2.1. Dimensional formulation

We consider a two-dimensional flow of a liquid of constant density $\rho$ , viscosity $\mu$ and thermal diffusivity $\alpha$ in a microchannel where the top wall is smooth and horizontally mobile, has a time-dependent horizontal (parallel to the $x^{\ast}$ coordinate) velocity $U^{\ast} (t^{\ast} )$ (and zero velocity parallel to the $y^{\ast}$ coordinate) and is separated from the (fixed) bottom ridged (superhydrophobic) surface by a distance $H^{\ast}$ , as per figure 2. (An asterisk indicates a dimensional variable.) The bottom surface is asymmetrically textured with alternating cold and hot ridges of length $L^{\ast}$ at temperatures $T^{\ast}_{c}$ and $T^{\ast}_{h}$ , respectively, separated by (flat) menisci. Due to the periodic nature of the problem, we focus on a $2D^{\ast}$ -long period window consisting of a single pair of ridges. The cold ridge begins at $x^{\ast}=-D^{\ast}$ and the length of the meniscus between the cold and hot ridges is $S^{\ast}$ .

Figure 2.

Schematic of a period window of the domain for (a) the dimensional problem, and (b) the dimensionless problem.

In general, the fluid pressure and velocity fields are governed by mass conservation and the Navier-Stokes equations and its temperature field by the thermal energy equation (see § 2.3). We define $\mathbf {u}^{\ast}=(u^{\ast},v^{\ast})$ as the velocity vector with components in the $x^{\ast}$ and $y^{\ast}$ directions, $p^{\ast}$ as pressure and $T^{\ast}$ as temperature. As shown below, the flow problem is one-way coupled to the temperature problem by stress balances along menisci.

The top wall is a no-slip, adiabatic surface whose velocity is coupled to the velocity field in the liquid per Newton’s second law via the following ODE:

(2.1) \begin{equation} m^{\ast}\frac {\mathrm{d}U^{\ast}}{\mathrm{d}t^{\ast}}=-\frac {1}{2D^{\ast}}\int _{-D^{\ast}}^{D^{\ast}}\mu \left .\frac {\partial u^{\ast}}{\partial y^{\ast}}\right |_{y^{\ast}=H^{\ast}}\mathrm{d}x^{\ast}-F^{\ast}_{load}, \end{equation}

where $m^{\ast}$ is the mass (per unit area) of the top wall and $F^{\ast}_{load}$ is the external load (per unit area) on the top wall. Two flat and adiabatic menisci span between the hot and cold ridges. A Marangoni stress balance along them equates the (spatial) gradient of surface tension to the viscous shear stress as per

(2.2) \begin{equation} \left .\mu \frac {\partial u^{\ast}}{\partial y^{\ast}}\right |_{y^{\ast}=H^{\ast}} = \beta \frac {\partial T^{\ast}}{\partial x^{\ast}}, \end{equation}

where $\beta =- \mathrm{d}\sigma / \mathrm{d}T^{\ast}$ is a positive constant that captures the temperature dependence of the liquid’s surface tension $\sigma$ , i.e. $\sigma = \sigma _0-\beta (T^{\ast} - T_0^{\ast})$ , where $\sigma _0$ and $T_0^{\ast}$ are inconsequential reference values. The final conditions are that the velocity, pressure and temperature fields are periodic in the streamwise direction.

It is useful to define three engineering quantities that can be calculated from the solution to the model problem. First, the output power per unit wall area is

(2.3) \begin{equation} \overline {P}^{\ast} = \frac {1}{2D^{\ast}}\int _{-D^{\ast}}^{D^{\ast}} \mu \frac {\partial u^{\ast}}{\partial y^{\ast}} U^{\ast}\ \mathrm{d}x, \end{equation}

i.e. the period-averaged product of the hydrodynamic friction force and wall speed. Secondly, the input power per unit wall area $\overline {Q}^{\ast}$ is determined by integrating the heat flux through the hot ridge and dividing by the period length as per

(2.4) \begin{equation} \overline {Q}^{\ast}= -\frac {1}{2D^{\ast}}\int _{{hot\ ridge}} k \frac {\partial T^{\ast}}{\partial y^{\ast}}\ \mathrm{d}x, \end{equation}

where $k$ is the thermal conductivity of the liquid. Finally, the device efficiency is defined as

