3.1 Lubrication equation

The draining base flow $H$ on a smooth cylindrical substrate is assumed to be invariant along the axial direction, which simplifies the lubrication (2.2) to

(3.1) $$\begin{eqnarray}\displaystyle H_{t}+\frac{1}{3}\left\{H^{3}\left[\underbrace{\frac{\unicode[STIX]{x1D6FF}^{4}}{Bo}(H_{\unicode[STIX]{x1D703}}+H_{\unicode[STIX]{x1D703}\unicode[STIX]{x1D703}\unicode[STIX]{x1D703}})}_{\text{I}}-\underbrace{\unicode[STIX]{x1D6FF}\cos \unicode[STIX]{x1D703}H_{\unicode[STIX]{x1D703}}}_{\text{II}}+\underbrace{\sin \unicode[STIX]{x1D703}}_{\text{III}}\right]\right\}_{\unicode[STIX]{x1D703}}=0. & & \displaystyle\end{eqnarray}$$

The considered initial condition is

(3.2) $$\begin{eqnarray}\displaystyle H_{0}(\unicode[STIX]{x1D703})=c_{1}\{1-\tanh [c_{2}(\unicode[STIX]{x1D703}-\unicode[STIX]{x1D719})]\}+b, & & \displaystyle\end{eqnarray}$$

where $c_{1}$ is the dimensionless amplitude yielding a dimensionless fluid’s area of unity, $\int _{0}^{\unicode[STIX]{x03C0}/2}[H_{0}(\unicode[STIX]{x1D703})-b]\,\text{d}\unicode[STIX]{x1D703}/\unicode[STIX]{x1D6FF}=1$ , $c_{2}=15$ sets the initial slope of the front, located for $t=0$ at $\unicode[STIX]{x1D703}=\unicode[STIX]{x1D719}=19^{\circ }$ , and $b$ is the precursor film thickness. Note that the fluid in the precursor film is removed in the measure of the volume. As will become clear in § 3.4, the base-flow solution at late times does not depend on the initial condition. The draining-flow lubrication (3.1) is solved over the full domain $\unicode[STIX]{x1D703}\in [0,\unicode[STIX]{x03C0}]$ with the boundary conditions $H_{\unicode[STIX]{x1D703}}=H_{\unicode[STIX]{x1D703}\unicode[STIX]{x1D703}\unicode[STIX]{x1D703}}=0$ at both ends, such that the flux vanishes at these locations. In order to avoid the unstable Rayleigh–Taylor regime, we only solve the lubrication equation until the advancing front reaches $\unicode[STIX]{x1D703}=\unicode[STIX]{x03C0}/2$ .

Figure 2.

Base flow temporal evolution for $Bo=15$ and $\unicode[STIX]{x1D6FF}=0.15$ , shown in polar coordinates (a) and cartesian coordinates (b). The parameters are taken such as to have the same dimensionless numbers as in the experiment of Takagi & Huppert (Reference Takagi and Huppert2010). The film thickness in (a) has been magnified by a factor of $3$ for visualization purposes.

A typical evolution of the base flow is shown in figure 2 (see appendix C for details of the numerical method). The drainage solution given by (3.1) can be decomposed into an outer region where the drainage term $\text{III}$ dominates and an inner region close to the front where the capillary pressure and variation of hydrostatic pressure terms $\text{I}$ and $\text{II}$ , respectively, are no longer negligible (Takagi & Huppert Reference Takagi and Huppert2010).

3.2 Drainage at the pole

In the outer region close to the pole at $\unicode[STIX]{x1D703}=0$ , the lubrication (3.1) with negligible capillary pressure and variation of hydrostatic pressure terms can be simplified assuming the leading-order approximation $\sin \unicode[STIX]{x1D703}\sim \unicode[STIX]{x1D703}$ for the drainage term. Following Takagi & Huppert (Reference Takagi and Huppert2010), the resulting equation can be solved analytically and the film thickness at the pole with initial value $e=H_{0}(\unicode[STIX]{x1D703}=0)$ evolves as

(3.3) $$\begin{eqnarray}\displaystyle H(\unicode[STIX]{x1D703}=0,t)=\frac{1}{\sqrt{\displaystyle \frac{1}{e^{2}}+\frac{2}{3}t}}. & & \displaystyle\end{eqnarray}$$

The drainage becomes independent of initial condition and follows the universal law $H\sim (2t/3)^{-1/2}$ for large times $t\gg \unicode[STIX]{x1D70F}_{i}=3/(2e^{2})$ , as visible in figure 3(a).

Figure 3.

(a) Comparison between the temporal film thickness evolution at the pole $\unicode[STIX]{x1D703}=0$ obtained numerically (black solid line) and the prediction (3.3) (red dashed line). The dotted line indicates the critical initial time $\unicode[STIX]{x1D70F}_{i}=3/(2e^{2})$ and the dashed line the crossover time at which $\ell _{c}=\ell$ . (b) Comparison between the film thickness profile at $t=133$ obtained numerically (black solid line) and the prediction (3.4) (red dashed line). Inset: zoom in the inner region. Here $Bo=15$ (and $Bo=8$ for the dot-dashed line in the inset) and $\unicode[STIX]{x1D6FF}=0.15$ .

