## Abstract

The flow of a viscous liquid layer over an inclined uneven wall heated from below is considered. The flow is assumed to occur at zero Reynolds number and the thermal Péclet number is taken to be sufficiently small that the temperature field inside the layer is governed by Laplace’s equation. With a prescribed wall temperature distribution and Newton’s Law of cooling imposed at the layer surface, the emphasis is placed on describing the surface profile of the liquid layer and, in particular, on studying how this is affected by wall heating. A linearized theory, valid when the amplitude of the wall topography is small, is derived and this is complemented by some nonlinear results computed using the boundary element method. It is shown that for flow over a sinusoidally shaped wall the liquid layer can be completely flattened by differential wall heating. For flow over a flat wall with a downwards step, it is demonstrated how the capillary ridge that has been identified by previous workers may be eliminated by suitable localized wall cooling in the vicinity of the step.

## 1. Introduction

A liquid film flowing steadily over an uneven surface deforms in response to any surface topography. Many practical applications involve flows of this type and examples include film evaporators, falling film reactors and spin coating devices used in the manufacture of microelectronic components. Depending upon the particular application, it may be desirable either to ameliorate or to promote disruptions to the film surface. For instance, in cooling applications wall corrugations may be used to enhance heat transfer across a film by inducing ripples and thereby increasing the surface area for heat transport. By way of contrast, heat is removed from microelectronic circuits by passing a film of water over a watertight plate mounted on top of the circuit board containing the hot computer chips. In film coating technologies, a liquid film is deposited onto a target surface, which may be irregular by design or owing to natural imperfections, with the aim of coating the surface as evenly as possible.

Controlling mechanisms which mitigate or magnify surface waves, ridges or troughs, are therefore of considerable practical interest, and a number of different mechanisms have been proposed to do this, including the use of insoluble surfactants (Pozrikidis 2003) or electric fields (Tseluiko *et al.* 2008). Here our concern is with the use of thermocapillary forces. These are established as a result of surface tension gradients which arise in direct response to a non-uniform temperature field inside the film generated by cooling or heating the wall appropriately. The role of thermocapillarity in film dynamics has been reviewed by Oron *et al.* (1997) and, more recently, by Craster & Matar (2009).

Numerous studies have addressed the thermal effects in liquid film flow. For flow down an inclined plane, various authors have used a number of different theoretical approaches in order to analyse the flow under different heating scenarios. Goussis & Kelly (1991) performed the seminal study of the stability of film flow down a uniformly heated plane wall. Kalliadasis *et al.* (2003*a*) examined thin film flow over a flat wall with a localized heat source using the integrated boundary-layer approximation while Kalliadasis *et al.* (2003*b*) and Trevelyan & Kalliadasis (2004) adopted the Schadov equations to investigate flow down a uniformly heated wall. Ruyer-Quil *et al.* (2005) and Scheid *et al.* (2005) appealed to boundary-layer-type approximations to model flow down an evenly heated wall. Trevelyan *et al.* (2007) also studied flow down a uniformly heated wall but applied a mixed heat flux boundary condition at the wall. Gatapova & Kabov (2008) studied film flow with local wall heating combined with the effect of a shearing gas flow above the film. It seems that the first work to examine the combined influence of bottom topography and thermal effects on film flow is that of Saprykin *et al.* (2007) who considered thermally induced flow in a liquid film coating a uniformly heated corrugated substrate by invoking the long-wave approximation.

Saprykin *et al.* (2007) assumed the substrate to be horizontal so that the flow is induced by thermal effects alone. They computed the time-evolution of a heated film from a white-noise initial condition and predicted that over time the film accumulates into drops of fluid centred over the wall troughs, connected to each other by thin lobes of fluid passing over the wall crests. Subsequently, D’Alessio *et al.* (2010) allowed for a tilted substrate and thereby appear to be the first to study the combined features of gravity-driven flow, topographic effects and thermally induced Marangoni forcing. They found that thermocapillary effects can have either a stabilizing or a destabilizing effect on the flow. Experiments conducted for liquid films over topography have been conducted in the absence of heating by a number of workers, including Fernandez-Parent *et al.* (1998), Decré & Baret (2003), Vlachogiannis & Bontozoglou (2002) and Argyriadi *et al.* (2006) for film flow over a wall with rectangular indentations, and by Wierschem *et al.* (2003, 2005) for flow down a sinusoidally shaped wall. However, to the best of our knowledge no experiments have yet been performed for such a flow in combination with topographic heating.