(2.5) \begin{equation} E=\overline {P}^{\ast}/\overline {Q}^{\ast}. \end{equation}

2.2. Discussion of forces and flow direction

With the geometry and flow-temperature problem defined, we return to discussion of the flow direction physics and forces in figure 2. First, it is useful to consider the physics when the load force $F_{load}^{\ast}$ is absent. The hot and cold parts of the boundary at the bottom of the domain result in thermocapillary stresses that induce fluid motion. Because of the negative relationship between surface tension and increasing temperature, thermocapillary stresses cause the cold and hot ridges to attract and repel the liquid, respectively. Therefore, a rather complicated flow develops where liquid moves to the left along the meniscus of length $S^{\ast}$ and to the right along the other meniscus (of length $2D^{\ast}-2L^{\ast}-S^{\ast}$ ). These flows interact locally, generally forming 1 or 2 vortices near the bottom (superhydrophobic) surface, but, globally, a net velocity is observed approximately 1 pitch away from this surface. In turn, due to no slip and no load, the top wall moves with constant velocity. The direction of this velocity is determined by the longer meniscus so that if $2D^{\ast}-2L^{\ast}-S^{\ast}\gt S^{\ast}$ or $L^{\ast}+S^{\ast}\lt D^{\ast}$ , as seen in figure 2(a), flow moves left-to-right and when $2D^{\ast}-2L^{\ast}-S^{\ast}\lt S^{\ast}$ flow moves right-to-left.

Inclusion of the load further complicates the physics. Of course, arbitrary assignment of $F^{\ast}_{load}$ yields a valid physical solution, but in the context of energy conversion we are particularly interested in situations where the external load is driven only by the flow generated by thermocapillary stresses. For this to be true (for a given geometry and temperature difference) two requirements of $F^{\ast}_{load}$ are needed. The first is that the drag generated by the thermocapillary stresses and the external load must have opposite signs to work against each other. The second is that $F^{\ast}_{load}$ must be less than or equal to in magnitude the largest possible drag force generated by the thermocapillary stresses, a quantity that can be determined exactly from solutions discussed later. This is because, if the applied external load exceeds this maximum force, the wall will move in the direction of the applied external load. In this case, the wall can be viewed as driving the liquid flow instead of the flow driving the wall. Then thermocapillary stresses may help to resist wall motion; however, the system can no longer be considered capable of providing power to external devices.

2.3. Non-dimensional formulation

It is natural to non-dimensionlise lengths with $H^{\ast}$ , velocities with a characteristic velocity scaling $U^{*}_{c}=\Delta T^{*}\beta /\mu$ , where $\Delta T^{\ast}=T^{\ast}_{h}-T^{\ast}_{c}$ , according to the shear stress balances along the menisci, and pressure with $\mu U_{c}^{*}/H^{*}$ . Since this problem is in a microfluidic configuration, the limiting temporal behaviour (i.e. the slowest time scale) is expected to be that of the upper wall. Therefore, time is rescaled with $m^{\ast}H^{\ast}/\mu$ , as follows from (2.1). The governing equations in the fluid (conservation of mass, momentum and thermal energy) become

(2.6) \begin{align} &\qquad\qquad\quad \nabla \cdot \mathbf {u} = 0, \end{align}

(2.7) \begin{align} &\textit {M}\frac {\partial \mathbf {u}}{\partial t}+\textit {Re}\left ( \mathbf {u}\cdot \nabla \mathbf {u}\right ) = -\nabla p + \nabla ^2 \mathbf {u}, \end{align}

(2.8) \begin{align} &\quad\textit{MPr}\frac {\partial T}{\partial t}+\textit{PrRe}\left (\mathbf {u}\cdot \nabla T\right )= \nabla ^2 T, \,\end{align}

where $\textit {Re}=\rho \Delta T^{\ast}\beta H^{\ast}/\mu ^2$ is the Reynolds number, $\textit {Pr}=\mu /(\alpha \rho )$ is the Prandtl number and $\textit {M}=(\rho H^{*})/m^{\ast}$ is a parameter that compares the mass of the liquid with that of the wall (per unit area in the $x$ and $z$ (out of plane) directions). The boundary conditions on the top wall are

(2.9) \begin{equation} \mathbf {u}=U(t), \quad \frac {\partial T}{\partial y} = 0\ \quad \ \text {at}\ y=1. \end{equation}