3.3 Spatial dependence of the outer region

The spatial variation of the film thickness in the outer region can be found by a regular perturbation expansion as described in Balestra et al. (Reference Balestra, Kofman, Brun, Scheid and Gallaire2017). Under the assumption of negligible capillary pressure and hydrostatic pressure contributions, the approximate solution for the film thickness up to the fourth order in $\unicode[STIX]{x1D703}$ reads

(3.4) $$\begin{eqnarray}\displaystyle & & \displaystyle H(\unicode[STIX]{x1D703},t)=\frac{1}{\sqrt{\displaystyle \frac{1}{e^{2}}+\frac{2}{3}t}}\left\{1+\frac{\unicode[STIX]{x1D703}^{2}}{16}\left[1-\frac{1}{\left(\displaystyle \frac{1}{e^{2}}+\frac{2}{3}t\right)^{4}}\right]\right.\nonumber\\ \displaystyle & & \displaystyle \quad +\,\left.\frac{43\unicode[STIX]{x1D703}^{4}}{10752}\left[1+\frac{315}{43\left(\displaystyle \frac{1}{e^{2}}+\frac{2}{3}t\right)^{8}}-\frac{288}{43\left(\displaystyle \frac{1}{e^{2}}+\frac{2}{3}t\right)^{7}}-\frac{70}{43\left(\displaystyle \frac{1}{e^{2}}+\frac{2}{3}t\right)^{4}}\right]\right\}+O(\unicode[STIX]{x1D703}^{6}).\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

Note that the equivalent expression for the draining flow on a spherical substrate has been proven to be in excellent agreement with experimental measurements by Lee et al. (Reference Lee, Brun, Marthelot, Balestra, Gallaire and Reis2016). The very good agreement between the numerical solution of the full lubrication (3.1) and the prediction (3.4) (see figure 3 b) clearly indicates that the capillary pressure and the variation of hydrostatic pressure are negligible in the outer region.

Figure 4.

Comparison between the evolution of the front position $\unicode[STIX]{x1D703}_{N}$ obtained numerically (black solid line), theoretically (red dashed line) and with the explicit theoretical expression (3.6) (blue dotted line). Here $Bo=15$ and $\unicode[STIX]{x1D6FF}=0.15$ . The black dotted line indicates the critical initial time $\unicode[STIX]{x1D70F}_{i}=3/(2e^{2})$ and the black dashed line the crossover time at which $\ell _{c}=\ell$ . The front location deduced from the numerical results is arbitrarily chosen to correspond to the location where the film thickness is $H_{N}=[H(\unicode[STIX]{x1D703}=0,t)+b]/2$ .

3.4 Front position

The viscous liquid spreads on the cylinder as time evolves, following the expression (3.4) in the outer region. Mass conservation imposes

(3.5) $$\begin{eqnarray}\displaystyle \int _{0}^{\unicode[STIX]{x1D703}_{N}}[H(\unicode[STIX]{x1D703},t)-b]\,\text{d}\unicode[STIX]{x1D703}/\unicode[STIX]{x1D6FF}=1, & & \displaystyle\end{eqnarray}$$

where the precursor film thickness is removed to avoid an increase in fluid volume when the ridge advances. Equation (3.5), together with the approximate expression for the draining flow $H$ , equation (3.4), can be solved to yield an implicit solution for the front position over time $\unicode[STIX]{x1D703}_{N}(t)$ (see figure 4). An explicit expression can be found if one assumes that the film has constant thickness $H_{N}=(2t/3)^{-1/2}\gg b$ . Mass conservation yields (Takagi & Huppert Reference Takagi and Huppert2010)

(3.6) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D703}_{N}(t)=\frac{\unicode[STIX]{x1D6FF}}{H_{N}}\sim \unicode[STIX]{x1D6FF}\sqrt{\frac{2t}{3}}, & & \displaystyle\end{eqnarray}$$

which is valid for $3/(2e^{2})\ll t\ll 3/(2b^{2})$ . Equation (3.6) evidences the square-root dependence of the front position with time. The implicit and explicit approximate estimates for the spreading length $\unicode[STIX]{x1D703}_{N}$ compare well with the result of the numerical solution of (3.1) (see figure 4). It has to be stressed that these estimates rely only on the conservation of mass for the outer region, neglecting the precise profile in the inner region close to the front.

The velocity of the front is given by $U_{N}=\text{d}\unicode[STIX]{x1D703}_{N}/\text{d}t/\unicode[STIX]{x1D6FF}\sim (6t)^{-1/2}$ . Although the front travels on a substrate whose local inclination increases, its velocity, which scales like $U_{N}\sim H_{N}^{2}\sin \unicode[STIX]{x1D703}_{N}$ , reduces with time. In fact, mass conservation imposes that the film gets thinner as $H_{N}\sim 1/\unicode[STIX]{x1D703}_{N}$ so that the increased viscous shear overcomes the effect of the increased gravity component driving the flow.

3.5 Length of the inner region

Surface-tension effects are only relevant in the vicinity of the ridge, where the outer film thickness has to be connected to the precursor film, inducing strong film-thickness variations. The film profile in the inner region presents a shape very similar to those found for other driven films, like the ones on an inclined plane (Spaid & Homsy Reference Spaid and Homsy1996) (see inset of figure 3 b).

