2 Statement of the problem and governing equations

We begin by considering a two-dimensional non-isothermal thin film of a Newtonian, incompressible liquid film with average thickness $h_{0}$ , density $\unicode[STIX]{x1D70C}$ , dynamic viscosity $\unicode[STIX]{x1D707}$ , kinematic viscosity $\unicode[STIX]{x1D708}=\unicode[STIX]{x1D707}/\unicode[STIX]{x1D70C}$ , thermal conductivity $\unicode[STIX]{x1D705}$ and thermal diffusivity $\unicode[STIX]{x1D712}$ . Neglecting gravity, fluid dynamics in the film domain is governed by

where $\boldsymbol{v}=(u,w)$ , $p$ and $\unicode[STIX]{x1D703}$ are the velocity, pressure and temperature fields, respectively. The spatial variables are $x$ and $z$ directed along and normally to the solid substrate supporting the film with $\boldsymbol{i}$ and $\boldsymbol{k}$ the respective unit vectors and $t$ is time.

At $z=0$ , the film is bounded by a substrate at which no-penetration and no-slip conditions enforce $\boldsymbol{v}=0$ and a spatially uniform temperature $\unicode[STIX]{x1D703}=\unicode[STIX]{x1D703}_{s}$ is maintained. The film thickness $h=h(x,t)$ , which may depend on both the spatial coordinate $x$ and time $t$ , is in thermal contact with a gas phase with spatially uniform temperature that undergoes a periodic excitation $\hat{\unicode[STIX]{x1D703}}_{g}\cos \unicode[STIX]{x1D714}t$ about a mean gas temperature $\bar{\unicode[STIX]{x1D703}}_{g}$ , so that $\tilde{\unicode[STIX]{x1D703}}_{g}(t)=\bar{\unicode[STIX]{x1D703}}_{g}+\hat{\unicode[STIX]{x1D703}}_{g}\cos \unicode[STIX]{x1D714}t$ . The dynamics of the liquid and gas phases are decoupled by assuming that the ratios between the liquid and gas phase densities, viscosities and thermal diffusivities are large. Hence, the normal and tangential stress balances that hold at the film interface $z$ = $h$ are simplified considerably, which are

respectively, with gas pressure $p_{g}$ , stress tensor $\mathbb{T}=-p\mathbb{I}+\unicode[STIX]{x1D707}(\unicode[STIX]{x1D735}\boldsymbol{v}+(\unicode[STIX]{x1D735}\boldsymbol{v})^{\text{T}})$ in the liquid phase, surface normal unit vector $\boldsymbol{n}=(-h_{x}\boldsymbol{i}+\boldsymbol{k})/(1+h_{x}^{2})^{1/2}$ , surface tangent unit vector $\boldsymbol{t}=(\boldsymbol{i}+h_{x}\boldsymbol{k})/(1+h_{x}^{2})^{1/2}$ and twice mean curvature $2{\mathcal{H}}=-\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{n}$ . The kinematic boundary condition at $z=h$ is,

The quadratic dependence of surface tension $\unicode[STIX]{x1D70E}$ on temperature is written as

where $\unicode[STIX]{x1D70E}_{0}$ is the surface tension, evaluated at $\unicode[STIX]{x1D703}=\unicode[STIX]{x1D703}_{0}$ . The parameter $\unicode[STIX]{x1D6FE}=(1/2)\text{d}^{2}\unicode[STIX]{x1D70E}/\text{d}\unicode[STIX]{x1D703}^{2}$ is positive for minima as in self-rewetting fluids. The problem is closed by assuming that heat transfer between the film and the gas at $z=h(x,t)$ follows Newton’s law of cooling, viz.,

where $q$ is the empirical rate of heat transfer between the interface and the gas phase.

Temperature variations along the film surface $z=h$ translate to tangential stresses via (2.2b ) which drive flow in the bulk domain. The nature of these stresses depend upon three temperature scales: the substrate heating/cooling $\unicode[STIX]{x1D703}_{s}-\bar{\unicode[STIX]{x1D703}}_{g}$ , the bias of $\unicode[STIX]{x1D703}_{s}$ with respect to $\unicode[STIX]{x1D703}_{0}$ and the amplitude of the gas temperature variation $\hat{\unicode[STIX]{x1D703}}_{g}$ . In the interest of focusing on the dynamics that results from the oscillatory modulation, we non-dimensionalize the equations using scales based on the magnitude of the thermocapillary stresses, $\unicode[STIX]{x1D6FE}\hat{\unicode[STIX]{x1D703}}_{g}^{2}$ . That is, velocity, length, time, gauge pressure with respect to $p_{g}$ and gauge temperature with respect to $\unicode[STIX]{x1D703}_{s}$ are rescaled with $\unicode[STIX]{x1D6FE}\hat{\unicode[STIX]{x1D703}}_{g}^{2}/\unicode[STIX]{x1D707}$ , $h_{0}$ , $h_{0}\unicode[STIX]{x1D707}/\unicode[STIX]{x1D6FE}\hat{\unicode[STIX]{x1D703}}_{g}^{2}$ , $\unicode[STIX]{x1D6FE}\hat{\unicode[STIX]{x1D703}}_{g}^{2}/h_{0}$ and $\hat{\unicode[STIX]{x1D703}}_{g}$ , respectively.