In the ODE governing the wall velocity (2.1) the load force (per unit area) is non-dimensionalised using $\mu U_{c}^{\ast}/H^{\ast}$ , yielding

(2.10) \begin{equation} \frac { \mathrm{d}U}{ \mathrm{d}t}=-\frac {1}{2\epsilon }\int _{-\epsilon }^\epsilon \left .\frac {\partial u}{\partial y}\right |_{y=1}\ \mathrm{d}x - F_{load}, \end{equation}

where $\epsilon =D^{\ast}/H^{\ast}$ is the aspect ratio of the domain. On the bottom superhydrophobic surface the boundary conditions are

(2.11) \begin{align} &\qquad\qquad\mathbf {u}=0,\ T=0 \quad \text {for } y=0,\ -\epsilon \lt x\lt -\epsilon (1-\delta ), \end{align}

(2.12) \begin{align} &\mathbf {u}=0,\ T=1 \quad \text {for }y=0,\ -\epsilon (1-\delta -\eta )\lt x\lt -\epsilon (1-2\delta -\eta ), \end{align}

(2.13) \begin{align} &\frac {\partial u}{\partial y}=\frac {\partial T}{\partial x},\ \frac {\partial T}{\partial y}=0 \quad \text {for }y=0,\ -\epsilon (1-\delta )\lt x\lt -\epsilon (1-\delta -\eta ), \end{align}

(2.14) \begin{align} &\qquad\frac {\partial u}{\partial y}=\frac {\partial T}{\partial x},\ \frac {\partial T}{\partial y}=0 \quad \text {for }y=0,\ -\epsilon (1-2\delta -\eta )\lt x\lt \epsilon , \end{align}

where $\delta =L^{\ast}/D^{\ast}$ is the solid fraction of the hot or cold ridges and $\eta =S^{\ast}/D^{\ast}$ is the fraction of a half-period attributable to the leftmost meniscus. In addition, periodicity is enforced for the velocity, pressure and temperature fields at $x=\pm \epsilon$ .

Finally, $\overline {P}^{\ast}$ is rescaled using $\mu U_{c}^{*2}/H^{\ast}$ and $Q^{\ast}$ using $k\Delta T^{\ast}/H^{\ast}$ so that

(2.15) \begin{equation} \overline {P} = \frac {1}{2\epsilon }\int _{-\epsilon }^\epsilon \frac {\partial u}{\partial y}U\ \mathrm{d}x, \end{equation}

and

(2.16) \begin{equation} \overline {Q} = -\frac {1}{2\epsilon }\int _{\textit{hot\ ridge}}\frac {\partial T}{\partial y}\ \mathrm{d}x. \end{equation}

The efficiency can be written in terms of the dimensionless quantities as

(2.17) \begin{equation} E = \Omega \frac {\ \overline {P}\ }{\overline {Q}}, \end{equation}

where $\Omega = \beta ^2\Delta T^{\ast}/(\mu k)$ .

3.1. Small period limit

A further assumption is needed to facilitate the analysis: that the period length is much smaller than the channel height, an oft-valid assumption for microchannels textured with superhydrophobic surfaces. This limit, corresponding to $\epsilon \rightarrow 0$ , is the same one recently considered by Hodes et al. (Reference Hodes, Kane, Bazant and Kirk2023) for the conventional transverse ridge problem, in the absence of thermocapillary stress, when the ridges are no-slip, diabatic (isoflux) and equally spaced, and the microchannel is symmetrically textured. To make the location of boundary conditions (2.11)–(2.14) independent of $\epsilon$ , we rescale $x$ according to $X=x/\epsilon$ . The governing equations transform to

(3.3) \begin{equation} \frac {1}{\epsilon ^2}\frac {\partial ^2 T}{\partial X^2} + \frac {\partial ^2 T}{\partial y^2} =0\quad \text {and} \quad \frac {1}{\epsilon ^4}\frac {\partial ^4 \psi }{\partial X^4} + \frac {2}{\epsilon ^2}\frac {\partial ^4 \psi }{\partial X^2\partial y^2} + \frac {\partial ^4 \psi }{\partial y^4} =0. \end{equation}