In the present work, we examine the flow of a liquid film down a shaped substrate which is inclined at an angle to the horizontal. The wall temperature is assumed to be given and may be spatially dependent. At the film surface, we apply a standard cooling law which states simply that the heat flux is proportional to the temperature difference between the film surface and the ambient environment. Despite the temperature variation within the film, the viscosity and density of the fluid are assumed to be constant, the surface tension is allowed to vary with temperature according to a simple linear law and evaporative effects are deliberately excluded. The flow is supposed to occur at a Reynolds number that is sufficiently low that the linear equations of Stokes flow hold, and it is presumed that thermal diffusive effects dominate those of convection. As some justification of this assertion, consider the flow of a liquid film of glycerol oil of density *ρ*=1.26×10^{3} kg m^{−3}, viscosity *μ*=1.41 Pa s and thermal diffusivity *D*=9×10^{−8} m^{2} s^{−1}. The Reynolds number, based on the classical Nusselt surface speed of a film of uniform thickness *h* flowing down a plane inclined at angle *α* to the horizontal, is *R*=*ρU*_{s}*h*/*μ*. The Péclet number measuring the relative importance of thermal convection to thermal diffusion is *Pe*=*U*_{s}*h*/*D*. For a film thickness of *h*=0.1 mm and a wall angle of inclination *α*=10^{°}, we have *R*=6.8×10^{−7} and *Pe*=8.5×10^{−3}, in which case the assumption of Stokes flow is valid, and neglecting convection in favour of diffusion is eminently reasonable.

We shall consider flow over a number of different topographies, including walls with a step, a sinusoidal profile and with rectangular corrugations. After detailing the structure of the model in §2, we work on the assumption that the amplitude of the wall topography is small in comparison with the thickness of the film, and thereby develop a linear theory in §3 that predicts the deflection of the film surface to a first-order approximation. Some nonlinear deformations of the free surface are computed for larger amplitude topography in §4 using the boundary element method and our work concludes in §5 with a few closing remarks.

## 2. Problem formulation

We consider the steady flow of a liquid layer over wall topography in the presence of thermocapillary Marangoni effects. The transport of heat within the layer is assumed to occur solely by conduction so that the temperature *T** satisfies the Laplace equation ∇^{2}*T**=0 where the Cartesian coordinate axes (*x**,*y**) are oriented so that *x** denotes the distance down the inclined slope. The thermal boundary condition imposed at the wall *y**=*S**(*x**) is *T**=*θ**(*x**) where the precise form of the temperature profile *θ**(*x**) will be specified later. At the free surface, *y**=*f**(*x**), we adopt Newton’s Law of cooling,
2.1
(Saprykin *et al.* 2007) where is the thermal conductivity of the liquid, *α*_{h} is a heat transfer coefficient, is the ambient temperature above the layer, **n** is the unit normal to the free surface pointing into the liquid and *T*^{*}_{s}≡*T**(*y**=*f**) is the surface temperature.

The flow in the liquid is governed by the Stokes momentum equation and the continuity equation
2.2
where **u***=(*u**, *v**) is the fluid velocity, *p** is the pressure, *μ* and *ρ* are the viscosity and density of the fluid, respectively. In addition, where *g* is the acceleration due to gravity and *α* is the angle of inclination to the horizontal. The no-slip and no-penetration conditions require that **u***=**0** on the wall *y*=*S*(*x*). At the free surface, *y*=*f*(*x*), under the present conditions of steady flow the kinematic condition demands that **u***⋅**n**=0. A balance of dynamic stresses at the free surface requires that
2.3
where ** σ*** is the fluid stress tensor,

*κ** is the curvature, which is taken to be positive when the surface is concave downwards,

*p*

^{*}

_{a}is the ambient air pressure above the layer, and

*l** is arc length along the surface which increases in the direction of the unit tangent surface vector,

**t**. The surface tension,

*γ**, is assumed to depend on the prevailing temperature at the free surface through the relation 2.4 where

*β** is a constant and

*γ*

^{*}

_{a}is the surface tension when the layer surface is at the ambient temperature

*T*

^{*}

_{a}. We note that

*β** is expected to be positive for most fluids of interest (Saprykin

*et al.*2007), but negative values may occur for some alcohols (Dijkstra 1990). We note that in any case surface tension must be everywhere positive.

To prepare the ground for our analysis, we introduce dimensionless variables
2.5
where is a characteristic wall temperature. Following this, the importance of
2.6
becomes apparent: here *C* is a capillary number, *β* is a thermocapillary parameter and *κ*_{h} is an inverse Biot number.

