2.1 Dimensional equations

We consider a 2-D flow, governed by the incompressible Navier–Stokes equations and with all bulk properties of the fluid (dynamic viscosity $\unicode[STIX]{x1D707}$ , density $\unicode[STIX]{x1D70C}$ , thermal conductivity $k$ and specific heat capacity $C_{p}$ ) independent of temperature. We use axes aligned with the wall, as illustrated in figure 2; the coordinate $x$ and velocity component $u$ are in the downslope direction, while the coordinate $y$ and velocity component $v$ are perpendicular to it. The free surface is defined by $y=h(x,t)$ , and the wall is inclined at an angle $\unicode[STIX]{x1D6FD}$ to the horizontal. We will suppose that the air temperature remains constant at a temperature $\unicode[STIX]{x1D703}_{A}$ , set $\unicode[STIX]{x1D703}_{A}+\unicode[STIX]{x1D703}(x,y,t)$ to be the temperature within the film, and $\unicode[STIX]{x1D703}_{A}+\unicode[STIX]{x1D6E9}(x,t)$ to be the wall temperature.

Figure 2.

Diagram of our system. A rigid wall is oriented at an angle $\unicode[STIX]{x1D6FD}$ to the horizontal. The coordinates $x$ and $y$ are directed down the slope and perpendicular to it, respectively. A layer of fluid occupies the region $0<y<h(x,t)$ . The temperature within the fluid is a function of space and time, but the fluid density and viscosity are constant.

Heat has no effect on the bulk properties, so within the fluid layer we have the usual momentum equations,

(2.1) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D70C}\left(\frac{\unicode[STIX]{x2202}u}{\unicode[STIX]{x2202}t}+u\frac{\unicode[STIX]{x2202}u}{\unicode[STIX]{x2202}x}+v\frac{\unicode[STIX]{x2202}u}{\unicode[STIX]{x2202}y}\right)=-p_{x}+\unicode[STIX]{x1D70C}g\sin \unicode[STIX]{x1D6FD}+\unicode[STIX]{x1D707}\unicode[STIX]{x1D6FB}^{2}u,\quad 0<y<h(x,t), & \displaystyle\end{eqnarray}$$

(2.2) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D70C}\left(\frac{\unicode[STIX]{x2202}v}{\unicode[STIX]{x2202}t}+u\frac{\unicode[STIX]{x2202}v}{\unicode[STIX]{x2202}x}+v\frac{\unicode[STIX]{x2202}v}{\unicode[STIX]{x2202}y}\right)=-p_{y}-\unicode[STIX]{x1D70C}g\cos \unicode[STIX]{x1D6FD}+\unicode[STIX]{x1D707}\unicode[STIX]{x1D6FB}^{2}v,\quad 0<y<h(x,t), & \displaystyle\end{eqnarray}$$

where $p$ is the fluid pressure and $g$ the acceleration due to gravity, along with conservation of mass in the form

(2.3) $$\begin{eqnarray}u_{x}+v_{y}=0,\quad 0<y<h(x,t).\end{eqnarray}$$

Within the fluid layer, the temperature $\unicode[STIX]{x1D703}(x,y,t)$ evolves according to the heat equation

(2.4) $$\begin{eqnarray}\unicode[STIX]{x1D70C}C_{p}\left(\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}{\unicode[STIX]{x2202}t}+u\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}{\unicode[STIX]{x2202}x}+v\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}{\unicode[STIX]{x2202}y}\right)=k\unicode[STIX]{x1D6FB}^{2}\unicode[STIX]{x1D703},\quad 0<y<h(x,t).\end{eqnarray}$$

On the rigid wall at $y=0$ , the fluid velocity vanishes, and the fluid temperature $\unicode[STIX]{x1D703}$ is prescribed:

(2.5a-c ) $$\begin{eqnarray}u=0,\quad v=0,\quad \unicode[STIX]{x1D703}=\unicode[STIX]{x1D6E9}(x,t),\quad \text{on }y=0.\end{eqnarray}$$

In the bulk equations, the velocity field can influence the temperature evolution, but there is no reverse coupling. The only feedback from the fluid temperature to the hydrodynamic fields is through the dynamic boundary condition,

(2.6) $$\begin{eqnarray}[\unicode[STIX]{x1D748}\boldsymbol{\cdot }\boldsymbol{n}]_{F}^{A}=\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D705}\boldsymbol{n}-\boldsymbol{t}(\boldsymbol{t}\boldsymbol{\cdot }\unicode[STIX]{x1D735})\unicode[STIX]{x1D6FE},\quad \text{on }y=h(x,t),\end{eqnarray}$$

where $\unicode[STIX]{x1D70E}$ is the stress tensor, $\boldsymbol{n}$ and $\boldsymbol{t}$ are unit vectors normal (directed into the air layer) and tangential to the free surface, respectively, and $\unicode[STIX]{x1D6FE}$ is the temperature-dependent coefficient of surface tension. For simplicity, we will model this temperature dependence by a linear relation, as appropriate for small variations in temperature, but a nonlinear function could be used without difficulty. Hence we write

(2.7) $$\begin{eqnarray}\unicode[STIX]{x1D6FE}=\unicode[STIX]{x1D6FE}_{A}-\hat{\unicode[STIX]{x1D6FE}}S,\end{eqnarray}$$

where the positive constants $\unicode[STIX]{x1D6FE}_{A}$ and $\hat{\unicode[STIX]{x1D6FE}}$ , respectively, are the coefficient of surface tension of the fluid at the temperature of the air and the strength of surface tension variation.

The hydrodynamic system also satisfies the kinematic boundary condition,

(2.8) $$\begin{eqnarray}h_{t}=v-uh_{x},\quad \text{on }y=h(x,t),\end{eqnarray}$$

and we will assume a simple linear relation for the heat flux into the air,

(2.9) $$\begin{eqnarray}\boldsymbol{n}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D703}=-L^{-1}\unicode[STIX]{x1D703},\quad \text{on }y=h(x,t),\end{eqnarray}$$

where the quantity $L$ has dimensions of length. The relationship (2.9) is commonly used in the heat transfer literature, and follows from the assumption that the air layer is always held at $\unicode[STIX]{x1D703}=0$ , combined with Newton’s law of cooling.

2.2 Non-dimensionalization

We will non-dimensionalize the system based on a typical film height $h_{s}$ and the corresponding surface speed $U_{s}=\unicode[STIX]{x1D70C}gh_{s}^{2}\sin \unicode[STIX]{x1D6FD}/(2\unicode[STIX]{x1D707})$ . Hence we define

(2.10a-f ) $$\begin{eqnarray}u=U_{s}\hat{u} ,\quad v=U_{s}\hat{v},\quad x=h_{s}\hat{x},\quad y=h_{s}{\hat{y}},\quad t=\frac{h_{s}}{U_{s}}\hat{t},\quad p=\frac{\unicode[STIX]{x1D707}U_{s}}{h_{s}}\hat{p}.\end{eqnarray}$$

We will scale $\unicode[STIX]{x1D703}$ , $\unicode[STIX]{x1D6E9}$ and $S$ on a typical temperature scale $\unicode[STIX]{x0394}\unicode[STIX]{x1D6E9}$ , which corresponds to a characteristic temperature variation between the heated wall and the ambient air. The bulk equations become (dropping hats), each holding in the region $0<y<h(x,t)$ :

(2.11) $$\begin{eqnarray}\displaystyle & \displaystyle Re\left(\frac{\unicode[STIX]{x2202}u}{\unicode[STIX]{x2202}t}+u\frac{\unicode[STIX]{x2202}u}{\unicode[STIX]{x2202}x}+v\frac{\unicode[STIX]{x2202}u}{\unicode[STIX]{x2202}y}\right)=-p_{x}+2+\frac{\unicode[STIX]{x2202}^{2}u}{\unicode[STIX]{x2202}x^{2}}+\frac{\unicode[STIX]{x2202}^{2}u}{\unicode[STIX]{x2202}y^{2}}, & \displaystyle\end{eqnarray}$$

(2.12) $$\begin{eqnarray}\displaystyle & \displaystyle Re\left(\frac{\unicode[STIX]{x2202}v}{\unicode[STIX]{x2202}t}+u\frac{\unicode[STIX]{x2202}v}{\unicode[STIX]{x2202}x}+v\frac{\unicode[STIX]{x2202}v}{\unicode[STIX]{x2202}y}\right)=-p_{y}-2\cot \unicode[STIX]{x1D6FD}+\frac{\unicode[STIX]{x2202}^{2}v}{\unicode[STIX]{x2202}x^{2}}+\frac{\unicode[STIX]{x2202}^{2}v}{\unicode[STIX]{x2202}y^{2}}, & \displaystyle\end{eqnarray}$$

(2.13) $$\begin{eqnarray}\displaystyle & \displaystyle Pe\left(\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}{\unicode[STIX]{x2202}t}+u\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}{\unicode[STIX]{x2202}x}+v\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}{\unicode[STIX]{x2202}y}\right)=\frac{\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D703}}{\unicode[STIX]{x2202}x^{2}}+\frac{\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D703}}{\unicode[STIX]{x2202}y^{2}} & \displaystyle\end{eqnarray}$$

and

(2.14) $$\begin{eqnarray}u_{x}+v_{y}=0.\end{eqnarray}$$

The dimensionless parameters in the bulk equations are the inclination angle $\unicode[STIX]{x1D6FD}$ , and the Reynolds, Péclet and Prandtl numbers, defined here as

(2.15a-c ) $$\begin{eqnarray}Re=\frac{\unicode[STIX]{x1D70C}U_{s}h_{s}}{\unicode[STIX]{x1D707}},\quad Pe=\frac{\unicode[STIX]{x1D70C}C_{p}U_{s}h_{s}}{k}=Pr\,Re,\quad Pr=\frac{\unicode[STIX]{x1D708}}{\unicode[STIX]{x1D705}}.\end{eqnarray}$$

The uniform state of an isothermal falling liquid film becomes unstable when $Re>1.25\cot \unicode[STIX]{x1D6FD}$ , and the initial instability is a long-wave variety. The Péclet number is related to the Reynolds number by the Prandtl number, which is a material constant, and is the ratio of diffusivities of momentum $\unicode[STIX]{x1D708}$ and heat $\unicode[STIX]{x1D705}=k/(\unicode[STIX]{x1D70C}C_{p})$ in the fluid. For water, $Pr\approx 7$ . This means that, if the Reynolds number is large enough to induce instabilities, then advection cannot be neglected when considering heat transfer through the film, and so for consistency we should consider regimes with moderate Péclet numbers.

