5.2.1. Long-wave instability mechanism

Considering the harmonic disturbance $\tilde {h}$ over a flat interface $\bar {h}$ (see § 4.1), the shear stress at the perturbed interface is given by a base state and a perturbation contribution. Expanding $D \hat {u}$ around the base state thickness $\bar {h}$, we obtain, at first order,

(5.3)\begin{equation} D \hat{u}(\hat{h}) = D\bar{u}(\bar{h}) + D\tilde{u}(\bar{h}) + D^2\bar{u}(\bar{h})\tilde{h} =0. \end{equation}

Since the base state is shear-free at the interface ($D\bar {u}(\bar {h})=0$) and knowing that $D^2\bar {u}(\bar {h})=-1$, (5.3) reduces to

(5.4)\begin{equation} D\tilde{u}(\bar{h}) - \tilde{h}=0. \end{equation}

The equilibrium of forces implies that the disturbance has to generate a positive shear stress $D\tilde {u}(\bar {h})$ to compensate for $\tilde {h}$. To analyse this mechanism, we considered a reference frame moving at the substrate velocity $U_p$ ($\hat {u}=-1$), and we expand the streamwise velocity amplitude $\acute {u}$ in a power series of $k$ assuming long-wave conditions (4.1) (Smith Reference Smith1990). The solution at $O(1)$ of (4.4) with boundary conditions (4.5) is given by a linear velocity amplitude $\acute {u}_{m0}(\hat {y})$ and a positive phase speed $c_{m0}$:

(5.5a,b)\begin{equation} \acute{u}_{m0}(\hat{y}) = \hat{y} + 1, \quad c_{m0} = \bar{h}^2. \end{equation}

The behaviour of $c_{m0}$ is solely determined by gravity. The thicker the base state, the more gravity is important, and the more the wave travels faster downwards along the $\hat {x}$ direction.

Figure 9(a) shows the flow field at leading order as a consequence of a harmonic displacement of the interface (red line). Since the flow field is in phase with the liquid-film displacement, the streamwise velocity is maximum at the peak and minimum at the trough. Considering a control volume in the range $\theta \in [0,{\rm \pi} /2]$, enclosed between a peak and a node at the interface, this has a positive net flow rate. The velocity field pushes liquid to the right at the crest, with a zero flow rate at the node. In accordance with the continuity equation, this implies a positive displacement of the film interface to accommodate this accumulation of mass, leading to a travelling wave in the positive $\hat {x}$ direction. The solution of (4.6a) gives the normal velocity at $O(k)$:

(5.6)\begin{equation} v_{1} ={-}i\hat{y}^2/2. \end{equation}

The link between the phase speed and the flow rate is given by the kinematic boundary condition at the interface (4.7b):

(5.7)\begin{equation} c_{m0} - \bar{u}_m(\hat{h}) = v_{1} =\int_{0}^{\hat{h}}u_{m0}-1\,{\rm d}\hat{y} = \hat{h}^2/2. \end{equation}

Figure 9.

Magnitude (greyscale) and vector field (arrows) of the perturbation's velocity field (a) at leading order and (b) at first order in the wavenumber $k$.

This initiating mechanism affects the development of the flow field at $O(k)$ through two inertial stresses in (4.6b):

(5.8a,b)\begin{equation} -{\rm i}Re(u_{m0}(\hat{y}) -1)(\bar{u}_m(\hat{y}) - c_{m0} - 1) \quad \text{and} \quad Re\, v_{1}(\hat{y})D\bar{u}_m(\hat{y}). \end{equation}

The first term corresponds to the advection of the reduced leading-order solution by the base state velocity with respect to the reduced leading order wave speed. The second term corresponds to the advection of the base state velocity by the first-order wall-normal velocity $v_1$. Since the phase speed $c_{m0}$ is larger than any values in $\bar {u}_m(\hat {y})$, and being $D\bar {u}_m(\hat {y})=-\hat {y} + \hat {h}$ positive for all $\hat {y}\in [0,\hat {h}]$, these two terms are both negative. This leads to a destabilising flow field given by the solution of the first-order equations (4.6) with boundary conditions (4.7):