The characteristic length of the ridge is found by balancing the dominant terms in (3.1). At small angles $\unicode[STIX]{x1D703}$ , the drainage term $\text{III}$ is weak compared to terms $\text{I}$ and $\text{II}$ due to the large film-thickness gradients at the front. Hence, the capillary pressure term $\text{I}$ has to balance the variation of the hydrostatic pressure term $\text{II}$ . The characteristic length of the inner region is therefore

(3.7) $$\begin{eqnarray}\displaystyle \ell _{c}=\left(\frac{\unicode[STIX]{x1D6FF}}{Bo\cos \unicode[STIX]{x1D703}_{N}}\right)^{1/2}, & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D703}_{N}$ is the front location. Expression (3.7) is nothing else than the capillary length defined with the gravity component in the radial direction. Written in dimensional quantities, it reads $\ell _{c}^{\ast }=\sqrt{\unicode[STIX]{x1D6FE}/(\unicode[STIX]{x1D70C}g\cos \unicode[STIX]{x1D703}_{N})}$ (Bertozzi & Brenner Reference Bertozzi and Brenner1997). Length $\ell _{c}$ is a monotonic increasing function of $\unicode[STIX]{x1D703}_{N}$ , and thus increases with time. For sufficiently large angles, the drainage term $\text{III}$ overcomes the variation of the hydrostatic pressure term $\text{II}$ and the film-thickness variations at the ridge occur on a length scale found by balancing the drainage term and the capillary pressure term, yielding

(3.8) $$\begin{eqnarray}\displaystyle \ell =\left(\frac{\unicode[STIX]{x1D6FF}H_{N}}{Bo\sin \unicode[STIX]{x1D703}_{N}}\right)^{1/3}=\left(\frac{\unicode[STIX]{x1D6FF}^{2}}{Bo\,\unicode[STIX]{x1D703}_{N}\sin \unicode[STIX]{x1D703}_{N}}\right)^{1/3}, & & \displaystyle\end{eqnarray}$$

where conservation of mass $H_{N}\unicode[STIX]{x1D703}_{N}=\unicode[STIX]{x1D6FF}$ has been used. For $3/(2e^{2})\ll t\ll 3/(2b^{2})$ , with the use of (3.6), one obtains $\ell \sim (2Bo\,t/3)^{-1/3}$ , indicating that $\ell$ decreases with time. Equation (3.8) is equivalent to the length of the capillary ridge for a film flowing on an inclined plane, $\ell =H_{N}/(3Ca)^{1/3}$ , when using $Ca=Bo/\unicode[STIX]{x1D6FF}U_{N}$ as capillary number, with $U_{N}$ the velocity at the front, given by $U_{N}=\sin \unicode[STIX]{x1D703}_{N}H_{N}^{2}/(3\unicode[STIX]{x1D6FF})$ .

For the parameters considered in the present study, the crossover between the two regimes is found to be at $t\simeq 44$ . The first regime corresponds to the relaxation of the initial film thickness profile over a relatively flat substrate, whereas the second regime corresponds to the well-developed gravity-driven ridge regularized by surface tension, where the initial condition is lost and the solution follows the scaling $H_{N}\sim (2t/3)^{-1/2}$ .

The two regimes have been identified also for the flow over an inclined plane depending on the inclination angle (Bertozzi & Brenner Reference Bertozzi and Brenner1997). Bertozzi & Brenner (Reference Bertozzi and Brenner1997) showed that the bump of the ridge vanishes for small inclination angles due to the component of gravity orthogonal to the substrate. This is also true for the flow over a cylinder, where the bump gains in thickness as the flow evolves downstream. It can be concluded that the length of the ridge always decreases with the Bond number, whereas it increases or decreases with the polar angle, and thus with time, for the first and second regime, respectively, as already observed in figures 2 and 3.

The most important scalings are summarized in table 1. The good agreement between the Bond-number-independent estimate for the film thickness $H$ given by (3.4) is only valid for the weak surface-tension limit, when the Bond number is sufficiently large. For very large Bond numbers, Ancey, Cochard & Andreini (Reference Ancey, Cochard and Andreini2009) showed that if surface tension is negligible everywhere, a very steep nose forms at the front. Otherwise, capillary forces will impact also the film thickness far from the ridge and the outer solution is no longer valid. Following Takagi & Huppert (Reference Takagi and Huppert2010), the present theory holds as long as $\unicode[STIX]{x1D703}_{N}\gg Bo^{-1/5}\unicode[STIX]{x1D6FF}$ , which is the case for the considered parameters.

Furthermore, the precursor film thickness has to be $b\ll e$ to avoid a non-negligible addition of mass when the ridge advances on the precursor film. In this work, we have set $b=8\times 10^{-3}$ , two orders of magnitude smaller than $e$ . With this value of  $b$ , the upper time limit for the theory to hold is $3/(2b^{2})\gg t_{f}$ , where $t_{f}=133$ is the time at which the ridge reaches $\unicode[STIX]{x1D703}=\unicode[STIX]{x03C0}/2$ with the considered parameters. It has to be pointed out, however, that the precise film-thickness profile of the inner region is dependent on the exact value of the precursor film (Bertozzi & Brenner Reference Bertozzi and Brenner1997). The height of the ridge is a weak (logarithmic) function of $b$ (Troian et al. Reference Troian, Herbolzheimer, Safran and Joanny1989; Tuck & Schwartz Reference Tuck and Schwartz1990). The smaller the precursor film thickness, the higher the bump and thus the stronger the growth of the perturbations. Yet, the selection of the most unstable spanwise wavenumber is not affected (Spaid & Homsy Reference Spaid and Homsy1996).

Table 1.

