2.1 Governing equations

The equations of mass and momentum conservation are given by

(2.3) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}=0, & \displaystyle\end{eqnarray}$$

(2.4) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D70C}[\boldsymbol{u}_{t}+\boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}]=-\unicode[STIX]{x1D735}p+\unicode[STIX]{x1D707}\unicode[STIX]{x1D6FB}^{2}\boldsymbol{u}+\unicode[STIX]{x1D70C}\boldsymbol{g}, & \displaystyle\end{eqnarray}$$

where $p$ is pressure and $\boldsymbol{g}=-g\sin (\unicode[STIX]{x1D703})\boldsymbol{e}_{r}-g\cos (\unicode[STIX]{x1D703})\boldsymbol{e}_{\unicode[STIX]{x1D703}}$ . For convenience, we define a new radial coordinate $z=r-R$ , so all governing equations are solved in the interval $0\leqslant z\leqslant h(\unicode[STIX]{x1D703},y,t)$ .

On the cylinder surface, $z=0$ , no-slip and no-penetration conditions are applied,

(2.5) $$\begin{eqnarray}u=\unicode[STIX]{x1D714}=v=0.\end{eqnarray}$$

At the liquid–air interface, $F(\unicode[STIX]{x1D703},y,z,t)=z-h(\unicode[STIX]{x1D703},y,t)=0$ , the liquid needs to satisfy the normal and tangential stress balances and the kinematic condition,

(2.6) $$\begin{eqnarray}\displaystyle & \displaystyle -p+2\unicode[STIX]{x1D707}(\boldsymbol{n}\boldsymbol{\cdot }\unicode[STIX]{x1D749}\boldsymbol{\cdot }\boldsymbol{n})=-\unicode[STIX]{x1D70E}\unicode[STIX]{x1D705}, & \displaystyle\end{eqnarray}$$

(2.7) $$\begin{eqnarray}\displaystyle & \displaystyle 2\unicode[STIX]{x1D707}(\boldsymbol{n}\boldsymbol{\cdot }\unicode[STIX]{x1D749}\boldsymbol{\cdot }\boldsymbol{t})=\unicode[STIX]{x1D735}_{s}\unicode[STIX]{x1D70E}, & \displaystyle\end{eqnarray}$$

(2.8) $$\begin{eqnarray}\displaystyle & \displaystyle \left.\frac{\text{D}F}{\text{D}t}\right|_{z=h(\unicode[STIX]{x1D703},y,t)}=0, & \displaystyle\end{eqnarray}$$

where $\boldsymbol{n}$ and $\boldsymbol{t}$ are the unit outward normal and tangent vectors, $\unicode[STIX]{x1D735}_{s}=(\unicode[STIX]{x1D644}-\boldsymbol{n}\boldsymbol{n})\boldsymbol{\cdot }\unicode[STIX]{x1D735}$ is the surface gradient operator, $\unicode[STIX]{x1D644}$ is the identity tensor, $\unicode[STIX]{x1D705}=\unicode[STIX]{x1D735}_{s}\boldsymbol{\cdot }\boldsymbol{n}$ is the surface curvature, $\unicode[STIX]{x1D749}=[\unicode[STIX]{x1D735}\boldsymbol{u}+(\unicode[STIX]{x1D735}\boldsymbol{u})^{\text{T}}]/2$ is the rate-of-strain tensor and $\text{D}/\text{D}t$ is the material derivative.

The surfactant concentration at the liquid–air interface is governed by the following convection–diffusion equation:

(2.9) $$\begin{eqnarray}\unicode[STIX]{x1D6E4}_{t}+\unicode[STIX]{x1D735}_{s}\boldsymbol{\cdot }(\boldsymbol{u}_{s}\unicode[STIX]{x1D6E4})+(\unicode[STIX]{x1D735}_{s}\boldsymbol{\cdot }\boldsymbol{n})\unicode[STIX]{x1D6E4}(\boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{u})=D\unicode[STIX]{x1D6FB}_{s}^{2}\unicode[STIX]{x1D6E4},\end{eqnarray}$$

where $D$ is the surface diffusivity.

2.2 Scaling

We consider the case where the characteristic film thickness, $H$ , is much smaller than the cylinder radius, $R$ , $\unicode[STIX]{x1D716}\equiv H/R\ll 1$ , so that lubrication theory can be applied. Following Evans et al. (Reference Evans, Schwartz and Roy2004) and Li & Kumar (Reference Li and Kumar2015), we scale velocity with $U=\unicode[STIX]{x1D70C}gh^{2}/\unicode[STIX]{x1D707}$ , pressure with $P=\unicode[STIX]{x1D707}U/H$ and time with $R/U$ , so that

(2.10) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle (z,h)=H(\tilde{z},\tilde{h}),\quad (r,y)=R(\tilde{r},{\tilde{y}})\quad t=T\tilde{t},\\ \displaystyle u=U\tilde{u} ,\quad w=\unicode[STIX]{x1D716}U\tilde{w},\quad v=U\tilde{v},\\ \displaystyle \unicode[STIX]{x1D6E4}=\unicode[STIX]{x1D6E4}_{m}\tilde{\unicode[STIX]{x1D6E4}},\quad \unicode[STIX]{x1D70E}=\unicode[STIX]{x1D70E}_{m}+\tilde{\unicode[STIX]{x1D70E}}\unicode[STIX]{x1D711},\quad p=P\tilde{p}.\end{array}\right\}\end{eqnarray}$$

Here, the dimensionless quantities are denoted with tildes. The terms $\unicode[STIX]{x1D6E4}_{m}$ and $\unicode[STIX]{x1D70E}_{m}$ are the mean surfactant concentration and the corresponding surface tension, respectively. Since the presence of surfactant lowers the surface tension, the spreading pressure, $\unicode[STIX]{x1D711}=\unicode[STIX]{x1D70E}_{0}-\unicode[STIX]{x1D70E}_{m}$ , is a positive constant. Order-of-magnitude of estimates for selected dimensional quantities are listed in table 1. For simplicity, we henceforth drop the tilde notation and all variables are taken to be dimensionless.

Table 1.

Order-of-magnitude estimates of dimensional quantities. The viscosity, density and surface tension are representative of a glycerol–water mixture. The values of surface diffusivities were obtained from the work of Campana & Saita (Reference Campana and Saita2006).