(5.9)\begin{equation} u_{m1}(\hat{y}) = \frac{Re}{24} {\rm i} (8\hat{h}^4\hat{y} - 4\hat{h}^2\hat{y}^3 + 12\hat{h}\hat{y} + \hat{h}\hat{y}^4 - 4\hat{y}^3). \end{equation}

Figure 9(b) shows the flow field calculated with (5.9). Since the velocity amplitude $u_{m1}(\hat {y})$ is imaginary and positive, the flow has a phase shift of ${\rm \pi} /2$ with respect to the wave displacement. Consequently, the velocity pushes fluid from the troughs to the peaks, sustaining the growth of the perturbation. This renders a growth mechanism based on the extraction of energy contained in the leading-order solution, which, in turn, extracts energy from the base state through the work done by the shear stress at the interface.

5.2.2. Vorticity perturbation at the free surface and in the bulk

In the previous subsection, we studied the general structure of the instability mechanism. The first-order solution in $k$, fed by leading-order inertial stresses, generates a destabilising flow from the troughs to the peaks due to a ${\rm \pi} /2$ phase shift to the free-surface displacement. In this section, we go into more detail, analysing how the phase shift and the magnitude of the perturbation change with the wavenumber and the flat liquid-film height. To this end, we look at the growth of the perturbation in terms of the vorticity field. The perturbation's shear stress correction in (5.4) can be seen as a source of vorticity at the free surface $\omega _{FS}$. We consider a reference frame moving with the wave speed $c_r$, i.e. $\theta =k(\hat {x}-c_r\hat {t})$, and we assume a sinusoidal displacement of the free surface:

(5.10)\begin{equation} \tilde{h} = \sin{(\theta)}, \end{equation}

with $\theta =0$ at 0, this implies through (4.12a), that

(5.11a,b)\begin{equation} \eta_r = 0,\quad \eta_i ={-}1. \end{equation}

By means of the kinematic boundary condition (2.22b), we scale the eigenfunction such that

(5.12a,b)\begin{equation} \phi_r(\hat{h}) = c_i, \quad \phi_i(\hat{h}) ={-}\hat{c}. \end{equation}

The shear stress condition (2.22d) imposes that

(5.13a,b)\begin{equation} D^2\phi_r(\hat{h}) ={-}k^2 c_i, \quad D^2\phi_i(\hat{h}) ={-}2 + k^2\hat{c}. \end{equation}

By replacing these expressions in (4.10d), we obtain the vorticity at the free surface:

(5.14)\begin{equation} \omega_{FS} = \gamma \sin{(\theta - \xi)}, \end{equation}

with the amplitude $\gamma$ and the phase shift $\xi$ given by

(5.15a,b)\begin{equation} \gamma = 2\sqrt{(1-k^2\hat{c})^2 + (k^2c_i)^2]}, \quad \xi = \arctan{(k^2c_i/(1-k^2\hat{c})}. \end{equation}

Depending on $\xi$, $\omega _{FS}$ stabilises ($\xi <0$) or destabilises ($\xi >0$) the free-surface displacement (Kelly et al. Reference Kelly, Goussis, Lin and Hsu1989).

Figure 10 shows (a,d) the growth rate $\omega _i$, (b,e) the amplitude $\gamma$ and (c,f) the phase shift $\xi$ of the free-surface vorticity as a function of the wavenumber $k$ and the liquid-film thickness $\hat {h}$ at $Re=30$, with the neutral curve (continuous red line) and the maximum growth rate (dashed red line) for (a–c) the corn oil ($Ka=4$) and (d–f) the liquid zinc ($Ka=11\,525$). For a fixed $k$, $\omega _i$ gently grows with $\hat {h}$ driven by a vorticity with a large amplitude and a very small positive phase shift. For $\hat {h}=1$, $\omega _i$ grows more sharply, accompanied by an increase in the phase shift and a decrease in the vorticity amplitude. In the corn oil ($Ka=4$) case, the growth rate and the phase shift reach a plateau for large $\hat {h}$, in line with the saturating mechanism introduced in § 5.1. The phase shift $\xi$ peaks at higher wavenumbers than the growth rate $\omega _i$. The decay of the vorticity amplitude $\gamma$ with $k$ compensates for the destabilising effect of larger phase shifts, decreasing the growth rate. This suggests that the instability mechanism is also driven by the vorticity in the bulk, thus confirming the results in the previous subsections.

