1)}(x)$ , and then substituting into (3.25), we find, after some algebra,

where

$\unicode[STIX]{x1D706}^{(0,\infty )}=a_{0}^{(0,\infty )}$ is (6.4), $a_{0}^{(1,\infty )}$ is in (6.10) and $\unicode[STIX]{x1D613}_{nm}$ is given by (A 1). Despite only $O(h^{-1})$ accuracy, these expressions are explicit, with only the sums requiring evaluation.

7.1 Apparent slip length

In the literature, results for the slip length are usually presented normalised by the period, so we will present the quantity $\unicode[STIX]{x1D706}^{\ast }/2d^{\ast }=\unicode[STIX]{x1D706}/2$ . Figure 4 shows plots of the slip lengths $\unicode[STIX]{x1D706}^{(0)}$ , $\unicode[STIX]{x1D706}_{1}^{(1)}$ , $\unicode[STIX]{x1D706}_{2}^{(1)}$ (normalised by ridge period) against the channel height $h$ , for various slip fractions $\unicode[STIX]{x1D6FF}$ . Also shown, as dashed lines, are the asymptotic solutions for $h\rightarrow \infty$ , given by (6.4), (6.10) and (6.11). The numerical and asymptotic solutions agree extremely well: less than 1 % difference is typically achieved by $h=1$ for $\unicode[STIX]{x1D706}^{(0)}$ , and already by $h=0.5$ for $\unicode[STIX]{x1D706}_{1}^{(1)}$ and $\unicode[STIX]{x1D706}_{2}^{(1)}$ . This attests to the exponential accuracy of the asymptotic solutions. The asymptotics should be exact in the limit $h\rightarrow \infty$ , and the slight difference seen to persist in the curves for $\unicode[STIX]{x1D706}_{1}^{(1)}$ is numerical error of the series solution due to the slow convergence with $N$ , the number of terms used. Hence, even for $N=7000$ , the number of terms used in figure 4, the asymptotic solution may actually be more accurate than the numerical solution for $h\gg 1$ .

The behaviour of $\unicode[STIX]{x1D706}^{(0)}$ is well known (see Teo & Khoo Reference Teo and Khoo2009; Maynes & Crockett Reference Maynes and Crockett2014), and $\unicode[STIX]{x1D706}_{1}^{(1)}$ and $\unicode[STIX]{x1D706}_{2}^{(1)}$ behave similarly to the case of ridges on only one wall (see Sbragaglia & Prosperetti Reference Sbragaglia and Prosperetti2007). It should be recalled that $\unicode[STIX]{x1D706}_{1}^{(1)}$ represents the effect of meniscus curvature on the velocity field. The asymptotic behaviour (6.10) of $\unicode[STIX]{x1D706}_{1}^{(1)}$ is of the form $\unicode[STIX]{x1D706}_{1}^{(1)}\sim -A(\unicode[STIX]{x1D6FF})+B(\unicode[STIX]{x1D6FF})h^{-1}$ , where $A$ and $B$ are positive (increasing) functions of $\unicode[STIX]{x1D6FF}$ . It is positive for $h\lesssim 1$ and negative for $h\gtrsim 1$ , tending to a constant negative value, $\unicode[STIX]{x1D706}_{1}^{(1,\infty )}=-A$ , as $h\rightarrow \infty$ . An estimate of when $\unicode[STIX]{x1D706}_{1}^{(1)}$ changes sign is given by $h\approx B/A$ . Physically, this means that the effect on the velocity field of the meniscus protruding slightly into ( $\unicode[STIX]{x1D716}>0$ ) the gas cavity is to reduce slip for thicker channels ( $h\gtrsim 1$ ) and increase slip for narrower channels ( $h\lesssim 1$ ). If the meniscus protrudes out of the cavity ( $\unicode[STIX]{x1D716}<0$ ), then the reverse is true.

Figure 4.

