2.1 Governing equations of vapour flow

The vapour has velocity components $u$ and $w$ , in the $x$ and $z$ directions, respectively. Viscosity ( $\unicode[STIX]{x1D707}_{v}$ ), density ( $\unicode[STIX]{x1D70C}_{v}$ ), thermal conductivity ( $k_{v}$ ) and heat capacity ( $c_{p,v}$ ) are all assumed constant. The governing equations for the vapour flow are the standard continuity, momentum and energy equations for incompressible flow (Kundu, Cohen & Dowling Reference Kundu, Cohen and Dowling2007),

(2.1) $$\begin{eqnarray}\displaystyle & \displaystyle u_{x}+w_{z}=0, & \displaystyle\end{eqnarray}$$

(2.2) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D70C}_{v}(u_{t}+uu_{x}+wu_{z})=-p_{x}+\unicode[STIX]{x1D707}_{v}(u_{xx}+u_{zz})-\unicode[STIX]{x1D719}_{x}, & \displaystyle\end{eqnarray}$$

(2.3) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D70C}_{v}(w_{t}+uw_{x}+ww_{z})=-p_{z}+\unicode[STIX]{x1D707}_{v}(w_{xx}+w_{zz})-\unicode[STIX]{x1D719}_{z}, & \displaystyle\end{eqnarray}$$

(2.4) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D70C}_{v}c_{p,v}(T_{t}+uT_{x}+wT_{z})=k_{v}(T_{xx}+T_{zz}), & \displaystyle\end{eqnarray}$$

where variable subscripts imply differentiation. Here $p$ is the pressure and $\unicode[STIX]{x1D719}$ is the body-force potential. The only difference from standard flow equations so far is that $\unicode[STIX]{x1D719}$ includes not only the gravity contribution, but also a film-thickness-dependent addition that represents van der Waals interactions between the liquid surface and the solid surface. This is called a disjoining pressure (Oron et al. Reference Oron, Davis and Bankoff1997), and gives a total potential of the form

(2.5) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D719}=\unicode[STIX]{x1D719}_{0}+\unicode[STIX]{x1D70C}_{v}gbz+{\displaystyle \frac{\tilde{A}}{6\unicode[STIX]{x03C0}h^{3}}}. & & \displaystyle\end{eqnarray}$$

Here $g$ is the gravitational acceleration, and $\tilde{A}$ is the effective Hamaker constant from van der Waals interaction theory. The constant $b=\pm 1$ is $+1$ for the liquid-above-solid configuration and $-1$ for the solid-above-liquid configuration. The constant $\unicode[STIX]{x1D719}_{0}$ is an arbitrary reference potential. The van der Waals interaction will only become significant on the sub-micrometre scale of film thickness. A derivation of the last term in (2.5) for the case of thin liquid films can be found in the work of Ruckenstein & Jain (Reference Ruckenstein and Jain1974), and here we assume that a term of the same form is valid for thin vapour films. Generally, the interaction may be either attractive ( $\tilde{A}>0$ ) or repulsive ( $\tilde{A}<0$ ).

2.2 Evaporation model

Due to the high temperature of the solid, evaporation occurs at the liquid–vapour interface, giving an evaporation heat flux $j$ . The only given temperatures are the controlled temperature in the solid bulk, $T_{w0}$ , and the saturation temperature known from thermodynamics, $T_{s}$ . The wall surface temperature, $T_{w}$ , will generally be a bit lower than $T_{w0}$ due to the finite thermal conductivity of the solid. Still, the temperature will be continuous at the wall. The situation at the liquid–vapour interface is more complicated. Classically, in the quasi-equilibrium limit, the interface temperature is assumed to be continuous and equal to $T_{s}$ . However, generally there is a temperature discontinuity at the interface, and neither side is necessarily equal to $T_{s}$ . However, they will both approach $T_{s}$ in the limit of weak evaporation. This situation is illustrated in figure 3.

Figure 3. Illustration of the local temperature profile in film boiling, on $x$ -scales much shorter than the wavelength seen in figure 2, so that the interface appears flat.

We label the vapour-side and liquid-side interface temperatures as $T_{iv}$ and $T_{il}$ , respectively. When evaporating, we always have that $T_{il}>T_{s}$ and $T_{il}>T_{iv}$ . The interface vapour temperature $T_{iv}$ may either be below $T_{s}$ (supersaturated) or above $T_{s}$ (superheated), depending on conditions (Ytrehus Reference Ytrehus1997). For moderate evaporation rates, we may neglect the effect of the discontinuity and consider a single interface temperature, $T_{i}=T_{il}\approx T_{iv}$ , which is superheated ( $T_{i}>T_{s}$ ). In these cases we may linearize the relationship between evaporation mass flux and $T_{i}$ of the form

(2.6) $$\begin{eqnarray}\displaystyle T_{i}-T_{s}=\tilde{K}j, & & \displaystyle\end{eqnarray}$$

as used by Burelbach et al. (Reference Burelbach, Bankoff and Davis1988). The interfacial thermal resistance can be estimated from kinetic gas theory and typically has the form

(2.7) $$\begin{eqnarray}\displaystyle \tilde{K}={\displaystyle \frac{\sqrt{2\unicode[STIX]{x03C0}R_{s}}T_{s}^{3/2}}{f(\unicode[STIX]{x1D6FC}_{e})\unicode[STIX]{x1D70C}_{v,s}L}}, & & \displaystyle\end{eqnarray}$$

where $R_{s}$ is the specific gas constant, $L$ is the latent heat of evaporation, $\unicode[STIX]{x1D70C}_{v,s}$ is the vapour density at the saturation temperature and $\unicode[STIX]{x1D6FC}_{e}$ is the evaporation coefficient. The function $f(\unicode[STIX]{x1D6FC}_{e})$ depends on the specific model. In the moderate-evaporation limit of the classical Hertz–Knudsen model, (Hertz Reference Hertz1882; Knudsen Reference Knudsen1915), we get

(2.8) $$\begin{eqnarray}\displaystyle f(\unicode[STIX]{x1D6FC}_{e})=\unicode[STIX]{x1D6FC}_{e}, & & \displaystyle\end{eqnarray}$$

which is what was used by Burelbach et al. (Reference Burelbach, Bankoff and Davis1988). A more recent refinement of this model is the Schrage formula, whose moderate-evaporation limit yields (Mills Reference Mills1995)

(2.9) $$\begin{eqnarray}\displaystyle f(\unicode[STIX]{x1D6FC}_{e})={\displaystyle \frac{\unicode[STIX]{x1D6FC}_{e}}{1-\frac{1}{2}\unicode[STIX]{x1D6FC}_{e}}}. & & \displaystyle\end{eqnarray}$$

Some more advanced evaporation models do exist (Ytrehus Reference Ytrehus1997), but quantitatively they reduce to something very similar to the Schrage formula for low-to-moderate evaporation rates.

Usually these models are stated in terms of density differences, not temperature differences like in the constitutive equation used here. Matching the form (2.6) may be achieved by applying the ideal gas law, linearizing the saturation line by the Clausius–Clapeyron relation and assuming that the differences between $T_{il}$ , $T_{iv}$ and $T_{s}$ are small.

The evaporation coefficient $\unicode[STIX]{x1D6FC}_{e}$ is the subject of much uncertainty, debate and active research to this date. It is typically assumed equal to the related condensation coefficient (Ytrehus Reference Ytrehus1997; Cheng et al. Reference Cheng, Lechman, Plimpton and Grest2011). This unknown coefficient is introduced through a boundary condition in kinetic theory, and cannot be determined from within kinetic theory itself. It represents the probability of an incoming vapour molecule sticking to the liquid, as opposed to reflecting back, and is thus by definition in the range of zero to one. The exact nature of this coefficient appears to be far from settled. Water is the only somewhat well-studied fluid, and even there the experiments show a large scatter from $0.1$ to $1.0$ , as seen in e.g. Tsuruta & Nagayama (Reference Tsuruta and Nagayama2004, Table 1). Besides experiments, a common way of estimating the coefficient is molecular dynamics simulations (MD). These methods show somewhat more consistent results, and generally give values quite close to unity. Overall, MD simulations from the last decade seem to generally agree on the following trends (Tsuruta & Nagayama Reference Tsuruta and Nagayama2004; Cao, Xie & Sazhin Reference Cao, Xie and Sazhin2011; Cheng et al. Reference Cheng, Lechman, Plimpton and Grest2011; Xie, Sazhin & Cao Reference Xie, Sazhin and Cao2011; Ishiyama et al. Reference Ishiyama, Fujikawa, Kurz and Lauterborn2013; Iskrenova & Patnaik Reference Iskrenova and Patnaik2017; Liang, Biben & Keblinski Reference Liang, Biben and Keblinski2017):

1. (i) For a given fluid, the evaporation/condensation coefficient decreases as liquid temperature is increased.

2. (ii) As long as the liquid temperature is less than 0.7 times $T_{c}$ (critical temperature), we can expect $\unicode[STIX]{x1D6FC}_{e}\in (0.7,1.0)$ for a considerable variety of fluids.