Figure 10.

(a,d) Growth rate, (b,e) free-surface vorticity amplitude and (c,f) phase shift as a function of $k$ and $\hat {h}$ with the neutral curve (continuous red line) and the positions of maximum growth rates (red dashed line) for $Re=30$, considering (a–c) corn oil ($Ka=4$) and (d–f) zinc ($Ka=11\,525$).

To investigate this behaviour, we analysed the vorticity perturbation $\tilde {\omega }(\theta,\hat {y})=\partial _{\hat {x}}\tilde {v} - \partial _{\hat {y}}\tilde {u}$ within the liquid film. The vorticity equation is obtained by taking the curl of the momentum equations (2.14b) and (2.14c), considering the continuity equations (2.14a) and the base state (2.11), which leads to

(5.16)\begin{equation} \partial_{\hat{t}}\tilde{\omega} + \bar{u}\partial_{\hat{x}}\tilde{\omega} - \tilde{v}D^2\bar{u} = \frac{1}{Re}(\partial_{\hat{x}\hat{x}}\tilde{\omega} + D^2\tilde{\omega}). \end{equation}

Considering the reference frame moving with the wave speed $c_r$ and knowing that $D^2\bar {u}=-1$, we obtain

(5.17)\begin{equation} k(\bar{u}-c_r)\partial_{\theta}\tilde{\omega} + \tilde{v} = \frac{1}{Re}(k^2\partial_{\theta\theta}\tilde{\omega} + D^2\tilde{\omega}). \end{equation}

On the left-hand side, the first term corresponds to the advection relative to the wave velocity, and the second term corresponds to the vorticity perturbation's advection of the base state. The term on the right-hand side corresponds to the viscous diffusion of vorticity. Inserting a normal mode solution for the vorticity with $k\in \mathbb {R}$ and $c = c_r + {\rm i}c_i$,

(5.18)\begin{equation} \tilde{\omega} = \acute{\omega}(\hat{y})\exp({\rm i}k(\hat{x} - c\hat{t})) = \acute{\omega}(\hat{y})\exp(kc_i\hat{t})\exp({\rm i}\theta), \end{equation}

and for the wall-normal velocity given by (4.1) in (5.17) leads to the expression for the free-surface amplitude $\acute {v}(\hat {y})$,

(5.19)\begin{equation} \acute{v}(\hat{y}) = {\rm i}k(c_r - \bar{u}(\hat{y}))\acute{\omega}(\hat{y}) + \frac{1}{Re}({-}k^2 + D^2)\acute{\omega}(\hat{y}). \end{equation}

Note that for $Re\gg 1$, the relation reduces at leading order in $k$ to

(5.20)\begin{equation} \acute{v}(\hat{y}) \approx {\rm i}k(c_r - \bar{u}(\hat{y}))\acute{\omega}(\hat{y}) , \end{equation}

which corresponds to neglecting viscous effects.

Approximating $c_r$ with the long-wave asymptotic solution at $O(1)$ in (B3), the term $c_r -\bar {u}$ is always positive $\forall \hat {y}\in [0,\hat {h}]$, implying that the vertical velocity amplitude changes linearly with the vorticity amplitude with a phase shift of ${\rm \pi} /2$ along $\theta$.

Figure 11 shows the perturbation (a, b, c, g, h and i) vorticity and (d, e, f, j, k and l) normal velocity fields at (a–f) $\hat {h}=0.7$ and (g–l) $\hat {h}=1.7$ for the corn oil ($Ka=4$) with $k=0.5$ (a, d, g and j), 0.93 (b, e, h and k) and 1.2 (c, f, i and l) with the maximum (red square) and minimum (red triangle) of vorticity at the interface with $Re=30$. For $k=0.5$, the vorticity contours deform towards the left, advected by the flow. Since the advection velocity strengthens, this tilting effect is more intense for large $\hat {h}$. This structure recalls the capillary separation eddy in a falling liquid film of finite amplitude in the capillary flow region (Dietze, Al-Sibai & Kneer Reference Dietze, Al-Sibai and Kneer2009). The induced normal velocity field changes sign at the peak of vorticity along $\theta$, leading to a positive (negative) net vertical flow rate under the crests (trough), which fosters the growth of the perturbation.