Slip lengths (normalised by period) against the channel height $h$ for increasing values of the slip fraction $\unicode[STIX]{x1D6FF}=1-\unicode[STIX]{x1D719}$ . Panels (a–c) correspond to the quantities (4.27)–(4.29) respectively. Solid lines are numerical solutions of the dual series equations; dashed lines are asymptotic solutions (6.4), (6.10) and (6.11) for $h\gg 1$ . Panel (d) shows the full first-order perturbation. The calculations used $N=7000$ .

On the other hand, the quantity $\unicode[STIX]{x1D706}_{2}^{(1)}$ , representing the change in flow rate due to the change in cross-sectional area, has asymptotic behaviour, $\unicode[STIX]{x1D706}_{2}^{(1)}\sim B(\unicode[STIX]{x1D6FF})h^{-1}$ (see (6.11)), which is positive, and vanishes in the limit $h\rightarrow \infty$ . Thus, for a meniscus protruding into the cavity, the increase in cross-sectional area always increases slip, but the effect vanishes in the large-channel limit.

Finally, the full first-order perturbation with $\unicode[STIX]{x1D706}_{1}^{(1)}$ and $\unicode[STIX]{x1D706}_{2}^{(1)}$ combined, $\unicode[STIX]{x1D706}^{(1)}=\unicode[STIX]{x1D706}_{1}^{(1)}+\unicode[STIX]{x1D706}_{2}^{(1)}\sim -A(\unicode[STIX]{x1D6FF})+2B(\unicode[STIX]{x1D6FF})h^{-1}$ , is shown in figure 4(d), with similar behaviour to $\unicode[STIX]{x1D706}_{1}^{(1)}$ , but changing sign at $h\approx 2B/A$ .

Figure 5.

(a–c) The first-order velocity field $\tilde{w}^{(1)}(x,y)$ for a solid fraction $\unicode[STIX]{x1D719}=0.5$ , and channel heights of $h=5$ , 1 and 0.5; note the $y$ scales. The black solid line marks the position of the ridge. (d) The velocity along the meniscus, $0\leqslant x\leqslant 0.5$ , $y=0$ . Here, $N=7000$ .

For an illustration of how the velocity field changes due to curvature, contour plots of the velocity correction $\tilde{w}^{(1)}$ throughout the domain cross-section, and along the meniscus, are shown in figure 5. For visualisations of $\tilde{w}^{(0)}$ see Teo & Khoo (Reference Teo and Khoo2009). It should be noted that $w^{(1)}(x,0)=h\tilde{w}^{(1)}(x,0)$ is the order- $\unicode[STIX]{x1D716}$ correction to the velocity on the meniscus, since $\unicode[STIX]{x2202}_{y}w^{(0)}(x,0)=0$ . As figure 5 shows, the flow is always increased in the vicinity of the triple contact line ( $\tilde{w}^{(1)}>0$ ), but flow can be either increased ( $\tilde{w}^{(1)}>0$ ) or decreased ( $\tilde{w}^{(1)}<0$ ) throughout the rest of the domain, depending on $h$ . Indeed, $\tilde{w}^{(1)}$ increases everywhere in the domain as $h$ decreases, giving rise to the behaviour of $\unicode[STIX]{x1D706}_{1}^{(1)}$ in figure 4(b), where it eventually becomes positive.

7.2 Leading-order Nusselt-number behaviour

The recent study by Maynes & Crockett (Reference Maynes and Crockett2014) considered the (isoflux) thermal problem for the case of a flat meniscus, corresponding to the $T^{(0)}$ problem in our boundary perturbation formulation, but they used the one-dimensional Navier slip velocity profile rather than the exact two-dimensional velocity profile, $w^{(0)}$ . The Navier slip profile corresponding to $w^{(0)}$ , i.e. with the same flow rate, is given by $w_{NS}^{(0)}=-(1/2)y^{2}+hy+h\unicode[STIX]{x1D706}^{(0)}$ . It is identical to $w^{(0)}$ without the infinite sum, which contributes nothing to the flow rate $Q^{(0)}$ . Therefore, the approximate solution of Maynes & Crockett corresponds exactly to replacing $(a_{0}^{(0)},a_{1}^{(0)},a_{2}^{(0)},\ldots )$ with $(a_{0}^{(0)},0,0,\ldots )$ in the expressions above. In addition, to save having to compute $a_{0}^{(0)}=\unicode[STIX]{x1D706}^{(0)}$ numerically, the value for a semi-infinite ( $h\rightarrow \infty$ ) channel, $\unicode[STIX]{x1D706}^{(0,\infty )}=(2/\unicode[STIX]{x03C0})\log [\sec (\unicode[STIX]{x1D6FF}\unicode[STIX]{x03C0}/2)]$ , was used.