According to (2.5), the dimensionless form of the dynamic stress condition (2.3) is
2.7
where is the dimensionless ambient pressure, and a prime denotes differentiation with respect to *x*. The dimensionless forms of (2.1) and (2.4) are
2.8
and on the wall *T*=*θ*_{w}*θ*(*x*). We note that the free-surface conditions in the problem consist of Newton’s Law of cooling, the kinematic condition and the stress balance equation. Note that although the governing equation (2.2) is linear, nonlinearity enters the problem through the boundary conditions (2.7) and (2.8), which are applied at the unknown location of the film surface so that the normal and tangent vectors **n** and **t** are themselves unknown.

When the wall is flat and held at the uniform temperature *θ*_{w}, the surface of the layer is simply *y*=1 so the flow adopts the classical Nusselt profile,
2.9
where *ψ*_{0} is a streamfunction defined in the usual way. The corresponding temperature profile is given by
2.10
where *λ*=(*θ*_{w}−1)/(1+*κ*_{h}) and this may be either positive or negative. Generally speaking, it is of most physical interest to take *λ*≥0, so that the wall temperature is greater than the ambient temperature above the liquid (Saprykin *et al.* 2007). Using (2.10), the corresponding surface tension profile for a flat wall is given by
2.11
Accordingly, in the basic state there is no Marangoni stress at the free surface, and the fluid motion is unaffected by the temperature field.

## 3. Flow over small amplitude topography

To examine flow over small amplitude topography, we assume that the wall is located at *y*=*S*(*x*)=*εs*(*x*), where *ε*≪1 and the form of the shape function *s*(*x*) is to be decided upon later. The solution for the stream function and pressure may be expanded by writing
3.1
the location of the free surface may be written
3.2
and the surface tension and temperature field may be expanded as
3.3
where *γ*_{0} and *T*_{0} are given by (2.11) and (2.10), respectively. Lastly, we assume that the wall temperature profile is expanded as
3.4
where *θ*_{1}(*x*) is a function that we shall prescribe.

Substituting the temperature expansion (3.3) into Laplace’s equation and retaining the first-order terms, we obtain
3.5
Using (2.10), the wall boundary condition *T*=*θ*_{w}*θ* becomes
3.6
on *y*=0, while at first order the cooling law (2.1) reduces to
3.7
at *y*=1. Eliminating the pressure from the momentum equation in the liquid, we obtain the biharmonic equation,
3.8
which is to be solved subject to the boundary conditions
3.9
at *y*=0 while the kinematic condition **u**⋅**n**=0 leads to
3.10
at *y*=1. In passing, it is interesting to note that the first-order flow problem feels a prescribed wall velocity, via the second condition in (3.9), which is dependent on the choice of the wall shape function, *s*(*x*).

The tangential and normal components of the dynamic stress condition (2.7) are
3.11
and
3.12
with both evaluated at *y*=1.

We proceed by taking the Fourier transform of (3.5)–(3.12) in the *x*-direction. The transform is defined by
3.13
and is generalized to encompass distributions such as the delta function. The transformed first-order temperature problem is
3.14
with
3.15
at *y*=0, and
3.16
at *y*=1. The Fourier transform of the fluid problem (3.8)–(3.12) is
3.17
subject to
3.18
at *y*=0, and
3.19
3.20
3.21
at *y*=1, where we have written . The general solutions for and may be cast as
3.22
and
3.23
and it is straightforward to determine the coefficients *C*_{j}, 1≤*j*≤4 and *B*_{1}, *B*_{2}. For our purposes, it will be sufficient to note that, once these coefficients are found, we obtain the transformed surface temperature
3.24
For the free-surface perturbation, we find
3.25
where
3.26
3.27
and
3.28
It will be useful to note that
3.29
where a bar denotes the complex conjugate. The first-order correction to the free-surface location in real space is obtained by inverting the transform,
3.30
while the first-order temperature correction, *T*_{1}, at the free surface is obtained similarly by inverting (3.24).

### (a) Flow over a small amplitude sinusoidal wall

To study flow over a sinusoidal wall with wavelength *L*, we set , where *m*=2*π*/*L* is the wall wavenumber. In Fourier *k*-space, this wall function becomes , where *δ*(*k*) is the standard Dirac delta function. Performing the inverse transform according to (3.30) and making use of property (3.29), we obtain the first-order correction to the free surface
3.31
In the absence of temperature effects, , we have computed the maximum value of *f*_{1} over one period, and the phase shift between the free surface and the wall for a sample case, and confirmed that the results coincide with the predictions of Pozrikidis (2003)—in particular, we have reproduced fig. 2 of that paper.