Summary of main scalings for the film thickness at the pole $H$ , the capillary-ridge length $\ell$ , the front location $\unicode[STIX]{x1D703}_{N}$ and its velocity $U_{N}$ ; the general expressions valid for any time and their simplified forms for $3/(2e^{2})\ll t\ll 3/(2b^{2})$ are specified.

4.1 Film-thickness decomposition

The numerically obtained one-dimensional draining flow discussed in § 3 is asymptotically linearly stable for $\unicode[STIX]{x1D703}\in [0,\unicode[STIX]{x03C0}/2]$ and can be considered as the base state for the linear stability analysis. The film thickness is decomposed into this drainage solution $H(\unicode[STIX]{x1D703},t)$ and small perturbations $\unicode[STIX]{x1D700}{\hat{h}}(\unicode[STIX]{x1D703},t)$ , with $\unicode[STIX]{x1D700}\ll 1$ (see figure 5):

(4.1) $$\begin{eqnarray}\displaystyle \bar{H}(\unicode[STIX]{x1D703},z,t)=\underbrace{H(\unicode[STIX]{x1D703},t)}_{drainage\;solution}+\,\underbrace{\unicode[STIX]{x1D700}{\hat{h}}(\unicode[STIX]{x1D703},z,t)}_{perturbation},\quad \unicode[STIX]{x1D700}\ll 1. & & \displaystyle\end{eqnarray}$$

Figure 5.

Decomposition of the total film thickness $\bar{H}$ (solid line) into draining solution $H$ (dotted line) and first-order perturbations $\unicode[STIX]{x1D700}h\cos (\unicode[STIX]{x1D6FD}z)$ for the optimal transient growth of initial conditions $h_{0}$ over a smooth substrate. The spanwise perturbations with wavenumber $\unicode[STIX]{x1D6FD}$ are shown in the three-dimensional sketch. Here $\unicode[STIX]{x1D703}\in [0,\unicode[STIX]{x03C0}/2]$ and $z\in [0,3\times 2\unicode[STIX]{x03C0}/\unicode[STIX]{x1D6FD}]$ .

Furthermore, we assume that the perturbations are periodic in the transverse direction and we proceed to a normal mode expansion in this direction

(4.2) $$\begin{eqnarray}\displaystyle {\hat{h}}(\unicode[STIX]{x1D703},z,t)=h(\unicode[STIX]{x1D703},t)\exp (\text{i}\unicode[STIX]{x1D6FD}z)+\text{c.c.}, & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D6FD}\in \mathbb{R}$ is the spanwise wavenumber. The one-dimensional perturbations $h(\unicode[STIX]{x1D703},t)$ have to be optimized for a given spanwise wavenumber $\unicode[STIX]{x1D6FD}$ .

4.2 Linear disturbance equation

The linear disturbance equation governing the dynamics of the first-order perturbations $h(\unicode[STIX]{x1D703},t)$ is found by entering the decomposition (4.1) together with (4.2) into the nonlinear lubrication equation (2.2) and considering terms of order $\unicode[STIX]{x1D700}$ :

(4.3) $$\begin{eqnarray}\displaystyle h_{t}+L(H,\unicode[STIX]{x1D6FD},Bo,\unicode[STIX]{x1D6FF})h=0. & & \displaystyle\end{eqnarray}$$

The linear operator $L(H,\unicode[STIX]{x1D6FD},Bo,\unicode[STIX]{x1D6FF})$ is a function of the drainage solution, the spanwise wavenumber, the Bond number and the film aspect ratio (see appendix A for its expression).

4.3 Optimization method

Due to the polar and temporal dependences of the base state $H$ in the linear operator, the classical normal mode ansatz for $h$ cannot be employed (Schmid Reference Schmid2007). Instead, we use a non-modal transient growth analysis to find the largest amplification of perturbations at finite times $T$ , and not in the limit $t\rightarrow \infty$ . Bertozzi & Brenner (Reference Bertozzi and Brenner1997) have shown that in spite of the linear stability of a flow over an inclined plane at low inclination angles, a transient growth can always occur. Given the base flow drainage on a substrate whose inclination increases in space, the employed procedure is therefore suitable.

The algorithm for the optimal transient growth of the optimal initial conditions is based on the one described in Balestra et al. (Reference Balestra, Brun and Gallaire2016), and only the main steps are presented hereafter.

The energy of the film-thickness perturbations can be defined as (Spaid & Homsy Reference Spaid and Homsy1996; Olsson & Henningson Reference Olsson and Henningson1995)

(4.4) $$\begin{eqnarray}\displaystyle E(t)=\frac{1}{2}\int _{\unicode[STIX]{x1D6FA}}h(\unicode[STIX]{x1D703},t)^{2}\,\text{d}\unicode[STIX]{x1D703}=\frac{1}{2}\langle h(t)\mid h(t)\rangle , & & \displaystyle\end{eqnarray}$$

where the scalar product for real quantities $a$ and $b$ on the geometry $\unicode[STIX]{x1D6FA}=\{\unicode[STIX]{x1D703}\mid \unicode[STIX]{x1D703}\in [0,\unicode[STIX]{x03C0}/2]\}$ is given by

(4.5) $$\begin{eqnarray}\displaystyle \langle a\mid b\rangle =\int _{\unicode[STIX]{x1D6FA}}a(\unicode[STIX]{x1D703})b(\unicode[STIX]{x1D703})\,\text{d}\unicode[STIX]{x1D703}. & & \displaystyle\end{eqnarray}$$