In component form, the dynamic boundary condition becomes

(2.16) $$\begin{eqnarray}p-p_{A}-2\frac{v_{y}-(u_{y}+v_{x})h_{x}+u_{x}h_{x}^{2}}{1+h_{x}^{2}}=\left(\frac{1}{Ca}-Ma\,S\right)\frac{h_{xx}}{(1+h_{x}^{2})^{3/2}},\quad \text{on }y=h(x,t)\end{eqnarray}$$

and

(2.17) $$\begin{eqnarray}\frac{(u_{y}+v_{x})(1-h_{x}^{2})-2(u_{x}-v_{y})h_{x}}{1+h_{x}^{2}}=-\frac{Ma\,S_{x}}{(1+h_{x}^{2})^{1/2}},\quad \text{on }y=h(x,t),\end{eqnarray}$$

where $Ca$ and $Ma$ are the capillary and Marangoni numbers, respectively,

(2.18a,b ) $$\begin{eqnarray}Ca=\frac{\unicode[STIX]{x1D707}U_{s}}{\unicode[STIX]{x1D6FE}_{A}},\quad Ma=\frac{\hat{\unicode[STIX]{x1D6FE}}\unicode[STIX]{x0394}\unicode[STIX]{x1D6E9}}{\unicode[STIX]{x1D707}U_{s}}.\end{eqnarray}$$

The kinematic condition is unchanged in these new variables, and the heat loss condition becomes

(2.19) $$\begin{eqnarray}\boldsymbol{n}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D703}=-Bi\,\unicode[STIX]{x1D703},\quad \text{on }y=h(x,t),\end{eqnarray}$$

where $Bi=h_{s}/L\geqslant 0$ is the Biot number. The Biot number condition is a modelling convenience, bypassing the conjugate heat transfer problem with the air and, given the low thermal conductivity of air, is often taken to be small.

4.1 Regular expansion in long-wave limit: a Benney-type model

The most obvious long-wave model is to obtain a small-wavenumber expansion about the known linear temperature profile for zero wavenumber ( $k=0$ or $\unicode[STIX]{x1D6FF}=0$ ). The resulting expansion is regular, and is derived in much the same way as the Benney model for thin-film hydrodynamics. We pose an expansion of $\unicode[STIX]{x1D703}(X,y,T)$ for small $\unicode[STIX]{x1D6FF}$ :

(4.4) $$\begin{eqnarray}\unicode[STIX]{x1D703}=\unicode[STIX]{x1D703}_{0}+\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D703}_{1}+\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D703}_{2}+O(\unicode[STIX]{x1D6FF}^{3}).\end{eqnarray}$$

The PDE (4.1) yields the sequence of problems:

(4.5a-c ) $$\begin{eqnarray}\unicode[STIX]{x1D703}_{0yy}=0,\quad \unicode[STIX]{x1D703}_{1yy}=Pe(\unicode[STIX]{x1D703}_{0T}+u(y)\unicode[STIX]{x1D703}_{0X}),\quad \unicode[STIX]{x1D703}_{2yy}=Pe(\unicode[STIX]{x1D703}_{1T}+u(y)\unicode[STIX]{x1D703}_{1X})-\unicode[STIX]{x1D703}_{0XX}.\end{eqnarray}$$

The boundary conditions come from (4.2) and (4.3) with the wall temperature $\unicode[STIX]{x1D6E9}(X,T)$ treated as a known $O(1)$ quantity, so that

(4.6a-c ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D703}_{0}(X,0,T)=\unicode[STIX]{x1D6E9}(X,T),\quad \unicode[STIX]{x1D703}_{1}(X,0,T)=0,\quad \unicode[STIX]{x1D703}_{2}(X,0,T)=0, & & \displaystyle\end{eqnarray}$$

with $\unicode[STIX]{x1D703}_{ny}=-Bi\,\unicode[STIX]{x1D703}_{n}$ at $y=0$ for $n=0,1,2,\ldots \,$ . We find that $\unicode[STIX]{x1D703}_{0}$ , $\unicode[STIX]{x1D703}_{1}$ and $\unicode[STIX]{x1D703}_{2}$ are respectively first-, fifth- and ninth-order polynomials in $y$ . With $\unicode[STIX]{x1D703}_{0}$ , $\unicode[STIX]{x1D703}_{1}$ and $\unicode[STIX]{x1D703}_{2}$ known, the surface temperature is simply $S=\unicode[STIX]{x1D703}(y=1)$ .

Retaining terms in the surface temperature up to and including $O(\unicode[STIX]{x1D6FF}^{2})$ , and then returning to the original variables, this model gives

(4.7) $$\begin{eqnarray}\displaystyle S(x,t) & = & \displaystyle \frac{1}{1+Bi}\unicode[STIX]{x1D6E9}-\frac{Pe}{(1+Bi)^{2}}\left[\left(\frac{1}{2}+\frac{Bi}{6}\right)\unicode[STIX]{x1D6E9}_{t}+\left(\frac{5}{12}+\frac{7Bi}{60}\right)\unicode[STIX]{x1D6E9}_{x}\right]\nonumber\\ \displaystyle & & \displaystyle +\,Pe^{2}\left(\frac{(7Bi^{2}+42Bi+75)}{360(1+Bi)^{3}}\unicode[STIX]{x1D6E9}_{tt}+\frac{(69Bi^{2}+454Bi+889)}{2520(1+Bi)^{3}}\unicode[STIX]{x1D6E9}_{xt}\right)\nonumber\\ \displaystyle & & \displaystyle +\,\left[\frac{(Bi+3)}{6(1+Bi)^{2}}+Pe^{2}\frac{(491Bi^{2}+3510Bi+7555)}{50\,400(1+Bi)^{3}}\right]\unicode[STIX]{x1D6E9}_{xx}.\end{eqnarray}$$

Thus we obtain $S$ explicitly through knowledge of $\unicode[STIX]{x1D6E9}$ and its derivatives with respect to space and time. We can write (4.7) schematically as

(4.8) $$\begin{eqnarray}S(x,t)={\mathcal{L}}_{B}\unicode[STIX]{x1D6E9}(x,t),\end{eqnarray}$$

where ${\mathcal{L}}_{B}$ is a constant-coefficient, second-order linear operator.

If $\unicode[STIX]{x1D6E9}=0$ , then $S=0$ , so property P3 is satisfied: in fact, the temperature distribution tends to zero instantaneously, which is not physically realistic. However, a more important problem arises for steady transmission: (4.7) is quantitatively accurate for small $k$ , but diverges for large $k$ , where the amplitude grows as $k^{2}$ rather than decaying exponentially. If we were to calculate further corrections in the long-wave limit to obtain an $n$ th-order-accurate model, the amplitude would grow as $k^{n}$ , and hence further long-wave corrections would lead to increasingly rapid divergence when evaluated in the short-wave limit. This means that the model fails properties P1 and P2. This explicit model is therefore ill-posed at short wavelengths, and of little use for finding $S(x,t)$ in practical cases. However, the explicit expansion (4.7) is useful for testing quantitative agreement in the strictly long-wave limit, and might also have some relevance for inverse problems, where we wish to determine $\unicode[STIX]{x1D6E9}(x,t)$ to deliver a particular surface temperature.

4.2 Weighted-residual approach for temperature equation

Inspired by the success of the weighted-residual approach developed by Ruyer-Quil & Manneville (Reference Ruyer-Quil and Manneville2000) in modelling thin-film hydrodynamic flow at moderate $Re$ by coupled evolution equations for $h(x,t)$ and $q(x,t)$ , a number of authors have similarly developed weighted-residual models for heat transfer through a flowing film. Previous papers usually present the models correct only to first order in the long-wave parameter $\unicode[STIX]{x1D6FF}$ (but note that, in their appendix, Kalliadasis et al. (Reference Kalliadasis, Ruyer-Quil, Scheid and Velarde2012) list a full second-order model, including Marangoni and inertial corrections to the velocity field). Furthermore, nearly all studies assume that heating is uniform or steady.

In a typical derivation of a weighted-residual temperature equation, the temperature is written in terms of basis functions, e.g.

(4.9) $$\begin{eqnarray}\unicode[STIX]{x1D703}(x,y,t)=\mathop{\sum }_{n=0}^{\infty }a_{n}(x,t)\unicode[STIX]{x1D713}_{n}\left(\frac{y}{h(x,t)}\right).\end{eqnarray}$$

Note that, in contrast to the hydrodynamic case, the individual functions $\unicode[STIX]{x1D713}_{n}$ do not satisfy the heat loss boundary condition at $y=1$ . The unknown coefficients $a_{n}$ are determined by requiring that the temperature must satisfy a combination of strong and weak conditions. Strong conditions always include that the temperature or flux matches its prescribed value at the wall. Typical weak conditions are derived by multiplying the heat equation by a suitable weighting function and integrating across the depth of the film.

