2.1 Normalisation, coordinates and translational invariance

Using $h$ as a length unit, we employ Cartesian coordinates with the $z$ -axis antiparallel to the applied-gradient direction, the liquid domain lying between $-1<y<1$ , and the plane $x=0$ passing through the centres of an opposing pair of menisci (see figure 1 b). Using $Gh$ as a temperature unit allows one to define the dimensionless temperature  $t$ ; with the temperature in the present problem being defined to within an arbitrary additive constant, its distribution in the solid substrate is taken simply as $t=-z$ . Since the liquid motion is driven by Marangoni stresses at the surfaces, we employ $\unicode[STIX]{x1D70F}=-G\unicode[STIX]{x1D70E}_{T}$ as a stress unit (positive in the common case of negative  $\unicode[STIX]{x1D70E}_{T}$ ). With the intrinsic length scale associated with interfacial temperature variations being  $l$ , it is natural to use ${\mathcal{U}}=l\unicode[STIX]{x1D70F}/\unicode[STIX]{x1D707}$ as the velocity unit. The dimensionless problem is governed by two geometric parameters, namely the dimensionless semi-period $\unicode[STIX]{x1D6EC}=l/h$ and the gas fraction  $\unicode[STIX]{x1D6E5}$ . In addition, it depends upon a single dimensionless group, namely the Marangoni number $Ma={\mathcal{U}}h/\unicode[STIX]{x1D6FC}$ ; upon substituting the derived scale  ${\mathcal{U}}$ , this number reads

(2.1) $$\begin{eqnarray}Ma=\frac{\unicode[STIX]{x1D70F}hl}{\unicode[STIX]{x1D707}\unicode[STIX]{x1D6FC}}.\end{eqnarray}$$

Because of translational invariance in the $z$ -direction, the temperature in the liquid domain may be written as

(2.2) $$\begin{eqnarray}t=-z+\unicode[STIX]{x1D703},\end{eqnarray}$$

where the excess temperature $\unicode[STIX]{x1D703}$ is independent of $z$ and is accordingly a function of $x$ and $y$ alone. Similarly, both the pressure $p$ and the velocity components $(u,v,w)$ are independent of  $z$ . (Note that the flow, while being fully developed, is not unidirectional.) Because of the resulting decoupling between the longitudinal flow in the $z$ -direction and the flow in the cross-sectional plane, it is convenient to define the cross-sectional velocity vector $\boldsymbol{u}=\hat{\boldsymbol{e}}_{x}u+\hat{\boldsymbol{e}}_{y}v$ ; the velocity vector is then given as

(2.3) $$\begin{eqnarray}\hat{\boldsymbol{e}}_{z}w+\boldsymbol{u}.\end{eqnarray}$$

Given the independence upon $z$ , we interpret the gradient operator $\unicode[STIX]{x1D735}$ as $\hat{\boldsymbol{e}}_{x}\unicode[STIX]{x2202}/\unicode[STIX]{x2202}x+\hat{\boldsymbol{e}}_{y}\unicode[STIX]{x2202}/\unicode[STIX]{x2202}y$ .

Owing to the obvious periodicity, it is sufficient to consider the liquid domain restricted to the ‘unit cell’ $|x|<\unicode[STIX]{x1D6EC}$ . Given the symmetry about the mid-plane $y=0$ , the domain is further restricted to $-1<y<0$ .

2.2 The thermal problem

The excess temperature in the ‘semi-cell’ defined above is governed by the energy equation

(2.4) $$\begin{eqnarray}\unicode[STIX]{x1D735}\boldsymbol{\cdot }(Ma\,\unicode[STIX]{x1D703}\boldsymbol{u}-\unicode[STIX]{x1D735}\unicode[STIX]{x1D703})=Ma\,w.\end{eqnarray}$$

At $y=0$ it satisfies the symmetry condition,

(2.5) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}{\unicode[STIX]{x2202}y}=0\quad \text{at}~y=0,\end{eqnarray}$$

while at $y=-1$ it satisfies a no-flux condition at the free surface,

(2.6) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}{\unicode[STIX]{x2202}y}=0\quad \text{for}~|x|<\unicode[STIX]{x1D6EC}\unicode[STIX]{x1D6E5},\end{eqnarray}$$

and the temperature-continuity condition at the edges of the solid slats,

(2.7) $$\begin{eqnarray}\unicode[STIX]{x1D703}=0\quad \text{for}~\unicode[STIX]{x1D6EC}\unicode[STIX]{x1D6E5}<|x|<\unicode[STIX]{x1D6EC}.\end{eqnarray}$$

In addition, it satisfies the periodicity conditions, which, given the symmetry about $x=0$ , simply read

(2.8) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}{\unicode[STIX]{x2202}x}=0\quad \text{at}~x=\pm \unicode[STIX]{x1D6EC}.\end{eqnarray}$$

2.3 Longitudinal and cross-sectional flow

Neglecting fluid inertia, the flow problem nicely decouples into two subproblems, which respectively govern the longitudinal and cross-sectional flows. The longitudinal velocity component $w$ is governed by Laplace’s equation,

(2.9) $$\begin{eqnarray}\unicode[STIX]{x1D6FB}^{2}w=0,\end{eqnarray}$$

representing the absence of a macroscopic pressure gradient. At the centreline of the cell, it satisfies a symmetry condition,

(2.10) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}w}{\unicode[STIX]{x2202}y}=0\quad \text{at}~y=0.\end{eqnarray}$$

At $y=-1$ it satisfies a Marangoni condition at the free surface,

(2.11) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}w}{\unicode[STIX]{x2202}y}=-\unicode[STIX]{x1D6EC}^{-1}\quad \text{for}~|x|<\unicode[STIX]{x1D6EC}\unicode[STIX]{x1D6E5},\end{eqnarray}$$

and a no-slip condition at the edges of the solid slats,

(2.12) $$\begin{eqnarray}w=0\quad \text{for}~\unicode[STIX]{x1D6EC}\unicode[STIX]{x1D6E5}<|x|<\unicode[STIX]{x1D6EC}.\end{eqnarray}$$

Finally, it satisfies the periodicity conditions,

(2.13) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}w}{\unicode[STIX]{x2202}x}=0\quad \text{at}~x=\pm \unicode[STIX]{x1D6EC}.\end{eqnarray}$$

Note that, beyond being uncoupled to the cross-sectional flow, the longitudinal velocity is independent of the excess temperature, and hence of the Marangoni number.