Also, in the limit that the average film thickness $h_{0}$ is much smaller than the typical disturbance wavelength in the longitudinal direction, $\ell$ , horizontal variations evolve much slower than those in the vertical, and the long-wave approximation can be applied (Oron, Davis & Bankoff Reference Oron, Davis and Bankoff1997). The approximation consists of making the following asymptotic rescaling: $X\sim \unicode[STIX]{x1D716}x$ , $T\sim \unicode[STIX]{x1D716}^{2}t$ , $U\sim \unicode[STIX]{x1D716}^{-1}u$ and $W\sim \unicode[STIX]{x1D716}^{-2}w$ , where $\displaystyle \unicode[STIX]{x1D716}=h_{0}/\ell \ll 1$ is a measure of the ratio of the mean film thickness $h_{0}$ to the characteristic horizontal length scale. The vertical coordinate $z$ , pressure $p$ , temperature $\unicode[STIX]{x1D703}$ and thickness $h$ remain $\mathit{O}(1)$ quantities with respect to $\unicode[STIX]{x1D716}.$

The long-wave analysis therefore begins by making the following substitutions into (2.1)–(2.5):

where the non-dimensional, long-wave variables are denoted by uppercase letters. Terms proportional to various powers of $\unicode[STIX]{x1D716}$ are collected, and our interest lies in the leading-order terms for each equation. From (2.1) we obtain, simply,

which are subject to boundary conditions $U=W=\unicode[STIX]{x1D6E9}=0$ at the substrate boundary $Z=0$ . The conditions at the film interface $Z=H$ are now which are written with several new dimensionless parameters. They are the Biot number $B$ , the substrate heating/cooling $\unicode[STIX]{x1D6E5}$ , dimensionless frequency $\unicode[STIX]{x1D6FA}^{\ast }$ , the bias $\unicode[STIX]{x1D6FF}$ and the crispation number $C$ , defined as

Observe $\unicode[STIX]{x1D6FA}^{\ast }$ and $C$ have been assumed to be order-one quantities in the limits that $\unicode[STIX]{x1D716}\rightarrow 0$ . Inclusion of $C$ at the lowest order of the long-wave expansion is necessary for the problem to be well posed, and the asymptotic requirement on $\unicode[STIX]{x1D6FA}^{\ast }$ ensures that dynamics induced by the excitation occurs on the long-wave time scale $T$ .

The domain equation for the temperature profile $\unicode[STIX]{x1D6E9}$ , (2.7d ), indicates that heat transfer within the film is strictly conductive at the given order of approximation, and admits linear temperature profiles that match the condition of $\unicode[STIX]{x1D6E9}=0$ at $Z=0$ to the interfacial heat flux condition (2.8a ). The solution is

If we consider the temperature at the surface, $\unicode[STIX]{x1D6E9}(Z=H)$ , and fixed $B$ , we see that the resulting coefficient $BH/(1+BH)$ varies with the local film thickness $H$ , and in turn leads to surface temperature variations for non-uniform $H$ . These variations generate interfacial Marangoni stresses that drive fluid flow via the tangential stress balance, (2.8b ), which evaluates to,

Because the vertical pressure gradient is equal to zero, equation (2.7b ), the horizontal pressure gradient appearing in (2.7a ) is independent of $Z$ and evaluated using the interfacial value specified by (2.8c ). The horizontal velocity $U$ and streamfunction $\unicode[STIX]{x1D713}$ may therefore be expressed as

and

respectively, where $\unicode[STIX]{x1D6F4}=-3C^{-1}H_{XXX}$ .

A nonlinear partial differential equation for the spatio-temporal evolution of $H$ is then obtained by integration of $U$ across the vertical coordinate $Z$ , which yields, via (2.8d ),

The explicit dependence of (2.14) on physical properties, as well as the dimensional control parameters, is obtained by substituting (2.6) and (2.9), yielding,

As expected, the surface tension $\unicode[STIX]{x1D70E}_{0}$ and the correction $\unicode[STIX]{x1D6FE}$ enter as coefficients to the stabilization and destabilization terms, respectively, and the viscosity $\unicode[STIX]{x1D707}$ acts to slow the growth/decay of any deflection. The fluid density $\unicode[STIX]{x1D70C}$ and gravity are notably absent from (2.15), which reminds us that effects due to buoyancy have been neglected. It is evident that, even in the case where $\unicode[STIX]{x1D703}_{s}=\unicode[STIX]{x1D703}_{0}$ , and $\unicode[STIX]{x1D703}_{s}=\bar{\unicode[STIX]{x1D703}}_{g}$ , or, $\unicode[STIX]{x1D6E5}=\unicode[STIX]{x1D6FF}=0$ , a time-averaged component arises from the third terms of (2.14) and (2.15) that may destabilize a flat film.

The system has been scaled so that the dimensionless parameter $C^{-1}$ is the ratio of the minimum surface tension $\unicode[STIX]{x1D70E}_{0}$ to the magnitude of the Marangoni stresses that are driven by the thermal modulation, i.e. $\unicode[STIX]{x1D6FE}{\hat{\unicode[STIX]{x1D703}}_{g}}^{2}$ . Therefore, generally, large/small values of $C^{-1}$ promote stabilization/destabilization of a flat film interface. However, the potential for destabilization is more intricately determined by the terms containing $B$ and the values of the parameters $\unicode[STIX]{x1D6E5}$ and $\unicode[STIX]{x1D6FF}$ . Finally, observe that (2.14) is invariant under the transformation $(\unicode[STIX]{x1D6E5},\unicode[STIX]{x1D6FF},\unicode[STIX]{x1D70F})\rightarrow (-\unicode[STIX]{x1D6E5},-\unicode[STIX]{x1D6FF},\unicode[STIX]{x1D70F}+\unicode[STIX]{x03C0}/\unicode[STIX]{x1D6FA}^{\ast })$ , which implies that the dynamics of a system during the positive/negative fluctuations of the excitation is identical to that during the negative/positive fluctuations, for systems in which the substrate heating/cooling and the bias of the mean substrate temperature are reversed.