As noted by Trevelyan et al. (Reference Trevelyan, Scheid, Ruyer-Quil and Kalliadasis2007) and Kalliadasis et al. (Reference Kalliadasis, Ruyer-Quil, Scheid and Velarde2012), even for a given order of long-wave accuracy, different choices of residual weights will lead to different model equations with different coefficients; the models will only necessarily be equivalent in the long-wave limit. In this section, we follow earlier studies by choosing weight functions for the $n$ th-order weak conditions as $y^{n-1}$ , but we apply only the first two weak conditions. We supplement the boundary conditions on $\unicode[STIX]{x1D703}$ at $y=0$ and $y=1$ ,

(4.10a-c ) $$\begin{eqnarray}\unicode[STIX]{x1D6E9}(x,t)=\unicode[STIX]{x1D703}(x,0,t),\quad S(x,t)=\unicode[STIX]{x1D703}(x,1,t),\quad \unicode[STIX]{x1D703}_{y}(x,1,t)=-Bi\,\unicode[STIX]{x1D703}(x,1,t),\end{eqnarray}$$

with two strong conditions: we require the residual $R(X,y,T)$ , defined as

(4.11a,b ) $$\begin{eqnarray}R(X,y,T)\equiv \unicode[STIX]{x1D6FF}Pe(\unicode[STIX]{x1D703}_{T}+u(y)\unicode[STIX]{x1D703}_{X})-\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D703}_{XX}-\unicode[STIX]{x1D703}_{yy},\quad u(y)=y(2-y),\end{eqnarray}$$

to be equal to zero at both the wall $y=0$ and the fluid surface $y=1$ . With the two weak conditions, the bulk residual yields four equations in total:

(4.12a-d ) $$\begin{eqnarray}R(X,0,T)=0,\quad R(X,1,T)=0,\quad \int _{0}^{1}R(X,y,T)\,\text{d}y=0,\quad \int _{0}^{1}yR(X,y,T)\,\text{d}y=0.\end{eqnarray}$$

We expand the temperature field as a polynomial in $y$ , with coefficients that depend on $X$ and $T$ :

(4.13) $$\begin{eqnarray}\unicode[STIX]{x1D703}(X,y,T)=\unicode[STIX]{x1D6E9}(X,T)+a_{1}(X,T)y+a_{2}(X,T)y^{2}+a_{3}(X,T)y^{3}+a_{4}(X,T)y^{4}+a_{5}(X,T)y^{5}.\end{eqnarray}$$

This expansion automatically satisfies $\unicode[STIX]{x1D703}=\unicode[STIX]{x1D6E9}$ at $y=0$ ; we wish to obtain an equation for $S(X,T)=\unicode[STIX]{x1D703}(X,1,T)$ . Five unknown coefficients are needed to obtain a model accurate to second order in the long-wave limit. We start by applying (4.10), which allows us to write $a_{1}$ and $a_{2}$ in terms of $S$ , $a_{3}$ , $a_{4}$ and $a_{5}$ ; likewise (4.12a ) does not require any derivatives of the unknown coefficients, allowing the algebraic elimination of $a_{3}$ . In contrast, the final three equations, (4.12b–d ), each involve derivatives of $S(x,t)$ , $a_{4}(x,t)$ and $a_{5}(x,t)$ . These three conditions yield coupled linear evolution equations for $S$ , $a_{4}$ and $a_{5}$ , each first-order in time. However, as the linear operators commute, we can take a suitable linear combination to obtain an equation linking $S$ and $\unicode[STIX]{x1D6E9}$ alone. This equation features up to third-order derivatives in time, and sixth-order in space, of both $S$ and  $\unicode[STIX]{x1D6E9}$ .

Once we have obtained a single evolution equation for $S(x,t)$ , we discard all terms in this third-order evolution equation that are smaller than $O(\unicode[STIX]{x1D6FF}^{2})$ (other terms at $O(\unicode[STIX]{x1D6FF}^{3})$ having already been neglected in our analysis), yielding the following governing equation:

(4.14) $$\begin{eqnarray}\displaystyle & & \displaystyle (1+Bi)S+Pe\left[\left(\frac{16}{35}+\frac{13Bi}{105}\right)S_{t}+\left(\frac{27}{70}+\frac{3Bi}{35}\right)S_{x}\right]\nonumber\\ \displaystyle & & \displaystyle \qquad +\,Pe^{2}\left[\left(\frac{3}{140}+\frac{Bi}{420}\right)S_{tt}+\left(\frac{143}{4410}+\frac{71Bi}{22\,050}\right)S_{xt}\right]\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\left[Pe^{2}\left(\frac{1627}{141\,120}+\frac{687Bi}{705\,600}\right)-\left(\frac{16}{35}+\frac{13Bi}{105}\right)\right]S_{xx}\nonumber\\ \displaystyle & & \displaystyle \quad =\unicode[STIX]{x1D6E9}-Pe\left(\frac{13}{420}\unicode[STIX]{x1D6E9}_{x}+\frac{3}{70}\unicode[STIX]{x1D6E9}_{t}\right)+\frac{3}{70}\unicode[STIX]{x1D6E9}_{xx}\nonumber\\ \displaystyle & & \displaystyle \qquad +\,Pe^{2}\left(\frac{1}{840}\unicode[STIX]{x1D6E9}_{tt}+\frac{11}{5880}\unicode[STIX]{x1D6E9}_{xt}+\frac{101}{141\,120}\unicode[STIX]{x1D6E9}_{xx}\right).\end{eqnarray}$$

The weighted-residual equation (4.14) involves second derivatives with respect to time and space of both $S$ and $\unicode[STIX]{x1D6E9}$ ; its compact form is

(4.15) $$\begin{eqnarray}{\mathcal{L}}_{W1}S(x,t)={\mathcal{L}}_{W2}\unicode[STIX]{x1D6E9}(x,t),\end{eqnarray}$$

where both ${\mathcal{L}}_{W1}$ and ${\mathcal{L}}_{W2}$ are second-order linear operators.

For steady heating, the solution to (4.14) is in agreement with (4.7) for small $\unicode[STIX]{x1D6FF}$ at up to second order. However, solutions to (4.14) do not always obey property P1 (surface amplitude always smaller than wall amplitude) and may also violate P2 (surface amplitude vanishes at short wavelength). For example, when $Pe=6$ and $Bi=0$ , the coefficient of $S_{xx}$ in (4.14) is close to zero and $S\sim -1.63\unicode[STIX]{x1D6E9}$ as $k\rightarrow \infty$ , thus violating properties P1 and P2. For general parameter values, the steady heat transmission amplitude tends to a non-zero constant for large $k$ (governed by the coefficients of the $\unicode[STIX]{x1D6E9}_{xx}$ and $S_{xx}$ terms), again contradicting property P2. However, all temporal eigenvalues $\unicode[STIX]{x1D706}$ have negative real part, and so the state $S=0$ is stable if $\unicode[STIX]{x1D6E9}=0$ ; hence P3 is satisfied.

4.3 Projection approach

The Benney model yields $S$ explicitly as a function of $\unicode[STIX]{x1D6E9}$ and its derivatives. In contrast, the weighted-residual model gives $S$ implicitly by solving an evolution equation, with a source term that involves $\unicode[STIX]{x1D6E9}$ and its derivatives. Both models are equivalent in the long-wave limit, but the weighted-residual model is better behaved in the short-wave limit in that the surface temperature is bounded. Here we go one step further, seeking an evolution equation for $S$ in which only $\unicode[STIX]{x1D6E9}$ , and not its derivatives, is the source term, so that the schematic form should be

(4.16) $$\begin{eqnarray}{\mathcal{L}}_{P}S(x,t)=\unicode[STIX]{x1D6E9}(x,t),\end{eqnarray}$$

where ${\mathcal{L}}_{P}$ is a second-order linear operator.

Such a model can be derived as the ‘inverse’ counterpart to the Benney equation. We pose an expansion for small $\unicode[STIX]{x1D6FF}$ :

(4.17) $$\begin{eqnarray}\unicode[STIX]{x1D703}(X,y,T)=\unicode[STIX]{x1D703}_{0}(X,y,T)+\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D703}_{1}(X,y,T)+\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D703}_{2}(X,y,T)+\cdots \,.\end{eqnarray}$$

However, this time we treat $S(x,t)$ as a known, $O(1)$ quantity, determined by the long-wave PDE (4.1) and the boundary conditions

(4.18a,b ) $$\begin{eqnarray}\unicode[STIX]{x1D703}(X,1,T)=S(X,T),\quad \unicode[STIX]{x1D703}_{y}(X,1,T)=-Bi\,S(X,T).\end{eqnarray}$$

We expand the PDE (4.1) and the two boundary conditions (4.18) for small $\unicode[STIX]{x1D6FF}$ to obtain the sequence of PDEs (4.5) with boundary conditions $\unicode[STIX]{x1D703}_{0}=S$ , $\unicode[STIX]{x1D703}_{0y}=-Bi\,S$ at $y=1$ , and $\unicode[STIX]{x1D703}_{1,2}=0$ , $\unicode[STIX]{x1D703}_{1,2y}=0$ at $y=1$ . Given that $u(y)=y(2-y)$ , the solutions for $\unicode[STIX]{x1D703}_{0}$ , $\unicode[STIX]{x1D703}_{1}$ and $\unicode[STIX]{x1D703}_{2}$ are polynomials in $y$ , with $\unicode[STIX]{x1D703}_{0}$ involving $S(X,T)$ , $\unicode[STIX]{x1D703}_{1}$ featuring $S_{X}$ and $S_{T}$ , and $\unicode[STIX]{x1D703}_{2}$ involving first and second derivatives of $S$ . With $\unicode[STIX]{x1D703}(x,y,t)$ known by truncating the small- $\unicode[STIX]{x1D6FF}$ asymptotic sequence after $O(\unicode[STIX]{x1D6FF})^{2}$ , we finally impose the boundary condition at the wall: $\unicode[STIX]{x1D703}=\unicode[STIX]{x1D6E9}(x,t)$ at $y=0$ (i.e. projecting the expansion onto $y=0$ ). This statement requires

(4.19) $$\begin{eqnarray}\displaystyle & & \displaystyle (1+Bi)S+Pe\left[\left(\frac{1}{2}+\frac{Bi}{6}\right)S_{t}+\left(\frac{5}{12}+\frac{7Bi}{60}\right)S_{x}\right]\nonumber\\ \displaystyle & & \displaystyle \quad +\,Pe^{2}\left[\left(\frac{1}{24}+\frac{Bi}{120}\right)S_{tt}+\left(\frac{23}{360}+\frac{29Bi}{2520}\right)S_{xt}\right]\nonumber\\ \displaystyle & & \displaystyle \quad +\,\left[Pe^{2}\left(\frac{239}{10\,080}+\frac{13Bi}{3360}\right)-\left(\frac{1}{2}+\frac{Bi}{6}\right)\right]S_{xx}=\unicode[STIX]{x1D6E9}(x,t),\end{eqnarray}$$

with $O(\unicode[STIX]{x1D6FF}^{3})$ error. The consistency condition (4.19) is in fact a single evolution equation for $S$ , second-order in space and time, with $\unicode[STIX]{x1D6E9}$ as a source term, with the compact form (4.16). The equation does not involve any derivatives of  $\unicode[STIX]{x1D6E9}$ .

For steady heating, it is possible to show that the model obeys both property P1 (surface amplitude always smaller than wall amplitude) and property P2 (surface amplitude vanishes at short wavelength). Furthermore, the amplitude of surface variation decays monotonically as the wavenumber increases. The temporal eigenvalues always have negative real part, and so the state $S=0$ is stable, thus satisfying property P3 (if the wall is held at fixed temperature, the fluid temperature will always evolve towards zero). We will discuss properties P4 and P5 later (see § 5) as these do not relate to individual Fourier modes.