In the cases considered here the liquid surface temperatures are very close to the saturation temperatures, and every liquid considered here has $T_{s}<0.7T_{c}$ . Thus we may expect that $\unicode[STIX]{x1D6FC}_{e}\in (0.7,1.0)$ .

2.3 Surface tension model

In order to capture the thermocapillary effect, it is essential to include the temperature dependence of surface tension ( $\unicode[STIX]{x1D70E}$ ). We follow Davis (Reference Davis1987) and model the variation as a linearization around its value at the saturation temperature, $\unicode[STIX]{x1D70E}_{0}$ ,

(2.10) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D70E}(T)=\unicode[STIX]{x1D70E}_{0}-\unicode[STIX]{x1D6FE}(T-T_{s}). & & \displaystyle\end{eqnarray}$$

Thus, the factor $\unicode[STIX]{x1D6FE}$ is

(2.11) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FE}=-{\displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D70E}}{\unicode[STIX]{x2202}T}}. & & \displaystyle\end{eqnarray}$$

For most liquids, $\unicode[STIX]{x1D6FE}$ is positive and often around $0.0002~\text{N}~\text{m}^{-1}~\text{K}^{-1}$ . As we shall demonstrate, $\unicode[STIX]{x1D6FE}$ will play a crucial role in the prediction of vapour film collapse.

2.4 Boundary conditions

2.4.1 Solid wall

The solid wall at $z=0$ is an impermeable no-slip surface. Also, as with any interface, there must be a continuity of energy flux. We represent the heat transfer inside the solid with a heat transfer coefficient $\unicode[STIX]{x1D6FC}_{w}$ . Since this is a solid, $\unicode[STIX]{x1D6FC}_{w}$ could of course be found from the thermal conductivity and a thermal boundary layer thickness, but for simplicity we keep the factor $\unicode[STIX]{x1D6FC}_{w}$ . Given the above, the wall surface boundary conditions are

(2.12) $$\begin{eqnarray}\displaystyle & \displaystyle u|_{z=0}=w|_{z=0}=0, & \displaystyle\end{eqnarray}$$

(2.13) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6FC}_{w}(T_{w0}-T_{w})=-k_{v}T_{z}|_{z=0}. & \displaystyle\end{eqnarray}$$

2.4.2 Liquid–vapour interface

The liquid–vapour interface is also no-slip, in the sense that the tangential velocity is continuous. In contrast to the solid surface, fluid may pass into this interface at a rate governed by the evaporation mass flux. The relation between the flow velocity at the interface, the velocity of the interface itself, and the evaporation rate is given by the kinematic boundary condition. Additionally, we must have continuity of stress and energy flux across the interface. Given the above, the interface boundary conditions are

(2.14) $$\begin{eqnarray}\displaystyle & \displaystyle (\boldsymbol{v}-\boldsymbol{v}_{l})\boldsymbol{\cdot }\hat{\boldsymbol{t}}|_{z=h}=0, & \displaystyle\end{eqnarray}$$

(2.15) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D70C}_{v}{\displaystyle \frac{(h_{t}+uh_{x}-w)|_{z=h}}{\sqrt{1+h_{x}^{2}}}}=j, & \displaystyle\end{eqnarray}$$

(2.16) $$\begin{eqnarray}\displaystyle & \displaystyle [j(\boldsymbol{v}_{l,e}-\boldsymbol{v}_{e})\boldsymbol{\cdot }\hat{\boldsymbol{n}}-([\unicode[STIX]{x1D64F}-\unicode[STIX]{x1D64F}_{l}]\boldsymbol{\cdot }\hat{\boldsymbol{n}})\boldsymbol{\cdot }\hat{\boldsymbol{n}}]_{z=h}=-\unicode[STIX]{x1D705}\unicode[STIX]{x1D70E}, & \displaystyle\end{eqnarray}$$

(2.17) $$\begin{eqnarray}\displaystyle & \displaystyle ([\unicode[STIX]{x1D64F}-\unicode[STIX]{x1D64F}_{l}]_{z=h}\boldsymbol{\cdot }\hat{\boldsymbol{n}})\boldsymbol{\cdot }\hat{\boldsymbol{t}}=\unicode[STIX]{x1D735}\unicode[STIX]{x1D70E}|_{z=h}\boldsymbol{\cdot }\hat{\boldsymbol{t}}, & \displaystyle\end{eqnarray}$$

(2.18) $$\begin{eqnarray}\displaystyle & \displaystyle -k_{v}\unicode[STIX]{x1D735}T\boldsymbol{\cdot }\hat{\boldsymbol{n}}|_{z=h}-\unicode[STIX]{x1D6FC}_{l}(T_{i}-T_{s})=jL. & \displaystyle\end{eqnarray}$$

Here the vectors $\boldsymbol{v}=[u,w]$ , $\hat{\boldsymbol{n}}$ and $\hat{\boldsymbol{t}}$ are the velocity, interface unit normal and interface unit tangent, respectively. The latter two are defined as shown in figure 2. The symbol $\unicode[STIX]{x1D705}$ is the interface curvature. The symbol $\unicode[STIX]{x1D64F}$ is the incompressible Newtonian flow stress tensor, $j$ is the evaporation mass flux, and $L$ is the fluid’s latent heat of evaporation. The efficiency of heat transfer from the interface to the liquid bulk is represented by a heat transfer coefficient $\unicode[STIX]{x1D6FC}_{l}$ . Overall, the subscript $l$ indicates the corresponding property on the liquid side.

2.5 Comparison with some previous models

The inclusion of a disjoining-pressure term in § 2.1 is identical to the treatment in Burelbach et al. (Reference Burelbach, Bankoff and Davis1988), though here presumably with a different value for the Hamaker constant due to the nature of the thin film. Similarly, the constitutive equation for evaporation in § 2.2 is similar, though here with a generalization allowing for different factors $f(\unicode[STIX]{x1D6FC}_{e})$ . The model of Burelbach et al. (Reference Burelbach, Bankoff and Davis1988) only uses the older Hertz–Knudsen model (2.8). The linearized surface tension model in § 2.3 is quite standard.

The differences to previous works become more nuanced when it comes to the boundary conditions in § 2.4. At the solid surface, the flow boundary conditions (2.12) are standard. However, an energy flux balance like (2.13) is not included in Burelbach et al. (Reference Burelbach, Bankoff and Davis1988), which simply assumes a constant given wall surface temperature. The interface boundary conditions (2.14)–(2.18) are essentially the same as the ones initially presented in Burelbach et al. (Reference Burelbach, Bankoff and Davis1988, equations (2.6)–(2.12)), besides some subtle sign changes due to the liquid–vapour role reversal. However, in Burelbach et al. (Reference Burelbach, Bankoff and Davis1988), the full boundary conditions are considerably simplified due to the negligible density, viscosity and conductivity of the bulk vapour phase outside the film. This cannot be done here as the outside bulk is liquid, and thus, the boundary conditions must remain in their complex form.

Some of the commonalities missing from Burelbach et al. (Reference Burelbach, Bankoff and Davis1988) are present in Panzarella et al. (Reference Panzarella, Davis and Bankoff2000). The latter considers a vapour film and does allow the solid surface temperature to vary. However, they include neither vapour thrust, thermocapillary nor van der Waals effects. In fact, they take the infinite liquid viscosity limit, which leads to setting the vapour interface velocity to zero. As we shall show, this limit has an important qualitative consequence, as it causes the model to decouple from the thermocapillary effect.

2.6 Scales and dimensionless numbers

We introduce a length scale $h_{0}$ for $z$ and $h$ in order to define the dimensionless equivalents $Z$ and $H$ . Similarly, we introduce a length scale $x_{0}$ for $x$ in order to define the dimensionless distance $X$ . The scales $h_{0}$ and $x_{0}$ are not arbitrary, and must be set similar to the typical film-thickness and interface disturbance wavelength, in order to ensure $\unicode[STIX]{x2202}/\unicode[STIX]{x2202}X\sim \unicode[STIX]{x2202}/\unicode[STIX]{x2202}Z\sim \mathit{O}(1)$ in the dimensionless equations. Here we choose $x_{0}=\unicode[STIX]{x1D706}/(2\unicode[STIX]{x03C0})$ , where $\unicode[STIX]{x1D706}$ is the wavelength of the disturbance. The ratio between the two scales is defined as

(2.19) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D716}={\displaystyle \frac{h_{0}}{x_{0}}}=2\unicode[STIX]{x03C0}{\displaystyle \frac{h_{0}}{\unicode[STIX]{x1D706}}}. & & \displaystyle\end{eqnarray}$$

We shall later take the long-wave approximation, which formally is the limit of small $\unicode[STIX]{x1D716}$ , i.e.  $\unicode[STIX]{x1D706}\gg h_{0}$ . We use a velocity scale $u_{0}$ to define the dimensionless tangential velocity, $U=u/u_{0}$ . Similarly we define the dimensionless perpendicular velocity $W=w/w_{0}$ , where continuity implies that $w_{0}=\unicode[STIX]{x1D716}u_{0}$ . The dimensionless time $\unicode[STIX]{x1D70F}$ is defined by the time scale $x_{0}/u_{0}$ . We scale the temperature according to its position on the scale between $T_{w0}$ and $T_{s}$ ,