The justification by Maynes & Crockett for using the Navier slip profile is that the velocity field becomes independent of $x$ very quickly as $y$ increases from zero, becoming indistinguishable from the Navier slip profile above a distance of approximately one ridge period (Teo & Khoo Reference Teo and Khoo2009). When the channel height is comparable to the wall period, $h=O(1)$ , they argued that heat diffusion away from the ridges ( $\widetilde{T}^{(0)}$ ) dominates over advection ( $T_{p}^{(0)}$ ), and so the exact form of the velocity profile is irrelevant. This was checked by trying other less accurate profiles, such as the no-slip profile, which produced a ${\leqslant}0.5\,\%$ difference in $\overline{Nu}^{(0)}$ . However, the $x$ variation of the velocity field $w^{(0)}$ does not just enter the wall temperature through the advective part ( $T_{p}^{(0)}$ ) but also through $\widetilde{T}^{(0)}$ – see (5.12) and (5.14). Thus, having $T_{p}^{(0)}$ small relative to $\widetilde{T}^{(0)}$ does not in and of itself imply that the effect of $x$ variation is negligible.

For domains with height much larger than the wall period ( $h\gg 1$ ), Maynes & Crockett argued that the advection terms dominate, in which case the Navier slip profile prevails throughout the vast majority of the channel, and thus is a good approximation. Checking against numerical solutions for each case showed a maximum error in $\overline{Nu}^{(0)}$ of 1.6 %. However, the numerical solutions did not account for $x$ variation of the velocity, and so no conclusion regarding $x$ variation can be made, despite the small errors stated. The check must be made against the full solution using the full two-dimensional profile, and this is carried out next.

The solution presented here accounts for the full velocity profile, and thus is valid for arbitrary channel heights $h$ , including $h=O(1)$ . We extend the results of Maynes & Crockett (Reference Maynes and Crockett2014) down to $h=O(1)$ , and show for what parameter values the Navier slip approximation is sufficient. We could consider the case of $h\ll 1$ , but in practice there is a lower limit to $h$ such that the surface structure stays dewetted, i.e. in the Cassie state (Cottin-Bizonne et al. Reference Cottin-Bizonne, Barentin, Charlaix, Bocquet and Barrat2004). For $h\rightarrow \infty$ , it can be seen from (5.8) and (5.11) that $T^{(0)}=O(h)$ , and so to connect with the relevant finite temperature field in the limit of large channels we consider $T^{(0)}/h$ . Moreover, this quantity is more relevant to the Nusselt number, with the local Nusselt number over the solid ridge given by $Nu^{(0)}(x)=4h/T_{s}^{(0)}(x)$ (and zero over the meniscus).

Figure 6.

The leading-order temperature, $T_{s}^{(0)}(x)/h$ , along the flat composite interface, $y=0$ , for channel heights $h=20$ , 5, 1 and 0.5. The solid lines are results calculated using the full velocity field and the dashed lines are calculated using the Navier slip profile with slip length $\unicode[STIX]{x1D706}^{(0,\infty )}=(2/\unicode[STIX]{x03C0})\log [\sec (\unicode[STIX]{x03C0}\unicode[STIX]{x1D6FF}/2)]$ , as in Maynes & Crockett (Reference Maynes and Crockett2014). The number of terms is $N=120$ . Panels (a) and (b) are for solid fractions $\unicode[STIX]{x1D719}=0.5$ and 0.1 respectively.