The cross-sectional flow is governed by the continuity equation,

(2.14) $$\begin{eqnarray}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}=0,\end{eqnarray}$$

and the Stokes equation, which implies that the vorticity (whose only non-zero component is in the $z$ -direction) is harmonic,

(2.15) $$\begin{eqnarray}\unicode[STIX]{x1D6FB}^{2}\left(\frac{\unicode[STIX]{x2202}v}{\unicode[STIX]{x2202}x}-\frac{\unicode[STIX]{x2202}u}{\unicode[STIX]{x2202}y}\right)=0.\end{eqnarray}$$

At $y=0$ the following symmetry conditions apply,

(2.16) $$\begin{eqnarray}v=\frac{\unicode[STIX]{x2202}u}{\unicode[STIX]{x2202}y}=0\quad \text{at}~y=0.\end{eqnarray}$$

At $y=-1$ the flow satisfies impermeability,

(2.17) $$\begin{eqnarray}v=0,\end{eqnarray}$$

as well as the Marangoni condition at the free surface,

(2.18) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}u}{\unicode[STIX]{x2202}y}=\unicode[STIX]{x1D6EC}^{-1}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}{\unicode[STIX]{x2202}x}\quad \text{for}~|x|<\unicode[STIX]{x1D6EC}\unicode[STIX]{x1D6E5},\end{eqnarray}$$

and the no-slip condition at the solid slats,

(2.19) $$\begin{eqnarray}u=0\quad \text{for}~\unicode[STIX]{x1D6EC}\unicode[STIX]{x1D6E5}<|x|<\unicode[STIX]{x1D6EC}.\end{eqnarray}$$

Last, the periodicity conditions read

(2.20) $$\begin{eqnarray}u=\frac{\unicode[STIX]{x2202}v}{\unicode[STIX]{x2202}x}=0\quad \text{at}~x=\pm \unicode[STIX]{x1D6EC}.\end{eqnarray}$$

2.4 General observations

Three observations readily follow. First, we note that the excess temperature and cross-sectional flow are mutually and nonlinearly coupled through both the Marangoni condition (2.18) and the energy equation (2.4). Upon substitution of the continuity equation (2.14), the latter adopts the more convenient advection–diffusion form,

(2.21) $$\begin{eqnarray}Ma(\boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D703}-w)=\unicode[STIX]{x1D6FB}^{2}\unicode[STIX]{x1D703}.\end{eqnarray}$$

The longitudinal problem, however, is neither affected by nor affects the thermal problem.

Second, the nonlinear problem formulated above possesses obvious symmetries. We have already exploited the symmetry about $y=0$ . In addition, it is readily observed that  $\unicode[STIX]{x1D703}$ , $w$ and $v$ are even functions of  $x$ , while $u$ is an odd function of it.

Third, we notice that the only inhomogeneity in the thermal problem enters through the energy equation (2.4). When $Ma=0$ that equation becomes homogeneous, and the solution of the thermal problem is the trivial one, $\unicode[STIX]{x1D703}\equiv 0$ . Since the only inhomogeneity in the cross-sectional flow problem enters through the Marangoni condition (2.18), it then follows that this problem also possesses the trivial solution, $\boldsymbol{u}\equiv \mathbf{0}$ .

2.5 Streamfunction formulation

With the pressure playing only a passive role in the cross-sectional flow problem, it is convenient to employ the streamfunction  $\unicode[STIX]{x1D713}$ , defined by

(2.22a,b ) $$\begin{eqnarray}u=\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}{\unicode[STIX]{x2202}y},\quad v=-\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}{\unicode[STIX]{x2202}x}.\end{eqnarray}$$

(Since $\unicode[STIX]{x1D713}$ is defined to within an arbitrary additive constant, it may be assumed – with no loss of generality – to be odd in  $x$ .) This function is biharmonic (cf. (2.15)),

(2.23) $$\begin{eqnarray}\unicode[STIX]{x1D6FB}^{4}\unicode[STIX]{x1D713}=0,\end{eqnarray}$$

and possesses a nil uniform value at the boundary of the semi-cell,

(2.24) $$\begin{eqnarray}\unicode[STIX]{x1D713}=0\quad \text{on both}~y=-1,0~\text{and}~x=\pm \unicode[STIX]{x1D6EC}.\end{eqnarray}$$

In addition, it satisfies the symmetry condition at the cell centreline,

(2.25) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D713}}{\unicode[STIX]{x2202}y^{2}}=0\quad \text{at}~y=0,\end{eqnarray}$$

the mixed conditions at the compound surface $y=-1$ ,

(2.26a,b ) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D713}}{\unicode[STIX]{x2202}y^{2}}=\unicode[STIX]{x1D6EC}^{-1}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}{\unicode[STIX]{x2202}x}\quad \text{for}~|x|<\unicode[STIX]{x1D6EC}\unicode[STIX]{x1D6E5},\quad \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}{\unicode[STIX]{x2202}y}=0\quad \text{for}~\unicode[STIX]{x1D6EC}\unicode[STIX]{x1D6E5}<|x|<\unicode[STIX]{x1D6EC},\end{eqnarray}$$

and the periodicity conditions

(2.27) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D713}}{\unicode[STIX]{x2202}x^{2}}=0\quad \text{at}~x=\pm \unicode[STIX]{x1D6EC}.\end{eqnarray}$$

4.1 Longitudinal flow

Consider first the longitudinal flow, animated by the Marangoni condition (2.11). This condition, which applies in the inner region, implies that $w$ is $O(1)$ there. We accordingly employ the asymptotic expansion

(4.2) $$\begin{eqnarray}w(x,y;\unicode[STIX]{x1D6EC})=W(X,Y)+\cdots \,.\end{eqnarray}$$

The leading-order velocity is governed by Laplace’s equation,

(4.3) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}^{2}W}{\unicode[STIX]{x2202}X^{2}}+\frac{\unicode[STIX]{x2202}^{2}W}{\unicode[STIX]{x2202}Y^{2}}=0,\end{eqnarray}$$

together with the mixed conditions at $Y=0$ ,

(4.4a,b ) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}W}{\unicode[STIX]{x2202}Y}=-1\quad \text{for}~|X|<\unicode[STIX]{x1D6E5},\quad W=0\quad \text{for}~\unicode[STIX]{x1D6E5}<|X|<1,\end{eqnarray}$$

and the periodicity conditions,

(4.5) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}W}{\unicode[STIX]{x2202}X}=0\quad \text{at}~X=\pm 1.\end{eqnarray}$$