Figure 11.

Perturbation (a, b, c, g, h and i) vorticity and (d, e, f, j, k and l) normal velocity fields at (a–f) $\hat {h}=0.7$ and (g–l) $\hat {h}=1.7$ for the corn oil ($Ka=4$) with $k=0.5$ (a, d, g and j), 0.93 (b, e, h and k) and 1.2 (c, f, i and l) with the maximum (red square) and minimum (red triangle) of vorticity at the interface with $Re=30$.

As the wavenumber increases, the vorticity lines bend towards the right near the substrate. This curvature stabilises the perturbation, enlarging the negative (positive) normal velocity area under the peak (trough). At the same time, the vorticity tends to concentrate near the free surface, approaching its maximum value at the interface and forming a boundary layer region with strong shear stresses. Despite the differences in the vorticities and normal velocities’ distributions, the stability mechanism at $\hat {h}=0.7$ and $\hat {h}=1.7$ for the corn oil ($Ka=4$) is very similar. As we have seen, the neutral curves and the dispersion relations (figure 6), the vorticity at the free surface (figure 10) and the order of magnitude of the vorticity in bulk are almost the same for the two conditions. This means that for small $Ka$, the instability growth is mainly driven by the vorticity at the free surface and in its proximity.

Things change as we move to larger $Ka$. Figure 12 shows the perturbation (a, b, c, g, h and i) vorticity and (d, e, f, j, k and l) normal velocity fields at (a–f) $\hat {h}=0.7$ and (g–l) $\hat {h}=1.7$ for the liquid zinc ($Ka=11\,525$) with $k$ equal to (a,d) 0.002, (g,j) 0.003, (b,e) 0.047, (c,f) 0.051, (h,k) 0.077 and (i,l) 0.15 with the maximum (red square) and minimum (red triangle) of vorticity at the interface at $Re=30$. For $\hat {h}=0.7$, the advection has a very small effect on the vorticity contour's curvature. The viscous effects almost completely dominate the vorticity distribution. As we have seen in figure 10, for $\hat {h}=0.7$, the instability grows mostly due to the vorticity magnitude rather than the phase shift, which is almost zero.

Figure 12.

Perturbation (a, b, c, g, h and i) vorticity and (d, e, f, j, k and l) normal velocity fields at (a–f) $\hat {h}=0.7$ and (g–l) $\hat {h}=1.7$ for the liquid zinc ($Ka=11\,525$) with $k$ equal to (a,d) 0.002, (g,j) 0.003, (b,e) 0.047, (c,f) 0.051, (h,k) 0.077 and (i,l) 0.15 with the maximum (red square) and minimum (red triangle) of vorticity at the interface at $Re=30$.

For $\hat {h}=1.7$, the vorticity lines tilt in the advection direction as we increase the wavenumber. Moreover, for $k =0.15$, a region of intense vorticity is created near the substrate, destabilising the flow, increasing the net flow rate under the crests, and creating a boundary layer near the substrate in addition to that near the free surface.

This region is the product of the streamwise pressure gradient induced by the surface tension. The normal vorticity flux at the substrate, also known as vorticity source strength (Lighthill Reference Lighthill1963), is given by(Morton Reference Morton1984)

(5.21)\begin{equation} -\partial_{\hat{y}}\omega = \partial_{\hat{x}}\tilde{p} - 1, \end{equation}

where the right-hand side represents the rate of vorticity production per unit area and comprises the perturbation's streamwise pressure gradient and the gravitational effect. Positive vorticity is generated when the streamwise pressure gradient outweighs the gravitational effect.