The gain of the film-thickness perturbations to be optimized for a given time horizon $T$ is defined by the ratio of the actual disturbance energy $E(T)$ and its initial value $E(0)=\langle h_{0}\mid h_{0}\rangle /2$ , where $h_{0}=h(0)$ is the initial condition of the perturbation:

(4.6) $$\begin{eqnarray}\displaystyle G(T)=\frac{E(T)}{E(0)}=\frac{\langle h(T)\mid h(T)\rangle }{\langle h_{0}\mid h_{0}\rangle }. & & \displaystyle\end{eqnarray}$$

The optimal growth analysis is based on the augmented Lagrangian, consisting of the objective function to be optimized, i.e. the gain, and the constraints, namely the governing (4.3) and the initial condition. The constraints are enforced via the Lagrange multipliers $h^{\dagger }$ and $h_{0}^{\dagger }$ , respectively:

(4.7) $$\begin{eqnarray}\displaystyle {\mathcal{L}}(h,h^{\dagger },h_{0},h_{0}^{\dagger };T)=\underbrace{G(T)}_{gain}-\underbrace{\int _{0}^{T}\langle h^{\dagger }\mid h_{t}+Lh\rangle \,\text{d}t}_{enforcing\;governing\;equation}-\underbrace{\langle h_{0}^{\dagger }\mid h(0)-h_{0}\rangle }_{enforcing\;initial\;condition}. & & \displaystyle\end{eqnarray}$$

The Lagrange multiplier $h^{\dagger }$ depends on space and on time, whereas the Lagrange multiplier $h_{0}^{\dagger }$ solely depends on space. The Lagrange multipliers are given by the adjoint equations, which are obtained by taking the Fréchet derivatives of the augmented Lagrangian with respect to $h$ and $h_{0}$ , respectively (Cossu Reference Cossu2014) (see Balestra et al. (Reference Balestra, Brun and Gallaire2016) for a detailed derivation):

(4.8) $$\begin{eqnarray}\displaystyle \displaystyle \left\langle \frac{\unicode[STIX]{x2202}{\mathcal{L}}}{\unicode[STIX]{x2202}h}\biggm\vert\unicode[STIX]{x1D6FF}h\right\rangle =0,\quad \forall \unicode[STIX]{x1D6FF}h\Leftrightarrow & & \displaystyle \!\!\!\!h^{\dagger }(T)=2\frac{h(T)}{\langle h_{0}|h_{0}\rangle },\end{eqnarray}$$

(4.9) $$\begin{eqnarray}\displaystyle & & \displaystyle \displaystyle \!\!\!\!h_{t}^{\dagger }-L^{\dagger }h^{\dagger }=0,\quad t\in [0,T],\end{eqnarray}$$

(4.10) $$\begin{eqnarray}\displaystyle & & \displaystyle \displaystyle \!\!\!\!h_{0}^{\dagger }=h^{\dagger }(0),\end{eqnarray}$$

and

(4.11) $$\begin{eqnarray}\displaystyle \left\langle \frac{\unicode[STIX]{x2202}{\mathcal{L}}}{\unicode[STIX]{x2202}h_{0}}\biggm\vert\unicode[STIX]{x1D6FF}h_{0}\right\rangle =0,\quad \forall \unicode[STIX]{x1D6FF}h_{0}\Leftrightarrow h_{0}=h_{0}^{\dagger }\frac{\langle h_{0}|h_{0}\rangle ^{2}}{2\langle h(T)|h(T)\rangle }. & & \displaystyle\end{eqnarray}$$

The boundary conditions for both the direct and the adjoint problem are $h_{\unicode[STIX]{x1D703}}(0)=h_{\unicode[STIX]{x1D703}\unicode[STIX]{x1D703}}(0)=0$ , $h_{\unicode[STIX]{x1D703}}^{\dagger }(0)=h_{\unicode[STIX]{x1D703}\unicode[STIX]{x1D703}}^{\dagger }(0)=0$ and $h(\unicode[STIX]{x03C0}/2)=h_{\unicode[STIX]{x1D703}}(\unicode[STIX]{x03C0}/2)=0$ , $h^{\dagger }(\unicode[STIX]{x03C0}/2)=h_{\unicode[STIX]{x1D703}}^{\dagger }(\unicode[STIX]{x03C0}/2)=0$ .

In order to find the optimal initial condition, one starts with an arbitrary initial condition $h_{0}$ and integrates the direct problem (4.3) forward in time from $t=0$ to the chosen time horizon $t=T$ (see appendix C for details of the numerical method). The adjoint (4.9) is then integrated backward in time, starting from the condition (4.8) at $t=T$ and using the adjoint operator $L^{\dagger }=L^{T}$ . The latter satisfies the property $\langle a\mid Lb\rangle =\langle L^{\dagger }a\mid b\rangle$ . The Lagrange multiplier $h_{0}^{\dagger }$ is given by (4.10), and the new guess for the optimal initial condition is given by (4.11). This algorithm (see figure 6) is repeated until the gain $G(T)$ converges, eventually yielding the optimal initial condition and the optimal gain for a particular time horizon $T$ , spanwise wavenumber  $\unicode[STIX]{x1D6FD}$ , Bond number $Bo$ and film aspect ratio $\unicode[STIX]{x1D6FF}$ . By repeating the same procedure for successive time horizons $T$ , the envelope of the optimal gains for the selected parameters can be found.