With the above scalings, equations (2.3), (2.4) and (2.9) and the corresponding boundary conditions can be simplified using the lubrication approximation (see appendix A and Evans et al. (Reference Evans, Schwartz and Roy2004), Evans et al. (Reference Evans, Schwartz and Roy2005) and Weidner (Reference Weidner2013) for details), leading to two coupled evolution equations for the film thickness and surfactant concentration. As in the works of Evans et al. (Reference Evans, Schwartz and Roy2004) and Evans et al. (Reference Evans, Schwartz and Roy2005), it is convenient to rescale all lengths with $R$ . This yields

(2.11) $$\begin{eqnarray}\displaystyle & \displaystyle (1+h)\frac{\unicode[STIX]{x2202}h}{\unicode[STIX]{x2202}t}=-MW\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}\left(h+\frac{h^{2}}{2}\right)+\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}\left[\left(\frac{h^{3}}{3}+\frac{h^{4}}{2}\right)\cos \unicode[STIX]{x1D703}\right]\hspace{74.00005pt} & \displaystyle \nonumber\\ \displaystyle & \displaystyle \qquad \qquad -\,\overline{\unicode[STIX]{x1D735}}\boldsymbol{\cdot }\left[\frac{h^{3}}{3Bo}\overline{\unicode[STIX]{x1D735}}(h+\overline{\unicode[STIX]{x1D6FB}}^{2}h)+\frac{h^{3}}{3}(W^{2}-\sin \unicode[STIX]{x1D703})\overline{\unicode[STIX]{x1D735}}h-\frac{1}{2}Mah^{2}\overline{\unicode[STIX]{x1D735}}\unicode[STIX]{x1D6E4}\right], & \displaystyle\end{eqnarray}$$

(2.12) $$\begin{eqnarray}\displaystyle & \displaystyle r\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6E4}}{\unicode[STIX]{x2202}t}+\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}(rMW\unicode[STIX]{x1D6E4}+u_{s}\unicode[STIX]{x1D6E4})+\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}y}[\unicode[STIX]{x1D6E4}v_{s}]+\frac{\unicode[STIX]{x2202}h}{\unicode[STIX]{x2202}t}\unicode[STIX]{x1D6E4}=\frac{1}{Pe}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}\left(\frac{1}{r}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6E4}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}\right)+\frac{1}{Pe}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}y}\left(r\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6E4}}{\unicode[STIX]{x2202}y}\right), & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

where

(2.13a,b ) $$\begin{eqnarray}\overline{\unicode[STIX]{x1D735}}=\boldsymbol{e}_{\unicode[STIX]{x1D703}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}+\boldsymbol{e}_{y}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}y}\quad \text{and}\quad \overline{\unicode[STIX]{x1D6FB}}^{2}=\frac{\unicode[STIX]{x2202}^{2}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}^{2}}+\frac{\unicode[STIX]{x2202}^{2}}{\unicode[STIX]{x2202}y^{2}}.\end{eqnarray}$$

Table 2 gives the definitions, physical meanings and typical values of the dimensionless parameters. Note that operators $\overline{\unicode[STIX]{x1D735}}$ and $\overline{\unicode[STIX]{x1D6FB}}^{2}$ are not the usual gradient and Laplacian in cylindrical coordinates.

In (2.11) and (2.12), we have neglected terms of $O(\unicode[STIX]{x1D716}^{2})$ and higher. The surface velocity, $(u_{s},v_{s})=(u,v)|_{z=h}$ , is

(2.14) $$\begin{eqnarray}\left.\begin{array}{@{}rcl@{}}\displaystyle u_{s}\ & =\ & \displaystyle -\frac{1}{2}h^{2}\cos \unicode[STIX]{x1D703}-\frac{5}{6}h^{3}\cos \unicode[STIX]{x1D703}-Mah\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6E4}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}\\ \ & \ & \displaystyle +\,\frac{1}{2}\left\{\frac{h^{2}}{Bo}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}(h+\overline{\unicode[STIX]{x1D6FB}}^{2}h)+h^{2}(W^{2}-\sin \unicode[STIX]{x1D703})\frac{\unicode[STIX]{x2202}h}{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}\right\},\\ v_{s}\ & =\ & \displaystyle \frac{1}{2}\left\{\frac{h^{2}}{Bo}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}y}(h+\overline{\unicode[STIX]{x1D6FB}}^{2}h)+h^{2}(W^{2}-\sin \unicode[STIX]{x1D703})\frac{\unicode[STIX]{x2202}h}{\unicode[STIX]{x2202}y}\right\}-Mah\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6E4}}{\unicode[STIX]{x2202}y}.\end{array}\right\}\end{eqnarray}$$

When Marangoni effects are absent ( $Ma=0$ ), equation (2.11) reduces to that obtained by Evans et al. (Reference Evans, Schwartz and Roy2004). In the limiting case where the cylinder is stationary ( $W=0$ ), equations (2.11) and (2.12) are consistent with the evolution equations obtained by Weidner (Reference Weidner2013).

Table 2.

Dimensionless problem parameters.

For a fixed set of problem parameters ( $W$ , $M$ , $Bo$ , $Pe$ and $Ma$ ), an alternating direction implicit (ADI) finite-difference method similar to that used in the works of Witelski & Bowen (Reference Witelski and Bowen2003) and Mata & Bertozzi (Reference Mata and Bertozzi2011) is applied to solve (2.11) and (2.12). Following the work of Li & Kumar (Reference Li and Kumar2015), we neglect the fourth term on the left-hand side of (2.12), $(\unicode[STIX]{x2202}h/\unicode[STIX]{x2202}t)\unicode[STIX]{x1D6E4}$ , to simplify our numerical scheme. This term arises due to surface stretching, and our simulations show that it only causes small changes in the resulting concentration profiles for the thin coatings considered here.

Second-order centred differences are used to discretize the spatial derivatives, and an iterative procedure is applied at each time step to improve solution accuracy (Mata & Bertozzi Reference Mata and Bertozzi2011). The $(\unicode[STIX]{x1D703},y)$ domain is discretized into a mesh of $n_{\unicode[STIX]{x1D703}}\times (n_{y}+1)$ grid points with grid spacing $\unicode[STIX]{x1D6E5}_{\unicode[STIX]{x1D703}}=\unicode[STIX]{x1D6E5}_{y}=2\unicode[STIX]{x03C0}/n_{\unicode[STIX]{x1D703}}$ and cylinder length $L=n_{y}\unicode[STIX]{x1D6E5}_{y}=2\unicode[STIX]{x03C0}n_{y}/n_{\unicode[STIX]{x1D703}}$ . Following the works of Evans et al. (Reference Evans, Schwartz and Roy2004) and Weidner (Reference Weidner2013), no-flux boundary conditions are imposed at the two ends of the cylinder ( $y=0$ and $y=L$ ), and periodicity is imposed in the $\unicode[STIX]{x1D703}$ direction ( $h(\unicode[STIX]{x1D703},y)=h(\unicode[STIX]{x1D703}+2\unicode[STIX]{x03C0},y)$ ). In our simulations, we choose $n_{\unicode[STIX]{x1D703}}=150$ – $200$ and $n_{y}=250$ – $500$ . For each set of $n_{\unicode[STIX]{x1D703}}$ and $n_{y}$ used to generate the results shown here, increasing the number of grid points did not change the results within graphical accuracy. Typical computation times for a given run ranged from one day to 1 week on a workstation with a 2.8 GHz Intel Xeon processor. We note that there are interesting questions concerning how surfactants influence the rate of convergence towards steady states and the behaviour of shock-like solutions (Badali et al. Reference Badali, Chugunova, Pelinovsky and Pollack2011; Burchard, Chugunova & Stephens Reference Burchard, Chugunova and Stephens2012), but a detailed investigation of these issues is beyond the scope of the present work.