Here, we introduce a normalization that reduces the number of independent parameters further. Rescaling time, the lateral coordinate, the film thickness and the excitation frequency according to

respectively, equation (2.14) becomes

and the corresponding streamfunction is

The Biot number and the dimensionless thickness have therefore been scaled out of the problem in favour of the parameter $\unicode[STIX]{x1D719}$ , which resembles a thermal thickness in the sense that it measures the relative effects of both the film thickness and the heat which is transferred between the film and the bounding gas. Because heat transfer is assumed to be fixed ( $q$ and $\unicode[STIX]{x1D705}$ are constant), $\unicode[STIX]{x1D719}$ may still be interpreted as a film profile thickness, and the effect of the Biot number in (2.17) is determined by considering different initial values of $\unicode[STIX]{x1D719}$ . Observe that the Biot number has also been absorbed into the definitions of $\unicode[STIX]{x1D709}$ , $\unicode[STIX]{x1D70F}$ and $\unicode[STIX]{x1D6FA}$ . Although this scaling complicates discussion of the model in dimensional parameters, the reduced dimensionless parameter space represents significant progress towards isolating the dynamic features of the model.

In the next two sections, the dynamics of (2.17) will be investigated on periodic domains of fundamental wavenumber $k$ . The problem therefore contains five independent dimensionless parameters: the mean film thickness $\unicode[STIX]{x1D6F7}$ , fundamental wavenumber $k$ , the bias of the substrate temperature $\unicode[STIX]{x1D6FF}$ , the substrate heating/cooling $\unicode[STIX]{x1D6E5}$ and the thermal wave frequency $\unicode[STIX]{x1D6FA}$ . Time will be measured in numbers of excitation periods $f\unicode[STIX]{x1D70F}$ , where $f=\unicode[STIX]{x1D6FA}/2\unicode[STIX]{x03C0}$ is the ordinary frequency. In this manner, the dynamics and physical processes that result from different values of $\unicode[STIX]{x1D6FA}$ are easily compared. In the next section, linear analysis is conducted for four separate cases of the thermal conditions: (i) $\unicode[STIX]{x1D6E5}=\unicode[STIX]{x1D6FF}=0$ , (ii) finite $\unicode[STIX]{x1D6FF}$ with $\unicode[STIX]{x1D6E5}=0$ , (iii) $\unicode[STIX]{x1D6FF}=0$ and finite $\unicode[STIX]{x1D6E5}$ and (iv) both finite $\unicode[STIX]{x1D6FF}$ and $\unicode[STIX]{x1D6E5}$ . Then, case (i), which considers modulation of the gas temperature about the mean film temperature, represents the primary scope of the numerical investigation of § 4. The main result that will be presented in this section is that the model admits saturated nonlinear flows that do not rupture in long time.

3 Linear stability analysis

We proceed with linear stability analysis of a laterally infinite, flat film of mean thermal thickness $\unicode[STIX]{x1D6F7}$ by considering the growth of perturbations to (2.17) with form $\unicode[STIX]{x1D719}=\unicode[STIX]{x1D6F7}+\hat{\unicode[STIX]{x1D719}}(\unicode[STIX]{x1D70F})\exp (ik\unicode[STIX]{x1D709})$ where the perturbation amplitude $\hat{\unicode[STIX]{x1D719}}\ll \unicode[STIX]{x1D6F7}$ and $k$ is its wavenumber. Note that since the flat interface corresponds to $H=1$ , in the rescaled version $\unicode[STIX]{x1D6F7}=B$ . Substitution yields the following equation for the evolution of the perturbation amplitude,

which may be integrated directly with the initial condition $\hat{\unicode[STIX]{x1D719}}(\unicode[STIX]{x1D70F}=0)=\hat{\unicode[STIX]{x1D719}}_{0}$ to obtain

where the time-averaged growth rate is

and

Figure 1.

Variation of (a) perturbation growth rates $\unicode[STIX]{x1D6FC}$ for an initial thickness $\unicode[STIX]{x1D6F7}=0.1$ with the wavenumber $k$ and (b) cutoff wavenumbers $k_{c}$ with $\unicode[STIX]{x1D6F7}$ . With the dash-dotted line indicating the $\unicode[STIX]{x1D6FC}=0$ level, the dashed line represents the case where $\unicode[STIX]{x1D6E5}=\unicode[STIX]{x1D6FF}=0$ and the solid lines each correspond to $\unicode[STIX]{x1D6E5}=1$ and $\unicode[STIX]{x1D6FF}=0.4$ , 0.2, 0, $-0.2$ and $-0.4$ (arrows indicate direction of increasing $\unicode[STIX]{x1D6FF}$ ).

Positive and negative values of $\unicode[STIX]{x1D6FC}$ naturally indicate instability and stability, respectively, and the cutoff wavenumber $k_{c}$ at which $\unicode[STIX]{x1D6FC}=0$ is

The time-averaged growth rates $\unicode[STIX]{x1D6FC}$ , for $\unicode[STIX]{x1D6F7}=0.1$ , are depicted in figure 1(a) and they exhibit a typical dependence on wavenumber, whereby stabilization due to surface tension and destabilization due to thermocapillary stresses enter as $k^{4}$ and $k^{2}$ terms, respectively. The parameters $\unicode[STIX]{x1D6E5}$ and $\unicode[STIX]{x1D6FF}$ clearly influence $\unicode[STIX]{x1D6FC}$ , as well as the cutoff wavelengths $k_{c}$ , which are presented as a function of the mean thermal thickness $\unicode[STIX]{x1D6F7}$ in figure 1(b). With $\unicode[STIX]{x1D6FF}=\unicode[STIX]{x1D6E5}=0$ , $k_{c}=2^{-1/2}$ at $\unicode[STIX]{x1D6F7}\rightarrow 0$ , and decays monotonically to zero due to diminished surface temperature variations at large $\unicode[STIX]{x1D6F7}$ . With $\unicode[STIX]{x1D6FF}=0$ , finite values of $\unicode[STIX]{x1D6E5}$ promote destabilization for all $k$ and $\unicode[STIX]{x1D6F7}$ . Stabilization, however, is promoted when $\unicode[STIX]{x1D6FF}$ has the opposite sign of $\unicode[STIX]{x1D6E5}$ , and small values of $\unicode[STIX]{x1D6F7}$ may even be stabilized completely, viz., $k_{c}$ for $\unicode[STIX]{x1D6FF}=-0.4$ and $-0.2$ in figure 1(b).