3.1.1. Outer problem

In the limit as $\epsilon \rightarrow 0$ , the outer region corresponds to fixing our location in the channel and keeping $X,y=O(1)$ . It is typical at this point to expand the streamfunction in the small parameter $\epsilon$ and determine a unique solution at each order of this parameter. However, solutions of the Laplace and biharmonic equations that exhibit small-scale periodicity (that is, not simply uniform) in $X$ will exhibit exponential variation in $y$ . To match with an inner solution near $y=0$ would necessitate this variation be exponentially small. As we are not concerned with these exponentially small orders, we only consider the solutions constant in $X$ , and posit the streamfunction expansion

(3.4) \begin{equation} \psi = \psi _{out}(y,\epsilon )+\text {e.s.t.}, \end{equation}

where e.s.t. refers to exponentially small terms. Substituting (3.4) into (3.3) this satisfies

(3.5) \begin{equation} \frac { \mathrm{d}^4 \psi _{out}}{ \mathrm{d} y^4} =0. \end{equation}

Furthermore, if we integrate the $x$ -momentum equation and apply periodicity of pressure (i.e. there is no linear component), then $ \mathrm{d}^3\psi _{out}/ \mathrm{d}y^3 = 0$ . Thus integrating this, applying the no-slip condition at the top ( $ \mathrm{d} \psi _{out}/ \mathrm{d} \rm y=U$ at $y=1$ ), and choosing $\psi _{out} = 0$ at $y=0$ , we find that

(3.6) \begin{equation} \psi _{out} = \frac {C(\epsilon )}{2}(2y-y^2)+Uy, \end{equation}

where $C(\epsilon )$ is an $\epsilon$ -dependent constant. A similar assumption for the temperature, $T = T_{out}(y,\epsilon ) + \text {e.s.t.}$ , leads to an outer solution independent of $y$ , that is, $T=D(\epsilon )+\text {e.s.t}$ , where $D(\epsilon )$ is, again an $\epsilon$ -dependent constant.

3.1.2. Inner problem

Near the bottom superhydrophobic surface, the flow and thermal problems are two-dimensional and require a different solution. Here, $x,y\sim \epsilon$ and all the terms in the Stokes equation balance. It is therefore natural to use inner variables $X=x/\epsilon$ and $Y=y/\epsilon = O(1)$ . Writing $\psi =\Psi (X,Y,\epsilon ) + \mbox {e.s.t}$ and $T=\Theta (X,Y,\epsilon ) + \mbox {e.s.t}$ , the inner problem for the temperature $\Theta$ is given by

(3.7) \begin{align} &\frac {\partial ^2 \Theta }{\partial X^2}+\frac {\partial ^2 \Theta }{\partial Y^2}=0, \quad Y\gt 0,\ -1\lt X\lt 1, \end{align}

(3.8) \begin{align} &\qquad\frac {\partial \Theta }{\partial Y} = 0,\quad \ Y=0,\ \text {on menisci}, \end{align}

(3.9) \begin{align} &\qquad \Theta = 1,\quad Y=0,\ \text {on hot ridge}, \end{align}

(3.10) \begin{align} &\qquad \Theta = 0,\quad Y=0,\ \text {on cold ridge}, \\[6pt] \nonumber \end{align}

with periodicity at $X=\pm 1$ , and the matching condition with the constant solution in the outer region, or, alternatively, the adiabatic matching condition

(3.11) \begin{equation} \frac {\partial \Theta }{\partial Y}\rightarrow 0,\quad Y\rightarrow \infty . \end{equation}

Given $\Theta$ , then the inner problem for $\Psi$ becomes

(3.12) \begin{align} & \frac {\partial ^4 \Psi }{\partial X^4}+2\frac {\partial ^4 \Psi }{\partial X^2Y^2}+\frac {\partial ^4 \Psi }{\partial Y^4}=0, \quad Y\gt 0,\ -1\lt X\lt 1, \end{align}

(3.13) \begin{align} &\quad\qquad \frac {\partial ^2 \Psi }{\partial Y^2} = \epsilon \frac {\partial \Theta }{\partial X},\quad \ Y=0,\ \text {on interfaces} \end{align}

(3.14) \begin{align} &\qquad \quad\qquad \frac {\partial \Psi }{\partial Y} = 0,\quad \ Y=0,\ \text {on ridges} \end{align}

(3.15) \begin{align} &\qquad\quad \qquad \Psi =0, \quad Y=0,\ -1\lt X\lt 1 \\[6pt] \nonumber \end{align}

with periodic conditions at $X=\pm 1$ . The final condition on $\Psi$ comes from matching, as $Y\to \infty$ , with the outer solution as $y\to 0$ as per