4.4 Other second-order models with special properties

We have highlighted above three long-wave models that each agree to second order in the long-wave limit. The Benney model and our new projected model are special in that they give $S(x,t)$ and $\unicode[STIX]{x1D6E9}(x,t)$ explicitly, respectively. The weighted-residual model is between these two extremes, and is one of a family of linear second-order models, of which the general form is

(4.20) $$\begin{eqnarray}aS+bS_{x}+cS_{xx}+dS_{t}+eS_{xt}+f\,S_{tt}=\unicode[STIX]{x1D6E9}+B\unicode[STIX]{x1D6E9}_{x}+C\unicode[STIX]{x1D6E9}_{xx}+D\unicode[STIX]{x1D6E9}_{t}+E\unicode[STIX]{x1D6E9}_{xt}+F\unicode[STIX]{x1D6E9}_{tt}.\end{eqnarray}$$

This equation has 11 unknown coefficients, which should be constant in space and time, but could depend on $Pe$ and  $Bi$ .

We can analyse the solution behaviour for slowly varying heating by considering solutions of the form

(4.21a,b ) $$\begin{eqnarray}\unicode[STIX]{x1D6E9}(x,t)=\exp (\text{i}\unicode[STIX]{x1D6FF}Kx+\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6EC}t),\quad S(x,t)={\hat{S}}\exp (\text{i}\unicode[STIX]{x1D6FF}Kx+\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6EC}t),\end{eqnarray}$$

with $\unicode[STIX]{x1D6FF}\ll 1$ . The solution of (4.20) for ${\hat{S}}$ is

(4.22) $$\begin{eqnarray}{\hat{S}}=\frac{1+\text{i}\unicode[STIX]{x1D6FF}KB-\unicode[STIX]{x1D6FF}^{2}K^{2}C+\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6EC}D+\unicode[STIX]{x1D6FF}^{2}\text{i}K\unicode[STIX]{x1D6EC}E+\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D6EC}^{2}F}{a+\text{i}\unicode[STIX]{x1D6FF}Kb-\unicode[STIX]{x1D6FF}^{2}K^{2}c+\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6EC}d+\unicode[STIX]{x1D6FF}^{2}\text{i}K\unicode[STIX]{x1D6EC}e+\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D6EC}^{2}f}.\end{eqnarray}$$

Expanding (4.22) for small $\unicode[STIX]{x1D6FF}$ , we obtain

(4.23) $$\begin{eqnarray}\displaystyle {\hat{S}} & = & \displaystyle \frac{1}{a}+\unicode[STIX]{x1D6FF}\left(\text{i}K\frac{Ba-b}{a^{2}}+\unicode[STIX]{x1D6EC}\frac{Da-d}{a^{2}}\right)\nonumber\\ \displaystyle & & \displaystyle +\,\unicode[STIX]{x1D6FF}^{2}\left(K^{2}\frac{-Ca^{2}+(Bb+c)a-b^{2}}{a^{3}}-\text{i}K\unicode[STIX]{x1D6EC}\frac{Ea^{2}-(Bc+Db+e)a+2bd}{a^{3}}\right.\nonumber\\ \displaystyle & & \displaystyle \left.+\,\unicode[STIX]{x1D6EC}^{2}\frac{Fa^{2}-(Dd+f)a+d^{2}}{a^{3}}\right)+O(\unicode[STIX]{x1D6FF}^{3}).\end{eqnarray}$$

For (4.20) to be second-order-accurate for long-wave heating, we require the expansion (4.23) to agree with that from the Benney equation for terms up to $\unicode[STIX]{x1D6FF}^{2}$ , for arbitrary $K$ and $\unicode[STIX]{x1D6EC}$ . Agreement only occurs if the coefficients match exactly, yielding six algebraic equations involving the 11 coefficients $a,b,\ldots ,F$ , but up to five parameters that can be freely chosen. The six conditions for second-order consistency cannot be satisfied by models with only first-order derivatives; at least one second-order derivative is required.

A particularly appealing model structure would be a diffusion-like equation, involving an $S_{xx}$ term but not $S_{xt}$ or $S_{tt}$ , and no second derivatives of $\unicode[STIX]{x1D6E9}$ . Such a model can be derived in two ways: either require that $D=E=F=e=f=0$ above, and choose the remaining coefficients to be second-order-accurate in the long-wave limit (this yields a unique solution for the coefficients); or differentiate the first-order counterpart of (4.19)

(4.24) $$\begin{eqnarray}(1+Bi)S+Pe\left[\left(\frac{1}{2}+\frac{Bi}{6}\right)S_{t}+\left(\frac{5}{12}+\frac{7Bi}{60}\right)S_{x}\right]=\unicode[STIX]{x1D6E9}(x,t)+O(\unicode[STIX]{x1D6FF}^{2}),\end{eqnarray}$$

with respect to $x$ and $t$ to eliminate $S_{xt}$ and $S_{tt}$ in the projected model. Either derivation yields the same result. For the particular case $Bi=0$ , the resulting equation is

(4.25) $$\begin{eqnarray}S+{\displaystyle \frac{5Pe}{12}}S_{t}+{\displaystyle \frac{43Pe}{120}}S_{x}-\left({\displaystyle \frac{1}{2}}+{\displaystyle \frac{Pe^{2}}{1680}}\right)S_{xx}=\unicode[STIX]{x1D6E9}-\frac{Pe}{12}\unicode[STIX]{x1D6E9}_{t}-\frac{7Pe}{120}\unicode[STIX]{x1D6E9}_{x}.\end{eqnarray}$$

This is a second-order, parabolic evolution equation for $S$ , asymptotically correct to second order, for both steady and time-dependent heating. When extended to non-zero $Bi$ , the coefficients of (4.25) become rational functions of $Pe$ and $Bi$ (note that for no choice of $Bi$ and $Pe$ are any of the coefficients equal to zero).

The parabolic model (4.25) satisfies properties P1–P3 for arbitrary $Bi\geqslant 0$ and $Pe\geqslant 0$ . For $\unicode[STIX]{x1D6E9}=\text{e}^{\text{i}kx}$ , $S/\unicode[STIX]{x1D6E9}$ decays as $O(k^{-1})$ for long-wave heating, which is better than the weighted-residual model but worse than the new projected model. We will show later that this model does not obey property P5; we can construct examples with negative surface temperature, see figure 5.

5.1 Fourier signature

The first test concerns the relationship between surface temperature and wall temperature for steady, sinusoidal heating. In this case, we set $\unicode[STIX]{x1D6E9}=\exp (\text{i}kx)$ and $S={\hat{S}}\exp (\text{i}kx)$ . Figure 3 shows results for ${\hat{S}}$ as a function of $k$ for the full 2-D system (3.3) and (3.4), the Benney model (4.7), the weighted-residual model (4.14), our new projected model (4.19) and the parabolic model (4.25). As expected, all results agree in the long-wave limit, but the Benney model rapidly diverges from the full 2-D results as $k$ increases. For $Pe=1$ , the projected, weighted-residual and parabolic models give similar results and are close to the full 2-D results, though at large $k$ , the projected model is in best agreement. At $Pe=10$ , only the projected model is still close to the full 2-D results and is in fact in very good agreement over the whole range of  $k$ .

Figure 4.

The real part of the eigenvalue $\unicode[STIX]{x1D706}$ for unforced time decay in the 2-D model as a function of $k$ for the models shown in figure 3. (Note that the Benney model has no eigenvalues, while the full system has infinitely many; the two shown here are the slowest-decaying.) As for figure 3, we take $Bi=0$ , with (a) $Pe=1$ and (b)  $Pe=10$ . At $k=0$ , the exact eigenvalues are transcendental ( $\unicode[STIX]{x1D706}Pe=-(\unicode[STIX]{x03C0}/2+n\unicode[STIX]{x03C0})^{2}$ ), and hence cannot be in perfect agreement with any of our long-wave models. However, the long-wave eigenvalues with real part closest to zero are all within 15 % of the exact value. The long-wave predictions for the second eigenvalue are in very poor agreement, but in all cases this eigenmode decays rapidly and should not be dynamically significant.

We can also analyse the time decay of a fluid layer with a sinusoidal temperature distribution and an unheated wall, corresponding to the relaxation to a uniform state. Hence we set $\unicode[STIX]{x1D6E9}=0$ and $S={\hat{S}}\exp (\text{i}kx+\unicode[STIX]{x1D706}t)$ , and solve for $\unicode[STIX]{x1D706}$ to give non-trivial solutions. Results for each system are shown for the case that $Bi=0$ and $Pe=1,10$ in figure 4. The Benney model is not shown, as decay is instantaneous. For each long-wave model shown in figure 4, all eigenvalues have negative real part and the eigenvalue corresponding to the slowest-decaying mode are in broad agreement (within 15 %) of the full system results at small $k$ . As $k$ increases, the temporal decay for each eigenmode becomes more rapid in all models. The full system has an infinite sequence of decaying eigenmodes for each $k$ , while the parabolic model has one, and the weighted-residual and parabolic equations have two eigenvalues for every $k$ . When comparing the decay rate of the slowest-decaying mode, the parabolic model is in the best quantitative agreement with the full system over a wide range of $k$ . The long-wave predictions for the second eigenvalue are quantitatively very poor, but in both weighted-residual and projected models these modes are predicted to decay rapidly, and the eigenvalues for the first and second mode for a given model do not cross for any $Pe$ , $Bi$ and  $k$ .

5.2 Response to steady strip-like localized heating

Although there are significant differences between the models at large $k$ , it is not obvious how important this will be in practice. We now consider the surface temperature that results from a localized, strip-like heat source, with width $2X$ and centred at $x=0$ , represented as

(5.1) $$\begin{eqnarray}\unicode[STIX]{x1D6E9}(x)={\textstyle \frac{1}{2}}[\tanh \{\unicode[STIX]{x1D70E}(x+X)\}-\tanh \{\unicode[STIX]{x1D70E}(x-X)\}],\end{eqnarray}$$

in an infinite (open) domain or as

(5.2) $$\begin{eqnarray}\unicode[STIX]{x1D6E9}(x)=\frac{1}{2}\left[1+\tanh \left\{\frac{\unicode[STIX]{x1D70E}(\cos (mx)-\cos (mX))}{m\sin mX}\right\}\right],\end{eqnarray}$$

in a periodic domain with period $L=2\unicode[STIX]{x03C0}/m$ . The sharpness of the strip is controlled by $\unicode[STIX]{x1D70E}$ . We take $\unicode[STIX]{x1D70E}=10$ and $X=2$ , so that the wall temperature is almost constant in a region of length 4, and positive but close to zero outside this region.