When the wall temperature is constant the integral in (3.31) vanishes. Thermal Marangoni effects then only come into play if *λ*≠0 and the leading order temperature profile in the layer is non-uniform (if *λ*=0 then there is no surface tension gradient at the free surface and the results are the same as those presented by Pozrikidis (2003)). The free-surface correction can now be written in the form , where *A*_{1} is the wave amplitude and *ν* is the phase shift between the free-surface profile and the wall profile. Figure 1 shows the dependence of the amplitude *A*_{1} and the phase shift *ν*/(*π*/2) for a sample set of parameter values. The results of Pozrikidis (2003) indicate that the free surface of a liquid layer flowing over a wavy wall will in general be shifted in phase from the wall profile. The present example demonstrates that by heating the wall to the correct level, it is possible to obtain a free-surface profile which is in phase with the wall. Heating the wall tends to reduce the amplitude of the free-surface deformation.

To quantify the heat transport from the layer, we define , the dimensionless net heat flux from one period of the deformed surface as a fraction of that found for a layer of uniform thickness on a flat wall, namely *κ*_{h}*λ*. We write
3.32
where *q*≡*κ*_{h}**n**⋅∇*T* is evaluated at the liquid surface. Then under the present linearized theory, we have *q*=*κ*_{h}*λ*+*εκ*_{h}*q*_{1}+*O*(*ε*^{2}), where for a sinusoidal wall held at a uniform temperature we find
3.33
and hence since the contribution to the net heat flux from *q*_{1} vanishes. It is of interest to proceed to the next order to examine whether the wall undulations effect a net increase or a net decrease in the heat transport relative to that found for a flat layer. In the interest of brevity, we omit the details of the next order calculation, which is a straightforward continuation of the procedure already described but which involves some very lengthy algebraic expressions. Particular results will be given in §4 and compared with the predictions of nonlinear calculations performed with the boundary-element method.

Next we consider the case when the wall is differentially heated so that the first-order correction to the wall temperature is sinusoidal with . In this case, the period of the temperature variations is the same as that of the wall. The first-order correction to the free-surface profile (3.31) is
3.34
This form suggests that it may be possible, by a judicious choice of parameters, to eliminate completely the free-surface disturbance, at least to first order. If, say, we fix all parameters but adjust the wall temperature scale *θ*_{w} and the wall shape/wall temperature phase shift *ϕ*, we find that the right-hand side of (3.34) can be made to vanish. A simple example is given by taking *θ*_{w}=1 in which case *λ*=0. The right-hand side of (3.34) then vanishes if
3.35
Selecting *m*=1 and *ϕ*=*π*/2, and taking *κ*_{h}=2, we find that irrespective of the angle of inclination, *α*. This prediction, which has been made on the basis of linear theory, will be confirmed in §4 by numerical solutions of the full equations. Evidently, the flattening of the free surface has been made possible by the contrivance of the Marangoni force at the free surface. In the present case, the Marangoni force in the direction of increasing arc length along the free surface is given at first order by
3.36
where we have made use of (3.35). So with the present choice of *ϕ*=*π*/2 the Marangoni force is exactly in phase with the wall shape. This surface force will tend to accelerate fluid in the free-surface peaks and decelerate fluid in the surface troughs, promoting the flattening of the free surface. Further discussion on this point will be given later.

### (b) Flow over a wall with small amplitude sawtooth protrusions

In this case, we consider a periodic wall with a sawtooth profile. One period of the wall of length *L* is described by the tent function
3.37
for which with zero average over one period. The complete wall function, *s*(*x*), is constructed by taking the periodic extension of *s*_{P}(*x*) in the obvious manner. First, we construct the Fourier series representation of *s*(*x*) and its transform. We obtain
3.38
where *m*=2*π*/*L* together with *a*_{0}=0 and *a*_{n}=((−1)^{n}−1)/(*nπ*)^{2} for *n*≠0.

Assuming a constant wall temperature, so that *θ*_{1}=0, performing the inverse transform and making use of the property (3.29), the first-order free-surface correction (3.30) becomes
3.39
We note that since *F*_{s}(*k*)∼*O*(e^{−k}) as the infinite sum in (3.39) may be curtailed after only a few terms for computational purposes. Using the fact that *F*_{s}(*nm*)→1 as *m*→0, we may show that *f*_{1}(*x*)→*s*(*x*) uniformly in the limit and so the first-order correction precisely coincides with the wall shape function in the limit of large wall period. Figure 2 shows how the amplitude of the first-order free-surface correction *f*_{1}(*x*) varies with wall frequency for sample values of the flow parameters. One point of interest is that, for a fixed value of *m*, the amplitude varies non-monotonically as *β* increases.