(3.16) \begin{equation} \frac {\partial \Psi }{\partial Y} \sim \epsilon U + \epsilon (1-\epsilon Y)C(\epsilon ),\quad Y\rightarrow \infty . \end{equation}

This problem has two inhomogeneous boundary conditions, the forcing due to temperature gradients along the interface and the matching condition as $Y\rightarrow \infty$ . Since it is linear in $\Psi$ , the solution can be decomposed into two parts

(3.17) \begin{equation} \Psi = \epsilon \left (\Psi _1 + \Psi _2\right ) ,\end{equation}

where $\Psi _1$ satisfies

(3.18) \begin{align} &\frac {\partial ^4 \Psi _1}{\partial X^4}+2\frac {\partial ^4 \Psi _1}{\partial X^2Y^2}+\frac {\partial ^4 \Psi _1}{\partial Y^4}=0, \quad Y\gt 0,\ -1\lt X\lt 1, \end{align}

(3.19) \begin{align} &\qquad \qquad\frac {\partial ^2 \Psi _1}{\partial Y^2} = \frac {\partial \Theta }{\partial X}\quad \ Y=0,\ \text {on interfaces}, \end{align}

(3.20) \begin{align} &\qquad \quad\qquad \frac {\partial \Psi _1}{\partial Y} = 0,\quad \ Y=0,\ \text {on ridges}, \end{align}

(3.21) \begin{align} &\qquad \quad\qquad\Psi _1 =0, \quad Y=0,\ -1\lt X\lt 1, \end{align}

(3.22) \begin{align} &\qquad \qquad\qquad\frac {\partial \Psi _1}{\partial Y} \sim U_{pump},\quad Y\rightarrow \infty , \end{align}

where $U_{pump}$ is a constant to be found, and $\Psi _2$ satisfies

(3.23) \begin{align} &\frac {\partial ^4 \Psi _2}{\partial X^4}+2\frac {\partial ^4 \Psi _2}{\partial X^2Y^2}+\frac {\partial ^4 \Psi _2}{\partial Y^4}=0, \quad -1\lt X\lt 1,0\lt Y \end{align}

(3.24) \begin{align} &\qquad \qquad\frac {\partial ^2 \Psi _2}{\partial Y^2} = 0,\quad \text {on interfaces},Y=0 \end{align}

(3.25) \begin{align} &\qquad \quad\qquad\frac {\partial \Psi _2}{\partial Y} = 0,\quad \text {on ridges},Y=0 \end{align}

(3.26) \begin{align} &\qquad \quad\qquad\Psi _2 =0, \quad -1\lt X\lt 1,Y=0 \end{align}

(3.27) \begin{align} &\,\,\frac {\partial \Psi _2}{\partial Y} \sim (U-U_{pump}) + (1-\epsilon Y)C(\epsilon ),\quad Y\rightarrow \infty , \end{align}

both with periodic conditions at $X=\pm 1$ . We remark that $U_{{pump}}$ can be interpreted physically as the far-field pump velocity when the upper wall is infinitely far away (see Crowdy et al. Reference Crowdy, Mayer and Hodes2023).

Figure 4.

Conformal mapping from an upper half-annulus $\rho \lt |\zeta | \lt 1, \textrm {Im}[\zeta ]\gt 0$ in a parametric $\zeta$ plane to a single period window of the inner problem.

The problem for $\Psi _1$ turns out to be exactly that already considered by Crowdy et al. (Reference Crowdy, Mayer and Hodes2023), meaning that the results there can be imported for reuse here. Defining a complex variable $Z=X+\textrm {i}Y$ , then $\Theta$ and $\Psi _1$ can be written in terms of a parametric conformal mapping variable $\zeta$ taking values in an upper half-annulus $\rho \lt |\zeta | \lt 1, \textrm {Im}[\zeta ]\gt 0$ . A correspondence between $\zeta$ and the physical variable $Z$ , as depicted in figure 4 using colour coding to reflect correspondences, is furnished by the following conformal mapping function:

(3.28) \begin{equation} Z=\mathcal {Z}(\zeta )=-\frac {\textrm { i}}{\pi }\log \left [R\frac {P(\zeta /\alpha ,\rho )P(\alpha \zeta ,\rho )}{P(\zeta /\overline {\alpha },\rho )P(\overline {\alpha }\zeta ,\rho )}\right ], \end{equation}

where

(3.29) \begin{equation} R=\exp (-\textrm {i}\pi )\frac {P(-1/\overline {\alpha },\rho )P(-\overline {\alpha },\rho )}{P(-1/\alpha ,\rho )P(-\alpha ,\rho )}. \end{equation}

This differs from that of Crowdy et al. (Reference Crowdy, Mayer and Hodes2023) only in the choice of the constant $R$ , which is chosen here to impose that $\mathcal {Z}(-1)=-1$ . The special function $P(\zeta ,\rho )$ is defined by the infinite product

(3.30) \begin{equation} P(\zeta ,\rho ) \equiv (1-\zeta ) \prod _{n=1}^\infty (1-\rho ^{2n}\zeta )(1-\rho ^{2n}/\zeta ), \end{equation}

which is convergent for $0 \leqslant \rho \lt 1$ ; this function is closely related to the so-called prime function of the annulus $\rho \lt |\zeta |\lt 1$ (Crowdy Reference Crowdy2020). The point $\alpha =\textrm {i}r$ lies on the imaginary axis, this being necessary to ensure that the two ridges are of the same length. The upper half-unit circle, denoted by $C_0^+$ , and the upper half-circle $|\zeta |=\rho$ , denoted by $C_1^+$ , are mapped to the two menisci. The above formulae depend on two real parameters, $\rho$ and $r$ , which are determined by the geometry of the ridge–menisci arrangement in each period window. Indeed, two nonlinear algebraic expressions enable the calculation of $\rho$ and $r$ for specified values of $\delta$ and $\eta$ as per

(3.31) \begin{equation} \mathcal {Z}(-\rho )= - 1+\delta ,\quad \mathcal {Z}(\rho )=-1+\delta +\eta . \end{equation}

These are easily solved using Newton’s method.

With this correspondence in place, Crowdy et al. (Reference Crowdy, Mayer and Hodes2023) determined that the temperature field is given in terms of the parametric variable $\zeta$ by

(3.32) \begin{equation} \Theta = 1-\frac {{\mathrm{arg}}\left [\zeta \right ]}{\pi }, \end{equation}

and the associated streamfunction by

(3.33) \begin{equation} \Psi _1=\mbox {Im}\left [(\overline {Z}-Z)f_1(\zeta )\right ], \end{equation}

where

(3.34) \begin{equation} f_1(\zeta ) = \frac {\textrm {i}}{4}\left [\frac {1}{2}+\sum _{n=1}^\infty \frac {2}{1+\rho ^n}\left (\frac {1-(-1)^n}{(\pi n)^2}\right )\left (\zeta ^n+\frac {\rho ^n}{\zeta ^n}\right )\right ]. \end{equation}

Moreover, from a far-field analysis as $Y\rightarrow \infty$ of $\partial \Psi _1/\partial Y$ , the pumping speed is found to be

(3.35) \begin{equation} U_{{pump}} = \frac {2}{\pi ^2}\left [\sum _{m=1}^\infty \frac {(-1)^{m+1}}{(2m-1)^2(1+\rho ^{2m-1})}\left (r^{2m-1} { -} \left (\frac {\rho }{r}\right )^{2m-1}\right )\right ]. \end{equation}

Conveniently, the solution for $\Psi _2$ can also be found in terms of the same parametric variable $\zeta$ . The boundary value problem determining $\Psi _2$ is similar to that of a linear shear flow over a superhydrophobic surface as resolved in a separate study by Crowdy (Reference Crowdy2011), who generalised solutions due to Philip (Reference Philip1972a ) for transverse shear flow over a periodic superhydrophobic surfaces with a single meniscus per period. The analysis of Crowdy (Reference Crowdy2011) allows for any finite number of menisci per period of the surface, including the case of two unequal menisci which is precisely the geometry relevant in the present problem. To cast it into a form whereby the results of Crowdy (Reference Crowdy2011) can be imported for use here, we define $\Psi _2=-\epsilon C(\epsilon )\tilde {\Psi }_2$ so that

(3.36) \begin{equation} \frac {\partial \tilde {\Psi }_2}{\partial Y}\sim Y -\frac {1}{\epsilon }\left (1+\frac {U-U_{pump}}{C(\epsilon )}\right ). \end{equation}