Figure 7.

The leading-order average Nusselt number, $\overline{Nu}^{(0)}$ , against the solid fraction $\unicode[STIX]{x1D719}$ for various values of $h$ . See the caption for figure 6. Panel (a) compares the present work (solid lines) with the approach of Maynes & Crockett (Reference Maynes and Crockett2014) (dashed lines), and panel (b) shows the relative percentage error.

The temperature, $T_{s}^{(0)}(x)/h$ , along the flat composite interface, $y=0$ , is shown in figure 6, for various channel heights, $h$ , from $h=20$ down to $h=0.5$ . Figure 6(a) is for a solid fraction $\unicode[STIX]{x1D719}=0.5$ , and figure 6(b) is for a lower value of $\unicode[STIX]{x1D719}=0.1$ . Results using the full velocity field, and using the Navier slip approximation with slip length $\unicode[STIX]{x1D706}^{(0,\infty )}=(2/\unicode[STIX]{x03C0})\log [\sec (\unicode[STIX]{x03C0}\unicode[STIX]{x1D6FF}/2)]$ as in Maynes & Crockett (Reference Maynes and Crockett2014) are both shown. The temperature appears to be approximately constant across the composite interface for $h>5$ , with the channel height showing little influence. As $h$ decreases, the temperature increases over the ridge, and decreases over the meniscus. Use of the Navier slip solution appears to be a good approximation for $h\gtrsim 5$ , with the temperatures indistinguishable, but the discrepancy with the full solution is visible for smaller $h$ , becoming larger as $h$ decreases. The Navier slip approximation underestimates the temperature over the ridge and most of the cavity, which could be because the slip profile overestimates the velocity just above the ridge (where the flow in reality satisfies no-slip) where the heat is entering the domain. Hence, the exact velocity field would convect heat downstream and away from its point of entrance more slowly, raising the temperature immediately above the ridge (and part way over the cavity by conduction). In addition, the Navier slip profile underestimates the velocity over the meniscus, but as no heat is entering through the meniscus (assumed adiabatic), this has less of an effect.

The effect of neglecting the $x$ variation in the velocity on the average Nusselt number, $\overline{Nu}^{(0)}$ , is shown in figure 7(a). The relative percentage error is shown in figure 7(b). It should be noted that only the temperature over the solid ( $1-\unicode[STIX]{x1D719}<x<1$ ) contributes to the calculation of $\overline{Nu}^{(0)}$ . In general, for any channel height, $\overline{Nu}^{(0)}$ decreases as the solid fraction decreases; its maximum value of $140/17\approx 8.235$ is attained when there is no meniscus – the classical uniform heating result (Kays, Crawford & Weigand Reference Kays, Crawford and Weigand2005). The average Nusselt number $\overline{Nu}^{(0)}$ stays close to this value for large $h$ (the asymptotics show $\overline{Nu}^{(0)}\sim 140/17+O(h^{-1})$ as $h\rightarrow \infty$ ), but decreases as $h$ decreases, signifying an overall increase in convective thermal resistance as the channel is made narrower. The Navier slip approximation is valid for sufficiently large $h$ , with an error in $\overline{Nu}^{(0)}$ of ${\leqslant}0.16\,\%$ for $h\geqslant 5$ . However, the discrepancy increases as $h$ decreases, with up to 6.48 % overestimation when $h=1$ , and up to 35.2 % overestimation when $h=0.5$ (when the channel height matches the period). The error is greatest for intermediate solid fractions $\unicode[STIX]{x1D719}$ , but disappears as $\unicode[STIX]{x1D719}\rightarrow 0$ (no ridge) and $\unicode[STIX]{x1D719}\rightarrow 1$ (no meniscus). This is because the Navier slip velocity coincides with the actual velocity in both limits. For $h=0.5$ , when the error is significant, figure 8 illustrates the wall temperature for various values of $\unicode[STIX]{x1D719}$ , with the greatest discrepancy for intermediate values of $\unicode[STIX]{x1D719}$ .

Figure 8.