By considering the ‘shifted’ field $W(X,Y)-Y$ , we readily observe that it satisfies a boundary-value problem analogous to that governing longitudinal shear flow about a no-slip plate that is decorated with an array of shear-free patches. The latter problem has been solved by Philip (Reference Philip1972) using conformal-mapping techniques; adopting Philip’s solution to the present notation gives

(4.6) $$\begin{eqnarray}W=\operatorname{Im}\left\{\frac{2}{\unicode[STIX]{x03C0}}\cos ^{-1}\frac{\cos (\unicode[STIX]{x03C0}Z/2)}{\cos (\unicode[STIX]{x03C0}\unicode[STIX]{x1D6E5}/2)}-Z\right\},\end{eqnarray}$$

in which $Z=X+\text{i}Y$ . In particular, at $Y=0$ ,

(4.7) $$\begin{eqnarray}W=\frac{2}{\unicode[STIX]{x03C0}}\cosh ^{-1}\frac{\cos (\unicode[STIX]{x03C0}X/2)}{\cos (\unicode[STIX]{x03C0}\unicode[STIX]{x1D6E5}/2)}\quad \text{for}~|X|<\unicode[STIX]{x1D6E5},\end{eqnarray}$$

while, as $Y\rightarrow \infty$ ,

(4.8) $$\begin{eqnarray}W\rightarrow -\frac{2}{\unicode[STIX]{x03C0}}\ln \cos \frac{\unicode[STIX]{x03C0}\unicode[STIX]{x1D6E5}}{2}.\end{eqnarray}$$

With an $O(1)$ inner flow that approaches a finite value at large  $Y$ , it is evident that the outer flow is $O(1)$ as well. We accordingly employ the outer expansion

(4.9) $$\begin{eqnarray}w(x,y;\unicode[STIX]{x1D6EC})=\bar{w}(X,y)+\cdots \,.\end{eqnarray}$$

At $O(\unicode[STIX]{x1D6EC}^{-2})$ , the momentum balance (2.9) implies that $\unicode[STIX]{x2202}^{2}\bar{w}/\unicode[STIX]{x2202}X^{2}=0$ ; this, together with the periodicity conditions (2.13), implies that $\bar{w}$ is independent of $X$ and is accordingly a function of $y$ alone. Its variation with $y$ is determined from solvability conditions, obtained in turn from (2.9) at $O(1)$ . These imply that $\bar{w}$ can at most vary linearly with  $y$ . Such variation, however, is excluded by the symmetry about $y=0$ ; see (2.10). We conclude that $\bar{w}$ is uniform. The value of this uniform velocity is obtained from asymptotic matching with the inner solution, $\bar{w}=\lim _{Y\rightarrow \infty }W$ ; making use of (4.8) thus gives

(4.10) $$\begin{eqnarray}\bar{w}\equiv -\frac{2}{\unicode[STIX]{x03C0}}\ln \cos \frac{\unicode[STIX]{x03C0}\unicode[STIX]{x1D6E5}}{2}.\end{eqnarray}$$

Note that: (i) this result has been originally obtained by Baier et al. (Reference Baier, Steffes and Hardt2010) using Lorentz’s reciprocal theorem; (ii) it is consistent with (3.7); and (iii) with a uniform outer leading-order velocity, it follows that the cross-sectional mean $\langle w\rangle$ is given by (4.10) at that order.

4.2 Outer cross-sectional flow

Considering now the cross-sectional flow in the outer region, we note that it is governed by the equation (cf. (2.23)),

(4.11) $$\begin{eqnarray}\left\{\!\frac{\unicode[STIX]{x2202}^{4}}{\unicode[STIX]{x2202}X^{4}}+O(\unicode[STIX]{x1D6EC})\!\right\}\unicode[STIX]{x1D713}=0.\end{eqnarray}$$

It follows that $\unicode[STIX]{x1D713}$ must vanish at any asymptotic order. Indeed, regardless of the Poincaré prefactor multiplying the leading-order term $\unicode[STIX]{x1D713}_{0}$ in the asymptotic expansion of  $\unicode[STIX]{x1D713}$ , we find from (4.11) that  $\unicode[STIX]{x1D713}_{0}$ , which must be odd in  $X$ , is of the form $A(y)X+B(y)X^{3}$ . Requiring that both $\unicode[STIX]{x1D713}_{0}$ and $\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D713}_{0}/\unicode[STIX]{x2202}X^{2}$ vanish at $X=1$ (cf. (2.24) and (2.27)) then implies that $\unicode[STIX]{x1D713}_{0}$ trivially vanishes, in contradiction to the postulated expansion. It follows that $\unicode[STIX]{x1D713}$ (and hence $u$ and  $v$ ) are exponentially small in the outer region.

The cross-sectional flow is accordingly confined to the inner region. (This could have been anticipated from the linear equations governing the cross-sectional flow, which suggest an exponential decay outside the inner region – see indeed (5.21).) Its determination there is complicated by the coupling with the thermal problem, which in turn introduces a dependence upon the Marangoni number; see (2.21). When considering the deep-channel scaling of the Marangoni number (see (2.1)), there are two plausible possibilities. The first is to assume that the Marangoni number is fixed. The second is to associate the limit $\unicode[STIX]{x1D6EC}\rightarrow 0$ with increasing  $h$ , suggesting in turn to consider the possibility of $O(\unicode[STIX]{x1D6EC}^{-1})$ Marangoni numbers. (A third possibility, where the limit $\unicode[STIX]{x1D6EC}\rightarrow 0$ is associated with decreasing  $l$ , actually suggests asymptotically small Marangoni numbers. This possibility, however, is already captured by the analysis of $O(1)$ Marangoni numbers.) In what follows, these two separate limits are considered in §§ 5 and 6, respectively. In both sections we start with the outer thermal problem, governed by (2.5)–(2.8) and (2.21). Then, we proceed to investigate the thermal and cross-sectional flow problems in the inner region.