3.1 Linear stability analysis

When gravitational forces are absent, a film of constant thickness with a uniform surfactant concentration is a solution of the governing equations. We examine the fate of small perturbations to this solution via a linear stability analysis (LSA), in which the following normal mode decomposition is used for $h$ and $\unicode[STIX]{x1D6E4}$ ,

(3.4) $$\begin{eqnarray}\displaystyle & \displaystyle h(x,y,t)=H_{b}+H_{0}\exp (\text{i}k_{\unicode[STIX]{x1D703}}\unicode[STIX]{x1D703}+\text{i}k_{y}y+st), & \displaystyle\end{eqnarray}$$

(3.5) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6E4}(x,y,t)=\unicode[STIX]{x1D6E4}_{b}+\unicode[STIX]{x1D6E4}_{0}\exp (\text{i}k_{\unicode[STIX]{x1D703}}\unicode[STIX]{x1D703}+\text{i}k_{y}y+st), & \displaystyle\end{eqnarray}$$

where $H_{b}$ and $\unicode[STIX]{x1D6E4}_{b}$ are the base-state values, and $H_{0}$ and $\unicode[STIX]{x1D6E4}_{0}$ are the perturbation amplitudes, which are assumed to have real-valued wavenumbers $k_{\unicode[STIX]{x1D703}}$ and $k_{y}$ and growth rate $s$ .

Substitution of (3.5) into the linearized versions of (3.1) and (3.2) leads to an eigenvalue problem of the form

(3.6) $$\begin{eqnarray}\boldsymbol{A}\left(\begin{array}{@{}c@{}}H_{0}\\ \unicode[STIX]{x1D6E4}_{0}\end{array}\right)=\left(s+\text{i}k_{\unicode[STIX]{x1D703}}\frac{\unicode[STIX]{x1D6FA}\unicode[STIX]{x1D707}R}{\unicode[STIX]{x1D70E}_{m}}\right)\left(\begin{array}{@{}c@{}}H_{0}\\ \unicode[STIX]{x1D6E4}_{0}\end{array}\right),\end{eqnarray}$$

where

(3.7) $$\begin{eqnarray}\boldsymbol{A}=\left(\begin{array}{@{}cc@{}}\displaystyle \frac{H_{b}^{3}}{3r_{b}}k^{2}[(1+S)-k^{2}] & \displaystyle -\frac{1}{2r_{b}}Mak^{2}H_{b}^{2}\\ \displaystyle \frac{1}{2r_{b}}H_{b}^{2}\unicode[STIX]{x1D6E4}_{b}k^{2}[(1+S)-k^{2}] & \displaystyle -\frac{1}{r_{b}}\left(MaH_{b}\unicode[STIX]{x1D6E4}_{b}k^{2}+\frac{1}{Pe_{s}}\frac{1}{r_{b}}k_{\unicode[STIX]{x1D703}}^{2}+\frac{1}{Pe_{s}}r_{b}k_{y}^{2}\right)\end{array}\right).\end{eqnarray}$$

Here, $k=(k_{\unicode[STIX]{x1D703}}^{2}+k_{y}^{2})^{1/2}$ and $r_{b}=1+H_{b}$ . The two values of the growth rate can be determined analytically, but we do not show the expressions for brevity. It turns out that both values of $s$ are purely real, but only one of them is positive over a range of wavenumbers and it is this value which is discussed below.

It is useful to first consider separately the cases of purely angular and purely axial perturbations when Marangoni stresses are absent ( $Ma=0$ ). Purely angular perturbations ( $k_{y}=0$ ) were considered by Evans et al. (Reference Evans, Schwartz and Roy2004), and the corresponding growth rate is

(3.8) $$\begin{eqnarray}s_{\unicode[STIX]{x1D703}}={\textstyle \frac{1}{3}}H_{b}^{3}((1+S)k_{\unicode[STIX]{x1D703}}^{2}-k_{\unicode[STIX]{x1D703}}^{4}).\end{eqnarray}$$

Angular perturbations with $k_{\unicode[STIX]{x1D703}}<k_{\unicode[STIX]{x1D703}}^{cutoff}$ have positive growth rates and grow in amplitude due to centrifugal forces, while those with $k_{\unicode[STIX]{x1D703}}>k_{\unicode[STIX]{x1D703}}^{cutoff}$ have negative growth rates and decay due to surface tension. The cutoff wavenumber, $k_{\unicode[STIX]{x1D703}}^{cutoff}$ , and the most dangerous wavenumber, $k_{\unicode[STIX]{x1D703}}^{max}$ , are

(3.9) $$\begin{eqnarray}\displaystyle & \displaystyle k_{\unicode[STIX]{x1D703}}^{cutoff}=[\sqrt{1+S}], & \displaystyle\end{eqnarray}$$

(3.10) $$\begin{eqnarray}\displaystyle & \displaystyle k_{\unicode[STIX]{x1D703}}^{max}=[\sqrt{(1+S)/2}], & \displaystyle\end{eqnarray}$$

where $[x]$ is the greatest integer less than or equal to $x$ . (Due to periodicity, $k_{\unicode[STIX]{x1D703}}$ is restricted to integer values.) Note that when $S=0$ (stationary cylinder), purely angular perturbations do not grow.

For purely axial perturbations ( $k_{\unicode[STIX]{x1D703}}=0$ ), it turns out that the growth rate has the same expression with $k_{y}$ in place of $k_{\unicode[STIX]{x1D703}}$ , i.e. $s_{y}=(1/3)H_{b}^{3}((1+S)k_{y}^{2}-k_{y}^{4})$ . The cutoff wavenumber is $k_{y}^{cutoff}=\sqrt{1+S}$ and the most dangerous wavenumber is $k_{y}^{max}=\sqrt{(1+S)/2}$ . In contrast to purely angular perturbations, purely axial perturbations can grow when $S=0$ , which corresponds to the Rayleigh–Plateau instability. When $S=0$ , perturbations having a wavelength longer than the cylinder circumference will tend to grow. When $S\neq 0$ , centrifugal forces further destabilize perturbations, causing the growth rates to increase and the band of unstable wavenumbers to widen.