(2.20) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D703}={\displaystyle \frac{T-T_{s}}{\unicode[STIX]{x0394}T}}, & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x0394}T=T_{w0}-T_{s}$ . We scale the remaining variables as

(2.21a-d ) $$\begin{eqnarray}p={\displaystyle \frac{\unicode[STIX]{x1D707}_{v}u_{0}}{\unicode[STIX]{x1D716}h_{0}}}P,\quad \unicode[STIX]{x1D719}={\displaystyle \frac{\unicode[STIX]{x1D707}_{v}u_{0}}{\unicode[STIX]{x1D716}h_{0}}}\unicode[STIX]{x1D6F7},\quad j={\displaystyle \frac{k_{v}\unicode[STIX]{x0394}T}{h_{0}L}}J,\quad \unicode[STIX]{x1D70E}=\unicode[STIX]{x1D70E}_{0}\unicode[STIX]{x1D6F4},\end{eqnarray}$$

where $P$ , $\unicode[STIX]{x1D6F7}$ , $J$ and $\unicode[STIX]{x1D6F4}$ are the dimensionless pressure, body-force potential, evaporation mass flux and surface tension, respectively. We define the following dimensionless numbers:

(2.22) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}Re={\displaystyle \frac{\unicode[STIX]{x1D70C}_{v}u_{0}h_{0}}{\unicode[STIX]{x1D707}_{v}}},\quad Pr={\displaystyle \frac{\unicode[STIX]{x1D707}_{v}c_{p,v}}{k_{v}}},\quad \unicode[STIX]{x1D6F9}={\displaystyle \frac{\unicode[STIX]{x1D707}_{v}}{\unicode[STIX]{x1D707}_{l}}},\quad D={\displaystyle \frac{\unicode[STIX]{x1D70C}_{v}}{\unicode[STIX]{x1D70C}_{l}}},\quad E={\displaystyle \frac{k_{v}\unicode[STIX]{x0394}T}{\unicode[STIX]{x1D70C}_{v}u_{0}h_{0}L}},\\ K={\displaystyle \frac{\tilde{K}k_{v}}{h_{0}L}},\quad Ca={\displaystyle \frac{\unicode[STIX]{x1D707}_{v}u_{0}}{\unicode[STIX]{x1D70E}_{0}}},\quad M={\displaystyle \frac{\unicode[STIX]{x0394}T\unicode[STIX]{x1D6FE}}{\unicode[STIX]{x1D707}_{v}u_{0}}},\quad A={\displaystyle \frac{\tilde{A}}{6\unicode[STIX]{x03C0}\unicode[STIX]{x1D707}_{v}u_{0}h_{0}^{2}}},\\ G_{v}={\displaystyle \frac{\unicode[STIX]{x1D70C}_{v}gh_{0}^{2}}{\unicode[STIX]{x1D707}_{v}u_{0}}},\quad G_{l}={\displaystyle \frac{\unicode[STIX]{x1D70C}_{l}gh_{0}^{2}}{\unicode[STIX]{x1D707}_{v}u_{0}}},\quad G={\displaystyle \frac{\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}gh_{0}^{2}}{\unicode[STIX]{x1D707}_{v}u_{0}}},\quad Bi_{w}={\displaystyle \frac{\unicode[STIX]{x1D6FC}_{w}h_{0}}{k_{v}}},\quad Bi_{l}={\displaystyle \frac{\unicode[STIX]{x1D6FC}_{l}h_{0}}{k_{v}}}.\end{array}\right\} & & \displaystyle\end{eqnarray}$$

2.7 Long-wave approximation

2.7.1 Approximate equations

We introduce the scales and dimensionless numbers of § 2.6 and make the assumptions of long waves and small Reynolds number, while retaining surface tension effects to leading order,

(2.23a-f ) $$\begin{eqnarray}\unicode[STIX]{x1D716}\ll 1,\quad \unicode[STIX]{x1D716}Re\ll 1,\quad D\ll 1,\quad Pr\sim \mathit{O}(1),\quad \unicode[STIX]{x1D716}^{3}Ca^{-1}\sim \mathit{O}(1),\quad \unicode[STIX]{x1D716}M\sim \mathit{O}(1).\quad\end{eqnarray}$$

We also want to retain the vapour thrust, van der Waals and gravitational effects to leading order, so we keep the terms $\unicode[STIX]{x1D716}ReE^{2}J^{2}$ , $\unicode[STIX]{x1D716}A/H^{3}$ and $\unicode[STIX]{x1D716}G_{v}$ . Given this, the governing equations of the vapour flow, equations (2.1)–(2.3) and (2.5) become

(2.24) $$\begin{eqnarray}\displaystyle & \displaystyle U_{X}+W_{Z}=0, & \displaystyle\end{eqnarray}$$

(2.25) $$\begin{eqnarray}\displaystyle & \displaystyle (P+\unicode[STIX]{x1D6F7})_{X}=U_{ZZ}, & \displaystyle\end{eqnarray}$$

(2.26) $$\begin{eqnarray}\displaystyle & \displaystyle (P+\unicode[STIX]{x1D6F7})_{Z}=0, & \displaystyle\end{eqnarray}$$

(2.27) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6F7}=\unicode[STIX]{x1D6F7}_{0}+\unicode[STIX]{x1D716}bG_{v}Z+{\displaystyle \frac{\unicode[STIX]{x1D716}A}{H^{3}}}, & \displaystyle\end{eqnarray}$$

respectively. The boundary conditions of the vapour flow (2.12), (2.14), (2.15), (2.16) and (2.17) become

(2.28) $$\begin{eqnarray}\displaystyle & \displaystyle [U]_{Z=0}=[W]_{Z=0}=0, & \displaystyle\end{eqnarray}$$

(2.29) $$\begin{eqnarray}\displaystyle & \displaystyle [U-U_{l}]_{Z=H}=0, & \displaystyle\end{eqnarray}$$

(2.30) $$\begin{eqnarray}\displaystyle & \displaystyle {\displaystyle \frac{E}{\unicode[STIX]{x1D716}}}J=[H_{\unicode[STIX]{x1D70F}}+UH_{X}-W]_{Z=H}, & \displaystyle\end{eqnarray}$$

(2.31) $$\begin{eqnarray}\displaystyle & \displaystyle [P-P_{l}]_{Z=H}+\unicode[STIX]{x1D716}ReE^{2}J^{2}=-H_{XX}\unicode[STIX]{x1D716}^{3}Ca^{-1}, & \displaystyle\end{eqnarray}$$

(2.32) $$\begin{eqnarray}\displaystyle & \displaystyle [U_{Z}-\unicode[STIX]{x1D6F9}^{-1}U_{l,Z}]_{Z=H}=-\unicode[STIX]{x1D716}M(\unicode[STIX]{x1D703}_{i})_{X}, & \displaystyle\end{eqnarray}$$

respectively. Similarly, the energy equation (2.4) becomes

(2.33) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D703}_{ZZ}=0, & & \displaystyle\end{eqnarray}$$

and the temperature boundary conditions (2.6), (2.13) and (2.18) become

(2.34) $$\begin{eqnarray}\displaystyle & \displaystyle KJ=\unicode[STIX]{x1D703}_{i}, & \displaystyle\end{eqnarray}$$

(2.35) $$\begin{eqnarray}\displaystyle & \displaystyle -\unicode[STIX]{x1D703}_{Z}|_{Z=0}=Bi_{w}(1-\unicode[STIX]{x1D703}_{w}), & \displaystyle\end{eqnarray}$$

(2.36) $$\begin{eqnarray}\displaystyle & \displaystyle J=-\unicode[STIX]{x1D703}_{Z}|_{Z=H}-Bi_{l}\unicode[STIX]{x1D703}_{i}, & \displaystyle\end{eqnarray}$$

respectively. The van der Waals effect is included in (2.27) ( ${\sim}A$ ), the vapour thrust effect is included in (2.31) ( ${\sim}ReE^{2}$ ), and the thermocapillary effect is included in (2.32) ( ${\sim}M$ ). These long-wave approximation equations have many similarities to the ones presented in Burelbach et al. (Reference Burelbach, Bankoff and Davis1988, equations (5.5)–(5.10)). However, there are significant differences. Besides some sign changes, these differences all relate to the fact that the bulk phase outside the thin film is different. In Burelbach et al. (Reference Burelbach, Bankoff and Davis1988), the normal-stress condition (here (2.31)) does not include a term for the outside pressure as it could be conveniently set constant and equal to zero. In the tangential-stress condition (here (2.32)), the bulk phase shear rate was set to zero, as the interface could be treated as a free surface. Neither simplification is possible in the present work, as the liquid and vapour have switched places. The equations for the temperature profile are also somewhat more complicated in the present work, since the wall surface temperature is allowed to vary (giving (2.35)) and since the bulk phase conductivity cannot be neglected (giving the final term of (2.36)). Overall, the main difference is that with this problem we cannot make a purely ‘one-sided’ model like in Burelbach et al. (Reference Burelbach, Bankoff and Davis1988).