5.1 Excess temperature

In the outer region we postulate the asymptotic expansion,

(5.1) $$\begin{eqnarray}\unicode[STIX]{x1D703}(X,y;\unicode[STIX]{x1D6EC})=\unicode[STIX]{x1D712}(\unicode[STIX]{x1D6EC})[\bar{\unicode[STIX]{x1D703}}(X,y)+\unicode[STIX]{x1D6EC}\unicode[STIX]{x1D703}_{1}(X,y)+\unicode[STIX]{x1D6EC}^{2}\unicode[STIX]{x1D703}_{2}(X,y)+\cdots \,],\end{eqnarray}$$

with the goal of determining the Poincaré prefactor $\unicode[STIX]{x1D712}$ and the leading-order term  $\bar{\unicode[STIX]{x1D703}}$ . With $\boldsymbol{u}$ being exponentially small in the outer region, it follows from (2.21) that the product of $\unicode[STIX]{x1D712}$ with the Laplacian of $\bar{\unicode[STIX]{x1D703}}$ must balance the constant term $-Ma\,\bar{w}$ , which represents the heat-flux divergence associated with the uniform longitudinal flow.

With $Ma=O(1)$ , any non-trivial $X$ dependence of $\bar{\unicode[STIX]{x1D703}}$ gives $\unicode[STIX]{x1D712}=\unicode[STIX]{x1D6EC}^{2}$ , implying in turn a balance of the constant divergence term with $\unicode[STIX]{x2202}^{2}\bar{\unicode[STIX]{x1D703}}/\unicode[STIX]{x2202}X^{2}$ . Such a balance, however, results in a quadratic variation of $\bar{\unicode[STIX]{x1D703}}$ with $X$ which contradicts the periodicity conditions (2.8). It follows that the leading-order term must be independent of  $X$ . The same argument can be repeated to show that $\unicode[STIX]{x1D703}_{1}$ is also independent of  $X$ . We conclude that $\unicode[STIX]{x1D712}=1$ , whereby (5.1) is refined to

(5.2) $$\begin{eqnarray}\unicode[STIX]{x1D703}(X,y;\unicode[STIX]{x1D6EC})=\bar{\unicode[STIX]{x1D703}}(X)+\unicode[STIX]{x1D6EC}\unicode[STIX]{x1D703}_{1}(X)+\unicode[STIX]{x1D6EC}^{2}\unicode[STIX]{x1D703}_{2}(X,y)+\cdots \,.\end{eqnarray}$$

Consider now the $O(1)$ balance of (2.21),

(5.3) $$\begin{eqnarray}-Ma\,\bar{w}=\frac{\text{d}^{2}\bar{\unicode[STIX]{x1D703}}}{\text{d}y^{2}}+\frac{\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D703}_{2}}{\unicode[STIX]{x2202}X^{2}}.\end{eqnarray}$$

Integration over $X$ from $-1$ to $+1$ in conjunction with the periodicity conditions (2.8) readily gives

(5.4) $$\begin{eqnarray}-Ma\,\bar{w}=\frac{\text{d}^{2}\bar{\unicode[STIX]{x1D703}}}{\text{d}y^{2}},\end{eqnarray}$$

revealing in turn that $\unicode[STIX]{x1D703}_{2}$ is also independent of  $X$ .

Equation (5.4) represents a balance between advection in the $z$ -direction and conduction in the $y$ -direction. Integration in conjunction with the symmetry condition (2.5) yields

(5.5) $$\begin{eqnarray}\bar{\unicode[STIX]{x1D703}}=\frac{Ma\,\bar{w}}{2}(1-y^{2}),\end{eqnarray}$$

where the additive constant was set to nullify $\bar{\unicode[STIX]{x1D703}}$ at $y=-1$ : a non-zero value there would imply sharp gradients in the inner region, which are incompatible with dominant conduction (see (5.7)). The parabolic profile (5.5) clearly illustrates the inadequacy of a semi-bounded idealisation. Physically, it represents a conductive heat flux towards the surfaces $y=\pm 1$ .

Heat-flux matching with the inner region implies that $\unicode[STIX]{x1D703}$ is $O(\unicode[STIX]{x1D6EC})$ there, say

(5.6) $$\begin{eqnarray}\unicode[STIX]{x1D703}(x,y;\unicode[STIX]{x1D6EC})=\unicode[STIX]{x1D6EC}\unicode[STIX]{x1D6E9}(X,Y)+\cdots \,.\end{eqnarray}$$

The cross-sectional velocity component $u$ , animated by the Marangoni condition (2.18), is accordingly only $O(1)$ there. From the continuity equation (2.14) we find that so is  $v$ . The advection–diffusion equation (2.21) is then dominated by conduction, implying that $\unicode[STIX]{x1D6E9}$ is governed by Laplace’s equation in the inner coordinates,

(5.7) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D6E9}}{\unicode[STIX]{x2202}X^{2}}+\frac{\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D6E9}}{\unicode[STIX]{x2202}Y^{2}}=0.\end{eqnarray}$$

In addition, it satisfies the homogeneous conditions at $Y=0$ ,

(5.8a,b ) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6E9}}{\unicode[STIX]{x2202}Y}=0\quad \text{for}~|X|<\unicode[STIX]{x1D6E5},\quad \unicode[STIX]{x1D6E9}=0\quad \text{for}~\unicode[STIX]{x1D6E5}<|X|<1,\end{eqnarray}$$

and the periodicity conditions,

(5.9) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6E9}}{\unicode[STIX]{x2202}X}=0\quad \text{at}~X=\pm 1.\end{eqnarray}$$

It is driven by the inhomogeneous flux-matching condition,

(5.10) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6E9}}{\unicode[STIX]{x2202}Y}\rightarrow Ma\,\bar{w}\quad \text{as}~Y\rightarrow \infty .\end{eqnarray}$$

Comparing with the problem (4.3)–(4.5) governing  $W$ , we observe that $\unicode[STIX]{x1D6E9}$ is readily expressed in terms of Philip’s solution,

(5.11) $$\begin{eqnarray}\unicode[STIX]{x1D6E9}=Ma\,\bar{w}(W+Y),\end{eqnarray}$$

wherein $W$ is provided by (4.6).