### (c) Flow over a wall with a small step

For a flat wall with a small downwards step of size *ε*, we choose *s*(*x*)=−*H*(*x*), where *H*(*x*) is the Heaviside function. The transformed wall function is then
3.40
and the first-order free-surface perturbation (3.30) becomes
3.41

Kalliadasis *et al.* (2000) computed the isothermal flow over a step topography. They showed that for thin film flow over a downwards step the film profile develops what is referred to as a capillary ridge which is manifest as a local bulge in the film thickness immediately prior to the step. Motivated by applications such as those discussed in §1, which require as smooth a film profile as possible, we investigate whether the ridge can be removed by localized heating of the wall around the step. To this end, we adopt a Gaussian form for the wall temperature distribution and its partner transform,
3.42
where *x*_{0}, *G* and *μ* are constants. Alternatively, and following Gramlich *et al.* (2002), we might model local heating using the top-hat function,
which is non-zero and of constant amplitude *G* over the range |*x*|≤*b*. In this case, the contribution to the first-order free-surface correction from the heating term in (3.41) is given by
3.43
To serve as a benchmark, figure 3*a* illustrates the first-order free-surface correction *f*_{1}(*x*) for both a Gaussian (with *μ*=0.01) and for a top-hat wall temperature distribution with semi-width fixed at so that the integral of the temperature profile over the *x*-line is the same in both cases. This corresponds to a relatively wide temperature distribution. The dashed line in figure 3*a* represents the free-surface deflection in response to the top-hat temperature distribution. This is broadly in line with the qualitative prediction of Gramlich *et al.* (2002) (see their figure 1*b* and the accompanying discussion) in that the step-up in temperature at *x*=−*b* induces a rise in the free-surface level, seen as the most pronounced peak in the dashed line. The step-down in temperature at *x*=*b* induces a dip in the free-surface level, seen as the clearest trough in the dashed line. The solid line, representing the response to the Gaussian temperature distribution, has similar features but is altogether smoother. As *μ* decreases, with , so the temperature change occurs over an increasingly short region, the solid and dashed lines tend to merge—for the parameter set used in figure 3*a*, they are almost identical when *μ*=0.01—moreover, the free-surface profiles obtained closely resemble those found by Kalliadasis *et al.* (2003) and shown in their figure 2.

For non-small values of *μ*, the relative smoothness of the free-surface response to the Gaussian temperature distribution over the top-hat distribution makes it the more appealing choice, particularly if a smooth free surface is the desired outcome. Working on this assumption, we show in figure 3*b* an example of how the capillary ridge encountered at a downwards step can be smoothed by localized Gaussian heating. The dashed curve shows the free-surface correction obtained in the absence of heating, *G*=0. The characteristic capillary ridge is seen as a bulge in the layer thickness above the step at *x*=0. In the presence of heating centred at the step, *x*=0, it is possible to tune the parameters to achieve a more-or-less smooth free-surface profile in which the ridge has been essentially eliminated.

## 4. Nonlinear calculations

When the amplitude of the topography is not small, the free-surface deformation may be computed numerically using the boundary element method. First, working with dimensional variables, we reformulate the problem for the temperature field following the well-established protocol for the boundary integral method (Pozrikidis 1997). Assuming periodic flow over a periodic wall geometry, we obtain an integral equation of the second kind for the temperature field at a point located inside the layer. In particular,
4.1
where **x***=(*x**,*y**) and is the singly periodic Green’s function for Laplace’s equation (e.g. Pozrikidis 2002, p. 212),
4.2
in which *L* denotes the extent of one wall period along the *x**-axis. In equation (4.1), *W* and *F* represent one period of the wall geometry and the free surface, respectively, **n** is the unit normal pointing into the liquid and *l* is arc distance along *W* or *F* according to context. The *PV* designation on some of the integrals in (4.1) indicates that the principal values should be taken.

For the fluid motion, the formulation is identical to that discussed by Pozrikidis (2003). In brief, the fluid velocity is decomposed into two parts; one corresponding to the base flow for a liquid film flowing down a plane wall in the absence of thermocapillary effects, and the other corresponding to a disturbance flow **u**^{D} produced either by wall topography or by Marangoni forces generated by a thermally induced surface tension gradient. Standard procedures lead to an integral equation for the disturbance components. The integral equations for the temperature and the fluid velocity and stress are solved numerically using a boundary-element method in which we discretize *W* and *F* using a sequence of connected straight line elements, employing *N*_{W} elements on *W* and *N*_{F} elements on *F*. The number of elements *N*_{W}, *N*_{F} used in the results presented below are quoted in the figure captions and in the text where relevant. The unit normal and tangent vectors, and the curvature and surface tension gradient, are approximated by using a periodic cubic-spline interpolation through the surface nodes.