2.7.2 Solution to temperature equations

We note how (2.33)–(2.36) for the temperature profile have no explicit time dependence, only implicitly through the variables $J(X,\unicode[STIX]{x1D70F})$ and $H(X,\unicode[STIX]{x1D70F})$ . Since $J$ is determined directly from the temperature profile through (2.34), the instantaneous value of $H$ determines the current temperature profile $\unicode[STIX]{x1D703}$ as well as the evaporation mass flux $J$ . The solution is

(2.37) $$\begin{eqnarray}\displaystyle & \displaystyle J(H)={\displaystyle \frac{1}{K+(Bi_{w}^{-1}+H)C}}, & \displaystyle\end{eqnarray}$$

(2.38) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D703}_{i}(H)={\displaystyle \frac{K}{K+(Bi_{w}^{-1}+H)C}}, & \displaystyle\end{eqnarray}$$

(2.39) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D703}_{w}(H)={\displaystyle \frac{K+HC}{K+(Bi_{w}^{-1}+H)C}}, & \displaystyle\end{eqnarray}$$

where we have defined the new constant

(2.40) $$\begin{eqnarray}\displaystyle C=1+C^{\prime }. & & \displaystyle\end{eqnarray}$$

Here $C^{\prime }=Bi_{l}K=\unicode[STIX]{x1D6FC}_{l}\tilde{K}/L$ and represents the effect of heat lost into the liquid bulk. Interestingly, $C^{\prime }$ is independent of $h_{0}$ , even though the interface temperature $\unicode[STIX]{x1D703}_{i}$ is not. It is instructive to look at a few special cases of this solution. In the quasi-equilibrium limit ( $K\rightarrow 0$ ), we get

(2.41) $$\begin{eqnarray}\displaystyle & \displaystyle J(K\rightarrow 0)={\displaystyle \frac{1}{Bi_{w}^{-1}+H}}, & \displaystyle\end{eqnarray}$$

(2.42) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D703}_{i}(K\rightarrow 0)=0, & \displaystyle\end{eqnarray}$$

(2.43) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D703}_{w}(K\rightarrow 0)={\displaystyle \frac{H}{Bi_{w}^{-1}+H}}. & \displaystyle\end{eqnarray}$$

As expected, the interface temperature is locked to $T_{s}$ . The evaporation rate is somewhat limited by the finite conductivity of the solid. If $H\rightarrow 0$ , $J$ does not diverge, due to the finite solid heat transfer efficiency. In the limit of a perfectly conducting solid ( $Bi_{w}\rightarrow \infty$ ) we get

(2.44) $$\begin{eqnarray}\displaystyle & \displaystyle J(Bi_{w}\rightarrow \infty )={\displaystyle \frac{1}{K+HC}}, & \displaystyle\end{eqnarray}$$

(2.45) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D703}_{i}(Bi_{w}\rightarrow \infty )={\displaystyle \frac{K}{K+HC}}, & \displaystyle\end{eqnarray}$$

(2.46) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D703}_{w}(Bi_{w}\rightarrow \infty )=1. & \displaystyle\end{eqnarray}$$

As expected, the wall surface temperature is locked to the bulk temperature, $T_{w0}$ . The evaporation rate is somewhat limited by the non-equilibrium effect ( $K\neq 0$ ) and liquid conduction ( $C>1$ ). If $H\rightarrow 0$ , $J$ does not diverge, due to the interface thermal resistance ( $K\neq 0$ ). Generally, if $H\rightarrow 0$ , we get

(2.47) $$\begin{eqnarray}\displaystyle & \displaystyle J(H\rightarrow 0)={\displaystyle \frac{1}{K+Bi_{w}^{-1}C}}, & \displaystyle\end{eqnarray}$$

(2.48) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D703}_{i}(H\rightarrow 0)=\unicode[STIX]{x1D703}_{w}(H=0)={\displaystyle \frac{K}{K+Bi_{w}^{-1}C}}, & \displaystyle\end{eqnarray}$$

i.e. the evaporation rate stays finite and the interface/surface temperature approaches an intermediate value between $T_{s}$ and $T_{w0}$ . However, note that in the $H\rightarrow 0$ limit, it is likely that the linearized relation in (2.6) for moderate evaporation rates becomes inaccurate.

We proceed by using the general solution in (2.37)–(2.39) in order to include both the non-equilibrium effect and the potential effects of heat transfer on both sides of the vapour film. Note that the non-equilibrium ( $K\neq 0$ ) effect is absolutely necessary for capturing the thermocapillary effect. If $K=0$ , $\unicode[STIX]{x1D703}_{i}$ becomes a constant, and the thermocapillary term in the tangential-stress condition (2.32) disappears.

2.7.3 Velocity profile

We define the reduced dimensionless pressure as $\bar{P}=P+\unicode[STIX]{x1D6F7}$ . From (2.26) we know that $\bar{P}$ is constant across the film, and thus, we may choose to evaluate it at $Z=H$ in (2.25), so that it reduces to

(2.49) $$\begin{eqnarray}\displaystyle U_{ZZ}=\bar{P}(X,H)_{X}. & & \displaystyle\end{eqnarray}$$

If we combine (2.31) and (2.27), we find that the gradient of reduced pressure is

(2.50) $$\begin{eqnarray}\bar{P}(X,H)_{X}=P_{l}(X,H)_{X}+\unicode[STIX]{x1D716}bG_{v}H_{X}-2\unicode[STIX]{x1D716}ReE^{2}JJ_{X}-\unicode[STIX]{x1D716}^{3}Ca^{-1}H_{XXX}-3\unicode[STIX]{x1D716}A{\displaystyle \frac{H_{X}}{H^{4}}}.\end{eqnarray}$$

The right-hand side of (2.49) is independent of $Z$ , and thus we may integrate the equation twice and use the no-slip wall boundary condition (2.28) to get the velocity profile

(2.51) $$\begin{eqnarray}\displaystyle U & = & \displaystyle {\textstyle \frac{1}{2}}\bar{P}_{X}(Z^{2}-2HZ)+U_{Z}|_{Z=H}Z,\end{eqnarray}$$

(2.52) $$\begin{eqnarray}\displaystyle & = & \displaystyle {\textstyle \frac{1}{2}}\bar{P}_{X}(Z^{2}-HZ)+U|_{Z=H}{\displaystyle \frac{Z}{H}},\end{eqnarray}$$

expressed in two different ways depending on whether one wants to use the interface shear rate or the interface velocity to define the $Z=H$ boundary. From this, we find the total flow rate to be

(2.53) $$\begin{eqnarray}\displaystyle \int _{0}^{H}U\,\text{d}Z & = & \displaystyle -{\textstyle \frac{1}{3}}H^{3}\bar{P}_{X}+{\textstyle \frac{1}{2}}H^{2}U_{Z}|_{Z=H},\end{eqnarray}$$

(2.54) $$\begin{eqnarray}\displaystyle & = & \displaystyle -{\textstyle \frac{1}{12}}H^{3}\bar{P}_{X}+{\textstyle \frac{1}{2}}HU|_{Z=H}.\end{eqnarray}$$

The two extremes of behaviour can be found by either setting the liquid velocity to zero at the boundary (corresponding to infinite liquid viscosity) or setting the liquid shear rate to zero at the boundary (treating the interface like a free surface). Thus, regardless of the specific liquid properties, we know that the flow rate must be within the range

(2.55) $$\begin{eqnarray}\displaystyle \int _{0}^{H}U\,\text{d}Z=\left\{\begin{array}{@{}ll@{}}\displaystyle -{\textstyle \frac{1}{12}}H^{3}\bar{P}_{X},\quad & U|_{Z=H}=0,\\ \displaystyle -{\textstyle \frac{1}{3}}H^{3}\bar{P}_{X}-{\textstyle \frac{1}{2}}H^{2}\unicode[STIX]{x1D716}M\unicode[STIX]{x1D703}_{i,X},\quad & U_{l,Z}|_{Z=H}=0,\end{array}\right. & & \displaystyle\end{eqnarray}$$

where we in the latter case have used the tangential-stress condition (2.32) to find the vapour shear rate. Generally, the interface velocity $U_{i}=U|_{Z=H}=U_{l}|_{Z=H}$ is

(2.56) $$\begin{eqnarray}\displaystyle U_{i}=-{\textstyle \frac{1}{2}}H^{2}\bar{P}_{X}+\unicode[STIX]{x1D6F9}^{-1}HU_{l,Z}|_{Z=H}-\unicode[STIX]{x1D716}MH\unicode[STIX]{x1D703}_{i,X}, & & \displaystyle\end{eqnarray}$$

and if we evaluate (2.51) at $Z=H$ , we get the following constraint on the boundary:

(2.57) $$\begin{eqnarray}\displaystyle [U-HU_{Z}]_{Z=H}=-{\textstyle \frac{1}{2}}H^{2}\bar{P}_{X}. & & \displaystyle\end{eqnarray}$$

Note that if we take the zero interface-velocity limit, the flow rate is fully determined by the first case of (2.55). Then there is no way to involve the tangential-stress condition (2.32), and therefore, any coupling to the thermocapillary effect is lost. Thus, the choice of velocity boundary condition made by Panzarella et al. (Reference Panzarella, Davis and Bankoff2000) is not an option here.