From Crowdy (Reference Crowdy2011), the solution for $\tilde {\Psi }_2$ can be found with the same machinery as used for $\Psi _1$ and, in particular, the solution can also be written as a function of the same parametric variable $\zeta$ . It is

(3.37) \begin{equation} \tilde {\Psi }_2 = \mbox {Im}\left [(\overline {Z}-Z)g(\zeta )\right ], \quad \text {where} \ g(\zeta ) = \frac {1}{4\pi } \log \left [\frac {P(\zeta /\alpha ,\rho )P(\zeta \overline {\alpha },\rho )}{P(\zeta /\overline {\alpha },\rho )P(\zeta {\alpha },\rho )}\right ], \end{equation}

and, as $Y\rightarrow \infty$

(3.38) \begin{equation} \frac {\partial \tilde {\Psi }_2}{\partial Y} \sim Y+\lambda _\perp \quad \text { where }\quad \lambda _\perp = \frac {1}{\pi }\log \left |\frac {P(\alpha ^2,\rho )}{P(|\alpha |^2,\rho )}\right |, \end{equation}

has an interpretation as the hydrodynamic slip length of the superhydrophobic surface (Crowdy Reference Crowdy2011). Equating the two far-field conditions, (3.36) and (3.38), necessitates that

(3.39) \begin{equation} \lambda _\perp = -\frac {1}{\epsilon }\left (1+\frac {U-U_{pump}}{C(\epsilon )}\right ). \end{equation}

Therefore $C(\epsilon ) =(U_{{pump}}-U)/(1+\epsilon \lambda _\perp )$ and

(3.40) \begin{equation} \Psi \sim \epsilon \Psi _1 +(U-U_{pump})\frac {\epsilon ^2}{1+\epsilon \lambda _\perp }\tilde {\Psi }_2. \end{equation}

3.1.3. Shear stress and a composite solution

With $C(\epsilon )$ known, the outer solution (3.4) becomes

(3.41) \begin{equation} \psi _{out} = \frac {U_{pump}-U}{1+\epsilon \lambda _\perp }\left (y-\frac {1}{2}y^2\right ), \end{equation}

which directly yields the shear stress at the top wall ( $y=1$ ) as

(3.42) \begin{equation} \frac {\partial u}{\partial y} ={\partial ^2 \psi _{out} \over \partial y^2} = \frac {U-U_{pump}}{1+\epsilon \lambda _\perp }. \end{equation}

A composite solution is constructed by adding the outer and inner solutions and subtracting that in the overlap region. Fortunately, the outer solution is a power series in $\epsilon$ and $Y$ ( $\lambda _\perp$ is at most O(1)), so it retains its form through the overlap region if all orders are included, meaning it is precisely the overlap solution. Thus, the composite solution is simply the inner solution, written in terms of outer variables,

(3.43) \begin{equation} \psi _{comp}=\Psi (x/\epsilon ,y/\epsilon ,\epsilon ). \end{equation}

3.2. Wall dynamics and power

Inserting the outer asymptotic solution for $\partial u/\partial y = \partial ^2 \psi /\partial y^2$ into the ODE governing the wall motion (2.10) yields

(3.44) \begin{equation} \frac { \mathrm{d}U}{ \mathrm{d}t}=\frac {U_{pump}-U}{1+\epsilon \lambda _\perp } - F_{load}. \end{equation}

Until now, $F_{{load}}$ has been deliberately left ambiguous to highlight the generality of this ODE. However, we now focus on the case when $F_{{load}}$ is a constant.

When the initial condition is $U(0)=U_{{pump}}$ (the steady solution without load), then the solution applying constant load $F_{{load}}$ for $t\gt 0$ is

(3.45) \begin{equation} U=U_{pump}-F_{load}\left (1+\epsilon \lambda _\perp \right )\left [1-\exp \left (-{\frac {t}{1+\epsilon \lambda _\perp }}\right )\right ]. \end{equation}