5.2 Transverse flow

With the cross-sectional velocity components being $O(1)$ , the streamfunction is $O(\unicode[STIX]{x1D6EC})$ . We accordingly employ the asymptotic expansions

(5.12a,b ) $$\begin{eqnarray}u(x,y;\unicode[STIX]{x1D6EC})=U(X,Y)+\cdots \,,\quad v(x,y;\unicode[STIX]{x1D6EC})=V(X,Y)+\cdots \,,\end{eqnarray}$$

and

(5.13) $$\begin{eqnarray}\unicode[STIX]{x1D713}(x,y;\unicode[STIX]{x1D6EC})=\unicode[STIX]{x1D6EC}\unicode[STIX]{x1D6F9}(X,Y)+\cdots \,,\end{eqnarray}$$

wherein $\unicode[STIX]{x1D6F9}$ is defined by (cf. (2.22))

(5.14a,b ) $$\begin{eqnarray}U=\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6F9}}{\unicode[STIX]{x2202}Y},\quad V=-\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6F9}}{\unicode[STIX]{x2202}X}.\end{eqnarray}$$

The linear dependence upon $Ma\,\bar{w}$ , implied by (5.11), further suggests the rescaling

(5.15) $$\begin{eqnarray}\unicode[STIX]{x1D6F9}=Ma\,\bar{w}\,\hat{\unicode[STIX]{x1D6F9}}.\end{eqnarray}$$

The rescaled streamfunction depends upon the single parameter  $\unicode[STIX]{x1D6E5}$ . It is governed by: (i) the biharmonic equation,

(5.16) $$\begin{eqnarray}\left(\frac{\unicode[STIX]{x2202}^{2}}{\unicode[STIX]{x2202}X^{2}}+\frac{\unicode[STIX]{x2202}^{2}}{\unicode[STIX]{x2202}Y^{2}}\right)^{2}\hat{\unicode[STIX]{x1D6F9}}=0;\end{eqnarray}$$

(ii) impermeability at the boundaries of the inner region,

(5.17) $$\begin{eqnarray}\hat{\unicode[STIX]{x1D6F9}}=0\quad \text{at both}~Y=0~\text{and}~X=\pm 1;\end{eqnarray}$$

(iii) the Marangoni and no-slip conditions at $Y=0$ ,

(5.18a,b ) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}^{2}\hat{\unicode[STIX]{x1D6F9}}}{\unicode[STIX]{x2202}Y^{2}}=\frac{\unicode[STIX]{x2202}W}{\unicode[STIX]{x2202}X}\quad \text{for}~|X|<\unicode[STIX]{x1D6E5},\quad \frac{\unicode[STIX]{x2202}\hat{\unicode[STIX]{x1D6F9}}}{\unicode[STIX]{x2202}Y}=0\quad \text{for}~\unicode[STIX]{x1D6E5}<|X|<1;\end{eqnarray}$$

(iv) the periodicity conditions,

(5.19) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}^{2}\hat{\unicode[STIX]{x1D6F9}}}{\unicode[STIX]{x2202}X^{2}}=0\quad \text{at}~X=\pm 1;\end{eqnarray}$$

and (v) velocity decay,

(5.20) $$\begin{eqnarray}\hat{\unicode[STIX]{x1D6F9}}~\text{bounded}\quad \text{for}~Y\rightarrow \infty .\end{eqnarray}$$

Using the eigenfunctions of the biharmonic equation, we readily find that the most general solution of (5.16) that is bounded at large $Y$ and satisfies the homogeneous conditions (5.17) and (5.19) is

(5.21) $$\begin{eqnarray}\hat{\unicode[STIX]{x1D6F9}}=\mathop{\sum }_{n=1}^{\infty }A_{n}Y\text{e}^{-n\unicode[STIX]{x03C0}Y}\sin n\unicode[STIX]{x03C0}X.\end{eqnarray}$$

Substitution into the mixed conditions (5.18) then gives

(5.22) $$\begin{eqnarray}\displaystyle & \displaystyle -2\mathop{\sum }_{n=1}^{\infty }n\unicode[STIX]{x03C0}A_{n}\sin n\unicode[STIX]{x03C0}X=\left.\frac{\unicode[STIX]{x2202}W}{\unicode[STIX]{x2202}X}\right|_{Y=0}\quad \text{for}~|X|<\unicode[STIX]{x1D6E5}, & \displaystyle\end{eqnarray}$$

(5.23) $$\begin{eqnarray}\displaystyle & \displaystyle \mathop{\sum }_{n=1}^{\infty }A_{n}\sin n\unicode[STIX]{x03C0}X=0\quad \text{for}~\unicode[STIX]{x1D6E5}<|X|<1. & \displaystyle\end{eqnarray}$$

The Fourier coefficients $\{A_{n}\}$ are determined following the procedure described by Lauga & Stone (Reference Lauga and Stone2003) and Teo & Khoo (Reference Teo and Khoo2009). Thus, making use of (4.7) we form the inner products of (5.23) with $\sin m\unicode[STIX]{x03C0}X$ for all $m\geqslant 1$ , thereby obtaining an infinite linear system of equations governing these coefficients. This system is solved by controlled truncation, whereby $A_{n}$ is set to zero for $n$ larger than a specified integer  $N$ , thereby transforming the infinite system to a finite one governing the $N$ coefficients $\{A_{n}\}_{n=1}^{\infty }$ . The ‘streamlines’ in the cross-sectional plane, given by $\hat{\unicode[STIX]{x1D6F9}}(X,Y)=\text{const.}$ , are portrayed in figure 2 for $\unicode[STIX]{x1D6E5}=1/2$ . For reference, we also present in that figure the excess-temperature isotherms, obtained using (4.6) and (5.11).

Figure 2.

Streamlines (closed blue) in the cross-sectional plane for $\unicode[STIX]{x1D6E5}=1/2$ , obtained using the eigenfunction solution (5.21). Also shown (thin red lines) are the isotherms $\unicode[STIX]{x1D6E9}(X,Y)=\text{const.}$ , obtained using (4.6) and (5.11).

While the exponential decay in (5.21) guarantees that very few low-order modes are necessary for an accurate calculation of $\hat{\unicode[STIX]{x1D6F9}}$ , convergence in the determination of the respective Fourier coefficients is actually attained only for large  $N$ , of approximately  $10^{4}$ . This feature is typical of solutions to mixed boundary-value problems when solved by dual series methods: it has to do with the inherent square-root singularity associated with the transitions in (5.18) between two types of boundary condition. Since the series method does not explicitly account for this singularity, the matrix representing the algebraic equations governing $\{A_{n}\}_{n=1}^{\infty }$ is dense. It is plausible that alternative complex-variable approaches, which do account for the singularity explicitly (Crowdy Reference Crowdy2011, Reference Crowdy2013), may be employed to solve the problem posed by (5.16)–(5.20).