2.7.4 Liquid dynamics

So far we have made no assumptions regarding the liquid flow outside the vapour film. However, in order to find the final vapour velocity profile we require a liquid pressure (as seen in (2.50)) and information regarding the liquid–vapour boundary (as seen in (2.51) and (2.52)).

First, we assume that the liquid pressure is purely hydrostatic,

(2.58) $$\begin{eqnarray}\displaystyle P_{l}=-\unicode[STIX]{x1D716}bG_{l}Z, & & \displaystyle\end{eqnarray}$$

similar to Panzarella et al. (Reference Panzarella, Davis and Bankoff2000). Note that the liquid layer is much thicker than the vapour layer, so the former does not have any disjoining-pressure contribution.

Second, we need to make an assumption regarding the liquid flow in order to find the interface velocity. The liquid is assumed to be stationary far away in the bulk, but close to the interface it will be pulled along with the vapour. From the perspective of the vapour film, the liquid slows down the vapour flow due to viscous drag. Generally, we expect the liquid velocity profile to monotonically decay from $U_{i}$ at $Z=H$ to zero at $Z=\infty$ . Regardless of the details of the liquid flow, we know that the interface velocity $U_{i}$ must be between the following two hypothetical extreme cases:

1. (i) Minimum interface velocity: $U_{i}^{min}=0$ (interface acts like a wall).

2. (ii) Maximum interface velocity: $U_{i}=U_{i}^{max}$ (interface acts like a free surface).

The second case corresponds to the case of zero liquid shear, i.e. when the liquid does not resist the vapour flow at all. If we set $U_{l,Z}|_{Z=H}=0$ in (2.56) we find that

(2.59) $$\begin{eqnarray}\displaystyle U_{i}^{max}=-{\textstyle \frac{1}{2}}H^{2}\bar{P}_{X}-\unicode[STIX]{x1D716}MH\unicode[STIX]{x1D703}_{i,X}. & & \displaystyle\end{eqnarray}$$

We then interpolate between the two known extreme cases by introducing the interpolation parameter $\unicode[STIX]{x1D702}\in [0,1]$ ,

(2.60) $$\begin{eqnarray}\displaystyle U_{i} & = & \displaystyle \unicode[STIX]{x1D702}U_{i}^{max}+(1-\unicode[STIX]{x1D702})U_{i}^{min}\nonumber\\ \displaystyle & = & \displaystyle \unicode[STIX]{x1D702}\left(-{\textstyle \frac{1}{2}}H^{2}\bar{P}_{X}-\unicode[STIX]{x1D716}MH\unicode[STIX]{x1D703}_{i,X}\right),\end{eqnarray}$$

which satisfies the constraint (2.57) for any value of $\unicode[STIX]{x1D702}$ . While the value of $\unicode[STIX]{x1D702}$ is unknown for now, we make the crucial assumption that it is independent of position $X$ , and thus only depends on constant fluid properties. Specifically, we expect $\unicode[STIX]{x1D702}$ to increase monotonically with the viscosity ratio $\unicode[STIX]{x1D6F9}$ , with the limits

(2.61a,b ) $$\begin{eqnarray}\displaystyle \lim _{\unicode[STIX]{x1D6F9}\rightarrow 0}[\unicode[STIX]{x1D702}(\unicode[STIX]{x1D6F9})]=0,\quad \lim _{\unicode[STIX]{x1D6F9}\rightarrow \infty }[\unicode[STIX]{x1D702}(\unicode[STIX]{x1D6F9})]=1 & & \displaystyle\end{eqnarray}$$

since the two extreme cases correspond to the theoretical limits $\unicode[STIX]{x1D6F9}\rightarrow 0$ and $\unicode[STIX]{x1D6F9}\rightarrow \infty$ , respectively. Since the driving force for flow is the vapour film pressure gradient and the almost stationary liquid just passively applies drag to this flow, we expect the average vapour velocity to be significantly larger than the interface velocity. This means that we can expect $\unicode[STIX]{x1D702}$ to be much closer to zero than one.

More detailed information on the value of $\unicode[STIX]{x1D702}$ requires more bold assumptions regarding the liquid velocity profile. One such assumption is shown in appendix A, which leads to the convenient approximation

(2.62) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D702}={\displaystyle \frac{1}{1+\unicode[STIX]{x1D6F9}^{-1}}}=\unicode[STIX]{x1D6F9}+\mathit{O}(\unicode[STIX]{x1D6F9}^{2}). & & \displaystyle\end{eqnarray}$$

For common values of $\unicode[STIX]{x1D6F9}$ this means that $\unicode[STIX]{x1D702}$ is in the range of 0.025–0.050. Note that while this is quite close to zero, taking the actual $\unicode[STIX]{x1D702}\rightarrow 0$ approximation is not an option as it would eliminate the thermocapillary effect.

No matter the specific model used to find a value for $\unicode[STIX]{x1D702}$ , we may insert (2.60) into (2.54) and find the mass flow rate to be

(2.63) $$\begin{eqnarray}\displaystyle \int _{0}^{H}U\,\text{d}Z=-{\displaystyle \frac{1}{12}}(1+3\unicode[STIX]{x1D702})H^{3}\bar{P}_{X}-{\displaystyle \frac{1}{2}}\unicode[STIX]{x1D702}\unicode[STIX]{x1D716}MH^{2}\unicode[STIX]{x1D703}_{i,X}, & & \displaystyle\end{eqnarray}$$

which as intended matches (2.55) in the limiting cases of $\unicode[STIX]{x1D702}=0$ and $\unicode[STIX]{x1D702}=1$ . The following derivation of a film thickness PDE, and the stability analysis thereof, is performed with a general unknown $\unicode[STIX]{x1D702}$ .

The problems addressed in this section represent a central modelling complication compared to the related works of Burelbach et al. (Reference Burelbach, Bankoff and Davis1988) and Panzarella et al. (Reference Panzarella, Davis and Bankoff2000). The former was able to ignore all bulk phase dynamics because it considered a liquid film with a free surface ( $\unicode[STIX]{x1D702}=1$ ). The latter made stationary liquid ( $\unicode[STIX]{x1D702}=0$ ) approximation, which eliminates the thermocapillary effect. Here it is necessary to have an actual intermediate value for the interface velocity in order to arrive at a one-sided model. The assumptions made for the effect of liquid shear in this section are admittedly somewhat bold. Ultimately their validity rests on the success of the resulting model for the Leidenfrost point.

2.7.5 Film-thickness PDE

If we integrate the continuity equation (2.24) across the film from $Z=0$ to $Z=H(\unicode[STIX]{x1D70F})$ , and apply the Leibniz integral rule, the kinematic boundary condition (2.30) and the wall boundary condition (2.28), we get the basic mass-conservation PDE

(2.64) $$\begin{eqnarray}\displaystyle H_{\unicode[STIX]{x1D70F}}+\left(\int _{0}^{H}U\,\text{d}Z\right)_{X}={\displaystyle \frac{E}{\unicode[STIX]{x1D716}}}J, & & \displaystyle\end{eqnarray}$$

with a flux term and a source term. We find the reduced pressure gradient by inserting (2.37) and (2.58) into (2.50):

(2.65) $$\begin{eqnarray}\displaystyle \bar{P}(X,H)_{X}=-\unicode[STIX]{x1D716}bGH_{X}+{\displaystyle \frac{2\unicode[STIX]{x1D716}ReE^{2}C}{[K+(Bi_{w}^{-1}+H)C]^{3}}}H_{X}-\unicode[STIX]{x1D716}^{3}Ca^{-1}H_{XXX}-3\unicode[STIX]{x1D716}A{\displaystyle \frac{H_{X}}{H^{4}}}. & & \displaystyle\end{eqnarray}$$

We can then insert $\bar{P}_{X}$ into (2.63) while using (2.38) for $\unicode[STIX]{x1D703}_{i}$ , in order to yield the flow rate. When we insert this flow rate into (2.64) and use (2.37) for $J$ in the source term, we get the final PDE for the film thickness $H$ :