For small $k$, the pressure is constant a long $\hat {y}$, with the most-sensitive zone to the surface tension close to the substrate. As we have shown in Appendix B.2, for high $Ka$, the surface tension influences the first-order solution for small $k$. The first-order streamwise velocity amplitude, derived by differentiating the stream function (B5) with respect to the wall-normal coordinate, is given by

(5.22)\begin{equation} \acute{u}_1 = D \varphi_1^{{\star}}(\hat{y}) = \frac{{\rm i} \hat{y}^2}{12}(\hat{h}\,Re\,\hat{y}^2 - 4\hat{h}^2Re\hat{y} + 12k^2\,Ca ). \end{equation}

Assuming $\hat {h}=O(1)$, the surface tension term is dominant for small $\hat {y}$ i.e. near the substrate.

Moreover, the surface tension also increases the vorticity near the free surface (figure 12e and f). The normal vorticity flux at the interface is given by (Dietze et al. Reference Dietze, Al-Sibai and Kneer2009; Wu Reference Wu1995)

(5.23)\begin{equation} -\partial_{\hat{n}}\tilde{\omega} = \frac{\partial_{\hat{x}}\tilde{p}}{\sqrt{1 + (\partial_{\hat{x}}\tilde{h})^2}}, \end{equation}

where $\hat {n}$ is the non-dimensional normal vector to the interface pointing toward the liquid. As for the substrate flux, it depends on the streamwise pressure gradient. At a peak (trough) location, the pressure gradient in the streamwise direction is negative (positive), so the film produces positive (negative) vorticity at the free surface.

The mechanism leading to the formation of this boundary layer zone is as follows: the pressure imposed by the surface tension pushes fluid down (up) at the crests (troughs), which, via the continuity equation, produces a positive (negative) streamwise flow near the substrate. The movement of the substrate increases the shear effects compared with the falling film case, leading to more intense vorticity regions.

These outcomes highlight the importance of $Ka$, even for long-wave perturbation (small $k$). This means that surface tension is crucial for stabilising short waves and storing perturbation energy and for the early development of instability and vorticity distribution in liquid film. Moreover, we reveal two mechanisms of instability involving viscous stresses associated with the boundary layer at the free surface for the corn oil ($Ka=4$) and the boundary layer at the substrate for the zinc ($Ka=11\,525$).

5.2.3. Energy balance of the perturbation

We have analysed the early stages of the long-wave instability mechanism and how the vorticity at the free surface and in bulk fosters the growth of unstable perturbations. Here, we study the terms forming the kinetic energy balance (4.8a) of an unstable perturbation, as presented in § 4.2.

The vorticity near the substrate also affects the dissipation terms defined in (4.8f) and (4.9). Figure 13 shows the elongational (${\rm DISSI}_1$ green dashed-dotted line with circles and ${\rm DISSI}_3$ orange dashed line with triangles) and shear (${\rm DISSI}_2$ blue dashed line with squares) stresses contributing to the total dissipative viscous effects (${\rm DISSI}_{tot}$ red continuous line) as a function of $k$ at $\hat {h}=1.7$ and $Re=30$ for (a) corn oil ($Ka=4$) and (b) liquid zinc ($Ka=11\,525$). Here ${\rm DISSI}_1$ is equal to ${\rm DISSI}_3$ due to the continuity equation. For small $k$, ${\rm DISSI}_2$, which is a function of the strain rate $\upsilon$, dominates the dissipative effects, with ${\rm DISSI}_1$ and ${\rm DISSI}_2$ slowly increasing with $k$. For larger $k$, ${\rm DISSI}_2$ decreases in magnitude. At $Ka=4$ (corn oil), it decreases monotonically with $k$, whereas for $Ka=11\,525$ (zinc), it reaches a minimum and increases again towards the cutoff wavenumber. This is due to the creation of a boundary layer near the substrate. Near the boundary, the strain rate $\upsilon$ is equal to the vorticity, and they are both generated with the same boundary flux densities (Morton Reference Morton1984). In addition to the boundary, the bulk is also a source of strain rate. The equation governing the strain rate is given by

(5.24)\begin{equation} \partial_{\hat{t}}\upsilon ={-}2 \partial_{\hat{x}\hat{y}}\tilde{p} + \frac{1}{Re}(\partial_{\hat{x}\hat{x}}\upsilon + \partial_{\hat{y}\hat{y}}\upsilon). \end{equation}

When the surface tension induces a pressure gradient in the liquid film, the bulk also generates a strain rate, feeding ${\rm DISSI}_2$.