Figure 6.

Sketch of the iterative procedure to find the optimal gain and initial condition for a given time horizon $T$ , spanwise wavenumber $\unicode[STIX]{x1D6FD}$ , Bond number $Bo$ and film aspect ratio $\unicode[STIX]{x1D6FF}$ .

5.1 Lubrication equation with a perturbed substrate

The substrate has been, so far, considered as perfectly cylindrical. However, it is also possible to consider topographical perturbations of the substrate, denoted as $\bar{H}^{o\ast }$ , that perturb the fluid. The free surface is then defined by $F=r^{\ast }-(R+\bar{H}^{\ast }+\bar{H}^{o\ast })=0$ . Since the derivation of a lubrication equation for this configuration is more involved, we will restrict our analysis to a particular case. First, we assume that the substrate perturbations are infinitesimally small such that the base flow $H$ discussed in § 3 is unaffected, and remains therefore invariant in the axial direction. Second, since a linear problem is eventually derived for the optimal substrate perturbations analysis, only linear terms in the substrate-topography perturbations are kept for consideration. The full derivation of the lubrication equation for this case can be found in appendix B. In dimensionless form it reads

(5.1) $$\begin{eqnarray}\displaystyle & & \displaystyle \bar{H}_{t}+\frac{1}{3}\left\{\bar{H}^{3}\left[\frac{\unicode[STIX]{x1D6FF}^{2}}{Bo}\bar{\unicode[STIX]{x1D705}}_{\unicode[STIX]{x1D703}}-\unicode[STIX]{x1D6FF}\cos \unicode[STIX]{x1D703}(\bar{H}_{\unicode[STIX]{x1D703}}+\bar{H}_{\unicode[STIX]{x1D703}}^{o})+\sin \unicode[STIX]{x1D703}\right]\right\}_{\unicode[STIX]{x1D703}}\nonumber\\ \displaystyle & & \displaystyle \quad +\,\frac{1}{3\unicode[STIX]{x1D6FF}}\left\{\bar{H}^{3}\left[\frac{\unicode[STIX]{x1D6FF}}{Bo}\bar{\unicode[STIX]{x1D705}}_{z}-\cos \unicode[STIX]{x1D703}(\bar{H}_{z}+\bar{H}_{z}^{o})-\unicode[STIX]{x1D6FF}\sin \unicode[STIX]{x1D703}\bar{H}\bar{H}_{\unicode[STIX]{x1D703}z}^{o}\right]\right\}_{z}=0.\end{eqnarray}$$

The curvature is given by $\bar{\unicode[STIX]{x1D705}}=-\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{n}$ with $\boldsymbol{n}$ the normal vector to the free surface (see expression (B 1)). It therefore includes contributions from the surface topography, as seen in expression (B 11) where only linear contributions in $\bar{H}^{o}$ have been kept in anticipation to the subsequent linearization (§§ 5.2 and 5.3). The curvature is eventually truncated at the second order in $\unicode[STIX]{x1D6FF}$ to keep only the leading-order terms.

5.2 Free-surface-elevation decomposition

When the substrate is perturbed, i.e.  $\bar{H}^{o}\neq 0$ , the free surface of the film is located at a distance $\bar{H}+\bar{H}^{o}$ from the originally smooth cylindrical substrate. Here, we consider only small time-independent substrate perturbations of order $\unicode[STIX]{x1D700}\ll 1$ : $\bar{H}^{o}=\unicode[STIX]{x1D700}{\hat{h}}^{o}$ . Hence, the elevation of the film free surface can be decomposed as (see figure 13)

(5.2) $$\begin{eqnarray}\displaystyle \bar{H}(\unicode[STIX]{x1D703},z,t)+\bar{H}^{o}(\unicode[STIX]{x1D703},z)=\underbrace{H(\unicode[STIX]{x1D703},t)}_{drainage\;solution}+\;\unicode[STIX]{x1D700}\left[\underbrace{{\hat{h}}(\unicode[STIX]{x1D703},z,t)}_{\;film\;perturbation}+\underbrace{{\hat{h}}^{o}(\unicode[STIX]{x1D703},z)}_{substrate\;perturbation}\right],\quad \unicode[STIX]{x1D700}\ll 1.\quad & & \displaystyle\end{eqnarray}$$

Figure 13.

Decomposition of the free-surface elevation $\bar{H}+\bar{H}^{o}$ starting from the smooth cylindrical substrate into draining solution $H$ and first-order perturbations $\unicode[STIX]{x1D700}(h+h^{o})$ for the optimal transient growth over a perturbed substrate with an initially uniform film. It is important to stress that the liquid thickness is only $\bar{H}=H+\unicode[STIX]{x1D700}h$ , whereas $\bar{H}^{o}=\unicode[STIX]{x1D700}h^{o}$ is the substrate-topography perturbation. The axial dependence as in figure 5 is not repeated here.