Figure 5.

The steady temperature distribution $S(x)$ for the full system (solving (2.13), (2.19) and (3.1) via Fourier transforms in $x$ and finite-differencing in $y$ ) and the four long-wave models for $Bi=0$ and (a) $Pe=1$ and (b) $Pe=10$ with $\unicode[STIX]{x1D6E9}(x)$ given by the strip-like profile (5.1) and shown here by the shaded region. The flow is from left to right.

The surface temperature as predicted by the full 2-D system and each long-wave model is shown in figure 5. We find that the Benney model gives nonsensical results for both $Pe=1$ and $Pe=10$ , predicting significant regions with negative temperature, and very large deviations from the true results. At $Pe=1$ , the other three long-wave models are all in good agreement with the full system. However, for $Pe=10$ , the parabolic and weighted-residual predictions exhibit extra turning points. The parabolic model predicts a region with negative surface temperature near the upstream end of the heating strip, thus violating property P5. By contrast, the projected model is in good agreement with the full results at both $Pe=1$ and $Pe=10$ . In the remainder of this study, we will focus our analysis on the new projected model (4.19), which has performed well at every test so far.

5.3 Sensitivity to a critical $Pe$

The equation for the projected model (4.19) takes the form

(5.3) $$\begin{eqnarray}aS+bS_{x}+cS_{xx}+dS_{t}+eS_{xt}+f\,S_{tt}=\unicode[STIX]{x1D6E9}(x,t),\end{eqnarray}$$

with coefficients

(5.4) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}a=1+Bi,\quad b=Pe\left({\displaystyle \frac{5}{12}}+{\displaystyle \frac{7Bi}{60}}\right),\quad c=Pe^{2}\left({\displaystyle \frac{239}{10\,080}}+{\displaystyle \frac{13Bi}{3360}}\right)-\left({\displaystyle \frac{1}{2}}+{\displaystyle \frac{Bi}{6}}\right),\\ d=Pe\left({\displaystyle \frac{1}{2}}+{\displaystyle \frac{Bi}{6}}\right),\quad e=Pe^{2}\left({\displaystyle \frac{23}{360}}+{\displaystyle \frac{29Bi}{2520}}\right),\quad f=Pe^{2}\left({\displaystyle \frac{1}{24}}+{\displaystyle \frac{Bi}{120}}\right).\end{array}\right\}\end{eqnarray}$$

The coefficients $a$ , $b$ , $d$ , $e$ and $f$ are always positive for $Pe>0$ and $Bi\geqslant 0$ . In contrast, $c$ changes sign at a critical value of $Pe$ , which we will denote as $Pe^{\ast }$ . This critical value is a function of $Bi$ (plotted in figure 6) and increases monotonically with $Bi$ , from $Pe^{\ast }=4.592$ at $Bi=0$ to $Pe^{\ast }=6.563$ as $Bi\rightarrow \infty$ . (Note that the weighted-residual equation (4.14) also has a critical $Pe^{\ast }$ , with $6.297<Pe^{\ast }(Bi)<11.277$ . The parabolic equation (4.25) has no such sensitivity.)

Figure 6.

The coefficient $c$ in (5.3) is equal to zero at $Pe=Pe^{\ast }$ . As a result, (5.3) is diffusion-dominated for $Pe<Pe^{\ast }$ in the sense that its two real characteristics travel in opposite directions, so that a localized heat source has both upstream and downstream influence on the surface temperature. For $Pe>Pe^{\ast }$ , the system is advection-dominated; both characteristics travel downstream, and a localized heat source has no upstream influence. At large $Pe$ and large $Bi$ , a third region arises (bordered by the condition $b^{2}-4ac=0$ ), where both characteristics travel downstream, but the solutions for $r$ in (5.5) are complex. It is in this region only that our model can predict negative surface temperatures arising from positive wall heating, potentially violating P5.

Viewed as a time-dependent PDE, (5.3) is hyperbolic if $Pe>0$ and $Bi\geqslant 0$ . For $Pe<Pe^{\ast }$ , its two families of characteristics travel in opposite directions, while for $Pe>Pe^{\ast }$ , both characteristics travel downstream. Hence if a localized disturbance is introduced, at small $Pe$ the influence will be felt both upstream and downstream, whereas at large $Pe$ , there is no upstream effect.

The steady influence of a localized heating strip has a similar dependence on $Pe$ . Steady solutions for $S(x)$ corresponding to a single strip heater in an infinite domain are shown in figure 7 as $Pe$ varies from 0 to 40. At $Pe=0$ , the maximum surface temperature occurs at $x=0$ , in the middle of the strip. As $Pe$ increases, the location of this maximum temperature moves downstream, the maximum value decreases, and the spatial decay is generally slower; in all cases $S(x)\geqslant 0$ everywhere. Figure 7 also shows full 2-D results computed from (4.2) and (2.13) using Fourier transforms in $x$ with a finite-difference discretization in $y$ . The agreement between the full system and the new projected model is generally good. The most significant discrepancies occur when $Pe$ is close to $Pe^{\ast }$ where the model becomes singular in that the coefficient of $S_{xx}$ vanishes, resulting in the surface temperature profile predicted by the long-wave model displaying an unphysical rapid change in gradient near the two ends of the smoothed heating strip.

Figure 7.

The surface temperature profile $S(x)$ corresponding to the strip heating given by (5.1) (shown by the black line) using the projected model (4.19) (dashed) and the full 2-D equations (solid lines), both computed in an open domain. Here $Bi=0$ , with solutions shown for $Pe=0,1,2,4,6,8,10,20$ and 40. Increasing $Pe$ decreases the maximum surface temperature and moves the location of this maximum downstream.

Variation of parameters allows us to construct solutions to the steady-state equation:

(5.5a,b ) $$\begin{eqnarray}aS+bS_{x}+cS_{xx}=\unicode[STIX]{x1D6E9}(x),\quad r=\frac{-b\pm \sqrt{b^{2}-4ac}}{2c},\end{eqnarray}$$

and hence to demonstrate the positivity of the solution. The complementary function involves the two solutions of the form $S=\exp (rx)$ , where $r$ is defined in (5.5b ). When $Pe<Pe^{\ast }$ , we have $a,b>0$ but $c<0$ ; the roots for $r$ are both real but of opposite sign, so we can take $r_{2}<0<r_{1}$ and the appropriate boundary conditions are that $S\rightarrow 0$ as $x\rightarrow \pm \infty$ . Supposing that $S=0$ except inside a strip $-z<x<z$ , we can write the solution for general $\unicode[STIX]{x1D6E9}(x)$ as

(5.6) $$\begin{eqnarray}S(x)=\left\{\begin{array}{@{}ll@{}}{\displaystyle \frac{\text{e}^{r_{1}x}}{c(r_{2}-r_{1})}}\displaystyle \int _{-z}^{z}\unicode[STIX]{x1D6E9}(x^{\prime })\text{e}^{-r_{1}x^{\prime }}\,\text{d}x^{\prime },\quad & x<-z,\\ {\displaystyle \frac{1}{c(r_{2}-r_{1})}}\displaystyle \int _{-z}^{x}\unicode[STIX]{x1D6E9}(x^{\prime })[\text{e}^{-r_{2}(x^{\prime }-x)}-\text{e}^{-r_{1}(x^{\prime }-x)}]\,\text{d}x^{\prime },\quad & -z<x<z,\\ {\displaystyle \frac{\text{e}^{r_{2}x}}{c(r_{2}-r_{1})}}\displaystyle \int _{-z}^{z}\unicode[STIX]{x1D6E9}(x^{\prime })\text{e}^{-r_{2}x^{\prime }}\,\text{d}x^{\prime },\quad & x>z.\end{array}\right.\end{eqnarray}$$

The temperature profile decays exponentially with respect to $x$ outside the localized heating region. We know that $c(r_{2}-r_{1})>0$ , so if $\unicode[STIX]{x1D6E9}(x)\geqslant 0$ , we must have $S(x)\geqslant 0$ everywhere. This proves property P5 for the case $Pe<Pe^{\ast }$ .

If $Pe>Pe^{\ast }$ , both $r_{1}$ and $r_{2}$ have negative real part. With boundary condition $S\rightarrow 0$ as $x\rightarrow -\infty$ , the solution is

(5.7) $$\begin{eqnarray}S(x)=\frac{1}{c(r_{2}-r_{1})}\int _{-\infty }^{x}\unicode[STIX]{x1D6E9}(x^{\prime })(\text{e}^{r_{2}(x-x^{\prime })}-\text{e}^{r_{1}(x-x^{\prime })})\,\text{d}x^{\prime }.\end{eqnarray}$$

If $r_{1}$ and $r_{2}$ are real, and $\unicode[STIX]{x1D6E9}(x)\geqslant 0$ for all $x$ , then the integral solution (5.7) means that $S(x)\geqslant 0$ for all $x$ . Hence for parameter values where $r_{1}$ and $r_{2}$ are real, regardless of $Pe$ , we have shown that for arbitrary positive substrate heating, the surface temperature is also positive; thus addressing property P5.

The only condition under which the projected model can predict negative surface temperature with non-negative wall temperature is if the roots for $r$ are complex, as then the integrand in (5.7) is oscillatory. This regime is indicated as ‘Complex conjugate roots’ in figure 6, and can only occur if both $Bi\geqslant 3.87$ and $Pe>18.9$ . Most physical studies involve small values of $Bi$ , and so this condition is not particularly restrictive. We note that none of the other models have any comparable positivity results; the integrands would involve linear combinations of $\unicode[STIX]{x1D6E9}$ and its derivatives with respect to $x$ , and could not be bounded using only conditions on the sign of $\unicode[STIX]{x1D6E9}$ .

The derivation of our new model supplies an approximation of the interior temperature profile in addition to an equation for the surface temperature. Contour plots of this interior temperature are shown in figure 8 for various $Pe$ with $Bi=0$ in response to strip heating. The predicted interior temperature profiles appear broadly similar to the full 2-D results. The biggest deviations are for $Pe=4$ and $Pe=6$ , which are closest to the critical value of $Pe^{\ast }=4.592$ . Even for these values of $Pe$ , the contours are in very good agreement away from the regions of rapid change.