The unknowns are approximated over the elements with constant values and the equations solved using an iterative scheme. If we designate the connection points between neighbouring elements as nodes, we begin by guessing the height above the surface nodes above the wall. To specify the volume of fluid in one period, we also fix the height of the first node throughout the calculation. Since the normal component of the free-surface traction is defined to within an arbitrary and dynamically unimportant constant, the solution to the linear system is not unique. To remove this problem, we arbitrarily discard one of the linear equations on *F* and prescribe the value of the normal disturbance stress on the first free-surface element in the period. The integral equation for the temperature field is solved first to yield estimates for *T** on the elements, and the integral equations for the fluid motion is then solved to produce estimates for the fluid velocity at the mid-points of the elements. The procedure is repeated iteratively by adjusting the *y**-location of the surface element nodes until the free-surface stress boundary condition is satisfied at the mid-point of each element to within a prescribed tolerance. Our numerical scheme was validated in the absence of thermal effects, *β*=0, by confirming agreement with the results of Pozrikidis (1988) for flow over a periodic wall. Successful comparison with the linear theory is presented in the next section. The computations presented below required from a few seconds to several minutes of CPU time on an iMac desktop computer with a 3.3 GHz Intel processor.

In describing our results, we make scales dimensionless following the prescription outlined in §2. In particular, we will make reference to the set of dimensionless parameters listed at the end of that section. When interpreting results, it is helpful to keep in mind that thermocapillary effects are present provided that (i) *θ*_{w}≠1, so that there is a temperature contrast between the wall and the ambient environment above the layer and (ii) *β*≠0, so that a surface tension gradient, and hence a Marangoni force, is present at the liquid surface. When discussing flow over periodic topography, for each calculation, we fix the fluid volume *V* (per unit length in the transverse direction) over one period of length *L* and use *h*=*V*/*L* as the characteristic length scale and report values of the capillary number using this length scale in the definition (2.6). Alternatively, this may be thought of as fixing the mean layer thickness over a period. In the calculations, the number of boundary elements was varied so as to produce reliable results and, unsurprisingly, more elements were required to obtain accurate solutions with highly deformed surface profiles.

Considering first flow over a sinusoidal wall, the dimensionless free-surface profiles in figure 4*a* for a typical example demonstrate that the nonlinear calculations agree well with the linear theory of §3*a*. The circular and triangular symbols in the figure represent the results of boundary element calculations for a sinusoidal wall of dimensionless amplitude 0.1 and for a sample set of parameter values. The solid and dashed lines correspond to the predictions of linear theory. A typical nonlinear calculation is shown in figure 4*b*. Evidently, the thermocapillary forces active at the surface of the layer act to shift the surface profile in the downstream direction. For the small value of the capillary number under consideration, *C*=0.1, the shift is only slight, even for flow over a large amplitude wall. Larger deflections are obtained for larger values of the capillary number, as will be demonstrated later.

Turning to the heat transport, we computed values of , defined in (3.32) for the sample case of flow over a sinusoidal wall of dimensionless amplitude 0.1 with *L*/*h*=2*π*, *θ*_{w}=2, *α*=*π*/4, *β*=1, *C*=0.4 and *κ*_{h}=1.0. From the linear theory (using *ε*=0.1), we obtain the value and by way of a boundary-element calculation (with *N*_{W}=*N*_{F}=81) we find , demonstrating excellent agreement between the two and, furthermore, confirming that the heat transport occurs at second order for small surface deflections. In all cases we examined, the net effect of the wall corrugations is to increase the heat flux from the layer.

Figure 5 shows results for a sawtooth wall, one period of which is described by *As*_{P}, where *s*_{P} is given in (3.37). The figure confirms the predictions of the linear theory, shown with dashed and solid lines in figure 5*a*, by comparison with the results of boundary element calculations, denoted by the circular and triangular symbols. The surface tension profile induced by the temperature field inside the layer is illustrated in figure 5*b*. For this example, according to (2.11), we have *γ*_{0}=0.5 when *β*=1 and so the mean surface tension is lower than its unit value for isothermal flow. The maximum deviation from the mean value is about 4 per cent. The positive sign of the gradient d*γ*/d*x* at *x*=0 corresponds to a Marangoni force accelerating fluid downstream, and this accounts for the slight thinning of the layer here compared with the isothermal case. The opposite is true around *x*=*π* where d*γ*/d*x*>0 and the Marangoni force is pushing fluid upstream.