6.1 Outer region

Considering first heat transport in the outer region we note that the general observations made at the beginning of § 5 are independent of the Marangoni number scaling. Thus, both the leading-order term in the outer expansion of  $\unicode[STIX]{x1D703}$ , as well as its two successive corrections (of relative magnitudes $\unicode[STIX]{x1D6EC}$ and  $\unicode[STIX]{x1D6EC}^{2}$ ) are independent of  $X$ . The $O(\unicode[STIX]{x1D6EC}^{-1})$ large longitudinal advection in the energy balance (2.21) can be balanced by conduction in the $y$ -direction only if the excess-temperature variations are of the same order. We accordingly postulate the outer expansion

(6.2) $$\begin{eqnarray}\unicode[STIX]{x1D703}(x,y;\unicode[STIX]{x1D6EC})=\unicode[STIX]{x1D6EC}^{-1}\bar{\unicode[STIX]{x1D703}}(y)+\cdots \,,\end{eqnarray}$$

where the subsequent $O(1)$ and $O(\unicode[STIX]{x1D6EC})$ corrections are also independent of $X$ . At leading order, the advection–diffusion equation (2.21) then reads

(6.3) $$\begin{eqnarray}-\widetilde{Ma}\,\bar{w}=\frac{\text{d}^{2}\bar{\unicode[STIX]{x1D703}}}{\text{d}y^{2}}.\end{eqnarray}$$

Integration in conjunction with the symmetry condition (2.5) gives (cf. (5.5))

(6.4) $$\begin{eqnarray}\bar{\unicode[STIX]{x1D703}}=\frac{\widetilde{Ma}\,\bar{w}}{2}(1-y^{2}),\end{eqnarray}$$

where, just as in (5.5), the additive constant has been set so as to nullify $\bar{\unicode[STIX]{x1D703}}$ at $y=-1$ .

6.2 Dominant advection in the inner region?

Naively, following the methodology used in the case of moderate Marangoni numbers, one may conclude from heat-flux matching that $\unicode[STIX]{x1D703}=O(1)$ in the inner region,

(6.5) $$\begin{eqnarray}\unicode[STIX]{x1D703}(x,y;\unicode[STIX]{x1D6EC})=\unicode[STIX]{x1D6E9}(X,Y)+\cdots \,,\end{eqnarray}$$

whereby the Marangoni condition (2.18) implies that the cross-sectional flow there is $O(\unicode[STIX]{x1D6EC}^{-1})$ ,

(6.6a,b ) $$\begin{eqnarray}u(x,y;\unicode[STIX]{x1D6EC})=\unicode[STIX]{x1D6EC}^{-1}U(X,Y)+\cdots \,,\quad v(x,y;\unicode[STIX]{x1D6EC})=\unicode[STIX]{x1D6EC}^{-1}V(X,Y)+\cdots \,.\end{eqnarray}$$

This scaling implies that $\unicode[STIX]{x1D6E9}$ is dominated by advection with the cross-sectional flow, namely

(6.7) $$\begin{eqnarray}\boldsymbol{U}\boldsymbol{\cdot }\unicode[STIX]{x1D735}_{\boldsymbol{X}}\unicode[STIX]{x1D6E9}=0,\end{eqnarray}$$

in which

(6.8a,b ) $$\begin{eqnarray}\boldsymbol{U}=\hat{\boldsymbol{e}}_{x}U+\hat{\boldsymbol{e}}_{y}V,\quad \unicode[STIX]{x1D735}_{\boldsymbol{X}}=\hat{\boldsymbol{e}}_{x}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}X}+\hat{\boldsymbol{e}}_{y}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}Y}.\end{eqnarray}$$

In addition to (6.7), $\unicode[STIX]{x1D6E9}$ is also governed by the homogeneous conditions (5.8) and (5.9), as well as the inhomogeneous flux-matching condition (cf. (5.10))

(6.9) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6E9}}{\unicode[STIX]{x2202}Y}\rightarrow \widetilde{Ma}\,\bar{w}\quad \text{as}~Y\rightarrow \infty .\end{eqnarray}$$

Equation (6.7) implies that $\unicode[STIX]{x1D6E9}$ is a constant on each of the cross-sectional ‘streamlines’ of  $\boldsymbol{U}$ . This is alternatively written as

(6.10) $$\begin{eqnarray}\unicode[STIX]{x1D6E9}=f(\unicode[STIX]{x1D6F9}),\end{eqnarray}$$

wherein $\unicode[STIX]{x1D6F9}$ , the streamfunction associated with $\boldsymbol{U}$ , is defined as in (5.14). The different temperature scaling in the outer and inner regions excludes ‘open’ streamlines in the inner region (namely lines that reach the outer region). A closed-streamline structure, as in the case of moderate  $Ma$ , is accordingly predicted. We now show that such a structure is inconsistent with (6.9), thus leading to a contradiction. The following arguments are based upon ideas devised in classical analyses of dominant advection along closed streamlines (Batchelor Reference Batchelor1956; Pan & Acrivos Reference Pan and Acrivos1968).

We start with the energy equation (2.4). Integrating over a two-dimensional domain ${\mathcal{D}}$ which lies entirely in the cross-sectional liquid domain yields, upon making use of the two-dimensional variant of Gauss’s theorem,

(6.11) $$\begin{eqnarray}\oint _{\unicode[STIX]{x2202}{\mathcal{D}}}\,\text{d}l\,\hat{\boldsymbol{n}}\boldsymbol{\cdot }(Ma\,\unicode[STIX]{x1D703}\,\boldsymbol{u}-\unicode[STIX]{x1D735}\unicode[STIX]{x1D703})=Ma\int _{{\mathcal{D}}}\,\text{d}x\,\text{d}y\,w,\end{eqnarray}$$

wherein $\unicode[STIX]{x2202}{\mathcal{D}}$ denotes the boundary of ${\mathcal{D}}$ , $\hat{\boldsymbol{n}}$ is an outward-pointing unit vector normal to $\unicode[STIX]{x2202}{\mathcal{D}}$ , and $\text{d}l$ is a differential length element. In particular, choosing ${\mathcal{D}}$ as the interior of an (arbitrary) streamline in the cross-sectional plane, for which $\hat{\boldsymbol{n}}\boldsymbol{\cdot }\boldsymbol{u}=0$ , gives

(6.12) $$\begin{eqnarray}-\!\oint _{\unicode[STIX]{x2202}{\mathcal{D}}}\,\text{d}l\,\hat{\boldsymbol{n}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D703}=Ma\int _{{\mathcal{D}}}\,\text{d}x\,\text{d}y\,w.\end{eqnarray}$$