The leading-order temperature, $T_{s}^{(0)}/h$ , along the flat composite interface, $y=0$ , for various values of the solid fraction $\unicode[STIX]{x1D719}$ , at a channel height of $h=0.5$ . The solid and dashed lines carry the same meanings as in figures 6 and 7.

In conclusion, the results of Maynes & Crockett (Reference Maynes and Crockett2014) are accurate for large channel heights ( $h\geqslant 5$ ), with very small error, but overestimate $\overline{Nu}^{(0)}$ and hence the overall convective heat transfer by up to 35 % when $h=O(1)$ . This means that, for flat menisci, the superhydrophobic structuring reduces convective heat transfer more than previously documented compared with classical uniform heating. However, the results are relatively unchanged for very small solid fractions $\unicode[STIX]{x1D719}$ , where lubrication is highest, since the error is ${\leqslant}2.5\,\%$ for $\unicode[STIX]{x1D719}<0.01$ .

7.3 First-order Nusselt-number behaviour

We now present the results for the first-order temperature problem, and hence the effect of small meniscus curvature on the average Nusselt number, $\overline{Nu}$ . First, we show the correction to the temperature (scaled by $h$ ) along the composite interface, given by $T_{s}^{(1)}(x)/h$ . We remark that even though $T_{s}^{(0)}(x)$ and $T_{s}^{(1)}(x)$ are evaluated at $y=0$ , the combination $T_{s}^{(0)}(x)+\unicode[STIX]{x1D716}T_{s}^{(1)}(x)$ corresponds to the temperature on the curved interface $y=-\unicode[STIX]{x1D716}\unicode[STIX]{x1D702}(x)$ when $0<x<1-\unicode[STIX]{x1D719}$ , correct to order $\unicode[STIX]{x1D716}$ . Moreover, it should be recalled that $\unicode[STIX]{x1D716}>0$ (or negative protrusion angle $\unicode[STIX]{x1D703}$ ) corresponds to the meniscus protruding into the cavity, and $\unicode[STIX]{x1D716}<0$ (or positive protrusion angle $\unicode[STIX]{x1D703}$ ) corresponds to the meniscus protruding into the liquid. Hence, if $\unicode[STIX]{x1D716}>0$ , then $T_{s}^{(1)}>0$ implies an increase in temperature compared with a flat meniscus, and $T_{s}^{(1)}<0$ implies a decrease. (The opposite is true if $\unicode[STIX]{x1D716}<0$ .)

Figure 9 shows $T_{s}^{(1)}/h$ , for the same $h$ and $\unicode[STIX]{x1D719}$ values as in figure 6. For any $h$ , $T_{s}^{(1)}>0$ over the centre of the meniscus, and, for most parameter values shown, $T_{s}^{(1)}<0$ over the entirety of the solid ridge. However, $T_{s}^{(1)}$ is most negative on the edges of the meniscus, just before the triple contact line. This means, if the meniscus curves into the cavity ( $\unicode[STIX]{x1D716}>0$ ), the temperature is increased over the centre of the meniscus, but decreased at its edges and over the solid. The temperature increase over the centre of the meniscus may surprise the reader, given its distance from the ridge, but the increase is just the consequence of the decrease near the ridge and enforcing zero-mixed-mean dimensionless temperature as per (2.12). Indeed, the same qualitative behaviour is seen for pure heat diffusion in the absence of advection, but we do not include the results here. As $h$ is reduced and the channel gets thinner, these effects are enhanced. However, for $h$ small enough ( $h\approx 0.5$ in 9 $a$ ), $T_{s}^{(1)}$ can become positive over the solid, and thus increase the temperature. This is surprising, and the physical cause is unclear.

The first-order correction, $\overline{Nu}^{(1)}$ , to the average Nusselt number is plotted in figure 10 against the solid fraction, $\unicode[STIX]{x1D719}$ , for the same values of $h$ as in figure 7. In general, decreasing $\unicode[STIX]{x1D719}$ increases $\overline{Nu}^{(1)}$ , meaning greater enhancement to convection for wider cavities. For $\unicode[STIX]{x1D719}>0.5$ , and $h$ small enough, we can see a local maximum emerge, but in the limit $\unicode[STIX]{x1D719}\rightarrow 1$ , we see $\overline{Nu}^{(1)}\rightarrow 0$ , as expected since the cavity and meniscus disappear.