When Marangoni stresses are present, we can no longer obtain analytical expressions for the cutoff and most dangerous wavenumbers. The case of purely angular perturbations ( $k_{y}=0$ ) was considered by Li & Kumar (Reference Li and Kumar2015), and it was found that surfactant-induced Marangoni stresses reduce the growth rates but do not significantly change the cutoff and most dangerous wavenumbers. This occurs because the surfactant concentration perturbations are in phase with the interface perturbations, lowering the surface tension at the interface crests relative to the troughs. The resulting Marangoni stresses then drive a flow back to the troughs, thereby reducing the instability growth rates. We have conducted a similar analysis for purely axial perturbations ( $k_{\unicode[STIX]{x1D703}}=0$ ) and observe the same behaviour. Rather than showing detailed results here, we refer the reader to the supplementary material available at https://doi.org/10.1017/jfm.2018.153.

3.2 Nonlinear evolution

We now turn to the nonlinear evolution of the instabilities discussed in § 3.1. The finite-difference scheme described in § 2 was validated by comparison with the LSA results of § 3.1. Excellent agreement was observed for both purely angular and purely axial perturbations for both $Ma=0$ and $Ma\neq 0$ (results not shown). Here, we examine what happens when perturbations in both directions grow simultaneously using a representative set of problem parameters. The following initial condition is used:

(3.11) $$\begin{eqnarray}h(\unicode[STIX]{x1D703},y,t=0)=h_{0}(1+\unicode[STIX]{x1D6FC}\cos (k_{\unicode[STIX]{x1D703}}\unicode[STIX]{x1D703})+\unicode[STIX]{x1D6FC}\cos (k_{y}y)),\end{eqnarray}$$

where $\unicode[STIX]{x1D6FC}$ denotes the perturbation amplitude and $h_{0}$ denotes the mean film thickness.

We first set $Ma=0$ and consider an initial perturbation with $k_{\unicode[STIX]{x1D703}}=2<k_{\unicode[STIX]{x1D703}}^{max}=3$ and $k_{y}=k_{y}^{max}=3$ . To gain insight into how the flow transports surfactant, we also calculate the concentration field $\unicode[STIX]{x1D6E4}$ even though it does not influence the film shape when $Ma=0$ . The evolution of the film follows two distinct stages, and we show in figure 2(a,c) the free-surface profiles and surfactant distributions in these two stages. For ease of visualization, the corresponding free-surface profiles around the cylinder circumference at $y=0$ (marked in figure 2 a) are shown in figure 2(b,d) with red solid lines.

Figure 2.

(a,c) Free-surface profiles and surfactant distributions in the zero-gravity case with $h_{0}=0.03162$ , $\unicode[STIX]{x1D6FC}=10^{-5}$ , $k_{y}=3$ , $k_{\unicode[STIX]{x1D703}}=2$ , $S=20$ , $Ma=0$ and $Pe=10^{5}$ at (a) $t=7\times 10^{3}$ and (c) $t=10^{4}$ . The corresponding free-surface profiles around the cylinder circumference at $y=0$ are plotted in (b,d) with red solid lines. The black dashed lines indicate the cylinder surface. For ease of visualization, the thickness plotted in (a–d) is five times larger than the actual value. The spacing between the dashed lines in (a,c) is $0.5\unicode[STIX]{x03C0}$ .

During the early stage (figure 2 a), 7.5 rings develop along the axis ( $k_{y}\times L/2\unicode[STIX]{x03C0}=7.5$ ), while the coating stays almost axisymmetric (figure 2 b). This is simply because the initial perturbation has a higher growth rate in the axial direction. As these rings grow in size, they become unstable to angular perturbations in the later stage and each of them breaks into two liquid droplets ( $k_{\unicode[STIX]{x1D703}}=2$ ) connected by relatively thin films as shown in figure 2(c,d). The growth of these droplets leads to a highly distorted free-surface shape, which significantly increases computational costs, and for this reason, we are unable to find a steady state with our lubrication model.

Figure 3.

(a,c) Free-surface profiles and surfactant distributions in the zero-gravity case with $h_{0}=0.03162$ , $\unicode[STIX]{x1D6FC}=10^{-5}$ , $k_{y}=2$ , $k_{\unicode[STIX]{x1D703}}=3$ , $S=20$ , $Ma=0$ and $Pe=10^{5}$ at (a) $t=7\times 10^{3}$ and (c) $t=10^{4}$ . The corresponding free-surface profiles around the cylinder circumference at $y=0$ are plotted in (b,d) with red dashed lines. The black dashed lines indicate the cylinder surface. For ease of visualization, the thickness plotted in (a–d) is five times larger than the actual value. The spacing between the dashed lines in (a,c) is $0.5\unicode[STIX]{x03C0}$ .

For an initial perturbation with $k_{\unicode[STIX]{x1D703}}=3$ and $k_{y}=2$ , the evolution of the coating also follows two distinct stages. The corresponding free-surface profiles and surfactant distributions at $t=7\times 10^{3}$ and $t=10^{4}$ are shown in figure 3, along with the free-surface profiles around the cylinder circumference at $y=0$ . In this case, three ridges ( $k_{\unicode[STIX]{x1D703}}=3$ ) aligned with the cylinder axis develop at the early stage (figure 3 b) with relatively small axial variations in film thickness (figure 3 a). As more liquid is driven into the ridges by centrifugal forces, the growth of axial perturbations is promoted and each ridge breaks into five droplets ( $k_{y}\times L/2\unicode[STIX]{x03C0}=5$ ) in the later stage, as shown in figure 3(c,d).

Figures 2(a,c) and 3(a,c) clearly show that the regions where the film is thicker also appear to be the high-concentration (red) regions. Thus, the convective flows triggered by centrifugal forces tend to drive surfactant to the thick-film regions, giving rise to concentration gradients at the free surface. This suggests that when $Ma\neq 0$ , the resulting Marangoni stresses will drive liquid away from the thick-film (lower-surface-tension) regions to the thin-film (higher-surface-tension) regions and thus suppress the growth rates of both the angular and axial perturbations. Results from our nonlinear simulations confirm this but are not shown here for brevity. We note that for $Ma=0$ , the surfactant concentration should eventually become uniform if the coating reaches a steady state in which all flows vanish. However, as mentioned earlier, we are unable to reach steady states in these simulations.

5.1 Rayleigh–Taylor instability

In the regime where the RT instability dominates (left side of the stable speed window in figure 4 a), the film sags due to gravity-driven drainage and forms a ridge of liquid aligned with the cylinder axis (Evans et al. Reference Evans, Schwartz and Roy2005). This ridge is unstable and tends to break up into droplets underneath the cylinder. These droplets grow in size until the coating reaches a steady state. We pick a representative rotation rate $W=0.1$ (in the RT-instability region in figure 4 a), and perform simulations with a layer of nearly uniform thickness with random noise added,

(5.4) $$\begin{eqnarray}h(\unicode[STIX]{x1D703},y,t=0)=h_{0}(1+0.01[R_{\unicode[STIX]{x1D703},y}-{\textstyle \frac{1}{2}}]),\end{eqnarray}$$

where $R_{\unicode[STIX]{x1D703},y}$ is a randomly chose value between $0$ and $1$ for each value of $\unicode[STIX]{x1D703}$ and $y$ .