Figure 13.

Distribution of the extensional stresses ${\rm DISSI}_1$ and ${\rm DISSI}_3$ and shear stress ${\rm DISSI}2$, with $\hat {h}=1.7$ and $Re=30$ for (a) the corn oil ($Ka=4$) and (b) the liquid zinc ($Ka=11\,525$).

5.3.1. AC threshold in the $Ka$–$Re$ space with $\hat {h}=1$

Figure 14 shows the AC threshold for the Derjaguin solution ($\hat {h}=1$) in (a) the $Ka$–$Re$ parameter space, with the trend line $Ka=0.15Re^{1.8}$ (red dashed line) and in (b) the $c_r$–$Re$ space. For $Ka<30$, the threshold has a minimum of $Ka=17$ at $Re=3$ and a maximum of $Ka=25$ at $Re=0.4$. For $Ka>30$ and $Re\gtrsim 10$, it gets to an asymptote, which goes as $Ka\sim Re ^{1.8}$ or in terms of capillary number as $Ca^{-1}\sim Re ^{1.5}$. These results show that for fluids with $Ka<17$, the Derjaguin solution is always convectively unstable regardless of the Reynolds number.

Figure 14.

AC threshold (black dotted line) of the Derjaguin flat film solution ($\hat {h}=1$) in (a) the $Ka$–$Re$ space with the trend line $Ka=0.15Re^{1.8}$ (red dashed line) and in (b) $c_r$–$Re$ space with the associated real wavenumber $k_r$ (red triangles) and non-dimensional capillary wavenumber $k_{\ell _{c}}$ (continuous blue line with blue squares).

The neutral perturbations ($\omega _i=0$) associated with the AC threshold have a negative phase speed for every $Re$. These travel faster than the substrate speed ($|c_r|>|U_p|$) for $Re\gtrsim 5.5$. The phase speed has a maximum at $Re=0.1$ and an asymptote at $c_r~2.7$ for $Re>300$. The real wavenumber $k_r$ of the neutral perturbations has a maximum at $k_r\approx 0.8$ in the range $100< Re<200$. For $Re>300$, $k_r$ exceeds $\hat {\ell }_{c}$, highlighting how, at these wavelengths, surface tension starts to dominate over gravity.

5.3.2. AC threshold in the $\hat {h}$–$Re$ space for different $Ka$

Moving to the AC threshold for the liquids in table 1, from figures 15–16 show the window of absolute instability in (a) the $\hat {h}$–$Re$ space bounded by a lower and an upper threshold (black dots) and in (b) the $c_r$–$Re$ space, with the real wavenumber $k_r$ of the upper (red empty triangular markers) and lower thresholds (red empty circular markers). We also show a second AC upper threshold (green empty markers) and the LLD solution $\mathop{h}\limits^{\circ}$ (dash-dotted orange line) for the water–glycerol ($Ka=195$) solution and corn oil ($Ka=4$). The absolute region's extrema and the upper threshold inflection points are reported in table 3.

Figure 15.

Upper and lower branches of the absolute instability region (black dots) in (a,c) the $\hat {h}$–$Re$ and (b,d) $c_r$–$Re$ spaces with the associated wavenumber $k_r$ for the lower (circles) and upper (triangles) thresholds and the curve $k_{\ell _c}$ (dashed blue line) for (a,b) liquid zinc ($Ka=11\,525$) and (c,d) water ($Ka=3400$).

Figure 16.

Upper and lower branches of the absolute instability region (black dots) in (a,c) the $\hat {h}$–$Re$ space with the second branch of the upper threshold (green empty triangles) and the LLD solution (orange dash- dotted line) and in (b,d) the $c_r$–$Re$ space with the associated wavenumber $k_r$ for the lower (circles) and upper (triangles) thresholds and the curve $k_{\ell _c}$ (dashed blue line) for (a,b) water–glycerol ($Ka=195$) and (c,d) corn oil ($Ka=4$).