Figure 2.

Dependence of maximum growth rates $\unicode[STIX]{x1D6FC}_{m}$ on the initial mean film thickness $\unicode[STIX]{x1D6F7}$ : (a) $-\unicode[STIX]{x1D6FF}=0$ with $\unicode[STIX]{x1D6E5}=0$ , 0.5, 1; (b) $-\unicode[STIX]{x1D6E5}=1.0$ with $\unicode[STIX]{x1D6FF}=0.4$ , 0.2, 0, $-0.2$ and $-0.4$ . The arrows in panels (a) and (b) indicate the directions of increasing $\unicode[STIX]{x1D6E5}$ and $\unicode[STIX]{x1D6FF}$ , respectively.

Next, we set $\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}/\unicode[STIX]{x2202}k=0$ and find that the fastest growing wavenumber is $k_{m}=2^{-1/2}k_{c}$ , which corresponds to a maximum growth rate of $\unicode[STIX]{x1D6FC}_{m}=(1/4)k_{c}^{4}\unicode[STIX]{x1D6F7}^{3}$ . For fixed values of $\unicode[STIX]{x1D6F7}$ , $\unicode[STIX]{x1D6FC}_{m}$ is presented in figure 2(a,b) for different values of $\unicode[STIX]{x1D6FF}$ and $\unicode[STIX]{x1D6E5}$ . Notably, for each case, $\unicode[STIX]{x1D6FC}_{m}$ approaches zero for small and large values of $\unicode[STIX]{x1D6F7}$ and, in between, attains a global maximum. This behaviour is a consequence of the form of the film temperature profile, which, using the current scaling, is

where $\unicode[STIX]{x1D701}=BZ$ . Accounting for this scaling and the definitions stated in (2.9), the dimensionless substrate temperature $\unicode[STIX]{x1D6E9}_{s}=0$ , the gas temperature $\tilde{\unicode[STIX]{x1D6E9}}_{g}=\bar{\unicode[STIX]{x1D6E9}}_{g}+\cos (\unicode[STIX]{x1D6FA}\unicode[STIX]{x1D70F})$ oscillates about the mean gas temperature $\bar{\unicode[STIX]{x1D6E9}}_{g}=\unicode[STIX]{x1D6E5}$ , and $\unicode[STIX]{x1D6E9}_{0}=-\unicode[STIX]{x1D6FF}$ corresponds to the temperature $\unicode[STIX]{x1D703}_{0}$ that yields the minimum surface tension $\unicode[STIX]{x1D70E}_{0}$ . As $\unicode[STIX]{x1D719}=\unicode[STIX]{x1D6F7}$ approaches zero, the surface temperature $\unicode[STIX]{x1D6E9}(\unicode[STIX]{x1D701}=\unicode[STIX]{x1D719})$ tends therefore uniformly to the substrate temperature. Hence, the thermocapillary stresses are negligible and the growth rate is zero. Likewise, $\unicode[STIX]{x1D6FC}_{m}$ is zero for large values of $\unicode[STIX]{x1D6F7}$ , under which conditions the film surface temperature is uniformly equal to the gas temperature $\tilde{\unicode[STIX]{x1D6E9}}_{g}=\unicode[STIX]{x1D6E5}+\cos (\unicode[STIX]{x1D6FA}\unicode[STIX]{x1D70F})$ . A trade-off therefore exists between the gas–liquid heat transfer rate and the film thickness, which leads to a global maximum growth rate $\unicode[STIX]{x1D6FC}_{g}$ at a specific value $\unicode[STIX]{x1D6F7}=\unicode[STIX]{x1D6F7}_{g}$ for any given values of $\unicode[STIX]{x1D6E5}$ and $\unicode[STIX]{x1D6FF}$ . The global $\unicode[STIX]{x1D6FC}_{g}$ occurs at $\unicode[STIX]{x1D6F7}_{g}=1$ for all cases where $\unicode[STIX]{x1D6FF}=0$ , and shifts to $\unicode[STIX]{x1D6F7}_{g}$ are introduced by setting finite $\unicode[STIX]{x1D6FF}$ and $\unicode[STIX]{x1D6E5}$ , as seen in panels 2(a) and 2(b), respectively.

Physical aspects of the preceding analysis will now be discussed by considering the evolution of the surface tension for local values of the film height $\unicode[STIX]{x1D719}$ . To do so, we first non-dimensionalize (2.4), which gives

and is valid at the interface $\unicode[STIX]{x1D701}=\unicode[STIX]{x1D719}$ . Then we substitute the temperature profile $\unicode[STIX]{x1D6E9}(\unicode[STIX]{x1D701}=\unicode[STIX]{x1D719})$ into (3.7), and define a non-dimensional correction to the local surface tension $S(\unicode[STIX]{x1D70F},\unicode[STIX]{x1D719})$ as

which has a time-averaged component

In the following discussion, the evolution of $S$ throughout an excitation period will be considered for different cases of ( $\unicode[STIX]{x1D6FF}$ , $\unicode[STIX]{x1D6E5}$ ) that are representative of cases (i)–(iv). To highlight the instantaneous and time-averaged Marangoni stresses that drive film dynamics, we consider the values of $S$ and $\bar{S}$ evaluated at $\unicode[STIX]{x1D719}=\unicode[STIX]{x1D719}_{1}=0.8$ (representative of a trough) and $\unicode[STIX]{x1D719}_{2}=1.0$ (crest). Accordingly, to simplify notation, we denote the trough and crest values of $S$ and $\bar{S}$ as $S_{1}=S(\unicode[STIX]{x1D719}_{1})$ and $\bar{S}_{1}=\bar{S(\unicode[STIX]{x1D719}_{1})}$ , and $S_{2}=S(\unicode[STIX]{x1D719}_{2})$ and $\bar{S}_{2}=\bar{S(\unicode[STIX]{x1D719}_{2})}$ , respectively. We note that the following discussion is highly nuanced, and it may be sufficient for the reader to only consider that pertaining to case (i), which is the focus of the nonlinear study in § 4. The discussion of cases (ii)–(iv) in this section can then be considered to infer behaviour in more complex experimental settings, which we discuss in § 5.