(2.66) $$\begin{eqnarray}\displaystyle & & \displaystyle H_{\tilde{\unicode[STIX]{x1D70F}}}+\underbrace{{\displaystyle \frac{b\unicode[STIX]{x1D709}G}{12E}}\unicode[STIX]{x1D716}^{2}[H^{3}H_{X}]_{X}}_{\mathit{gravity}}-\underbrace{{\displaystyle \frac{\unicode[STIX]{x1D709}ReEC}{6}}\unicode[STIX]{x1D716}^{2}[F^{3}(H)H_{X}]_{X}}_{\mathit{vapour}\,\mathit{thrust}}+\underbrace{{\displaystyle \frac{\unicode[STIX]{x1D709}}{12CaE}}\unicode[STIX]{x1D716}^{4}[H^{3}H_{XXX}]_{X}}_{\mathit{capillary}}\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\underbrace{{\displaystyle \frac{\unicode[STIX]{x1D709}A}{4E}}\unicode[STIX]{x1D716}^{2}\left[{\displaystyle \frac{H_{X}}{H}}\right]_{X}}_{\mathit{vdW}}+\underbrace{{\displaystyle \frac{\tilde{M}KC}{2E}}\unicode[STIX]{x1D716}^{2}[F^{2}(H)H_{X}]_{X}}_{\mathit{thermocapillary}}=\underbrace{{\displaystyle \frac{F(H)}{H}}}_{\mathit{evaporation}}.\end{eqnarray}$$

Here we have changed to the evaporative time scale,

(2.67) $$\begin{eqnarray}\displaystyle \tilde{t}_{0}={\displaystyle \frac{h_{0}}{(j_{0}/\unicode[STIX]{x1D70C}_{v})}}={\displaystyle \frac{\unicode[STIX]{x1D70C}_{v}h_{0}^{2}L}{k_{v}\unicode[STIX]{x0394}T}}, & & \displaystyle\end{eqnarray}$$

with the corresponding dimensionless time $\tilde{\unicode[STIX]{x1D70F}}$ , and we have defined the shorthands

(2.68) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D709}=1+3\unicode[STIX]{x1D702}, & \displaystyle\end{eqnarray}$$

(2.69) $$\begin{eqnarray}\displaystyle & \displaystyle \tilde{M}=\unicode[STIX]{x1D702}M, & \displaystyle\end{eqnarray}$$

(2.70) $$\begin{eqnarray}\displaystyle & \displaystyle F(H)={\displaystyle \frac{H}{C(H+Bi_{w}^{-1})+K}}. & \displaystyle\end{eqnarray}$$

The function $F(H)$ will in most cases stay close to unity, since $K\ll 1$ , $Bi_{w}^{-1}\ll 1$ and $C\approx 1$ . The constants $C$ , $G/E$ , $ReE$ , $CaE$ , $A/E$ and $\tilde{M}K/E$ , as well as the function $F$ , are all independent of the unknown scale $u_{0}$ , and thus (2.66) is also independent of it.

3.1 Basic solution

We consider a spatially uniform time-dependent base solution to (2.66), $\bar{H}(\tilde{\unicode[STIX]{x1D70F}})$ . We define the scale $h_{0}$ as the initial film thickness so that $\bar{H}(0)=1$ . The analytical solution is

(3.1) $$\begin{eqnarray}\displaystyle \bar{H}(\tilde{\unicode[STIX]{x1D70F}})={\displaystyle \frac{\sqrt{2C\tilde{\unicode[STIX]{x1D70F}}+(Bi_{w}^{-1}C+K+C)^{2}}-(Bi_{w}^{-1}C+K)}{C}}. & & \displaystyle\end{eqnarray}$$

The initial growth rate of this basic solution is reduced by every non-ideal effect, $K>0$ , $Bi_{w}^{-1}>0$ and $C>1$ . If all these effects are negligible, we get the upper-bound ideal solution $\bar{H}=\sqrt{1+2\tilde{\unicode[STIX]{x1D70F}}}$ . In any case, we see that the basic solution will grow monotonically, and thus, any vapour film collapse must be initiated by instabilities of this uniform solution.

3.2 Linear stability of basic solution

We now propose a solution which is a sum of the base solution and a spatially periodic perturbation with a small time-dependent amplitude,

(3.2) $$\begin{eqnarray}H(X,\tilde{\unicode[STIX]{x1D70F}})=\bar{H}(\tilde{\unicode[STIX]{x1D70F}})+{\hat{H}}(\tilde{\unicode[STIX]{x1D70F}})\exp \left(\text{i}{\displaystyle \frac{k}{\unicode[STIX]{x1D716}}}X\right).\end{eqnarray}$$

Here $k/\unicode[STIX]{x1D716}$ is the dimensionless wavenumber on the scale $x_{0}$ , and thus $k$ is the dimensionless wavenumber on the scale $h_{0}$ . If we insert (3.2) into (2.66) and reduce to first order in the perturbation, we get the following ordinary differential equation for the perturbation amplitude,

(3.3) $$\begin{eqnarray}{\displaystyle \frac{{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\tilde{\unicode[STIX]{x1D70F}}}}{\hat{H}}}{{\hat{H}}}}={\displaystyle \frac{\unicode[STIX]{x1D709}Gb\bar{H}^{3}k^{2}}{12E}}+{\displaystyle \frac{\unicode[STIX]{x1D709}Ak^{2}}{4E\bar{H}}}+{\displaystyle \frac{C\bar{F}^{2}K\tilde{M}k^{2}}{2E}}-{\displaystyle \frac{\unicode[STIX]{x1D709}C\bar{F}^{3}E\text{Re}k^{2}}{6}}-{\displaystyle \frac{\unicode[STIX]{x1D709}\bar{H}^{3}k^{4}}{12E\text{Ca}}}-{\displaystyle \frac{C\bar{F}^{2}}{\bar{H}^{2}}}.\end{eqnarray}$$

The recurring factor $\bar{F}$ , which appears in every term directly related to the temperature profile, is simply $F(H)$ from (2.70) with $\bar{H}$ substituted for $H$ . The last $k$ -independent term in (3.3) will only have an algebraic contribution to the exponential instability for the same reasons as the ones stated by Burelbach et al. (Reference Burelbach, Bankoff and Davis1988), and may thus be disregarded in the following analysis. All the remaining terms are $\mathit{O}(k^{2})$ except the capillary term, which is $\mathit{O}(k^{4})$ . The latter will simply provide a cutoff in $k$ and stabilize the shorter wavelengths. We may then consider the stability of long waves by only comparing the $\mathit{O}(k^{2})$ terms.

If we have initial stability, the film will grow according to (3.1). If it later turns unstable after growing somewhat, we might re-scale $h_{0}$ and reset the time parameter, and consider it a new stability problem from $\bar{H}=1$ . Thus we simply consider initial stability at $\tilde{\unicode[STIX]{x1D70F}}=0$ , and investigate the terms’ dependence on film thickness $h_{0}$ . Stability of long waves may then be analysed by considering the sign of

(3.4) $$\begin{eqnarray}\displaystyle S=\underbrace{{\displaystyle \frac{\unicode[STIX]{x1D709}Gb}{12E}}}_{\mathit{gravity}}+\underbrace{{\displaystyle \frac{CF_{0}^{2}K\tilde{M}}{2E}}}_{\mathit{thermocap.}}-\underbrace{{\displaystyle \frac{\unicode[STIX]{x1D709}CF_{0}^{3}E\text{Re}}{6}}}_{\mathit{vapour}\,\mathit{thrust}}+\underbrace{{\displaystyle \frac{\unicode[STIX]{x1D709}A}{4E}},}_{\mathit{vdW}} & & \displaystyle\end{eqnarray}$$

where $F_{0}$ is $\bar{F}$ evaluated at $\bar{H}=1$ . Here $S>0$ indicates a growing perturbation (instability). We can make the following observations about the terms in (3.4):

1. (i) Gravity: this term is destabilizing (if $b>0$ ).

2. (ii) Thermocapillary: this term is destabilizing, which is also the case for evaporating liquid films (Burelbach et al. Reference Burelbach, Bankoff and Davis1988).

3. (iii) Vapour thrust: this term is stabilizing, in contrast to its destabilizing influence in evaporating liquid films (Burelbach et al. Reference Burelbach, Bankoff and Davis1988).

4. (iv) Van der Waals: this term is destabilizing (if $A>0$ ).

The main qualitative difference compared to the stability analysis of evaporating liquid films lies in the vapour thrust term, which here is found to be stabilizing. In the analysis of Burelbach et al. (Reference Burelbach, Bankoff and Davis1988), every $\mathit{O}(k^{2})$ term is found to have a destabilizing influence, which means that an evaporating liquid film is always unstable if sufficiently large wavelengths are allowed. Film boiling appears to be different in that it has a stabilizing $\mathit{O}(k^{2})$ term, which means that the stability of long waves depend on specific conditions. This is the key to the vapour film collapse prediction in the following section.

3.3 Influence of non-ideal effects

We now briefly investigate the influence on stability by the following non-ideal effects:

1. (i) Non-equilibrium evaporation: $K\neq 0$ .

2. (ii) Heat transfer to liquid bulk: $C\neq 1$ .

3. (iii) Imperfect wall temperature control: $Bi_{w}^{-1}\neq 0$ .