Figure 8.

Contour plots of temperature profile $\unicode[STIX]{x1D703}(x,y)$ within the film in an open domain, with wall temperature given by the strip profile (5.1),  $Bi=0$ , and calculated using our new model (4.19) (red) and the full 2-D system (black), solving (2.13) with (3.1) using Fourier transforms in $x$ and a finite-difference discretization in $y$ . The aspect ratio is $1:1$ . The contours correspond to $\unicode[STIX]{x1D703}=0.05,0.15,0.25,\ldots ,0.85,0.95$ .

Figure 9.

Contour plot of $\unicode[STIX]{x1D703}(x,y)$ within the film, as predicted by the new projected model (4.19), with wall temperature (5.1). Here $Pe=4.5$ and $Bi=0$ , so $a=1$ , $b=1.9875$ and $c=-0.01987$ . For $Bi=0$ , $c=0$ at $Pe^{\ast }=4.592$ , so this case is close to the change in criticality. Although $S(x)\geqslant 0$ here, the internal temperature field has a region where $\unicode[STIX]{x1D703}<0$ (inside the blue contours near $x=-2$ ), and also a local maximum (inside the red contour near $x=2$ ).

Our final thermodynamic consistency check comes from property P4; there should be no extrema of the temperature inside the fluid layer. For the cases shown in figure 8, this property is satisfied. However, if we repeat the calculations from figure 8 with $Pe$ close to the critical value $Pe^{\ast }$ , we can construct cases with negative internal temperature. In figure 9, we show a contour plot of the internal temperature for one such case, with $Bi=0$ and $Pe=4.5$ . Although the surface temperature is non-negative, there is a region of negative temperature inside the layer just upstream of the heating strip and also a local maximum of $\unicode[STIX]{x1D703}$ inside the fluid layer above the downstream end of the heating strip.

5.4 Time-periodic heating

We have shown that our new model (4.19) successfully predicts steady heat transfer through a uniform flow, at least for cases that are not too near to the critical Péclet number. However, time-dependent heating is also important, particularly for applications involving real-time control. The simplest test of time dependence is to take the applied heating and resulting temperature profile to be sinusoidal in time. For the long-wave models, we can obtain solutions straightforwardly using Fourier transforms in $x$ . Fourier transforms are also appropriate for the full 2-D system, though we must numerically solve an ordinary differential equation (ODE) in $y$ at each $k$ . The results of this calculation for a periodically oscillating strip are shown in figure 10, and show very good agreement between the projected and full system for these parameter values. Note that, in a full initial-value problem, the periodic response should be combined with a full start-up problem, but the decay of transients in this initial problem has been addressed in § 5.1.

Figure 10.

Time-dependent response of surface temperature $S$ to heating of the form $\unicode[STIX]{x1D6E9}(x,t)=\text{Re}\{f(x)\exp (\text{i}\unicode[STIX]{x1D714}t)\}$ with $\unicode[STIX]{x1D714}=0.5$ and $f(x)$ given by (5.2), at a selection of instants over one period ( $T=2\unicode[STIX]{x03C0}/\unicode[STIX]{x1D714}$ ). (Here $\unicode[STIX]{x1D6E9}(x,0)=f(x)$ is shown by the black solid line.) Results are shown for the full system (solid lines) using Fourier transforms of (2.13) in $x$ and finite-differencing in $y$ , and the new projected model (4.19) (dashed), with $Bi=0$ and (a) $Pe=1$ and (b) $Pe=10$ . For $Pe=1$ , $S(x,t)$ feels the influence of the heating strip even upstream, whereas for $Pe=10$ , the influence is felt strictly downstream.

6.1 Effect of surface temperature on film equations

We use a two-equation, weighted-residual model for the film dynamics, coupling the film height $h(x,t)$ and the depth-integrated flux $q(x,t)$ , which naturally satisfy

(6.1a,b ) $$\begin{eqnarray}q(x,t)=\int _{0}^{h(x,t)}u(x,y,t)\,\text{d}y,\quad h_{t}+q_{x}=0,\end{eqnarray}$$

where the second equation follows from mass conservation.

We follow the standard weighted-residual approach, seeking to include surface tension, inertia, axial diffusion and Marangoni stresses. In order to retain second-order terms relating to Marangoni effects, and including some interactions between inertia and Marangoni stresses, we set $Re=O(\unicode[STIX]{x1D6FF})$ , $Ma=O(\unicode[STIX]{x1D6FF}^{-1})$ , $Ca=O(\unicode[STIX]{x1D6FF}^{2})$ and reach

(6.2) $$\begin{eqnarray}\displaystyle q+\frac{2}{5}Re\,h^{2}q_{t} & = & \displaystyle \frac{h^{3}}{3}\left(2-2h_{x}\cot \unicode[STIX]{x1D6FD}+\frac{[(1-Ma\,Ca\,S)h_{xx}]_{x}}{Ca}\right)+Re\left(\frac{18q^{2}h_{x}}{35}-\frac{34hqq_{x}}{35}\right)\nonumber\\ \displaystyle & & \displaystyle +\,\left(\frac{8}{5}qh_{x}^{2}-\frac{9}{5}hh_{x}q_{x}-\frac{12}{5}qhh_{xx}+\frac{9}{5}q_{xx}h^{2}\right)-\frac{1}{2}Ma\,h^{2}S_{x}\nonumber\\ \displaystyle & & \displaystyle +\,Ma\,Re\left(\frac{qh^{2}h_{x}S_{x}}{56}+\frac{19h^{3}S_{x}q_{x}}{840}+\frac{3h^{3}qS_{xx}}{112}+\frac{h^{4}S_{xt}}{120}\right)\nonumber\\ \displaystyle & & \displaystyle +\,Ma\left(-\frac{h^{4}S_{xxx}}{10}-\frac{3h_{x}S_{xx}h^{3}}{10}+\frac{3h_{x}^{2}S_{x}h^{2}}{10}-\frac{h_{xx}S_{x}h^{3}}{15}\right)\nonumber\\ \displaystyle & & \displaystyle -\,Ma^{2}Re\left(\frac{19}{1680}h^{4}S_{x}^{2}h_{x}+\frac{5h^{5}S_{x}S_{xx}}{672}\right).\end{eqnarray}$$

In this case the Marangoni number appears both through tangential stresses and as variable surface tension from the normal stress balance.

6.2 Heat transfer through a non-uniform film

The heat equation is affected by film non-uniformity in two ways: the boundary conditions should be applied at $y=h(x,t)$ , and the velocity field $(u,v)$ appears directly in the heat equation. The first effect here is easy to deal with, but, for the second, we need a suitable characterization of the velocity field. Noting that the derivation of the equation for surface temperature makes use of $u$ and its derivatives at the surface, it is reasonable to use the flux-parametrized velocity profile, as is also used in the derivation of the weighted-residual hydrodynamic equations:

(6.3a,b ) $$\begin{eqnarray}u=\frac{3qy(2h-y)}{2h^{3}},\quad v=\unicode[STIX]{x1D6FF}\hat{v}=\unicode[STIX]{x1D6FF}\left(\frac{3qh_{X}y^{2}(2h-y)}{2h^{4}}-\frac{q_{X}y^{2}(3h-y)}{2h^{3}}\right).\end{eqnarray}$$

This velocity field is incompressible, and obeys $u_{y}=0$ and $h_{T}=-q_{X}=\hat{v}-uh_{X}$ at $y=h$ .

As for the uniform film case, we derive the projected heat equation by posing an expansion of $\unicode[STIX]{x1D703}(X,y,T)$ for small $\unicode[STIX]{x1D6FF}$ in the form (4.4). The boundary conditions on $\unicode[STIX]{x1D703}_{0}$ , $\unicode[STIX]{x1D703}_{1}$ and $\unicode[STIX]{x1D703}_{2}$ are obtained from expanding the two conditions

(6.4a,b ) $$\begin{eqnarray}\unicode[STIX]{x1D703}=S(X,T),\quad \unicode[STIX]{x1D703}_{y}-\unicode[STIX]{x1D6FF}^{2}h_{X}\unicode[STIX]{x1D703}_{X}=-Bi\,S(1+\unicode[STIX]{x1D6FF}^{2}h_{X}^{2})^{1/2},\quad \text{on }y=h(X,T),\end{eqnarray}$$

in powers of $\unicode[STIX]{x1D6FF}$ . Similarly, the PDEs obeyed by $\unicode[STIX]{x1D703}_{0}$ , $\unicode[STIX]{x1D703}_{1}$ and $\unicode[STIX]{x1D703}_{2}$ are determined from a small- $\unicode[STIX]{x1D6FF}$ expansion of

(6.5) $$\begin{eqnarray}\unicode[STIX]{x1D6FF}Pe(\unicode[STIX]{x1D703}_{T}+u\unicode[STIX]{x1D703}_{X}+\hat{v}\unicode[STIX]{x1D703}_{y})=\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D703}_{XX}+\unicode[STIX]{x1D703}_{YY}.\end{eqnarray}$$

We close the model with the condition that $\unicode[STIX]{x1D703}(X,0,T)=\unicode[STIX]{x1D6E9}(X,T)$ .