Figure 3.

Evolution of non-dimensional surface tension at a trough, $S_{1}$ (solid lines in both panels), and at a crest, $S_{2}$ , with respect to (a) surface temperature and (b) time for $(\unicode[STIX]{x1D6FF},\unicode[STIX]{x1D6E5})=(0,0)$ . In panel (a), $S_{1}$ traces the path along $S_{2}$ and its endpoints have been marked by circles. In panel (b), $S_{1}$ is given by the dashed line. In both panels, dash-dotted lines are used to indicate time-averaged values of $\bar{S}_{1}=0.099$ and $\bar{S}_{2}=0.125$ as labelled.

The evolution of $S_{1}$ and $S_{2}$ for case (i), with $\unicode[STIX]{x1D6FF}=\unicode[STIX]{x1D6E5}=0$ , is presented throughout the excitation in figure 3(a,b) as functions of the temporally evolving surface temperature $\unicode[STIX]{x1D6E9}_{\unicode[STIX]{x1D719}}$ and the time coordinate $f\unicode[STIX]{x1D70F}$ , respectively. At time $f\unicode[STIX]{x1D70F}=0$ , the gas temperature $\tilde{\unicode[STIX]{x1D6E9}}_{g}=1$ is higher than the substrate temperature $\unicode[STIX]{x1D6E9}_{s}=0$ , and, via (3.8), $S$ is an increasing function of $\unicode[STIX]{x1D719}$ . Therefore, instantaneously, a surface traction drives flow from a thinner location of the interface to a thicker one, e.g. from $\unicode[STIX]{x1D719}_{1}$ to $\unicode[STIX]{x1D719}_{2}$ . As time proceeds, $\tilde{\unicode[STIX]{x1D6E9}}_{g}$ and $S_{2}-S_{1}$ decrease until they vanish at $f\unicode[STIX]{x1D70F}=1/4$ , at which point the system temperature is uniformly equal to $\tilde{\unicode[STIX]{x1D6E9}}_{g}=0$ . The reverse process occurs during $1/4\leqslant f\unicode[STIX]{x1D70F}\leqslant 1/2$ , whereby $S_{2}-S_{1}$ increases from 0 to its maximal value and $\tilde{\unicode[STIX]{x1D6E9}}_{g}$ continues to decrease from 0 to $-1$ . Then, the evolution of $S_{2}-S_{1}$ during the gas cooling cycle is repeated during the subsequent warming cycle that occurs for $1/2\leqslant f\unicode[STIX]{x1D70F}\leqslant 1$ as $\tilde{\unicode[STIX]{x1D6E9}}_{g}$ increases from $-1$ to 1. At no point during the excitation are the surface tractions directed from crest to trough for case (i), which would promote stabilization. As a result, there exists a time-averaged flow from crest to trough that destabilizes domains of wavenumbers $k<k_{c}$ as depicted in figure 1(a). The destabilization process is $2\unicode[STIX]{x1D6FA}$ -periodic for case (i), due to the symmetry of the surface tension with respect to the thermal fluctuations. In general, the destabilizations associated with cases (ii)–(iv) contain both $\unicode[STIX]{x1D6FA}$ and $2\unicode[STIX]{x1D6FA}$ components.

Figure 4.

Evolution of non-dimensional surface tension at a trough, $S_{1}$ , and at a crest, $S_{2}$ , with respect to (a) surface temperature and (b) time for $(\unicode[STIX]{x1D6FF},\unicode[STIX]{x1D6E5})=(0.2,0)$ . All curve and point designations are as in figure 3. In both panels, the dash-dotted lines indicate time-averaged values of $\bar{S}_{1}=0.139$ and $\bar{S}_{2}=0.165$ as labelled.

Case (ii) describes the configuration where the substrate and mean gas temperatures are held equal ( $\unicode[STIX]{x1D6E5}=0$ ) to a value that may either be higher (positive $\unicode[STIX]{x1D6FF}$ ) or lower (negative $\unicode[STIX]{x1D6FF}$ ) than $\unicode[STIX]{x1D703}_{0}$ , which, in scaled units, is equal to $-\unicode[STIX]{x1D6FF}$ . Positive and negative values of $\unicode[STIX]{x1D6FF}$ therefore bias the excitation towards higher and lower temperatures than that described by case (i). Figure 4(a,b) depicts the evolution of $S_{2}$ and $S_{1}$ during an excitation period where $\unicode[STIX]{x1D6E5}=0$ and $\unicode[STIX]{x1D6FF}=0.2$ . Clearly, the bias breaks the symmetry of the excitation with respect to $\unicode[STIX]{x1D703}_{0}$ , which leads to the introduction of an $\unicode[STIX]{x1D6FA}$ linear response frequency in addition to the $2\unicode[STIX]{x1D6FA}$ response. Nevertheless, this bias does not change the difference between $\bar{S}$ for two different values of $\unicode[STIX]{x1D719}$ and as a result the cutoff wavelength $k_{c}$ does not depend upon $\unicode[STIX]{x1D6FF}$ for $\unicode[STIX]{x1D6E5}=0$ , as follows from equation (3.5).

Figure 5.

Evolution of non-dimensional surface tensions at a trough, $S_{1}$ , and at a crest, $S_{2}$ , with respect to (a) surface temperature and (b) time for $(\unicode[STIX]{x1D6FF},\unicode[STIX]{x1D6E5})=(0,2)$ . All curve and point designations are as in figure 3. In both panels, the dash-dotted lines indicate time-averaged values of $\bar{S}_{1}=0.89$ and $\bar{S}_{2}=1.13$ as labelled.