Among the terms in (3.4), only the thermocapillary and vapour thrust terms are influenced by these effects. Besides the thermocapillary factor $K\tilde{M}$ , all dependencies on these non-idealities are collected into the factor

(3.5) $$\begin{eqnarray}F_{0}={\displaystyle \frac{1}{(1+C^{\prime })(1+Bi_{w}^{-1})+K}}.\end{eqnarray}$$

Out of the three factors, $K$ and $Bi_{w}^{-1}$ are dependent on film-thickness scale. They both decrease towards zero as $h_{0}$ increases ( ${\sim}h_{0}^{-1}$ ). Generally we will have $K\lll 1$ and $Bi_{w}^{-1}\ll 1$ when $h_{0}>1~\unicode[STIX]{x03BC}\text{m}$ . The third factor $C^{\prime }$ is actually independent of $h_{0}$ , but as explained in appendix B it can be expected to be very close to zero. In other words, the energy transferred into the liquid bulk is negligible compared to the energy spent on evaporation, no matter the film thickness.

Overall, this means that for moderate to large film thicknesses ( $h_{0}>1~\unicode[STIX]{x03BC}\text{m}$ ) the influence of these non-ideal effects are negligible, and we have $F_{0}\approx 1$ . For very thin films, $F_{0}<1$ . For such films, the reduction in $F_{0}$ reduces the vapour thrust term ( ${\sim}F_{0}^{3}$ ) more than it reduces the thermocapillary term ( ${\sim}F_{0}^{2}$ ), which means that the non-ideal effects have a destabilizing influence, if any.

3.4 Predicting vapour film collapse

In (3.4) we have three destabilizing terms (assuming $b>0$ and $A>0$ ) working against the sole stabilizing vapour thrust term. Their typical dependencies on film-thickness scale are illustrated in figure 4. We note the following features:

1. (i) For large $h_{0}$ , the destabilizing influence of gravity will dominate.

2. (ii) For $h_{0}<100~\text{nm}$ , the destabilizing influence of van der Waals forces will dominate

3. (iii) For intermediate $h_{0}$ , there is a remarkably even struggle between the destabilizing influence of the thermocapillary effect and the stabilizing influence of the vapour thrust effect.

We see that the vapour film is always predicted to be unstable at very small or very large thickness scales due to the van der Waals and gravity terms, respectively. However, at the intermediate thickness scales the vapour thrust and thermocapillary terms are of similar magnitude but approximately two orders of magnitude larger than the other two. This means that the thermocapillary effect is the only destabilizing effect that is capable of cancelling out the stabilizing vapour thrust in the intermediate thickness range. While the gravity and van der Waals (vdW) terms also work against vapour thrust, their effect is negligible in comparison. In summary, the model suggests the following:

1. (i) The very small and very large thickness scales are always unstable.

2. (ii) The intermediate thickness scale can only be unstable if the thermocapillary term overpowers the vapour thrust term.

We may combine these two observations with the following hypothesis:

1. (i) Hypothesis: observed vapour film collapse (Leidenfrost transition) occurs when there is instability on every thickness scale.

The hypothesis implies that a necessary condition for film boiling collapse is that all three regions indicated in figure 4 are unstable. As stated above, the very small and very large scales are always unstable. This leaves the intermediate scales, which are dominated by the thermocapillary and vapour thrust terms. To be even more specific, film boiling collapse would require instability in the $h_{0}>1~\unicode[STIX]{x03BC}\text{m}$ part of the intermediate region. On these scales $F_{0}$ approaches unity, as discussed in § 3.3.

Figure 4. Sample values for the terms in (3.4) in the case of water film boiling with $\unicode[STIX]{x0394}T=225~\text{K}$ (above the Leidenfrost point), as a function of film-thickness scale $h_{0}$ . The different shaded parts have labels indicating the dominant influence(s) in the given region.

In summary, the above hypothesis together with the behaviour of the terms in (3.4) implies that a theoretical predictor for the Leidenfrost point may be found from the balance between the thermocapillary and vapour thrust terms in the $F_{0}\rightarrow 1$ limit. Based on this we find the following $h_{0}$ -independent criterion for vapour film collapse:

(3.6) $$\begin{eqnarray}\displaystyle {\displaystyle \frac{K\tilde{M}}{2E}}>{\displaystyle \frac{\unicode[STIX]{x1D709}E\text{Re}}{6}}. & & \displaystyle\end{eqnarray}$$

The above condition depends on fluid properties as well as the superheat $\unicode[STIX]{x0394}T$ . We interpret the $\unicode[STIX]{x0394}T$ that satisfies (3.6) as an equality as the Leidenfrost point, $\unicode[STIX]{x0394}T_{L}$ . This is the superheat below which film boiling collapse is observed.

Note that the vapour density and conductivity contained in $K$ , $E$ , and $Re$ are supposed to be evaluated at the average film temperature, $T_{f}=T_{s}+\unicode[STIX]{x0394}T/2$ , which is initially unknown. We seek an explicit expression for $\unicode[STIX]{x0394}T_{L}$ that depends on known saturation properties only. When we insert expressions for the dimensionless constants in (3.6), we get

(3.7) $$\begin{eqnarray}\displaystyle {\displaystyle \frac{\unicode[STIX]{x0394}T_{L}}{T_{s}}}=\left({\displaystyle \frac{3\unicode[STIX]{x1D702}}{1+3\unicode[STIX]{x1D702}}}\right){\displaystyle \frac{\sqrt{2\unicode[STIX]{x03C0}R_{s}T_{s}}}{f(\unicode[STIX]{x1D6FC}_{e})k_{v,s}}}\unicode[STIX]{x1D6FE}\left[{\displaystyle \frac{\unicode[STIX]{x1D70C}_{v}}{\unicode[STIX]{x1D70C}_{v,s}}}{\displaystyle \frac{k_{v,s}}{k_{v}}}\right]. & & \displaystyle\end{eqnarray}$$

The left-hand side is a convenient dimensionless quantity which we call the ‘relative Leidenfrost temperature’. We see that (3.7) is an implicit equation for the relative Leidenfrost temperature, since the square bracket also depends on it. For ideal gases at constant pressure we know that $\unicode[STIX]{x1D70C}_{v}\sim 1/T$ and $k_{v}\sim \sqrt{T}$ , and thus, we may collect all $\unicode[STIX]{x0394}T_{L}$ dependence on the left-hand side, giving

(3.8) $$\begin{eqnarray}\displaystyle \left(1+{\displaystyle \frac{1}{2}}{\displaystyle \frac{\unicode[STIX]{x0394}T_{L}}{T_{s}}}\right)^{3/2}{\displaystyle \frac{\unicode[STIX]{x0394}T_{L}}{T_{s}}}=\left({\displaystyle \frac{3\unicode[STIX]{x1D702}}{1+3\unicode[STIX]{x1D702}}}\right)\left({\displaystyle \frac{\sqrt{2\unicode[STIX]{x03C0}}}{f(\unicode[STIX]{x1D6FC}_{e})}}\right)\left({\displaystyle \frac{\sqrt{R_{s}T_{s}}}{k_{v,s}}}\right)\unicode[STIX]{x1D6FE}\equiv \unicode[STIX]{x1D6E9}. & & \displaystyle\end{eqnarray}$$

Here the right-hand side, labelled $\unicode[STIX]{x1D6E9}$ for short, may be evaluated solely from known saturation properties. Its value is usually considerably less than unity. For small $\unicode[STIX]{x0394}T_{L}/T_{s}$ , we may make (3.8) explicit, as

(3.9) $$\begin{eqnarray}\displaystyle {\displaystyle \frac{\unicode[STIX]{x0394}T_{L}}{T_{s}}}\approx {\displaystyle \frac{2}{3}}[\sqrt{1+3\unicode[STIX]{x1D6E9}}-1]. & & \displaystyle\end{eqnarray}$$

It turns out that the third parenthesis in $\unicode[STIX]{x1D6E9}$ is essentially fluid independent because we generally have that $k_{v}\sim \sqrt{R_{s}T}$ , as known from ideal kinetic theory. When we define the (almost constant) variable

(3.10) $$\begin{eqnarray}\displaystyle c_{k}={\displaystyle \frac{\sqrt{RT}}{k_{v}}}, & & \displaystyle\end{eqnarray}$$

the expression for $\unicode[STIX]{x1D6E9}$ becomes

(3.11) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6E9}\approx \left({\displaystyle \frac{3\unicode[STIX]{x1D702}}{1+3\unicode[STIX]{x1D702}}}\right)\left({\displaystyle \frac{c_{k}\sqrt{2\unicode[STIX]{x03C0}}}{f(\unicode[STIX]{x1D6FC}_{e})}}\right)\unicode[STIX]{x1D6FE}. & & \displaystyle\end{eqnarray}$$

If we apply the expression (2.62) for $\unicode[STIX]{x1D702}(\unicode[STIX]{x1D6F9})$ , we find that $3\unicode[STIX]{x1D702}/(1+3\unicode[STIX]{x1D702})=3/(4+\unicode[STIX]{x1D6F9}^{-1})$ , which gives

(3.12) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6E9}\approx \left({\displaystyle \frac{3}{4+\unicode[STIX]{x1D6F9}^{-1}}}\right)\left({\displaystyle \frac{c_{k}\sqrt{2\unicode[STIX]{x03C0}}}{f(\unicode[STIX]{x1D6FC}_{e})}}\right)\unicode[STIX]{x1D6FE}. & & \displaystyle\end{eqnarray}$$

Equation (3.9) with (3.12) constitute the final and relatively simple practical result which may be used to predict the relative Leidenfrost temperature. Given that fluids generally have the same values for $\unicode[STIX]{x1D6F9}$ , $c_{k}$ and $\unicode[STIX]{x1D6FC}_{e}$ , this model predicts that the relative Leidenfrost temperature depends almost solely on $\unicode[STIX]{x1D6FE}$ and that this relationship is approximately linear.