The steady film-thickness profile at $t=4\times 10^{6}$ and the corresponding axial flux and its contributions are plotted in figure 5(a,c,e,g), respectively, as functions of angular position at $y=\unicode[STIX]{x03C0}$ . Figure 5(b) shows the full free-surface profile observed from the cylinder front. Figure 5(d,f,h) shows the same free-surface profile but only at the lower half of the cylinder, where large variations in free-surface curvature occur.

As shown in figure 5(b), two distinct half-droplets of almost the same size develop near the bottom at the two ends of the cylinder. The corresponding film thickness around the circumference of the cylinder at $y=\unicode[STIX]{x03C0}$ is shown in figure 5(a). The thickness profile on the downward-moving side of the droplet ( $\unicode[STIX]{x03C0}<\unicode[STIX]{x1D703}<1.5\unicode[STIX]{x03C0}$ ) is steeper than that on the upward-moving side of the droplet ( $1.5\unicode[STIX]{x03C0}<\unicode[STIX]{x1D703}<2\unicode[STIX]{x03C0}$ ) due to the effects of cylinder rotation, and the thickness reaches its maximum (denoted with a dashed line) slightly above the bottom of the cylinder ( $\unicode[STIX]{x1D703}=1.5\unicode[STIX]{x03C0}$ ).

For $W=0.1$ , the centrifugal term in (5.1) is negligible relative to the gravitational term, and the flux arising due to gravity ( $Q_{y}^{G/\unicode[STIX]{x1D6FA}}$ shown in figure 5 c,d) is negative around cylinder circumference, indicating that gravitational forces tend to drive liquid to the thick-film regions (i.e. droplets). The strongest flux occurs at the point of maximum film thickness, which also corresponds to the point where gravitational forces acting on the liquid are the strongest.

The flux arising due to surface-tension forces ( $Q_{y}^{\unicode[STIX]{x1D70E}}$ shown in figure 5 e,f) is positive on the downward-moving side of the droplet, but changes sign in a small region on the upward-moving side. This suggests that surface tension acts mainly as a stabilizing force, driving liquid to the thin-film region (i.e. middle of the cylinder). The strongest flux occurs on the downward-moving side of the droplet, slightly ahead of the point of maximum film thickness.

The resulting total flux ( $Q_{y}=Q_{y}^{G/\unicode[STIX]{x1D6FA}}+Q_{y}^{\unicode[STIX]{x1D70E}}$ shown in figure 5 g,h) changes sign at the point of maximum film thickness, and is positive on the upward-moving side of the droplet and negative on the downward-moving side. This implies that gravitational forces dominate on the upward-moving side, while surface-tension forces dominate on the downward-moving side. The competition between them determines whether the initial axial perturbation will grow or decay. The areas of regions $I$ and $II$ marked in figure 5(e), $A_{I}$ , and $A_{II}$ , measure how much flux crosses over the $y=\unicode[STIX]{x03C0}$ plane on the upward-moving side and on the downward-moving side, respectively. For $W=0.1$ , our numerical results (not shown here for brevity) reveal that $A_{II}$ is always greater than $A_{I}$ during the early transient stage, so droplets tend to grow in size under the action of gravity until the coating reaches a steady state (figure 5 b) where $A_{I}=A_{II}$ .

An increase in the value of $W$ causes a decrease in the size of the ridge (Evans et al. Reference Evans, Schwartz and Roy2004; Li & Kumar Reference Li and Kumar2015), which corresponds to a smaller amount of liquid inside the ridge and therefore weaker destabilizing gravitational forces acting on the ridge. In this case, the flux arising due to gravity-driven drainage is not strong enough to compensate the liquid loss in the droplet caused by surface-tension-induced flow. This explains why the coating becomes more stable (larger cutoff wavelength) to axial perturbations with an increase in the rotation rate (near $W=0.186$ in figure 4 a) in the region where the RT instability dominates.

Further calculations reveal that $(h_{\unicode[STIX]{x1D703}\unicode[STIX]{x1D703}})_{y}$ is the dominant term in (5.2); this term arises from both angular and axial variations in the angular curvature. To understand why the effects of surface tension are especially strong on the downward-moving side of the droplet (figure 5 e,f), we plot in figure 6( $h$ ) and $h_{\unicode[STIX]{x1D703}\unicode[STIX]{x1D703}}$ for $\unicode[STIX]{x03C0}<\unicode[STIX]{x1D703}<2.5\unicode[STIX]{x03C0}$ (marked in figure 6 a) at different positions along the cylinder axis for the simulation shown in figure 5. We choose the region $\unicode[STIX]{x03C0}<\unicode[STIX]{x1D703}<2.5\unicode[STIX]{x03C0}$ because large variations in $h$ and $h_{\unicode[STIX]{x1D703}\unicode[STIX]{x1D703}}$ occur in this region.

Figure 6.

Film thickness ( $h$ ) and its second derivative in the angular direction ( $h_{\unicode[STIX]{x1D703}\unicode[STIX]{x1D703}}$ ) at (a) $y=0$ and (b,c) at other values of $y$ with $Bo=1$ , $h_{0}=0.03162$ , $M=0.007$ , $L=5\unicode[STIX]{x03C0}$ , $Ma=0$ and $W=0.1$ at $t=4\times 10^{6}$ .

For $y=0$ , $h_{\unicode[STIX]{x1D703}\unicode[STIX]{x1D703}}$ (red dashed line in figure 6 a) is almost zero near the cylinder top ( $2\unicode[STIX]{x03C0}<\unicode[STIX]{x1D703}<2.5\unicode[STIX]{x03C0}$ ) where the film is very thin due to gravity-driven drainage. In the thick-film region ( $1.25\unicode[STIX]{x03C0}<\unicode[STIX]{x1D703}<2\unicode[STIX]{x03C0}$ ), $h_{\unicode[STIX]{x1D703}\unicode[STIX]{x1D703}}$ decreases on the downward-moving side of the droplet ( $1.25\unicode[STIX]{x03C0}<\unicode[STIX]{x1D703}<1.5\unicode[STIX]{x03C0}$ ), increases on the upward-moving side ( $1.5\unicode[STIX]{x03C0}<\unicode[STIX]{x1D703}<2\unicode[STIX]{x03C0}$ ), and reaches both its maximum and minimum on the downward-moving side.

Figure 6(b) shows that the maximum film thickness around cylinder circumference at a given value of $y$ decreases with an increase in $y$ (closer to the middle of the cylinder), since gravitational forces draw liquid to both ends of the cylinder ( $y=0$ ). The increase in $y$ also causes the point of maximum thickness to rise higher (larger  $\unicode[STIX]{x1D703}$ ). This phenomenon can be explained by noticing that the point of maximum thickness is determined by the interplay between viscous drag and gravity. A smaller maximum thickness corresponds to weaker gravitational forces there, and as a result, viscous-drag forces push the point of maximum thickness upward, closer to the cylinder symmetry plane ( $\unicode[STIX]{x1D703}=0$ ).