For case (iii), the substrate temperature is held at $\unicode[STIX]{x1D703}_{0}$ ( $\unicode[STIX]{x1D6FF}=0$ ) and is heated (negative $\unicode[STIX]{x1D6E5}$ ) or cooled (positive $\unicode[STIX]{x1D6E5}$ ) with respect to the mean gas temperature. As in case (ii), the film surface is biased towards temperatures that are either warmer or cooler than $\unicode[STIX]{x1D703}_{0}$ . However, via (3.6), heating or cooling magnifies the temperature differences between a crest and a trough throughout the excitation. As a result, $\unicode[STIX]{x1D6FC}$ and $k_{c}$ are increased for all $\unicode[STIX]{x1D6F7}$ . Also observe that the film surface temperature can be set to reside entirely above (or below) $\unicode[STIX]{x1D703}_{0}$ by increasing the magnitude of heating/cooling larger than the modulation amplitude, i.e. $|\unicode[STIX]{x1D6E5}|>1$ . The evolution of $S$ is depicted for such a configuration in figure 5 with $(\unicode[STIX]{x1D6FF},\unicode[STIX]{x1D6E5})=(0,2)$ and shows that the $2\unicode[STIX]{x1D6FA}$ response is diminished significantly because the surface tension minimum is not traversed.

Figure 6.

Evolution of non-dimensional surface tensions at a trough, $S_{1}$ , and at a crest, $S_{2}$ , with respect to (a) surface temperature and (b) time for $(\unicode[STIX]{x1D6FF},\unicode[STIX]{x1D6E5})=(-2,1)$ . All curve and point designations are as in figure 3. In both panels, the dash-dotted lines indicate time-averaged values of $\bar{S}_{1}=2.52$ and $\bar{S}_{2}=2.38$ as labelled.

Finally, for case (iv), both $\unicode[STIX]{x1D6FF}$ and $\unicode[STIX]{x1D6E5}$ take finite values, which represents the combination of thermal modulation with steady heating/cooling, for substrate temperatures set away from $\unicode[STIX]{x1D703}_{0}$ . From (3.3), it is clear that positive/negative values of $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6E5}$ promote instability/stability via the second term in the parentheses. This term describes the thermocapillary effects due to steady heating/cooling that depend linearly on the surface temperature, and arises when the system is biased towards temperatures above or below $\unicode[STIX]{x1D703}_{0}$ . For example, when $\unicode[STIX]{x1D6FF}<0$ , the system is biased towards temperatures that are less than  $\unicode[STIX]{x1D703}_{0}$ , where surface tension is a decreasing function of temperature. In this region, uniform substrate heating (negative $\unicode[STIX]{x1D6E5}$ ) is destabilizing, as is the case for typical fluids whose surface tension decreases (linearly) with temperature, see Oron et al. (Reference Oron, Davis and Bankoff1997). Contrary to typical fluids, films with surface tension minima may also be destabilized by uniform substrate cooling (negative $\unicode[STIX]{x1D6E5}$ ) when the system is biased towards temperatures higher than $\unicode[STIX]{x1D703}_{0}$ (positive $\unicode[STIX]{x1D6FF}$ ), where surface tension is an increasing function of temperature. The contribution of these linear thermocapillary effects must nevertheless compete with those given by the first term in the parentheses of (3.3), which depends quadratically on the surface temperature. It is worth noting that, at small values of $\unicode[STIX]{x1D6F7}$ , the linear thermocapillary contribution to $\unicode[STIX]{x1D6FC}$ is $\mathit{O}(\unicode[STIX]{x1D6F7}^{2})$ and therefore dominates both the quadratic thermocapillary and surface tension contributions, which are $\mathit{O}(\unicode[STIX]{x1D6F7}^{3})$ . As a result, we find that the cutoff wavenumber diverges as $\unicode[STIX]{x1D6F7}\rightarrow$ 0 for increasing positive values of $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6E5}$ , see $k_{c}$ for $(\unicode[STIX]{x1D6E5},\unicode[STIX]{x1D6FF})=(1,0.2)$ and (1,0.4) in figure 1(b). Alternatively, when the product $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6E5}$ is negative, steady heating/cooling can stabilize the film completely as $\unicode[STIX]{x1D6F7}\rightarrow$ 0, as depicted by $k_{c}$ for $(\unicode[STIX]{x1D6E5},\unicode[STIX]{x1D6FF})=(1,-0.2)$ and ( $1,-0.4$ ). An example of such a configuration, which leads to a time-averaged value of $\bar{S}_{2}<\bar{S}_{1}$ , is given in figure 6.

To summarize, modulation of the gas temperature alone can destabilize a film that is otherwise held at isothermal conditions, due to quadratic dependence of surface tension on temperature. The cutoff wavelengths $k_{c}$ and time-averaged growth rates $\unicode[STIX]{x1D6FC}$ depend upon the arrangement of the thermal boundary conditions via $\unicode[STIX]{x1D6FF}$ and $\unicode[STIX]{x1D6E5}$ as well as the mean thermal thickness $\unicode[STIX]{x1D6F7}$ . Considering that system dynamics is equivalent under the transformation $(\unicode[STIX]{x1D6E5},\unicode[STIX]{x1D6FF},f\unicode[STIX]{x1D70F})\rightarrow (-\unicode[STIX]{x1D6E5},-\unicode[STIX]{x1D6FF},f\unicode[STIX]{x1D70F}+1)$ , figure 1 conclusively presents the behaviour of $k_{c}$ with respect to this parameter space. Although $\unicode[STIX]{x1D6FA}$ does not affect linear stability it will be seen in the next section to bear critical importance on the emergence of saturated nonlinear flows.