Table 3.

Maximum and minimum of the absolute instability windows in the $\hat {h}$–$Re$ space for different liquids.

For the four liquids, the lower threshold stems from $\hat {h}=1$ as $Re\rightarrow 0$ and develops mainly in the thin film domain, with a minimum at intermediate $Re$ values. The threshold extends into the thick film domain, with a plateau at $\hat {h}\approx 1.4$ for water–glycerol ($Ka=195$) and corn oil ($Ka=4$). The lower threshold of the corn oil extends solely into the thick film, with thin film base states ($\hat {h}<1$) always convectively unstable. In the $c_r$–$Re$ space, the neutral waves associated with the lower threshold travel upward against gravity ($c_r<0$). The associated wavenumber $k_r$ monotonically increases with $Re$. Here $k_r$ overtakes $k_{\ell _{c}}$ for the liquid zinc ($Ka=11\,525$) and water, showing that the surface tension prevails over gravity for large $Re$. Due to surface tension effects, which support the entrainment action against gravity for intermediate $Re$, the minimum of the lower threshold $\hat {h}_{min}$ decreases with $Ka$, following the relations:

(5.25a,b)\begin{equation} Re_{min} = 1.5496Ka^{0.4677}, \quad \hat{h}_{min} ={-} 0.0022Re_{min} + 0.9403, \end{equation}

where $Re_{min}$ is the $Re$ associated to $\hat {h}_{min}$.

The upper thresholds stem from $\hat {h}=1$ and extend into the thick film domain ($\hat {h}>1$) with a maximum of $\hat {h}\approx 1.65$ at $Re\approx 10$ and a plateau at $\hat {h}\approx 1.5$ for $Re>100$. In the $c_r$–$Re$ space, the neutral waves, associated with the upper threshold, travel downwards ($c_r>0$) with a maximum at $Re\approx 1$ for corn oil ($Ka=4$) and $Re\approx 100$ for the other liquids. As for the lower branch, based on the liquid's properties, we define the value of $\hat {h}_{max}$ and the associated $Re$ ($Re_{max}$) as a function of $Ka$ via the relations

(5.26a,b)\begin{equation} Re_{max} = 1.9372Ka^{0.2732}, \quad \hat{h}_{max} ={-} 0.0008Re_{max} + 1.6663. \end{equation}

Considering the operational range of the hot-dip galvanising process, using liquid zinc with $Re\in [437,2273]$ and $\hat {h}<0.2$, the flat liquid film is always convectively unstable.

In addition to the saddle point defining the upper branch, we found other valid saddle points for water–glycerol ($Ka=195$). Figure 17 shows two saddle points (red circle and red square) (a) in the complex wavenumber space $k_r$–$k_i$ with the $\omega _i$ colour map and (b) in the complex frequency space $\omega _r$–$\omega _i$ with the spatial branches along the real $k$ axis with $k_i=0$ for water–glycerol ($Ka=195$) at $\hat {h}=1.48$ and $Re=100$. The two saddle points satisfy the collision criterion since an odd number of temporal branches surmount them. Based on the second saddle point (red square), which appears around $Re\approx 40$, we traced another AC threshold (green empty triangles in figure 16a). Interestingly, this threshold is always below the first and approaches the lower branch for larger $Re$, closing the window of absolute instability. This is also visible in the $c_r$–$Re$ space (green triangles in figure 16b), where the phase speed of the two branches converges. Moreover, the wavenumber increases with $Re$ reaching 0.5 for $Re=3000$.

Figure 17.

Position of two saddle points (red circle and red square) in (a) the $k_r$–$k_i$ space with the $\omega _i$ colour plot and in (b) the $\omega _r$–$\omega _i$ space with the temporal branches along the real wavenumber axis with $k_i=0$ (continuous coloured lines) for water–glycerol ($Ka=195$) with $\hat {h}=1.48$ and $Re=100$.