The resulting equation involves time derivatives of the hydrodynamic variables $h$ and $q$ . We can use $h_{T}=-q_{X}$ to eliminate time derivatives of $h$ , to reach (in terms of the original variables)

(6.6) $$\begin{eqnarray}\displaystyle & & \displaystyle \hspace{-5.0pt}Pe^{2}\left(\frac{1}{24}+\frac{Bi\,h}{120}\right)h^{4}S_{tt}+Pe^{2}\left(\frac{161}{1680}+\frac{29Bi\,h}{1680}\right)qh^{3}S_{xt}\nonumber\\ \displaystyle & & \displaystyle \hspace{-5.0pt}\qquad +\,\left[Pe^{2}q^{2}\left(\frac{239}{4480}+\frac{39Bi\,h}{4480}\right)-\left(\frac{1}{2}+\frac{Bi\,h}{6}\right)\right]h^{2}S_{xx}\nonumber\\ \displaystyle & & \displaystyle \hspace{-5.0pt}\qquad +\,Pe\left[\left(\frac{1}{2}+\frac{Bi\,h}{6}\right)h^{2}+\left(\frac{3}{40}+\frac{17Bi\,h}{560}\right)qh^{2}h_{x}Pe-\left(\frac{13}{120}+\frac{73Bi\,h}{1680}\right)h^{3}q_{x}Pe\right]S_{t}\nonumber\\ \displaystyle & & \displaystyle \hspace{-5.0pt}\qquad +\,\left[\left(\frac{5}{8}+\frac{7Bi\,h}{40}\right)Pe\,q+\left(Pe^{2}q^{2}\left(\frac{239}{4480}+\frac{117Bi\,h}{4480}\right)-(1+Bi\,h)\right)h_{x}\right.\nonumber\\ \displaystyle & & \displaystyle \hspace{-5.0pt}\qquad \left.+\,Pe^{2}\left\{-\left(\frac{173}{4480}+\frac{27Bi\,h}{896}\right)hqq_{x}+\left(\frac{7}{120}+\frac{3Bi\,h}{280}\right)h^{2}q_{t}\right\}\right]hS_{x}\nonumber\\ \displaystyle & & \displaystyle \hspace{-5.0pt}\qquad +\,\left[(1+Bi\,h)+Pe\,Bi\left(\frac{7}{40}hh_{x}q-\frac{9}{40}h^{2}q_{x}\right)+Bi\,h\left(Pe^{2}\frac{39}{4480}q^{2}-\frac{1}{2}\right)(h_{x}^{2}+hh_{xx})\right.\nonumber\\ \displaystyle & & \displaystyle \hspace{-5.0pt}\qquad +\,Bi\,Pe^{2}h^{2}\left.\left(\frac{1331}{40\,320}hq_{x}^{2}-\frac{97}{3360}h_{x}qq_{x}-\frac{271}{13\,440}hqq_{xx}+\frac{3}{280}hh_{x}q_{t}-\frac{1}{84}h^{2}q_{xt}\right)\right]S\nonumber\\ \displaystyle & & \displaystyle \hspace{-5.0pt}\quad =\unicode[STIX]{x1D6E9}(x,t).\end{eqnarray}$$

A substantial simplification arises if the system is steady, as then all derivatives of $q$ vanish, as do time derivatives of $S$ , leaving

(6.7) $$\begin{eqnarray}\displaystyle & & \displaystyle \hspace{-10.00002pt}\left[Pe^{2}q^{2}\left(\frac{239}{4480}+\frac{39Bi\,h}{4480}\right)-\left(\frac{1}{2}+\frac{Bi\,h}{6}\right)\right]h^{2}S_{xx}\nonumber\\ \displaystyle & & \displaystyle \hspace{-10.00002pt}\qquad +\,\left[Pe\,q\left(\frac{5}{8}+\frac{7Bi\,h}{40}\right)+h_{x}\left(Pe^{2}q^{2}\left(\frac{239}{4480}+\frac{117Bi\,h}{4480}\right)-(1+Bi\,h)\right)\right]hS_{x}\nonumber\\ \displaystyle & & \displaystyle \hspace{-10.00002pt}\qquad +\,\left(1+Bi\,h+\frac{7Pe\,Bi}{40}hh_{x}q+Bi\,h\left(Pe^{2}q^{2}\frac{39}{4480}-\frac{1}{2}\right)(h_{x}^{2}+hh_{xx})\right)S\nonumber\\ \displaystyle & & \displaystyle \hspace{-10.00002pt}\quad =\unicode[STIX]{x1D6E9}(x).\end{eqnarray}$$

We can also include the effect of Marangoni stresses on the flow field used to derive the heat equation:

(6.8a,b ) $$\begin{eqnarray}u=\frac{3qy(2h-y)}{2h^{3}}+\unicode[STIX]{x1D6FF}\frac{Ma\,S_{X}}{4}\frac{y(2h-3y)}{h},\quad \hat{v}=-\int _{0}^{y}u_{X}(x,y^{\prime })\,\text{d}y^{\prime }.\end{eqnarray}$$

For steady states, and assuming that $Ma$ and $Pe$ are both $O(1)$ , the Marangoni stresses have the effect of adding the term

(6.9) $$\begin{eqnarray}Pe\,Ma\left(-\frac{h^{3}S_{x}^{2}}{48}+Bi\left(\frac{1}{80}SS_{xx}h^{4}+\frac{1}{240}h^{4}S_{x}^{2}+\frac{7}{240}h^{3}SS_{x}h_{x}\right)\right),\end{eqnarray}$$

to the left-hand side of (6.7). (The expression required to describe Marangoni corrections to the unsteady equation (6.6) is considerably longer.)

7.1 Small displacement solutions

In the limit of small $Ma$ , we can obtain a solution by linearizing about a flat film. We let

(7.1a-d ) $$\begin{eqnarray}h=1+Ma\,H\text{e}^{\text{i}kx},\quad q={\textstyle \frac{2}{3}},\quad S={\hat{S}}\text{e}^{\text{i}kx}+O(Ma),\quad \unicode[STIX]{x1D6E9}=\text{e}^{\text{i}kx}.\end{eqnarray}$$

We can analyse the linear solution via a heat transfer problem, which relates $\unicode[STIX]{x1D6E9}$ and $S$ , and a hydrodynamic problem, which relates $S$ and $h$ . We solve the hydrodynamic problem (6.2) at $O(Ma)$ to obtain

(7.2) $$\begin{eqnarray}H=\frac{\text{i}k{\hat{S}}}{4}\frac{1-{\displaystyle \frac{\text{i}kRe}{28}}-{\displaystyle \frac{k^{2}}{5}}}{1-{\displaystyle \frac{\text{i}k\cot \unicode[STIX]{x1D6FD}}{3}}+{\displaystyle \frac{4\text{i}kRe}{35}}+{\displaystyle \frac{4k^{2}}{5}}-{\displaystyle \frac{\text{i}k^{3}}{6Ca}}}.\end{eqnarray}$$

The heat transfer problem, to determine ${\hat{S}}$ , is exactly the linear problem discussed in § 5.

For small displacements of the fluid surface, we can use an Orr–Sommerfeld analysis to solve both the Navier–Stokes and heat equations and hence determine the equivalent full 2-D surface displacement. The results are shown in figure 11 for comparison with the long-wave results, in a case with strong surface tension and low Reynolds number, so that the base flow is stable. The long-wave results are in very good agreement with those from the linearized 2-D system. Note that the half-width of the strip heating profile has been increased from $X=2$ to $X=5$ ; this leads to increased deflection of $h(x)$ and makes the surface shape easier to visualize.

Figure 11.

Solutions at small $Ma$ (with $h(x)=1+Ma\,{\hat{h}}(x)+O(Ma^{2})$ ) for the strip heating profile (5.1) with $Re=1$ , $Ca=0.01$ , $Bi=0$ , $\unicode[STIX]{x1D6FD}=\unicode[STIX]{x03C0}/4$ , with (a) $Pe=1$ and (b) $Pe=10$ , in a periodic domain with period 100. The full 2-D result is shown by the black solid line, the red dashed line is the long-wave result, and the blue line indicates the smoothed strip heater profile with half-width $X=5$ . Note that the heating strip leads to the formation of a capillary ridge above the upstream edge of the heater.

7.2 Finite-amplitude solutions

At small $Ma$ , the heat problem is exactly the same as that for flat films. Hence, to test the validity of the dependence of (6.7) on $h$ and $q$ , we must increase the deformation, most easily achieved by increasing $Ma$ . For large-amplitude solutions, we solve numerically in a periodic domain, applying the strip-like distribution (5.2) for $\unicode[STIX]{x1D6E9}$ and obtain steady states by continuation in $Ma$ from $h=1$ at $Ma=0$ , subject to the constraint that the mean layer height is fixed at 1. Results using the long-wave equations (6.2) and (6.7), computed using the bifurcation and continuation transfer package Auto 07p (Doedel & Oldman Reference Doedel and Oldman2009), are shown in figure 12. With all other parameters fixed, we find that increasing $Ma$ increases the surface deformation $h$ , accompanied by only small changes in the surface temperature profile  $S$ .

Figure 12.

Long-wave solutions for $Ma=1,5,10,15,10$ and $25$ at $Pe=1,5,10$ and $Bi=0,1$ , with $\unicode[STIX]{x1D6E9}$ given by (5.1) with $X=5$ and $\unicode[STIX]{x1D70E}=10$ . The equations solved are (6.2) and (6.7), excluding the $Pe\,Ma$ terms in (6.9). Dots show oomph-lib solutions of the 2-D Navier–Stokes equation and heat equation, while solid lines represent the long-wave model. All solutions are obtained by continuation from $Ma=0$ , except for the dashed solutions shown for the two $Pe=5$ cases, which are obtained by continuation in $Pe$ and $Ma$ from solutions at $Pe=1$ , $Ma=25$ . We cannot obtain long-wave solutions in the interval $14.67<Ma<14.88$ for $Bi=0$ , $Pe=5$ . The interval without solutions for the case $Bi=1$ , $Pe=5$ is much larger, with no converged long-wave solutions in $7.79<Ma<17.67$ .

Solutions are available for a wide range of $Ma$ (at least up to $Ma=25$ ), except for the two cases with $Pe=5$ . This value of $Pe=5$ is close to the critical $Pe$ for a uniform film with $h=1$ , $q=2/3$ for both $Bi=0$ (where $Pe^{\ast }=4.592$ ) and $Bi=1$ (where $Pe^{\ast }=4.917$ ), so that the coefficient of $S_{xx}$ , at $h=1$ and $q=2/3$ , is small but non-zero in both cases. Therefore, it is possible that the coefficient of $S_{xx}$ in the nonlinear calculations,

(7.3) $$\begin{eqnarray}\left[Pe^{2}q^{2}\left(\frac{239}{4480}+\frac{39Bi\,h}{4480}\right)-\left(\frac{1}{2}+\frac{Bi\,h}{6}\right)\right]h^{2},\end{eqnarray}$$

may vanish as $q$ and $h$ vary.

For $Bi=0$ , the sign of (7.3) is a function only of $Pe$ and $q$ , and would vanish at $q=0.6123$ for $Pe=5$ , for example. As $q$ is spatially constant in steady calculations, in fact this would mean that the coefficient of $S_{xx}$ vanishes everywhere in the domain simultaneously if $q$ exactly coincides with the critical value, but the equation would be non-singular for all other $q$ . In the case $Pe=5$ , $Bi=0$ , continuation from $Ma=0$ yields solutions up to $Ma=14.68$ , where $q=0.614208$ . In order to obtain solutions for larger $Ma$ , we use continuation from $Pe=1$ , $Bi=0$ , first increasing $Pe$ and then decreasing $Ma$ . Using this method, the Auto code yields converged solutions down to $Ma=14.88$ .