On increasing *β*, the base surface tension level decreases according to (2.11). The surface tension profile is qualitatively similar to that in figure 5*b* for larger values of *β* than those used in the computations for figures 4 and 5 (note that if *β* is increased beyond a certain value the surface tension becomes negative which is physically inadmissible). Consistent with this remark, further calculations showed that the free-surface profile is shifted further downstream as *β* increases; however, only minor shifts are observed beyond those seen in figures 4*b* and 5*b* before *β* is increased outside the physically relevant range.

We noted in §3*a* that differential heating of a sinusoidal wall could be used to completely flatten the free surface, to within a second-order correction. The same flattening effect may be achieved even for a large amplitude wall. The results shown in figure 6 confirm the prediction of linear theory for a small amplitude wall, and illustrate flattening for a wall of dimensionless amplitude 0.5. The parameter values for figure 6*a* were chosen on the basis of the linear theory discussed in §3*a*. The parameter values in figure 6*b* were tuned to produce as flat a free surface as possible. Also shown in figure 6 are the streamline patterns for the disturbance velocity field **u**^{D} (figure 6*a*) and the total velocity field **u** (figure 6*b*). The vertical arrow in figure 6*a* indicates the direction of the disturbance flow. We have confirmed that the disturbance streamline pattern in figure 6*a* is consistent with the prediction of the linear theory. The disturbance flow tends to accelerate fluid just beneath the free surface over the first half-period, and it tends to decelerate fluid just beneath the free surface in the second half-period. These effects almost perfectly counterbalance the flow disturbance produced by the sinusoidal wall topography and leads to a free surface which is flat (to within imperceptible variations) in both cases. In the absence of heating, the disturbance flow is characterized by two counter-rotating cells (one in each half-period) similar to those seen in figure 6*a*, but the cells are skewed (so that the vertical axis of symmetry for each cell shown in figure 6*a* is rotated somewhat in the counter-clockwise direction). The thermal Marangoni force at the free surface is given by (3.36) to a linear approximation. The direction of the Marangoni forcing is indicated by the horizontal arrows above the free surface in figure 6*a*.

Sample calculations for periodic flow over a wall with periodic protrusions are shown in figure 7. The layer surface profiles are shown for walls with two different protrusion types, a rectangular bump and a semicircular bump, for both heated and isothermal flow. It is striking that the surface profiles for heated flow undergo a larger amplitude displacement than those for isothermal flow. If the same calculations are repeated at the lower capillary number *C*=0.1, then the isothermal and thermal surface profiles are small perturbations of each other, as in figures 4*b* and 5*b*. For both calculations, the average heat transport from the surface over one period represents an increase over that for a flat layer with the same fluid volume. Further calculations revealed that, in the case of a rectangular bump, the average relative heat flux is an increasing function of the bump width *wL* (with the bump height fixed so that the wall surface area in contact with the fluid remains the same) over the tested range 0.05≤*w*≤0.75.

Finally, we examine flow over a trench with a particular focus on the capillary ridge, identified by Kalliadasis *et al.* (2000), which appears above the downwards step under certain flow conditions. According to the linear theory presented in §3 for small amplitude topography, local cooling of the wall is effective in ironing out the ridge. We have conducted a number of boundary element computations to see if the smoothing of the ridge may be achieved when the height of the step is not small. In performing the calculations, we have used the formulation described above and computed flow over a periodic wall, each period of which is indented with a rectangular trench. By taking the wall period to be sufficiently large, for a fixed trench width, we are able to study what is effectively unidirectional Nusselt flow approaching an effectively isolated step or isolated trench. Figure 8 shows the result of a boundary element calculation for flow over a wall of large period with a trench whose depth is comparable to the thickness of the oncoming liquid layer. In the absence of heating, the layer profile exhibits the anticipated capillary ridge just above the downward step. The maximum height of the layer occurs at the top of the ridge which lies at the point (*x*,*y*)=(11,1.1) ahead of the downward step which is at *x*=14.8.