The effect of channel height, $h$ , is more easily discerned in figure 11, where $\overline{Nu}^{(1)}$ is plotted with $h$ as the dependent variable; (a) shows the very-low- $h$ behaviour (included for completeness) and (b) the range $h\geqslant 0.5$ (more physically relevant). In figure 11(b), the solid lines are $\overline{Nu}^{(1)}$ and the dashed lines are $\overline{Nu}^{(0)}$ , which are included for reference. As $h\rightarrow \infty$ , we have $\overline{Nu}^{(1)}=O(h^{-1})\rightarrow 0$ , which can be seen from the asymptotic solution, (6.14). Then, as $h$ decreases from $\infty$ , $\overline{Nu}^{(1)}$ increases to a maximum, typically between $h=1$ and $h=10$ , then falls again as $h$ decreases further. In other words, $\overline{Nu}^{(1)}$ behaves non-monotonically with respect to $h$ , with the curvature presenting the greatest enhancement for $1\lesssim h\lesssim 5$ .

Figure 9.

The first-order temperature correction, $T_{s}^{(1)}/h$ , along the composite interface for values of the channel height of $h=20$ , 5, 1 and 0.5. Panels (a) and (b) are for solid fractions of $\unicode[STIX]{x1D719}=0.5$ and 0.1 respectively. In all cases, the number of terms used is $N=500$ .

Figure 10.

The first-order average Nusselt number correction, $\overline{Nu}^{(1)}$ , against the solid fraction, $\unicode[STIX]{x1D719}$ , for various values of $h$ . The number of terms is $N=500$ .

Figure 11.

The first-order average Nusselt number correction, $\overline{Nu}^{(1)}$ (solid lines), against the channel height, $h$ , for various values of $\unicode[STIX]{x1D719}$ . Panel (a) shows small- $h$ behaviour and (b) shows $h\geqslant 0.5$ . The dashed lines are $\overline{Nu}^{(0)}$ . Here, $N=500$ .

To understand this non-monotonicity, consider the two regimes $h\gg 1$ , $h\ll 1$ . When the channel height is large but finite, $h\gg 1$ , the heat transfer is approximately diffusive on the scale of the ridges, with advection only significant in the main body of the channel, where the velocity is approximately one-dimensional (a Navier slip profile). In this regime, a downward protruding meniscus affects convection in two competing ways: (i) by decreasing the lubrication as seen by the Navier slip profile, and (ii) increasing the cross-sectional area into which heat can diffuse, near the triple contact line in particular. These effects appear as the two $O(h^{-1})$ terms in the expression (6.14) for $\overline{Nu}^{(1)}$ as $h\rightarrow \infty$ . Comparison of the two terms shows that effect (ii) is always greater than effect (i), and so convection is enhanced, $\overline{Nu}^{(1)}>0$ . Both effects vanish for a semi-infinite channel, $h\rightarrow \infty$ , and so $\overline{Nu}^{(1)}\rightarrow 0^{+}$ . For the case when the channel height is small but finite, $h\ll 1$ , the ridges on opposing walls are almost touching. In the region between opposing ridges, the fluid velocity decreases as $h$ decreases (no-slip is imposed on both ridges). However, flow in the region between opposing menisci does not degrade as $h$ decreases (for example, $w^{(0)}\rightarrow (\unicode[STIX]{x1D6FF}^{2}-x^{2})/2$ as $h\rightarrow 0$ ). Therefore, the heat must diffuse from the ridges through the stagnant region between opposing ridges into the region between menisci. Menisci protruding into the cavities ( $\unicode[STIX]{x1D716}>0$ ) would then increase the flow rate between menisci, and the area for heat diffusion from the ridge–ridge region, enhancing convection, $\overline{Nu}^{(1)}>0$ . In the limit $h\rightarrow 0$ , the ridges touch, the region between the ridges vanishes and the heat has no way of reaching the region of flow, and so $\overline{Nu}^{(1)}\rightarrow 0^{+}$ , regardless of meniscus curvature.