4 Numerical study

We proceed with a numerical investigation of initially flat films described by (2.17) on spatially periodic domains, aiming to provide insight into strongly nonlinear regimes. The numerical method is identical to that used by Batson et al. (Reference Batson, Agnon and Oron2015) in their investigation of hanging films subject to thermal excitation by acoustic waves. In summary, a Fourier collocation method is used to discretize space and build appropriate differentiation operators (Boyd Reference Boyd2001). An explicit integration scheme for the evaluation of the increment in the interface profile $\unicode[STIX]{x1D719}(\unicode[STIX]{x1D709},\unicode[STIX]{x1D70F})$ in time is used via a first-order forward Euler method with time step $\unicode[STIX]{x1D6E5}\unicode[STIX]{x1D70F}$ and Newton–Kantorovich linearization of the nonlinearities (Haimovich & Oron Reference Haimovich and Oron2010). Numerical analysis of time series at the excitation frequency is supported by setting the number of time steps per excitation period to an integer number. Of particular interest are saturated flows, which we classify as those numerical solutions for which the $L_{2}$ norm of $\unicode[STIX]{x1D719}$ changes by less than $1\times 10^{-8}$ after an excitation period. It will be demonstrated that saturated, non-rupturing flows emerge when $\unicode[STIX]{x1D6FA}$ is the same order as the linear growth rate $\unicode[STIX]{x1D6FC}$ . Mainly we focus on these values of $\unicode[STIX]{x1D6FA}$ , and, in the results that follow, convergence is typically obtained by using 128 Fourier modes and between 300 and 400 time steps per period. Whereas some integrations of (2.17) are initialized with small perturbations to flat films, for example, $\unicode[STIX]{x1D719}(\unicode[STIX]{x1D709},\unicode[STIX]{x1D70F}=0)=\unicode[STIX]{x1D6F7}+0.001\cos k\unicode[STIX]{x1D709}$ , solution branches are also tracked by continuation with respect to small changes in the independent parameters by initializing computations with nearby saturated solutions. Finally, we restrict our interest in the following numerical study to case (i), that is, a film that is thermally modulated about its mean temperature, which is fixed to $\unicode[STIX]{x1D703}_{0}$ . In doing so, the parameter space is reduced significantly and permits a thorough investigation of the dynamics induced solely by modulation about $\unicode[STIX]{x1D703}_{0}$ .

5 Potential experimental settings

We will now use the model to predict behaviour of self-rewetting films in potential experimental settings, with initial intent to determine the conditions at which one may observe the periodic flow in figures 7–9 where $(\unicode[STIX]{x1D6E5},\unicode[STIX]{x1D6FF},\unicode[STIX]{x1D6F7},k,\unicode[STIX]{x1D6FA})=$ (0, 0, 0.1, 0.55, $7.06\times 10^{-5}$ ). Specifically, we consider films of a 0.7 molar aqueous solution of butanol, for which (Savino et al. Reference Savino, Cecere, VanVaerenbergh, Abe, Pizzirusso, Tzevelecos, Mojahed and Galand2013) have reported the surface tension as a function of temperature. Analysing their data, we find $\unicode[STIX]{x1D703}_{0}=66.3\,^{\circ }$ C, $\unicode[STIX]{x1D70E}_{0}=0.0263~\text{N}~\text{m}^{-1}$ and $\unicode[STIX]{x1D6FE}=2.98\times 10^{-6}~\text{N}~\text{m}^{-1}$  K $^{-2}$ . The viscosity and thermal conductivity of the dilute mixture are set equal to that of water, $\unicode[STIX]{x1D707}=4.3\times 10^{-4}~\text{Pa}~\text{s}$ and $\unicode[STIX]{x1D705}=0.66~\text{W}~\text{m}^{-1}~\text{K}^{-1}$ .

The temperature differences $\unicode[STIX]{x1D6E5}$ and $\unicode[STIX]{x1D6FF}$ remain defined as in (2.9), and, for this case where $\unicode[STIX]{x1D6E5}=\unicode[STIX]{x1D6FF}=0$ , $\unicode[STIX]{x1D703}_{s}$ and $\bar{\unicode[STIX]{x1D703}}_{g}$ will both be set to $\unicode[STIX]{x1D703}_{0}$ . Via (2.16), (2.9) and (2.6), the remaining dimensionless parameters are defined as

where $k^{\ast }$ is the dimensional wavenumber, and provide freedom to set all but three parameters of the dimensional system. Although, in an experiment, one is far more likely to have control of $h_{0}$ , $\unicode[STIX]{x1D714}$ , and $\hat{\unicode[STIX]{x1D703}}_{g}$ , with $k^{\ast }$ as an output, we will fix $\hat{\unicode[STIX]{x1D703}}_{g}$ $a$ $priori$ because the relations for $\unicode[STIX]{x1D6F7}$ , $k$ and $\unicode[STIX]{x1D6FA}$ in (5.1a–c ) are linear in $h_{0}$ , $k^{\ast }$ and $\unicode[STIX]{x1D714}$ . We therefore set $\hat{\unicode[STIX]{x1D703}}_{g}=6\,^{\circ }$ C, a thermal excitation characteristic of a high power acoustic wave in air at standard temperature and pressure.

Of greatest uncertainty is prescription of $q$ , the rate at which heat is exchanged between the liquid and gas phases at the film interface. Several thin film works, such as Kalliadasis, Kiyashko & Demekhin (Reference Kalliadasis, Kiyashko and Demekhin2003) and Nepomnyashchy, Velarde & Colinet (Reference Nepomnyashchy, Velarde and Colinet2001), state that $q$ , or, likewise, the Biot number, is often unknown and therefore models can be used to make qualitative predictions at best. To proceed with a representative value for $q$ , we observe that Holman (Reference Holman2010) has provided approximate values for free convection heat transfer that range in the interval 4.5– $890~\text{W}\,\text{m}^{-2}\,\text{K}^{-1}$ . Additionally, Kalliadasis et al. (Reference Kalliadasis, Ruyer-Quil, Scheid and Velarde2012) have discussed the relevance of thin film heat transfer coefficients ranging from 100– $500~\text{W}\,\text{m}^{-2}\,\text{K}^{-1}$ . Conceivably, one also could determine $q$ experimentally by measuring the instability wavelength at criticality for a given threshold value of $\hat{\unicode[STIX]{x1D703}}_{g}$ , see (3.5).