The above relation is exact. We consider now the limit $\unicode[STIX]{x1D6EC}\rightarrow 0$ with the Marangoni-number scaling (6.1). Applying (6.12) to a closed streamline in the inner region, one obtains

(6.13) $$\begin{eqnarray}-\!\oint _{\unicode[STIX]{x2202}{\mathcal{D}}}\,\text{d}L\,\hat{\boldsymbol{n}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}_{\boldsymbol{X}}(\unicode[STIX]{x1D6E9}+\cdots \!)=\unicode[STIX]{x1D6EC}\,\widetilde{Ma}\int _{{\mathcal{D}}}\,\text{d}X\,\text{d}Y\,(W+\cdots \,),\end{eqnarray}$$

where $\text{d}L=\unicode[STIX]{x1D6EC}^{-1}\,\text{d}l$ is a properly scaled differential length element. At leading order, where $\unicode[STIX]{x2202}{\mathcal{D}}$ coincides with an isoline of  $\unicode[STIX]{x1D6F9}$ , we accordingly obtain

(6.14) $$\begin{eqnarray}\oint _{\unicode[STIX]{x1D6F9}=\text{const.}}\,\text{d}L\,\hat{\boldsymbol{n}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}_{\boldsymbol{X}}\unicode[STIX]{x1D6E9}=0.\end{eqnarray}$$

At this stage it is convenient to employ $\unicode[STIX]{x1D6F9}$ itself as a curvilinear coordinate. The other curvilinear coordinate, say  $\unicode[STIX]{x1D709}$ , is chosen such that $\unicode[STIX]{x1D6F9}$ and $\unicode[STIX]{x1D709}$ constitute a family of orthogonal coordinates. With that choice we have

(6.15) $$\begin{eqnarray}(\text{d}L)^{2}=h_{\unicode[STIX]{x1D6F9}}^{2}(\text{d}\unicode[STIX]{x1D6F9})^{2}+h_{\unicode[STIX]{x1D709}}^{2}(\text{d}\unicode[STIX]{x1D709})^{2},\end{eqnarray}$$

wherein $h_{\unicode[STIX]{x1D6F9}}$ and $h_{\unicode[STIX]{x1D709}}$ are the appropriate metric coefficients, which are generally functions of $\unicode[STIX]{x1D6F9}$ and  $\unicode[STIX]{x1D709}$ . In particular, on the isoline $\unicode[STIX]{x1D6F9}=\text{const.}$ we have $\text{d}L=h_{\unicode[STIX]{x1D709}}|\text{d}\unicode[STIX]{x1D709}|$ , in which $h_{\unicode[STIX]{x1D709}}$ is a function of $\unicode[STIX]{x1D709}$ alone. Moreover, making use of (6.10) we find that $\hat{\boldsymbol{n}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}_{\boldsymbol{X}}\unicode[STIX]{x1D6E9}=-(h_{\unicode[STIX]{x1D6F9}})^{-1}\,\text{d}\unicode[STIX]{x1D6E9}/\text{d}\unicode[STIX]{x1D6F9}$ on that curve. Substituting this expression into relation (6.14) and noting that $\text{d}\unicode[STIX]{x1D6E9}/\text{d}\unicode[STIX]{x1D6F9}$ may be taken out of the integral sign, we eventually obtain

(6.16) $$\begin{eqnarray}\frac{\text{d}\unicode[STIX]{x1D6E9}}{\text{d}\unicode[STIX]{x1D6F9}}\oint _{\unicode[STIX]{x1D6F9}=\text{const.}}|\text{d}\unicode[STIX]{x1D709}|\,\frac{h_{\unicode[STIX]{x1D709}}}{h_{\unicode[STIX]{x1D6F9}}}=0.\end{eqnarray}$$

Since both metric coefficients are positive, so must be the integral. It follows that $\text{d}\unicode[STIX]{x1D6E9}/\text{d}\unicode[STIX]{x1D6F9}=0$ on the curve $\unicode[STIX]{x1D6F9}=\text{const}$ . Since the constant is arbitrary, this is true for all closed streamlines and hence in the entire inner region. It follows that $\unicode[STIX]{x1D6E9}$ is uniform there. The said uniformity, however, is incompatible with (6.9). The preceding construction, starting with expansion (6.5), must be abandoned.

6.3 Nested boundary-layer structure

The presumed $O(\unicode[STIX]{x1D6EC})$ extent of the inner region was set by the unit-cell geometry. In postulating the naive expansion (6.5) we have assumed that the $\unicode[STIX]{x1D6EC}$ scale is relevant to the near-boundary transport in both the $x$ - and $y$ -directions. While this assumption is consistent at moderate Marangoni numbers, it fails at large Marangoni numbers. This suggests that the small scale pertinent to transport in the $y$ -direction, say $\unicode[STIX]{x1D6E5}$ , is determined in that case by a balance between advection and conduction, rather than by the geometry. (Owing to periodicity, $x$ -variations still take place on the $\unicode[STIX]{x1D6EC}$ scale.) Since the case $\unicode[STIX]{x1D6E5}=\unicode[STIX]{x1D6EC}$ resulted in dominant advection (see (6.7)), we anticipate that $\unicode[STIX]{x1D6E5}\ll \unicode[STIX]{x1D6EC}$ is needed for conduction to enter the dominant balance. It is important to note that the newly identified boundary layer is relevant only to the advection–diffusion equation governing  $\unicode[STIX]{x1D703}$ : since the cross-sectional flow is coupled to $\unicode[STIX]{x1D703}$ only through the boundary condition (2.18) at $y=-1$ , it is ‘unaware’ of the existence of that layer. The velocity components $u$ and $v$ thus vary on the inner $\unicode[STIX]{x1D6EC}$ scale in both $x$ - and $y$ -directions.