Since the minimum value of $h_{\unicode[STIX]{x1D703}\unicode[STIX]{x1D703}}$ occurs near the point of maximum thickness (figure 6 a), the location of this minimum also rises higher with an increase in $y$ (figure 6 c). As a result, the value of $h_{\unicode[STIX]{x1D703}\unicode[STIX]{x1D703}}$ increases with an increase in $y$ (yielding a positive value of $(h_{\unicode[STIX]{x1D703}\unicode[STIX]{x1D703}})_{y}$ ) on the downward-moving side of the droplet ( $1.25\unicode[STIX]{x03C0}<\unicode[STIX]{x1D703}<1.75\unicode[STIX]{x03C0}$ ), while it decreases (yielding a negative value of $(h_{\unicode[STIX]{x1D703}\unicode[STIX]{x1D703}})_{y}$ ) in a small region on the upward-moving side ( $1.75\unicode[STIX]{x03C0}<\unicode[STIX]{x1D703}<2\unicode[STIX]{x03C0}$ ). The variation in $h_{\unicode[STIX]{x1D703}\unicode[STIX]{x1D703}}$ is more significant on the downward-moving side than elsewhere, and this gives rise to a strong flux there in the positive $y$ -direction (figure 5 e). This flux stabilizes the coating by driving liquid back to the middle of the cylinder.

Figure 7.

(a) Free-surface profile at $t=4\times 10^{6}$ with $L=5\unicode[STIX]{x03C0}$ , $Bo=1$ , $h_{0}=0.03162$ , $M=0.007$ , $Ma=0$ and $W=0.3$ . (b) Film thickness and total axial flux ( $Q_{y}$ ) at $y=\unicode[STIX]{x03C0}$ (marked in a). For ease of visualization, the thickness plotted in (a) is 50 times larger than the actual value.

5.2 Rayleigh–Plateau instability

We now turn to the regime where the RP instability dominates (right side of the stable speed window in figure 4 a). In this regime, a Moffatt-type smooth coating, which is thicker on the upward-moving side of the cylinder and thinner on the downward-moving side, may break up into ring-shaped liquid bulges wrapped around the cylinder. Simulations similar to those described in § 5.1 indicate that these rings tend to slowly shift along the cylinder axis, and as a consequence, we are unable to find steady states in this regime. For a representative case where $W=0.3$ , we show in figure 7 the free-surface profile, and the film thickness and axial flux ( $Q_{y}$ ) at $y=\unicode[STIX]{x03C0}$ , at a relatively late time ( $t=4\times 10^{6}$ ).

Figure 7(a) shows the appearance of one full ring (located near the middle of the cylinder) and one half-ring (located near the end of the cylinder). These rings are non-axisymmetric, in contrast to the zero-gravity case (figure 2 a), and are much thicker on the upward-moving side of the cylinder (dark red region). In this regime, centrifugal forces play a more significant role (relative to gravity) in the axial flux (5.1), and they are especially strong in the rings since more liquid is located there. Liquid is drawn from the surrounding areas to the rings, and as a result, the coating is destabilized.

There are three terms in the surface-tension contribution to the axial flux (5.2). The first term, $h_{y}$ , represents axial variations in the angular curvature, and gives rise to surface-tension forces that tend to minimize area. As a result, liquid is driven to the rings, which in turn strengthens the centrifugal forces there. The second term, $h_{yy}$ , represents axial variations in axial curvature ( $h_{yy}$ ), and the third term, $(h_{\unicode[STIX]{x1D703}\unicode[STIX]{x1D703}})_{y}$ , arises from both angular and axial variations in the angular curvature. The latter two terms give rise to surface-tension forces that tend to smooth out curvature gradients at the free surface and thus stabilize the coating. Note that the $(h_{\unicode[STIX]{x1D703}\unicode[STIX]{x1D703}})_{y}$ term does not exist in the standard RP instability where the film is axisymmetric. Analysis of our results indicates that this term dominates the other two surface-tension terms, similar to what is observed in the regime where the RT instability dominates (§ 5.1).

Figure 7(b) shows that the total axial flux (red dashed line) at $y=\unicode[STIX]{x03C0}$ is negative in a small region slightly ahead of the point of maximum film thickness, indicating that stabilizing surface-tension forces are stronger than destabilizing centrifugal forces in this region, driving liquid away from the ring (in the negative $y$ -direction). However, centrifugal forces dominate elsewhere and the resulting value of $\int _{0}^{2\unicode[STIX]{x03C0}}Q_{y}\,\text{d}\unicode[STIX]{x1D703}$ is positive for the case shown in figure 7(b), leading to growth of the rings. A decrease in the value of $W$ weakens the flux triggered by centrifugal forces and thus causes the coating to be more stable to axial perturbations. This explains the significant increase in the cutoff wavelength as $W$ is decreased to a value of $0.242$ (figure 4 a).

We now explain the mechanism behind the stable speed window shown in figure 4(a). For moderate cylinder rotation rates ( $0.186<W<0.242$ ), the liquid film stays axially uniform, which leads to $Q_{y}=0$ . Based on the results shown in figures 5 and 7, we conclude that surface-tension forces arising due to the term $(h_{\unicode[STIX]{x1D703}\unicode[STIX]{x1D703}})_{y}$ will smooth out any small perturbations in the axial direction since both gravitational forces (weak due to small ridge size) and centrifugal forces (weak due to small rotation rate) are not able to continuously draw liquid to the droplets or rings in this regime. In the following section, we will discuss how the presence of insoluble surfactant causes this stable speed window to disappear.

6.1 Rayleigh–Taylor instability

6.1.1 Surfactant concentration distribution

In the regime where the RT instability dominates (left side of figure 4 b), it is useful to first examine the surfactant concentration distribution in the absence of Marangoni stresses. We again choose a representative rotation rate of $W=0.1$ .

For $W=0.1$ and $Ma=0$ , the time evolution of surfactant concentration follows three distinct stages. We show in figures 8 and 9 the corresponding free-surface profiles and concentration distributions on the downward-moving side of the cylinder, where the surfactant concentration reaches its maximum.

Figure 9.

(a,c) Free-surface and concentration profiles, and film thickness (black solid lines in (b,d)) and concentration (red dashed lines in (b,d)) at $\unicode[STIX]{x1D703}=1.5\unicode[STIX]{x03C0}$ with $Bo=1$ , $h_{0}=0.03162$ , $M=0.007$ , $Ma=0$ , $Pe=10^{5}$ , $L=5\unicode[STIX]{x03C0}$ and $W=0.1$ at (a,b) $t=2\times 10^{6}$ and (c,d) $t=4\times 10^{6}$ . For ease of visualization, the thickness plotted in (a,c) is five times larger than the actual value.