Moving forward, with $q=100~\text{W}\,\text{m}^{-2}\,\text{K}^{-1}$ , we use (5.1a–c ) to determine the set of conditions ( $\unicode[STIX]{x1D703}_{s}$ , $\bar{\unicode[STIX]{x1D703}}_{g}$ , $h_{0}$ , $\unicode[STIX]{x1D706}^{\ast }$ , $f^{\ast }$ ) $=$ ( $66.3\,^{\circ }$ C, $66.3\,^{\circ }$ C, 0.66 mm, 1.18 m, $1.73\times 10^{-6}$  Hz) at which the dimensionless dynamics presented in figures 7–9 is applicable to films of aqueous butanol. Here, $\unicode[STIX]{x1D706}^{\ast }$ and $f^{\ast }$ are the dimensional wavelength and ordinary frequency, respectively. Although the film thickness is experimentally reasonable, the wavelength, and even more so, the frequency, are not. Conceivably one could increase $h_{0}$ , so that larger values of $\unicode[STIX]{x1D6F7}$ , and, accordingly, larger values of $\unicode[STIX]{x1D6FA}$ are relevant, however, continued increase in $h_{0}$ will render invalid the assumption that buoyancy is negligible. Furthermore, the wavelength is independent of $h_{0}$ .

Instead, it appears that introduction of substrate heating/cooling and bias, by considering finite $\unicode[STIX]{x1D6E5}$ and $\unicode[STIX]{x1D6FF}$ , is the most practical means by which non-rupturing flows can be generated in experimental settings. By doing so, the growth rates and wavenumbers can be increased significantly. For example, with $\unicode[STIX]{x1D6E5}=2$ and $\unicode[STIX]{x1D6FF}=1$ , we find via (3.5) that $k_{c}=4.46$ and $\unicode[STIX]{x1D6FC}_{m}=0.0991$ . We then find that the evolution equation generates a non-rupturing flow for the parameters ( $\unicode[STIX]{x1D6E5}$ , $\unicode[STIX]{x1D6FF}$ , $\unicode[STIX]{x1D6F7}$ , $k$ , $\unicode[STIX]{x1D6FA}$ ) $=$ (2, 1, 0.1, 4.3, 0.0991). We note that the resulting periodic flow strongly resembles that presented in figure 9, with the exception that destabilization cells are more tightly constrained to the vicinity of the major film trough. For aqueous butanol with an excitation amplitude of $\hat{\unicode[STIX]{x1D703}}_{g}=6^{\circ }$ C, the corresponding dimensional parameters are ( $\unicode[STIX]{x1D703}_{s}$ , $\bar{\unicode[STIX]{x1D703}}_{g}$ , $h$ , $\unicode[STIX]{x1D706}^{\ast }$ , $f^{\ast }$ ) $=$ ( $72.3\,^{\circ }$ C, $84.3\,^{\circ }$ C, 0.66 mm, 0.151 m, 0.00244 Hz), which are much more feasible. Here, the wavelength remains large. Inspection of (5.1a–c ) reveals that the wavelength can be decreased by increasing the excitation amplitude, however, one must bear in mind that the range of applicable dimensional temperatures is bounded by phase transitions at which model assumptions will lose validity. For example, with $\unicode[STIX]{x1D6E5}=2$ and $\unicode[STIX]{x1D6FF}=1$ , an increase of $\hat{\unicode[STIX]{x1D703}}_{g}$ to $10\,^{\circ }$ C corresponds to $\unicode[STIX]{x1D703}_{s}=76.3\,^{\circ }$ C and $\bar{\unicode[STIX]{x1D703}}_{g}=96.3\,^{\circ }$ C. Hence, the gas temperature will rise above the saturation temperature of pure water (at atmospheric pressure) during the warm cycle of the excitation and phase change processes will become important.

A different choice of an operating fluid can shift the value of $\unicode[STIX]{x1D703}_{0}$ and increase the range of applicable temperatures. For example, using data reported by Legros (Reference Legros1986), we find for aqueous heptanol that $\unicode[STIX]{x1D703}_{0}=40.7\,^{\circ }$ C, $\unicode[STIX]{x1D70E}_{0}=0.0375~\text{N}~\text{m}^{-1}$ and $\unicode[STIX]{x1D6FE}=3.42\times 10^{-6}~\text{N}~\text{m}^{-1}~\text{K}^{-2}$ . Using these values, and increasing of the excitation amplitude to $\hat{\unicode[STIX]{x1D703}}_{g}=15^{\circ }$ , the dimensionless parameters ( $\unicode[STIX]{x1D6E5}$ , $\unicode[STIX]{x1D6FF}$ , $\unicode[STIX]{x1D6F7}$ , $k$ , $\unicode[STIX]{x1D6FA}$ ) $=$ (2, 1, 0.1, 4.3, 0.496) correspond to dimensional values of ( $\unicode[STIX]{x1D703}_{s}$ , $\bar{\unicode[STIX]{x1D703}}_{g}$ , $h_{0}$ , $\unicode[STIX]{x1D706}^{\ast }$ , $f^{\ast }$ ) $=$ ( $55.7\,^{\circ }$ C, $85.7\,^{\circ }$ C, 0.66 mm, 0.0673 m, 0.0175 Hz). Here, the wavelength is measurable, and the mean gas temperature is low enough such that the effects of phase change are mitigated. Here we have considered substrate cooling as a means to drive instability and increase the growth rates, however, substrate heating could also be used if one were to operate at temperatures lower than $\unicode[STIX]{x1D703}_{0}$ . In such situations, the operating temperature range could then be expanded by choosing shorter carbon-chained alcohols, which are characterized by larger values of $\unicode[STIX]{x1D703}_{0}$ .