In contrast, for $Bi\neq 0$ , the dependence of the sign of the coefficient of (7.3) on $h$ means that the coefficient can vanish at some points in the domain but not others, and hence, for sufficiently large surface deformation, we expect the ODE (6.7) to have one or more singular points. Fixing $Pe=5$ , $Bi=1$ and $q=2/3$ , (7.3) would vanish at points where $h=1.326$ . However, the calculations shown in figure 12 are performed with a constraint on layer height, so that $q$ decreases slightly as $Ma$ increases. For $Bi=1$ , $Pe=5$ , the last converged solution when increasing $Ma$ from zero is for $Ma=7.7899$ , where $q=0.66237$ and the maximum value of $h$ is $1.15829$ . Using a continuation method from solutions at large $Ma$ for $Pe=1$ , we can also obtain solutions at $Pe=5$ , $Bi=1$ for $Ma>17.64$ , where $q=0.64321$ and the minimum value of $h$ is 0.684856. Comparison of these values of $Pe$ , $Bi$ , $h$ and $q$ with (7.3) suggests that the border of the region where we can compute solutions does indeed correspond to conditions where the coefficient of $S_{xx}$ locally vanishes.

The finite-amplitude calculations require 2-D Navier–Stokes simulations for validation. Steady solutions are calculated using the finite-element library oomph-lib (Heil & Hazel Reference Heil, Hazel, Schäfer and Bungartz2006). In this computation, Marangoni effects are created by specifying the variation of surface tension with temperature, leading to effects in both the normal and tangential components of the dynamic boundary condition, consistent with the boundary conditions (2.16) and (2.17) used to derive our long-wave equations. The air layer is treated as a region of constant pressure and temperature, and heat loss is parametrized by the same Biot-number condition as we used to derive the model. The Navier–Stokes calculations are performed in closed conditions: a periodic domain with mean layer height 1.

Several of the long-wave calculations are in remarkably good quantitative agreement with the full nonlinear Navier–Stokes results. In particular, for $Bi=1$ , the long-wave results shown in figure 12 are in excellent agreement for $h(x)$ across the range of $Ma$ where they exist. The predictions for $S(x)$ are also in good agreement at $Bi=1$ , capturing the notable increase in $S$ at the downstream end of the heater as $Ma$ increases. The agreement for $Bi=0$ is less good than for $Bi=1$ ; this is likely due to the generally reduced surface temperatures for larger $Bi$ leading to smaller variations in $h(x)$ at the same $Ma$ and hence less significant nonlinearity. For $Ma\gtrapprox 10$ and $Bi=0$ at the larger $Pe$ , the predictions for the upstream fluid ridge have the wrong shape when compared to the 2-D nonlinear results. This may be related to the observation that, while the long-wave results correctly predict an increasingly rapid decay of $S(x)$ downstream of the heater as $Ma$ increases, they do not capture the slowing of the upstream decay, which should lead to increasingly important upstream influence regions.

Figure 13.

Long-wave solutions (solid lines) and Navier–Stokes calculations (dotted lines) for $Ma=1,5,10,15,20$ and $25$ at $Pe=1,5,10$ and $Bi=0,1$ , with $\unicode[STIX]{x1D6E9}$ given by (5.1) with $X=5$ and $\unicode[STIX]{x1D70E}=10$ . These results include the $Pe\,Ma$ interactions in (6.9). All solutions shown here are computed by continuation from $Ma=0$ at fixed $Pe$ and $Bi$ . This continuation method does not yield long-wave solutions above $Ma=9.388$ for $Pe=5$ , $Bi=0$ , above $Ma=18.00$ for $Pe=10$ , $Bi=0$ , and above $Ma=8.240$ for $Pe=5$ , $Bi=1$ ; in each case this last converged solution is plotted with dashed lines.

Similar calculations including the Marangoni influences on the heat equation itself via (6.9) are shown in figure 13. For $Pe=1$ , $Bi=0$ , these correction terms lead to better agreement for $S(x)$ , particularly just upstream of the heating strip. There is also slightly improved agreement for $h(x)$ . However, for larger $Pe$ , the inclusion of $Pe\,Ma$ terms via (6.9) reduces the range of $Ma$ over which solutions can be obtained. In particular, note that for $Bi=0$ , we obtain solutions at $Pe=5$ up to $Ma=9.39$ and at $Pe=10$ up to $Ma=18.0$ . In both cases, the last converged solution is associated with strong deviations from the Navier–Stokes results for $S(x)$ at the upstream edge of the heating strip. The full 2-D solutions persist without qualitative changes in behaviour until at least $Ma=25$ .

We can isolate the effects of $Pe\,Ma$ interactions in a much simpler problem: the zero- $Ca$ , zero- $Re$ limit of strong surface tension, where the normal component of the dynamic boundary condition implies simply that the surface is flat, but Marangoni stresses drive an internal flow due to the tangential stress balance. If $S(x)$ is known, the full equations for $u$ and $v$ are linear:

(7.4a-c ) $$\begin{eqnarray}u=y(2-y)-\unicode[STIX]{x1D713}_{y},\quad v=\unicode[STIX]{x1D713}_{x},\quad \unicode[STIX]{x1D6FB}^{4}\unicode[STIX]{x1D713}=0,\quad \text{in }0<y<1,\end{eqnarray}$$

subject to $u=v=0$ on $y=0$ , and $u_{y}=-Ma\,S_{x}$ , $v=0$ on $y=1$ . The unique solution for $u$ and $v$ with $S$ given can be obtained by Fourier transforms. However, the coupled problem is nonlinear, as the surface temperature $S(x)=\unicode[STIX]{x1D703}(x,1)$ is determined by the heat equation, which itself involves the velocity field

(7.5) $$\begin{eqnarray}Pe\left(u\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}{\unicode[STIX]{x2202}x}+v\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}{\unicode[STIX]{x2202}y}\right)=\frac{\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D703}}{\unicode[STIX]{x2202}x^{2}}+\frac{\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D703}}{\unicode[STIX]{x2202}y^{2}},\end{eqnarray}$$

subject to $\unicode[STIX]{x1D703}=\unicode[STIX]{x1D6E9}(x)$ on $y=0$ and $\unicode[STIX]{x1D703}_{y}=-Bi\,\unicode[STIX]{x1D703}$ on $y=1$ .

Within the long-wave limit, the structure of the equations for this limiting case is similar to that of the full problem. The steady velocity field induced by a surface temperature $S(x)$ is given by

(7.6a,b ) $$\begin{eqnarray}u=y(2-y)+\frac{Ma}{4}y(2-3y)S_{x},\quad v=\frac{Ma}{4}S_{xx}y^{2}(y-1).\end{eqnarray}$$

It is straightforward to verify that the ‘projected’ heat equation can also be obtained by setting $h=1$ and $q=2/3$ in the equivalent non-uniform equation (6.7) with the addition of (6.9):

(7.7) $$\begin{eqnarray}\displaystyle & & \displaystyle \left[Pe^{2}\left(\frac{239}{10\,080}+\frac{13Bi}{3360}\right)-\left(\frac{1}{2}+\frac{Bi}{6}\right)\right]S_{xx}+\left[Pe\left(\frac{5}{12}+\frac{7Bi}{60}\right)\right]S_{x}\nonumber\\ \displaystyle & & \displaystyle \quad +\,\left(1+Bi\right)S+Pe\,Ma\left(-\frac{S_{x}^{2}}{48}+Bi\left(\frac{SS_{xx}}{80}+\frac{S_{x}^{2}}{240}\right)\right)=\unicode[STIX]{x1D6E9}(x).\end{eqnarray}$$

Without the $Pe\,Ma$ term, the velocity field is independent of $S$ , and so if $h$ is known, $u$ and $v$ are also known, the heat equation is linear, and the full and long-wave heat equations each have a unique solution for $S$ . However, with non-zero $Pe\,Ma$ , there is no reason to expect a unique solution in either the full or long-wave system.

Figure 14.

Calculations for $S(x)$ in the strong surface tension limit, as $Ma$ increases from zero at $Pe=10$ with (a) $Bi=0$ and (b) $Bi=1$ , with $\unicode[STIX]{x1D6E9}$ given by (5.1) with $X=5$ and $\unicode[STIX]{x1D70E}=10$ . The long-wave results (solid lines) are obtained by solving (7.7) directly, while the 2-D results (dotted lines) are from Navier–Stokes calculations with $Ca=10^{-5}$ . For $Bi=0$ , we show additionally the 2-D results for $Ma=100$ .

We can apply the same strip heating profile as in figure 13 and calculate the surface temperature profile in the strong surface tension limit. The long-wave and full 2-D results (calculated using the full Navier–Stokes code with $Ca=10^{-5}$ ) are shown in figure 14. We find that the long-wave solution branch for $Pe=10$ , $Bi=0$ persists to $Ma\approx 40$ , but the maximum value of $S$ increases with increasing $Ma$ and in fact crosses 1 at $Ma\approx 32$ , thus violating property P5, which should still hold in this limit. For relatively small $Ma$ (up to $Ma=20$ for $Bi=0$ and up to $Ma=40$ for $Bi=1$ ), the full 2-D results are quite similar to the long-wave results, with a broadening region of constant temperature above the strip. Beyond these Marangoni numbers, the 2-D results predict that this region continues to broaden, and the surface is always hottest above the downstream end of the strip. At the very largest $Ma$ ( $Ma=100$ for $Bi=1$ ), we start to see the appearance of a reversal in the direction of the temperature gradient, in qualitative agreement with the long-wave results. However, the long-wave model predicts that this feature should emerge at much lower $Ma$ and be much larger in magnitude by $Ma=100$ . The long-wave models hence lose accuracy at around $Ma=20$ , with $Pe=10$ , by which point the combination $Pe\,Ma$ is clearly not small. Hence, if better agreement is required at these parameters, the equations should be rederived under the assumption of large $Pe\,Ma$ with respect to the long-wave parameter $\unicode[STIX]{x1D6FF}$ , likely introducing further nonlinear interactions.