For $Re=3000$, we discovered a collection of saddle points near the imaginary wavenumbers’ axis. Figure 18 shows the position of six saddle points with (a,c) red markers in the $k_r$–$k_i$ space with the $\omega _i$ colour map and with (b,d) coloured markers in the $\omega _r$–$\omega _i$ space and the temporal branches along the real wavenumber axis (coloured lines) with $\hat {h}=1.41$ and $Re=3000$ for (a,b) water–glycerol ($Ka=195$) and (c,d) $Ka=0$. The first saddle point is used to construct the AC threshold described previously. For plotting convenience, we do not show the saddle point of the second upper branch because it is at a much larger real wavenumber. Only the first four of the six saddle points respect the collision criterion, with the first being the last to have a negative growth rate as we increase $\hat {h}$. Since the AC associated with this point represents the upper bound for the region of absolute stability, we did not investigate the AC for the other saddle points. The location of these saddle points is invariant with $Ka$ since their positions are the same for the case with water–glycerol and $Ka=0$. This implies that this part of the AC upper branch is not affected by the surface tension and is solely given by a balance of gravity and inertia.

Figure 18.

Multiple saddle points in (a,c) the complex wavenumber space $k_r$–$k_i$ (numbered red markers) and (b,d) in the complex frequency domain $\omega _r$–$\omega _i$ (empty coloured markers) with the temporal branches along the real wavenumber axis (coloured continuous line) at $Re=3000$ and $\hat {h}=1.41$ for (a,b) water–glycerol ($Ka=195$) and (c,d) $Ka=0$.

The LLD solution $\mathop{h}\limits^{\circ}$ (defined in (2.10)) is always convectively unstable for zinc ($Ka=11\,525$), water ($Ka=3400$) and water-glycerol ($Ka=195$). For convenience, we did not report this curve for liquid zinc and water because it was much smaller $\hat {h}$ compared with the window of absolute instability. For corn oil ($Ka=4$), $\mathop{h}\limits^{\circ}$ crosses the area of absolute instability in the range $100< Re<500$. However, since this intersects the lower branch at a $Re\approx 300$, it is unrealistic to see it in an experiment. In addition, given that the wavenumber of the lower branch is tiny, this would require a very long domain for substrate velocities in the range $U_p\in [0.1$–$1]$ (m s$^{-1}$).

We rescale the upper and lower AC thresholds in the $\hat {h}$–$Re$ space with ($\hat {h}_{max},Re_{max}$) and ($\hat {h}_{min},Re_{min}$), respectively, for liquid zinc ($Ka=11\,525$), water ($Ka=3400$) and water–glycerol ($Ka=195$). Figure 19 shows the scaled (a,b) upper and (c,d) lower thresholds in (a,c) the $\hat {h}$–$Re$ and (c,d) the $\breve {h}$–$Re$ spaces for the liquid zinc ($Ka=11\,525$) (red circle), water ($Ka=3400$) (blue squares) and water–glycerol ($Ka=195$) (green triangles), where $\breve {h}$ is the non-dimensional film thickness based on the viscous length $\ell _{\nu }$ defined in (2.4). The curve matches perfectly apart from the lower bound in the $\hat {h}$–$Re$ space at small and large $Re$. Moreover, we found a simple approximation to the lower AC threshold. Figure 19(d) shows a trend line (black loosely dashed line) for the lower branch, given by

(5.27)\begin{equation} \breve{h}/\breve{h}_{min}=(Re/Re_{min})^{4/9}. \end{equation}

This approximation enables the construction of the absolute instability window for any fluid within the range $Ka\in [195,11\,525]$ without the need for further simulations.

Figure 19.

Scaled absolute instability window for liquid zinc ($Ka=11\,525$; red circles), water ($Ka=3400$; blue squares) and water-glycerol ($Ka=195$; green triangles), with the (a,b) upper and (c,d) lower AC thresholds rescaled with their values at the maximum and the minimum in (a,c) the $\hat {h}$–$Re$ and (b,d) the $\acute {h}$–$Re$ space with the trend line $\breve {h}/\breve {h}_{min}=(Re/Re_{min})^{4/9}$ (dashed black line).