Due to the initial gravity-driven drainage, surfactant tends to accumulate at the bottom of the cylinder in a liquid ridge with little concentration variation in the axial direction (red region in figure 8 a). The corresponding film thickness and concentration at $y=0$ are shown in figure 8(b). Surfactant concentration increases significantly on the downward-moving side of the ridge ( $1.3\unicode[STIX]{x03C0}<\unicode[STIX]{x1D703}<1.5\unicode[STIX]{x03C0}$ ), and reaches its maximum several degrees below the point of maximum thickness.

As axial perturbations start to grow in amplitude (figure 9 a), axial variations in both film thickness and surfactant concentration become significant, so we plot in figure 9(b) the film thickness and concentration along the cylinder axis at $\unicode[STIX]{x1D703}=1.5\unicode[STIX]{x03C0}$ (marked in figure 8 a). Figure 9(b) shows that film thickness and surface concentration evolve out of phase: liquid is driven to the ends of the cylinder forming two half-droplets at $y=0$ and $y=5\unicode[STIX]{x03C0}$ , while surfactant is driven to the middle of the cylinder, giving rise to a high-concentration region in the thin-film area ( $y=2.5\unicode[STIX]{x03C0}$ ). This happens because the region of high surfactant concentration in figure 8(a) ( $1.3\unicode[STIX]{x03C0}<\unicode[STIX]{x1D703}<1.5\unicode[STIX]{x03C0}$ ) occurs where the axial flux drives liquid to the middle of the cylinder (similar to the case shown in figure 5). As in § 5.1, the negative axial flux arises because the contribution from surface tension is stronger than the contribution from gravity and centrifugal forces. Again, the term $(h_{\unicode[STIX]{x1D703}\unicode[STIX]{x1D703}})_{y}$ dominates the surface-tension contribution.

The steady free-surface and concentration profiles are shown in figure 9(c). The corresponding film thickness and concentration at $y=1.5\unicode[STIX]{x03C0}$ are shown in figure 9(d). As the two half-droplets grow in size, a satellite droplet forms in the middle of the cylinder (small bump in between $2\unicode[STIX]{x03C0}<y<3\unicode[STIX]{x03C0}$ in figure 9 d). This triggers an additional capillary flow that drives surfactant away from the middle of the cylinder. As a result, the surfactant concentration in the middle of the cylinder decreases dramatically and forms maxima in between the main droplets and the satellite droplet.

Figure 10.

Free-surface and surfactant concentration profiles with $Bo=1$ , $h_{0}=0.03162$ , $L=5\unicode[STIX]{x03C0}$ , $M=0.007$ , $Ma=0.001$ , $Pe=10^{5}$ and $W=0.1$ at (a) $t=8.5\times 10^{4}$ (b) $t=2.5\times 10^{5}$ (steady state). For ease of visualization, the thickness plotted in this figure is five times larger than the actual value.

6.1.2 Marangoni stresses

We now include the influence of Marangoni stresses by performing simulations with $Ma=0.001$ . Figure 10(a,b) shows the free-surface and surfactant concentration profiles on the downward-moving side of the cylinder during the transient stage and at steady state, respectively. Notably, a third droplet forms in the middle of the cylinder during the transient stage, corresponding to a decrease in the spacing between droplets relative to the zero- $Ma$ case (figure 9 c). The third droplet grows in size together with the two half-droplets at two ends. The decrease in the spacing between droplets when $Ma\neq 0$ is consistent with the decrease in the cutoff wavelength in the RT regime shown in figure 4.

The results shown in figure 9 can be used to understand the growth of the third droplet. The accumulation of surfactant in between the two main droplets (figure 9 b) lowers the local surface tension, and the resulting Marangoni stresses then drive liquid away from that area to those droplets. The surfactant concentration in the middle of the cylinder becomes especially low when the satellite droplet forms (figure 9 d). The resulting Marangoni stresses then drive flow to the middle of the cylinder, leading to the formation of a third droplet (figure 10). The flows driven by the Marangoni stresses enhance the growth rate of the droplets, so steady state is reached sooner.

The Marangoni flows also lead to disappearance of the stable speed window (figure 4). Recall that the term $(h_{\unicode[STIX]{x1D703}\unicode[STIX]{x1D703}})_{y}$ drives flow to the middle of the cylinder and thus opposes the RT instability, leading to the large increase in the cutoff wavelength as the rotation rate is increased (figure 4 a). The Marangoni stresses initially drive flow towards the ends of the cylinder and thus oppose the flow driven by the $(h_{\unicode[STIX]{x1D703}\unicode[STIX]{x1D703}})_{y}$ term. As a consequence, this term becomes much less effective at opposing the RT instability and the stable speed window disappears (figure 4 b).

Figure 11.

(a,c) Free-surface and surfactant concentration profiles for two different values of $Ma$ with $Bo=1$ , $h_{0}=0.03162$ , $L=5\unicode[STIX]{x03C0}$ , $M=0.007$ , $Pe=10^{5}$ and $W=0.3$ at $t=4\times 10^{6}$ . For ease of visualization, the thickness plotted in (a,c) is $50$ times larger than the actual value. (b,d) Corresponding film thickness (black solid lines) and concentration (red dashed lines) at $\unicode[STIX]{x1D703}=0$ (marked in a).

6.2 Rayleigh–Plateau instability

In the regime where the RP instability dominates (right side of figure 4 b), we again choose a representative rotation rate of $W=0.1$ . Figure 11(a) shows the free-surface and surfactant concentration profiles for the simulation shown in figure 7 where $Ma=0$ . The corresponding film thickness and surfactant concentration along the cylinder axis at $\unicode[STIX]{x1D703}=0$ are shown in figure 11(b). The case where $Ma=0.001$ is also considered, and results are shown in figure 11(c,d) for comparison.

For $Ma=0$ , the liquid film breaks into one-and-a-half rings wrapped circumferentially around the cylinder due to the RP instability (figure 11 a). Surfactant tends to accumulate on the upward-moving side of the rings (red regions in figure 11 a). Recall that when the rotation rate is smaller, surfactant tends to accumulate in a liquid ridge at the cylinder bottom (figure 8). As the rotation rate is increased, the ridge and point of maximum concentration shift to the upward-moving side of the cylinder (Li & Kumar Reference Li and Kumar2015). As the ridge breaks up due to the RP instability, liquid and surfactant get driven into the rings, and as a consequence, the film thickness and surfactant concentration are in phase at $\unicode[STIX]{x1D703}=0$ , which is on the upward-moving side (figure 11 b).

For $Ma=0.001$ , the gradients in film thickness and surfactant concentration are much smaller than those when $Ma=0$ and the coating stays almost axially uniform (figures 11 c and 11 d). Since surfactant tends to accumulate at the surface of the rings, the resulting Marangoni stresses drive flow from the rings to the thin-film areas, leading to the more uniform coating and concentration profile. In addition, these Marangoni flows slow the growth of the instability but do not appear to change the spacing between the rings.