This results corresponds to the application of a load when the top wall is already moving at velocity $U_{pump}$ and leads to some relevant physical parameters. First, we define a maximum load that can be supported by the thermocapillary pump by constraining $F_{load}$ . In particular, we impose that $F_{load}$ is in the opposite direction of the thermocapillary-induced pumping velocity $U_{pump}$ (thus acting as a resistance) and $|F_{load}|\leqslant |U_{pump}|/ (1+\epsilon \lambda _\perp )$ so that $U\gt 0$ in all cases. If there is equality, then $|F_{load}|= |U_{pump}|/(1+\epsilon \lambda _\perp )$ , and (3.45) tends to the steady-state $U=0$ as $t\to \infty$ . Hence, we refer to this as the maximum load that can be supported – if it is any larger, the wall will move against the pumping flow at steady state. Alternatively, $|U_{pump}|/(1+\epsilon \lambda _\perp )$ can be interpreted as the fluid force on an immobile (or pinned) wall.

We now consider power generation. Denoting the average shear stress from the fluid on the wall as $F_{{fluid}}$ and multiplying (3.44) by $U$ , conservation of energy for the wall takes the form

(3.46) \begin{align} \frac { \mathrm{d}}{ \mathrm{d}t}\left ( \frac {1}{2}U^2\right ) &= F_{fluid} U - F_{load}U = \overline {P} - \overline {P}_{load}. \end{align}

The term $\overline {P}$ is the (average) power from the thermocapillary flow delivered to the wall (defined earlier in (2.15)), and $\overline {P}_{load}$ is the mechanical power output from the wall to the load. (Note that no averaging of the asymptotic solution is actually needed, since the outer solution is uniform in $x$ .) During the transient, $\overline {P}$ and $\overline {P}_{load}$ will not be equal in general, as some energy is going to/coming from the changes in kinetic energy of the wall. But at steady state the powers must balance; therefore, $\overline {P}=\overline {P}_{load}=F_{load}U$ . Thus, $\overline {P}$ is the “steady output power”.

Inserting the asymptotic approximation for shear stress into the definition (2.15) of (instantaneous) fluid power gives

(3.47) \begin{equation} \overline {P} \sim \frac {U(U_{pump}-U)}{1+\epsilon \lambda _\perp },\ \epsilon \ll 1 . \end{equation}

Notably, this implies that the maximum fluid power will always occur when $U=U_{pump}/2$ , taking the value

(3.48) \begin{equation} \overline {P}_{max} \sim \frac {U_{pump}^2}{4(1+\epsilon \lambda _\perp )},\ \epsilon \ll 1 . \end{equation}

If at steady state (set (3.44) to zero and substitute $U=U_{pump}/2$ ), this requires that $F_{load}=U_{pump}/[2(1+\epsilon \lambda _\perp )]$ . Alternatively, since $\overline {P}=\overline {P}_{load}$ at steady state, one can substitute this $F_{load}$ expression into $F_{load}U$ and derive the same maximum power.

From an expression for the heat rate through the hot ridge from Crowdy et al. (Reference Crowdy, Mayer and Hodes2023) the thermal power input to the fluid from the hot ridge is given by

(3.49) \begin{equation} \overline {Q}=\frac {1}{2\pi \epsilon }\log \rho , \end{equation}

where $\rho$ is a geometric parameter related to $\delta$ and $\eta$ (but not $\epsilon$ ). It is notable that $\overline {P}$ is only weakly dependent on $\epsilon$ , but $\overline {Q}$ scales with $1/\epsilon$ . The second observation can be easily explained because, for a fixed channel height, making $\epsilon$ smaller is akin to making the period smaller. This brings the cold and hot ridges closer together; therefore, more heat is required to maintain the imposed temperature difference between the hot and cold ridges. Power only weakly depending on $\epsilon$ stems from the fact that thermocapillary pumping also depends weakly on $\epsilon$ . This is not the case for calculated slip lengths for shear flows over a mixed no-slip/no-shear surface, where the far-field velocity increment scales linearly with $\epsilon$ (Philip Reference Philip1972b ). The difference here is the inclusion of thermocapillary stress which itself scales like $1/\epsilon$ , growing as $\epsilon$ gets smaller and the ridges approach each other. This, combined with the fact that the menisci lengths scale with $\epsilon$ , yields an $O(1)$ quantity and a power output with only a weak dependence on $\epsilon$ . Finally, an expression for efficiency can be derived as

(3.50) \begin{equation} E \sim 2\pi \epsilon \Omega \frac {U(U_{pump}-U)}{\left (1+\epsilon \lambda _\perp \right )\log \rho }. \end{equation}