For the two above physical limits, i.e. $\overline{Nu}^{(1)}\rightarrow 0^{+}$ as $h\rightarrow 0$ and $h\rightarrow \infty$ , to be achieved, $\overline{Nu}^{(1)}$ is necessarily non-monotonic in $h$ . Bridging the aforementioned regimes is the case of $h=O(1)$ , where there is a non-trivial interaction between advection, meniscus curvature and channel geometry, as presented in figure 11(b).

Indeed, we also found that for a sufficiently high solid fraction (see figure 11 $a$ ), $\overline{Nu}^{(1)}$ becomes negative for an interval of $h$ less than 0.45. The values of solid fraction $\unicode[STIX]{x1D719}$ and channel height $h$ for which this occurs (within the region $\unicode[STIX]{x1D719}>0.402$ and $h<0.45$ of parameter space) are not practical for applications, and we have not been able to explain it physically, but this behaviour is mathematically interesting. Given the two asymptotic behaviours $\overline{Nu}^{(1)}\rightarrow 0^{+}$ (i.e. from above) as $h\rightarrow 0$ and $h\rightarrow \infty$ , the unusual sign changes possible in $\overline{Nu}^{(1)}$ would have been difficult to anticipate, and display the complex nonlinear dependence of the problem on both  $h$ and  $\unicode[STIX]{x1D719}$ .

Figure 12.

Comparison between the numerical solution (solid lines) for $\overline{Nu}^{(0)},\overline{Nu}^{(1)}$ and the asymptotic solution (dotted lines) as $h\rightarrow \infty$ , given by (6.13) and (6.14).

Comparison between the numerical solutions for $\overline{Nu}^{(0)}$ and $\overline{Nu}^{(1)}$ and asymptotic expressions (6.13) and (6.14) as $h\rightarrow \infty$ is shown in figure 12. Although not as accurate as our asymptotic slip length solutions, since the error is estimated to be $O(h^{-2})$ , the $O(h^{-1})$ behaviour for $\overline{Nu}^{(0)}$ and $\overline{Nu}^{(1)}$ is confirmed. The asymptotic approximation for $\overline{Nu}^{(0)}$ does much better than that for $\overline{Nu}^{(1)}$ , but the approximation for $\overline{Nu}^{(1)}$ improves for smaller solid fractions, which is surprising.

The aforementioned results show that meniscus curvature can enhance/reduce the overall heat transfer, but the role of changes in advection alone is difficult to assess. One would expect the behaviour to be apparent in the correction to the velocity field, $\tilde{w}^{(1)}$ . However, as figure 5 shows, the flow is always increased in the vicinity of the triple contact line ( $\tilde{w}^{(1)}>0$ ), but flow can be either increased ( $\tilde{w}^{(1)}>0$ ) or decreased ( $\tilde{w}^{(1)}<0$ ) throughout the rest of the domain, depending on $h$ . As $h$ is decreased, $\tilde{w}^{(1)}$ becomes more positive, first over the ridge (figure 5 $b$ ), then eventually the whole cross-section (figure 5 $c$ ). This could explain the initial increase in $\overline{Nu}^{(1)}$ , but not its subsequent decrease for $h\lesssim 1$ , as $\tilde{w}^{(1)}$ continues to increase everywhere. Indeed, $T^{(1)}$ and $\overline{Nu}^{(1)}$ depend not only on the velocity change, $\tilde{w}^{(1)}$ , but on the base velocity and temperature, $\tilde{w}^{(0)}$ and $T^{(0)}$ , all of which can interact to produce inscrutable sign changes as shown in figure 11.