We accordingly look for a dominant balance of the advection–diffusion equation (2.21). Balancing the $O(\unicode[STIX]{x1D6EC}^{-1})$ heat flux associated with the outer scaling (6.2) implies that the boundary-layer excess temperature is of order $\unicode[STIX]{x1D6E5}/\unicode[STIX]{x1D6EC}$ . The cross-sectional flow, driven by the Marangoni condition (2.18), is then of order $\unicode[STIX]{x1D6E5}/\unicode[STIX]{x1D6EC}^{2}$ . (Recall that $u$ varies on the scale $\unicode[STIX]{x1D6EC}$ , rather than  $\unicode[STIX]{x1D6E5}$ .) Since $\unicode[STIX]{x1D6E5}\ll \unicode[STIX]{x1D6EC}$ , the conduction term in (2.21) is dominated by $y$ -derivatives, and is hence of order $(\unicode[STIX]{x1D6EC}\unicode[STIX]{x1D6E5})^{-1}$ . This term must be balanced by advective terms. With the Marangoni-number scaling (6.1), the longitudinal advection term in (2.21) is $O(\unicode[STIX]{x1D6EC}^{-1})$ , while the cross-sectional advection terms are of order $\unicode[STIX]{x1D6E5}^{2}/\unicode[STIX]{x1D6EC}^{5}$ . The only possible dominant balance is between conduction perpendicular to the surface and cross-sectional advection, giving

(6.17) $$\begin{eqnarray}\unicode[STIX]{x1D6E5}=\unicode[STIX]{x1D6EC}^{4/3},\end{eqnarray}$$

which is indeed much less than $\unicode[STIX]{x1D6EC}$ , as required. With both the conduction and cross-sectional advection terms in (2.21) being $O(\unicode[STIX]{x1D6EC}^{-7/3})$ , the $O(\unicode[STIX]{x1D6EC}^{-1})$ longitudinal advection term is subdominant.

We accordingly define the boundary-layer coordinate

(6.18) $$\begin{eqnarray}\unicode[STIX]{x1D702}=\frac{y+1}{\unicode[STIX]{x1D6EC}^{4/3}}\end{eqnarray}$$

and employ the associated boundary-layer expansion

(6.19) $$\begin{eqnarray}\unicode[STIX]{x1D703}(x,y;\unicode[STIX]{x1D6EC})=\unicode[STIX]{x1D6EC}^{1/3}\unicode[STIX]{x1D6E9}(X,\unicode[STIX]{x1D702})+\cdots \,.\end{eqnarray}$$

In addition, we employ the following expansions in the velocity components in the inner region, where they are functions of $X$ and $Y$ (cf. (5.12)),

(6.20a,b ) $$\begin{eqnarray}u(x,y;\unicode[STIX]{x1D6EC})=\unicode[STIX]{x1D6EC}^{-2/3}U(X,Y)+\cdots \!,\quad v(x,y;\unicode[STIX]{x1D6EC})=\unicode[STIX]{x1D6EC}^{-2/3}V(X,Y)+\cdots \,.\end{eqnarray}$$

At leading order, then, the advection–diffusion balance (2.21) reads:

(6.21) $$\begin{eqnarray}\widetilde{Ma}\left[U(X,0)\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6E9}}{\unicode[STIX]{x2202}X}+\unicode[STIX]{x1D702}\frac{\unicode[STIX]{x2202}V}{\unicode[STIX]{x2202}Y}(X,0)\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6E9}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}}\right]=\frac{\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D6E9}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}^{2}}.\end{eqnarray}$$

This equation is supplemented by the flux-matching condition,

(6.22) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6E9}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}}\rightarrow \widetilde{Ma}\,\bar{w}\quad \text{as}~Y\rightarrow \infty ;\end{eqnarray}$$

the homogeneous conditions at $\unicode[STIX]{x1D702}=0$ ,

(6.23a,b ) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6E9}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}}=0\quad \text{for}~|X|<\unicode[STIX]{x1D6E5},\quad \unicode[STIX]{x1D6E9}=0\quad \text{for}~\unicode[STIX]{x1D6E5}<|X|<1;\end{eqnarray}$$

and the periodicity conditions,

(6.24) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6E9}}{\unicode[STIX]{x2202}X}=0\quad \text{at}~X=\pm 1.\end{eqnarray}$$

The equations governing the cross-sectional flow components follow from (2.14)–(2.20). They consist of: (i) the continuity and Stokes equations,

(6.25a,b ) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}U}{\unicode[STIX]{x2202}X}+\frac{\unicode[STIX]{x2202}V}{\unicode[STIX]{x2202}Y}=0,\quad \left(\frac{\unicode[STIX]{x2202}^{2}}{\unicode[STIX]{x2202}X^{2}}+\frac{\unicode[STIX]{x2202}^{2}}{\unicode[STIX]{x2202}Y^{2}}\right)\left(\frac{\unicode[STIX]{x2202}V}{\unicode[STIX]{x2202}X}-\frac{\unicode[STIX]{x2202}U}{\unicode[STIX]{x2202}Y}\right)=0;\end{eqnarray}$$

(ii) conditions at $Y=0$ , including impermeability and Marangoni forcing at the free surface,

(6.26a,b ) $$\begin{eqnarray}V=0,\quad \frac{\unicode[STIX]{x2202}U}{\unicode[STIX]{x2202}Y}=\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6E9}}{\unicode[STIX]{x2202}X}\quad \text{for}~|X|<\unicode[STIX]{x1D6E5},\end{eqnarray}$$

as well as impermeability and no-slip conditions at the solid patches,

(6.27) $$\begin{eqnarray}U=V=0\quad \text{for}~\unicode[STIX]{x1D6E5}<|X|<1;\end{eqnarray}$$

(iii) the periodicity conditions,

(6.28) $$\begin{eqnarray}U=\frac{\unicode[STIX]{x2202}V}{\unicode[STIX]{x2202}X}=0\quad \text{at}~X=\pm 1;\end{eqnarray}$$

and (iv) large- $Y$ attenuation,

(6.29) $$\begin{eqnarray}U,V\rightarrow 0\quad \text{as}~Y\rightarrow \infty .\end{eqnarray}$$

Note that the derivative $\unicode[STIX]{x2202}\unicode[STIX]{x1D6E9}/\unicode[STIX]{x2202}X$ appearing in the Marangoni condition (6.26b ) is understood to be evaluated at $\unicode[STIX]{x1D702}=0$ (since $\unicode[STIX]{x1D6E9}$ is a function of  $\unicode[STIX]{x1D702}$ , rather than  $Y$ ).

The flow and excess-temperature problems are mutually coupled through both (6.21) and (6.26b ). This coupling involves the distributions of the pertinent fields only at the respective boundaries $Y=0$ and $\unicode[STIX]{x1D702}=0$ . Unfortunately, the inherent nonlinearity in (6.21) appears to thwart the possibility of a closed-form solution. This nonlinearity also implies a complicated dependence of both the excess temperature and cross-sectional flow upon the Marangoni number.