In the linear calculations, the ridge was smoothed by focusing a Gaussian wall cooling around the downwards step. The dashed line in figure 8 shows the change in the free-surface profile when the boundary element calculation is repeated under identical conditions but this time with the dimensionless Gaussian wall temperature profile
4.3
and *θ*_{w}=1 so that far upstream and downstream the wall temperature approaches the same value as the ambient temperature. According to (4.3), the wall is locally cooled in the vicinity of the downwards step. In order to interpret how the smoothing is effected, it is useful to refer to the long-wave model developed by Kalliadasis *et al.* (2000). These authors derived a model equation in which the topographic forcing appears as the third spatial derivative of the wall shape function (see their eqn 1.2*a*). Approximating the step down into the trench with a steep but smoothed-out, differentiable function to mimic the sharp right-angled corner, we see that the topography forces the fluid upstream just prior to the step, it accelerates the fluid downstream at the step itself and forces it upstream just downstream of the step. The upstream forcing prior to the step is responsible for a slowing of the fluid and the appearance of the capillary ridge. The thermocapillary forcing resulting from the wall heating (4.3) is evident from the surface tension profile shown in figure 8*b*. The peak in the profile occurs at the step, *x*=14.8. To the left of the step the Marangoni force, which is proportional to d*γ*/d*x*, accelerates the fluid upstream. This counteracts the deceleration effect present due to the topography and smoothes out the ridge. As has been kindly pointed out to us by a referee, conformal mapping methods have been used to study surface water waves over large amplitude topography (Nachbin 2003). The change of co-ordinates within the mapping tends to generate a smoothed topography profile.

## 5. Discussion

We have examined the gravity-driven flow of a liquid layer over topography when the underlying surface is heated in a prescribed manner. The motion is assumed to take place at zero Reynolds number, so that the flow can be described using the linear Stokes equations, and the thermal Péclet number is assumed to be so small that convection can be ignored in comparison to diffusion and the temperature field inside the layer described using Laplace’s equation.

First, we carried out a linearized analysis working on the basis that the amplitude of the wall topography is small in comparison to the thickness of the liquid layer and the wall heating is only slight. We have derived explicit expressions involving Fourier inversion integrals for the first-order correction to an undisturbed free surface for a number of different wall topographies, including a sinusoidal wall, a wall with a periodic sawtooth profile and flow over an otherwise flat wall with an isolated step. For a small amplitude sinusoidal profile, it has been shown previously that the surface of the layer is in general out of phase with the wall. Our results have demonstrated that the surface profile can be brought into phase with the wall if the wall is heated uniformly. This might have useful implications for coating applications if a wavy surface is to be coated with a layer which mimics the wall profile as closely as possible. Alternatively in such applications, it may be desirable to achieve a layer of uniform height over a corrugated substrate. We have shown that the surface profile of a layer flowing over a sinusoidal wall may be flattened, even for large amplitude corrugations, by heating the wall sinusoidally. Such a differential wall heating might possibly be achieved by placing local heaters at equally spaced distances beneath the wall and out of phase with the wall peaks.

For all of the topographies studied, the heat transport from the surface of the layer was found to increase when the surface profile is deformed. In the particular case of flow over a rectangular bump, the heat flux increases as the width of the bump increases. Such topographic profiles may be encountered in microelectronic cooling technologies (Krauss *et al.* 2006), where spray-jet devices are used to produce a thin liquid film over a plate mounted onto the circuit board (in some cases, electronic chips are themselves immersed in liquid). In some devices, rectangular-shapes fins may be mounted on a hermetic surface to increase the area for heat transport, creating a profile similar to that seen in figure 7*a*. In fact, in such cooling processes, the principal mechanism of heat loss is via evaporation, and this motivates reconsidering the cooling law applied at the liquid surface in the current work. This is left as a subject for future study.

Flow over sharp topography, such as a rectangular step or trench produces interesting features in the free-surface profile, and these have been well studied. In particular, flow over a downwards step induces in the surface profile the capillary ridge identified by Kalliadasis *et al.* (2000) seen as a localized bulge in the layer thickness as the liquid negotiates the turn down over the step. The ridge originates from the topographic forcing which works to force fluid upstream immediately prior to the step before helping to accelerate it over. The upstream forcing slows down the fluid locally and increases the layer thickness. We have shown how the ridge may be effectively removed, so that the liquid flows smoothly over the step with a near-monotonic decrease in height as it passes over, by locally cooling the wall in the region of the step. The thermocapillary force generated at the surface of the liquid is sufficient to counteract the topographic forcing and smooth out the surface profile.

## Acknowledgements

The referees are thanked for a number of helpful comments that led to an improved presentation of the work.

- Received July 9, 2012.
- Accepted September 3, 2012.

- This journal is © 2012 The Royal Society