7.4 Remark on convergence of series

The solutions developed in this paper are based on truncating infinite series at $n=N$ , where $N$ is chosen large enough for the solutions to have sufficiently converged. The rate of convergence will thus depend on the decay of the terms in the series. The terms in each series solution considered here all decay exponentially with $n$ if $y>0$ , i.e. in the interior of the domain. This decay becomes algebraic when $y=0$ , i.e. on the composite interface. For example, the $a_{n}^{(0)}$ are the Fourier coefficients of $w^{(0)}(x,0)$ , which can be shown to have a square root singularity in its derivative at the triple contact line, i.e. $w^{(0)}(x,0)=O(|x-\unicode[STIX]{x1D6FF}|^{1/2})$ , as $x\rightarrow \unicode[STIX]{x1D6FF}^{-}$ (Philip Reference Philip1972). This implies $a_{n}^{(0)}=O(n^{-3/2})$ as $n\rightarrow \infty$ . The decay in the series for $w^{(1)}(x,0)$ is governed by $a_{n}^{(1)}$ , and the behaviour of $w^{(1)}(x,0)$ as $x\rightarrow \unicode[STIX]{x1D6FF}^{-}$ can be determined from a local analysis of the full solution, $w$ , close to the singularity. Close to the singularity $x=\unicode[STIX]{x1D6FF}$ , the domain locally takes the form of Laplace’s equation in a wedge with interior angle $\unicode[STIX]{x03C0}-\unicode[STIX]{x1D703}$ (see figure 1). The problem can be easily solved by conformally mapping to the first quadrant, from which we find, on the meniscus, $w=O(r^{\unicode[STIX]{x03C0}/2(\unicode[STIX]{x03C0}-\unicode[STIX]{x1D703})})$ as $r\rightarrow 0$ , where $r$ is the distance from the singularity. An expansion for small angle $\unicode[STIX]{x1D703}$ should then yield the local behaviour of $w^{(0)}$ and $w^{(1)}$ ,

This suggests $w^{(1)}=O(r^{1/2}\log r)$ , and that its Fourier coefficients behave as $a_{n}^{(1)}=O(n^{-3/2}\log n)$ as $n\rightarrow \infty$ , which is confirmed in figure 13(a). It should be noted that the expansion in $\unicode[STIX]{x1D703}$ above breaks down for $r$ sufficiently small, i.e. $r\sim \text{e}^{-1/\unicode[STIX]{x1D703}}$ , when the terms become disordered. This suggests that a small-angle expansion does not capture the behaviour exponentially close to the singularity.

Figure 13.

Decay rates as $n\rightarrow \infty$ for the coefficients $a_{n}^{(0)},a_{n}^{(1)},b_{n}^{(0)},b_{n}^{(1)}$ for $N=10^{4}$ , $h=10$ , $\unicode[STIX]{x1D719}=0.5$ . Velocity coefficients are shown in (a) and temperature coefficients in (b).

Using these dependences for $a_{n}^{(0)}$ and $a_{n}^{(1)}$ , the expressions in § 5.1 give $b_{n}^{(0)}=O(n^{-2})$ for the series terms in $\widetilde{T}^{(0)}$ , and $O(n^{-5/2})$ for those in $T_{p}^{(0)}$ . Hence, the $b_{n}^{(0)}$ limits the convergence of $T^{(0)}$ . At the next order, our numerics suggest (see figure 13) that $b_{n}^{(1)}=O(n^{-2}\log n)$ . Similarly to $w$ , the behaviour of $T$ near $x=\unicode[STIX]{x1D6FF}$ , and hence $T^{(0)}$ and $T^{(1)}$ , can also be found via a local analysis.

All of this implies that the first-order series converge slower than the leading-order series, by a factor of $\log n$ , and the velocity series converge slower than the temperature series, by a factor of $n^{1/2}$ . This is reflected in the results, where, to achieve graphical accuracy with a laptop computer, we needed $N=7000$ (with several minutes of computation time) for the first-order slip lengths, but only $N=120$ and $N=500$ (with less than a second per plot) for $\overline{Nu}^{(0)}$ and $\overline{Nu}^{(1)}$ .