As for the study of the optimal initial conditions of § 4, given the invariance of the base state in the $z$ -direction, we consider periodic perturbations in the transverse direction:

(5.3) $$\begin{eqnarray}\displaystyle & \displaystyle {\hat{h}}(\unicode[STIX]{x1D703},z,t)=h(\unicode[STIX]{x1D703},t)\exp (\text{i}\unicode[STIX]{x1D6FD}z)+\text{c.c.}, & \displaystyle\end{eqnarray}$$

(5.4) $$\begin{eqnarray}\displaystyle & \displaystyle {\hat{h}}^{o}(\unicode[STIX]{x1D703},z)=h^{o}(\unicode[STIX]{x1D703})\exp (\text{i}\unicode[STIX]{x1D6FD}z)+\text{c.c.}, & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D6FD}\in \mathbb{R}$ is the spanwise wavenumber. It is important not to confuse $h_{0}$ , the initial condition of the perturbation, with $h^{o}$ , the substrate profile. The one-dimensional perturbations $h^{o}(\unicode[STIX]{x1D703})$ have to be optimized for a given spanwise wavenumber $\unicode[STIX]{x1D6FD}$ .

5.3 Linear disturbance equation

The governing equation for the first-order disturbances is obtained by entering the film-thickness decomposition (5.2) together with (5.3) and (5.4) into the nonlinear lubrication (5.1) and considering terms at order $\unicode[STIX]{x1D700}$ :

(5.5) $$\begin{eqnarray}\displaystyle h_{t}+L(H,\unicode[STIX]{x1D6FD},Bo,\unicode[STIX]{x1D6FF})h=-L^{o}(H,\unicode[STIX]{x1D6FD},Bo,\unicode[STIX]{x1D6FF})h^{o}. & & \displaystyle\end{eqnarray}$$

The linear operators $L(H,\unicode[STIX]{x1D6FD},Bo,\unicode[STIX]{x1D6FF})$ , which is the same as in § 4, and $L^{o}(H,\unicode[STIX]{x1D6FD},Bo,\unicode[STIX]{x1D6FF})$ are a function of the drainage solution, the spanwise wavenumber, the Bond number and the film aspect ratio (see appendix A for their expressions). Thus, the linear operators are space dependent and are not autonomous.

As expected, the perturbation of the substrate topography results in the appearance of a time-dependent – through $H(\unicode[STIX]{x1D703},t)$ – forcing term: $L^{o}(H,\unicode[STIX]{x1D6FD},Bo,\unicode[STIX]{x1D6FF})h^{o}$ . Note that a change of variable in order to homogenize (5.5), making the optimal substrate and the optimal initial condition problems mathematically similar (the state variable being only the difference $h-h^{o}$ ), was found not to be possible due to the time dependency of the forcing term.

5.4 Optimization method

We will now derive the equations leading to the optimization algorithm for the substrate-topography perturbations. The gain to be optimized at a time horizon $T$ is defined as the ratio of the film disturbance energy $E(T)$ and the substrate disturbance energy $E^{o}=\langle h^{o}\mid h^{o}\rangle /2$ :

(5.6) $$\begin{eqnarray}\displaystyle G(T)=\frac{E(T)}{E^{o}}=\frac{\langle h(T)\mid h(T)\rangle }{\langle h^{o}\mid h^{o}\rangle }. & & \displaystyle\end{eqnarray}$$

The choice of the gain (5.6) requires a vanishing initial free-surface disturbance, $h_{0}=0$ , otherwise the algorithm will bring $h^{o}$ to zero naturally, leading to an infinite gain as soon as $h(T)$ is a non-zero number. Perturbing the substrate – the forcing term in the state (5.5) – is the only way to perturb the base flow. This choice also has a practical relevance since it is sufficient to let the base flow evolve on the perturbed substrate, without having to experimentally impose an initial condition for the fluid’s perturbation. Another possible choice for the gain is $\langle h(T)\mid h(T)\rangle -\unicode[STIX]{x1D712}\langle h^{o}\mid h^{o}\rangle$ . Such a choice would relax the condition $h_{0}=0$ , to the price of depending on the value of the penalization coefficient $\unicode[STIX]{x1D712}$ . We have therefore not retained this gain definition hereafter.

The augmented Lagrangian reads

(5.7) $$\begin{eqnarray}\displaystyle {\mathcal{L}}(h,h^{\dagger },h^{o};T)=\underbrace{G(T)}_{gain}-\underbrace{\int _{0}^{T}\langle h^{\dagger }\mid h_{t}+Lh+L^{o}h^{o}\rangle \,\text{d}t}_{enforcing\;governing\;equation}. & & \displaystyle\end{eqnarray}$$

By proceeding in the same manner as for the derivation of the algorithm for the optimal initial condition of § 4.3, the adjoint equations for the Lagrange multiplier $h^{\dagger }$ are given by setting to zero the first variations of ${\mathcal{L}}$ with respect to any variation $\unicode[STIX]{x1D6FF}h$ :

(5.8) $$\begin{eqnarray}\displaystyle \displaystyle \left\langle \frac{\unicode[STIX]{x2202}{\mathcal{L}}}{\unicode[STIX]{x2202}h}\biggm\vert\unicode[STIX]{x1D6FF}h\right\rangle =0,\quad \forall \unicode[STIX]{x1D6FF}h\Leftrightarrow & & \displaystyle \!\!\!\!h^{\dagger }(T)=2\frac{h(T)}{\langle h^{o}|h^{o}\rangle },\end{eqnarray}$$

(5.9) $$\begin{eqnarray}\displaystyle & & \displaystyle \displaystyle \!\!\!\!h_{t}^{\dagger }-L^{\dagger }h^{\dagger }=0,\quad t\in [0,T].\end{eqnarray}$$

The optimality condition for the substrate perturbation is obtained by imposing a zero Fréchet derivative of ${\mathcal{L}}$ with respect to $h^{o}$ (see derivation in appendix D):