5.1 Simplified fluid mechanical models

A semi-empirical model for the Leidenfrost point was developed by Berenson (Reference Berenson1961, equation (40)), who developed a model for the film boiling heat transfer coefficient based on classical Rayleigh–Taylor stability analysis and conservation equations in a simplified geometry. When he combined this with the minimum heat flux model by Zuber (Reference Zuber1959), which also employs simplified fluid mechanical considerations, this resulted in an expression for $\unicode[STIX]{x0394}T_{L}$ ,

(5.1) $$\begin{eqnarray}\displaystyle {\displaystyle \frac{\unicode[STIX]{x0394}T_{L}}{T_{s}}}=0.127{\displaystyle \frac{\unicode[STIX]{x1D70C}_{v}L}{k_{v}T_{s}}}\left({\displaystyle \frac{g\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}}{\unicode[STIX]{x1D70C}_{l}+\unicode[STIX]{x1D70C}_{v}}}\right)^{2/3}\left({\displaystyle \frac{\unicode[STIX]{x1D70E}_{0}}{g\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}}}\right)^{1/2}\left({\displaystyle \frac{\unicode[STIX]{x1D707}_{v}}{g\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}}}\right)^{1/3}. & & \displaystyle\end{eqnarray}$$

Note that (5.1) is semi-empirical. The exponents are theoretically derived, but the pre-factor $0.127$ stems from an experimental fit to film boiling data.

5.2 Leidenfrost point from superheat limit

A different class of models is based on the simple hypothesis that the Leidenfrost point corresponds to the liquid superheat limit, also called the homogeneous nucleation temperature. The superheat limit is commonly estimated in two different ways. The first method is by calculating the spinodal temperature from an equation of state. The spinodal is the theoretical absolute maximum superheat temperature, where the vapour nucleation barrier goes to zero. However, homogeneous nucleation will usually proceed spontaneously before the barrier reaches zero, and the temperature where this happens may be approximated by classical nucleation theory (CNT). This is the second method. Both methods are purely theoretical and do not have any fitted empirical parameters. See Aursand et al. (Reference Aursand, Gjennestad, Aursand, Hammer and Wilhelmsen2017) for further discussion on nucleation theory and the spinodal.

Superheat limit from spinodal. Using the spinodal to estimate the Leidenfrost point was first suggested by Spiegler et al. (Reference Spiegler, Hopenfeld, Silberberg, Bumpus and Norman1963). They used the van der Waals equation of state to analytically relate the spinodal ( $T_{sp}$ ) to the critical temperature, $T_{sp}=(27/32)T_{c}$ . This implies that the relative Leidenfrost temperature is simply

(5.2) $$\begin{eqnarray}\displaystyle {\displaystyle \frac{\unicode[STIX]{x0394}T_{L}}{T_{s}}}={\displaystyle \frac{27}{32}}{\displaystyle \frac{T_{c}}{T_{s}}}-1. & & \displaystyle\end{eqnarray}$$

Superheat limit from nucleation theory. Alternatively, one may use classical nucleation theory to predict the vapour nucleation rate at a given degree of liquid superheating. In combination with high accuracy equations of state, using this to predict the experimental superheat limit has been found to be quite accurate (Aursand et al. Reference Aursand, Gjennestad, Aursand, Hammer and Wilhelmsen2017). Going one step further and using this to represent the Leidenfrost point is less established but has been suggested by authors such as Yao & Henry (Reference Yao and Henry1978) and Sakurai et al. (Reference Sakurai, Shiotsu and Hata1990). In classical nucleation theory (Aursand et al. Reference Aursand, Gjennestad, Aursand, Hammer and Wilhelmsen2017) the nucleation rate $\unicode[STIX]{x1D6EC}$ ( $\text{s}-1~\text{m}^{-3}$ ) is expressed as an Arrhenius rate law,

(5.3) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6EC}=\unicode[STIX]{x1D6EC}_{0}\exp \left(-{\displaystyle \frac{\unicode[STIX]{x0394}G}{k_{B}T}}\right), & & \displaystyle\end{eqnarray}$$

with the activation energy being

(5.4) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x0394}G={\displaystyle \frac{16\unicode[STIX]{x03C0}\unicode[STIX]{x1D70E}^{3}(T)}{3(p_{s}(T)-p)^{2}}}, & & \displaystyle\end{eqnarray}$$

and the rate at zero activation barrier being

(5.5) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6EC}_{0}={\displaystyle \frac{\unicode[STIX]{x1D70C}_{l}}{m^{3/2}}}\sqrt{{\displaystyle \frac{2\unicode[STIX]{x1D70E}}{\unicode[STIX]{x03C0}}}}. & & \displaystyle\end{eqnarray}$$

Here, $m$ is the mass of a single molecule and $p_{s}$ is the thermodynamic saturation pressure. The specific expression for $\unicode[STIX]{x1D6EC}_{0}$ may vary a little between authors, but this has a negligible effect on the final result for the superheat limit.

The expression in (5.3) simply gives the nucleation rate as a function of fluid properties and temperature. In order to find the superheat limit, one must define a critical nucleation rate $\unicode[STIX]{x1D6EC}_{c}<\unicode[STIX]{x1D6EC}_{0}$ , which corresponds to sudden macroscopic phase change. It turns out that due to the rapid growth of the exponential in (5.3), the result is quite insensitive to the specific choice of $\unicode[STIX]{x1D6EC}_{c}$ . Here, we use the value of $\unicode[STIX]{x1D6EC}_{c}=1\times 10^{12}~\text{s}^{-1}~\text{m}^{-3}$ , as seen in previous works (Bernardin & Mudawar Reference Bernardin and Mudawar1999; Aursand et al. Reference Aursand, Gjennestad, Aursand, Hammer and Wilhelmsen2017). Thus, in order to predict the superheat limit, we simply have to solve the implicit equation

(5.6) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6EC}(T)=\unicode[STIX]{x1D6EC}_{c} & & \displaystyle\end{eqnarray}$$

for $T$ . Note that it is absolutely essential to include the temperature dependence of $\unicode[STIX]{x1D70E}$ in (5.4), as it is one of the major sources of temperature dependence in $\unicode[STIX]{x0394}G$ . In order to obtain a model of comparable simplicity and avoid having to iteratively solve for the saturation line using an equation of state, we use the Clausius–Clapeyron relation to estimate the saturation pressure as

(5.7) $$\begin{eqnarray}\displaystyle p_{s}(T)=p\exp \left[{\displaystyle \frac{L}{R_{s}}}\left({\displaystyle \frac{1}{T_{s}(p)}}-{\displaystyle \frac{1}{T}}\right)\right]. & & \displaystyle\end{eqnarray}$$

5.3 Performance comparison

We now seek to compare the predictive performance of the present model with the three alternative models presented in § 5.1 (Berenson model) and § 5.2 (Spiegler model and CNT model). Since $\unicode[STIX]{x1D6FC}_{e}$ can generally be anywhere in the range of $(0,1)$ , well-defined prediction by the present model requires the choice of a specific value. Here, we choose $\unicode[STIX]{x1D6FC}_{e}=0.85$ , which is the centre point of the expected range identified in § 2.2. The fluid properties necessary to evaluate the other models were mainly found from the NIST database (Linstrom & Mallard Reference Linstrom and Mallard2017; Dean Reference Dean1998). Missing mercury properties were found in Skapski (Reference Skapski1948), Epstein & Powers (Reference Epstein and Powers1953), Vinogradov (Reference Vinogradov1981) and Huber, Laesecke & Friend (Reference Huber, Laesecke and Friend2006).

The evaluation of predictive performance is shown in figure 6, where we see that only the present model can accurately predict the relative Leidenfrost point within an error of $10\,\%$ for every fluid. This is further discussed in § 6.2.

Figure 6. Comparison of model predictions with experimental data for the relative Leidenfrost point for (a) the present model (with $\unicode[STIX]{x1D6FC}_{e}=0.85$ ), (b) the Berenson model (5.1), (c) the Spiegler model (5.2) and (d) the CNT model (5.6). See figure 5 for data point legend. Grey bands show the range of a $\pm 10\,\%$ error in prediction of $T_{L}$ , relative to $T_{s}$ . Data points that fall outside this band are marked with red circles.