(5.10) $$\begin{eqnarray}\displaystyle \left\langle \frac{\unicode[STIX]{x2202}{\mathcal{L}}}{\unicode[STIX]{x2202}h^{o}}\biggm\vert\unicode[STIX]{x1D6FF}h^{o}\right\rangle =0,\quad \forall \unicode[STIX]{x1D6FF}h^{o}\Leftrightarrow \frac{h^{o}}{\langle h^{o}|h^{o}\rangle ^{2}}=\frac{1}{-2\langle h(T)\mid h(T)\rangle }\int _{0}^{T}L^{o\dagger }h^{\dagger }\,\text{d}t. & & \displaystyle\end{eqnarray}$$

By deriving the augmented Lagrangian with respect to the multiplier one retrieves the constraint. Both liquid film direct and adjoint perturbations, as well as the substrate perturbations satisfy the following boundary conditions: $h_{\unicode[STIX]{x1D703}}(0)=h_{\unicode[STIX]{x1D703}\unicode[STIX]{x1D703}}(0)=0$ , $h_{\unicode[STIX]{x1D703}}^{\dagger }(0)=h_{\unicode[STIX]{x1D703}\unicode[STIX]{x1D703}}^{\dagger }(0)=0$ and $h_{\unicode[STIX]{x1D703}}^{o}(0)=h_{\unicode[STIX]{x1D703}\unicode[STIX]{x1D703}}^{o}(0)=0$ , and $h(\unicode[STIX]{x03C0}/2)=h_{\unicode[STIX]{x1D703}}(\unicode[STIX]{x03C0}/2)=0$ , $h^{\dagger }(\unicode[STIX]{x03C0}/2)=h_{\unicode[STIX]{x1D703}}^{\dagger }(\unicode[STIX]{x03C0}/2)=0$ and $h^{o}(\unicode[STIX]{x03C0}/2)=h_{\unicode[STIX]{x1D703}}^{o}(\unicode[STIX]{x03C0}/2)=0$ .

In order to find the optimal substrate, one starts with an arbitrary substrate profile $h^{o}$ and integrates the direct problem (5.5) from $t=0$ to the chosen time horizon $t=T$ , with $h_{0}=0$ (see appendix C for details of the numerical method). Equation (5.8) gives the initial condition for the adjoint field, which is then integrated backward in time using the adjoint (5.9) from $t=T$ to $t=0$ . The new guess for the optimal substrate can be found with the help of (5.10). First, the new guess is approximated by

(5.11) $$\begin{eqnarray}\displaystyle \tilde{h}^{o}=\frac{1}{-2\langle h(T)\mid h(T)\rangle }\int _{0}^{T}L^{o\dagger }h^{\dagger }\,\text{d}t, & & \displaystyle\end{eqnarray}$$

which is then normalized by $\langle \tilde{h}^{o}|\tilde{h}^{o}\rangle ^{2}$ according to equation (5.10) to get $h^{o}$ . Second, a low-pass filter is applied to $h^{o}$ , whose meaning is explained in the following. The algorithm (see figure 14) is repeated until the gain $G(T)$ converges. The optimal substrate as well as the optimal gain for the time horizon $T$ , the wavenumber $\unicode[STIX]{x1D6FD}$ , the Bond number $Bo$ and the film aspect ratio $\unicode[STIX]{x1D6FF}$ are obtained. The envelope of the optimal gains for the selected parameters can be found by repeating the procedure for different time horizons $T$ .

Figure 14.

Sketch of the iterative procedure to find the optimal gain and substrate perturbation for a given time horizon $T$ , spanwise wavenumber $\unicode[STIX]{x1D6FD}$ , Bond number $Bo$ and film aspect ratio $\unicode[STIX]{x1D6FF}$ .

The necessity of a low-pass filter for the optimal substrate perturbation can be argued on the grounds of both numerical and physical reasons. In fact, we have found that the optimal substrate after an iteration, before filtering, presents high-frequency perturbations. The latter would result in very large oscillating derivatives in the linear operators $L$ and $L^{o}$ , compromising therefore the numerical stability. This point is even more dramatic since derivatives up to the fourth order have to be computed. Note that no filter is necessary for the study of § 4 since the optimal initial conditions are regularized by surface-tension effects. The high-frequency perturbations of the optimal substrate perturbations for a given time horizon $T$ are situated in the region close to the front location reached by the draining flow at $t=T$ . In fact, as already demonstrated by Bertozzi & Brenner (Reference Bertozzi and Brenner1997), the roughness of the substrate at the contact line is an effective amplifier of film perturbations. It is therefore not surprising that the optimal substrate found by the algorithm presents such a rough pattern. However, as will become clear in the following, these localized high-frequency perturbations of the substrate, which dominate at short time horizons $T$ , are not the most efficient perturbations. Other perturbations at larger times $T$ being of much larger characteristic size and therefore insensitive to the low-pass filter are even more effective. Furthermore, our numerical discretization does not allow for describing a realistic high-frequency random roughness distribution. Therefore, a low-pass filter is applied to smooth out these perturbations (see appendix C for details) and to only consider low-frequency perturbations, which can be resolved accurately. The filter can also be pragmatically justified by noting that, in practice, one can only fabricate an optimal perturbed substrate for experiments with a relatively large characteristic size. The smoothed-out high-frequency perturbations would naturally correspond to the intrinsic roughness of the material surface.