The Marangoni flows again lead to disappearance of the stable speed window (figure 4). Recall that the term $(h_{\unicode[STIX]{x1D703}\unicode[STIX]{x1D703}})_{y}$ drives flow to the middle of the cylinder and thus opposes the RP instability (§ 5.2), leading to the large increase in the cutoff wavelength as the rotation rate is decreased (figure 4 a). Because the Marangoni flows lead to a smoother free surface, the $(h_{\unicode[STIX]{x1D703}\unicode[STIX]{x1D703}})_{y}$ term is much less effective at opposing the RP instability and the stable speed window disappears (figure 4 b).

7.1 Experimental set-up

The experimental apparatus is shown in figure 12. A smooth steel cylinder is mounted in a container whose front gate can be lifted, and the cylinder is rotated using an electric motor. In all of our experiments, liquid was first coated on the cylinder by rotating it with the container partially filled. The glass gate was then quickly lifted to drain the liquid from the container into a tray, leaving liquid on the rotating cylinder. Excess liquid would immediately drip off, which sometimes triggered visible axial variations in coating thickness or even dry patches. As a result, we sometimes had to repeat this process until a sufficiently uniform initial coating was obtained.

The steel cylinder has radius $R=0.2~\text{cm}$ , which is consistent with the value used in our simulations (table 1), and length $L=30~\text{cm}$ . The liquid is a glycerol/water mixture (96 wt % glycerol) having surface tension $\unicode[STIX]{x1D70E}_{0}=66~\text{dyn}~~\text{cm}^{-1}$ , density $\unicode[STIX]{x1D70C}=1.23~\text{g}~\text{cm}^{-3}$ and viscosity $\unicode[STIX]{x1D707}=2.6~\text{P}$ . To study the influence of surfactant, we use sodium dodecyl sulphate (SDS; Thermo Scientific). For a bulk concentration $c=0.1$  wt %, the surface tension drops to $50~\text{dyn}~~\text{cm}^{-1}$ , while the density and viscosity remain unchanged. An increase in $c$ further reduces surface tension, suggesting that $c=0.1~\text{wt}\%$ is still smaller than the critical micelle concentration.

The values of key dimensionless parameters in the experiments are summarized in table 3. Note that $Bo$ in our experiments is around unity, consistent with the simulations. The dimensionless viscosity in our experiments ( $M=0.74$ ) is approximately 100 times higher than that in our simulations ( $M=0.007$ ) so that a much thicker coating can be maintained on the cylinder for ease of flow visualization. As noted in § 4, if $M$ or the initial film thickness are too large, a stable speed window may not be observed. In the experiments, the smallest rotation rates ( $W\sim 0.03{-}0.06$ ) were comparable to those in the simulations, while the largest rotation rates ( $W\sim 0.2{-}0.3$ ) were smaller. The Marangoni number in our experiments ( $Ma=0.35$ ) is much larger than that in our simulations ( $Ma=0.001$ ), so we expect to observe much stronger Marangoni effects in the experiments.

Figure 12.

Experimental apparatus. The steel cylinder has length $L=30~\text{cm}$ and radius $R=0.2~\text{cm}$ .

Table 3.

Dimensionless problem parameters.

7.2 Results and discussion

For the surfactant-free solution (which corresponds to $Ma=0$ ), images of the free surface on the upward-moving side of the cylinder at several different times are shown in figure 13 for $W=0.056$ and $W=0.22$ .

Figure 13.

Images of the free surface on the upward-moving side of the cylinder for the surfactant-free solution at several different times for (a) $W=0.056$ and (b) $W=0.22$ with $Bo=0.73$ , $M=0.74$ and $Ma=0$ .

For $W=0.056$ (figure 13 a), axial perturbations grow in amplitude due to gravity, and become visible near the underside of the cylinder within approximately 7 s (1 s corresponds to approximately 100 dimensionless time units). At later times, the coating breaks up into small droplets which are approximately evenly spaced along the cylinder axis, similar to the simulation result shown in figure 5 for the Rayleigh–Taylor instability. The sizes of the droplets reach a steady state within about 35 s, with the droplets tending to slowly shift along the cylinder axis.

With an increase in rotation rate to $W=0.22$ , there is a transition from the Rayleigh–Taylor instability to the Rayleigh–Plateau instability (figure 13 b). Initial perturbations grow into small ring-shaped liquid bulges under the action of centrifugal forces, similar to the simulation result shown in figure 7 for the Rayleigh–Plateau instability. These bulges are not stable, and adjacent ones tend to merge to form a droplet (marked by arrows in figure 13 b). We do observe some shifting of rings along the cylinder axis in our simulations (§ 5.2), but are not able to run simulations for a long enough time to observe merging. The formation of large bulges enhances the strength of gravitational forces and appears to draw liquid from the surrounding areas, which leads to a droplet. The sizes of the rings and droplet reach a steady state at a later time ( ${\sim}$ 70 s) compared to the low rotation rate case (figure 13 a).

Figure 14.

Images of the free surface on the upward-moving side of the cylinder for the surfactant-laden solution at several different times for (a) $W=0.056$ and (b) $W=0.22$ with $Bo=1.0$ , $M=0.74$ and $Ma=0.35$ .

For the surfactant-laden solution (which corresponds to $Ma=0.35$ ), images of the free surface on the upward-moving side of the cylinder at several different times are shown in figure 14 for $W=0.056$ and $W=0.22$ . Similar to the surfactant-free case, droplets form at the lower rotation rate (figure 14 a) while rings form at the higher rotation rate (figure 14 b). However, the time required for axial perturbations to become visible is much longer compared to the surfactant-free case for both rotation rates. As a consequence, the sizes of the droplets and rings reach steady state at much later times, ${\sim}5$ min. for the droplets and ${\sim}10$ min. for the rings. In addition, the shifting and merging of rings and droplets seen in the surfactant-free case (figure 13 b) appears to be considerably suppressed when surfactant is present.

The slower development of free-surface perturbations at the higher rotation rate when $Ma\neq 0$ is consistent with the simulation results in § 6.2 for the RP instability. However, the simulation results for the RT instability in § 6.2 predict a faster development of free-surface perturbations when $Ma\neq 0$ , in contrast to the experimental observations at the lower rotation rate. We believe that this situation arises because the initial coating in the experiments is non-axisymmetric, in contrast to the initial condition used in the simulations. The non-axisymmetric initial condition in the experiments arises due to liquid dripping. When surfactant is present, the resulting Marangoni stresses tend to suppress this dripping (Weidner Reference Weidner2013), leading to a slower development of free-surface perturbations. Nevertheless, this slower development created by surfactant may be very useful for applications, allowing for a longer time period to solidify the coating before free-surface perturbations become prominent.

Figure 15.

Most dangerous wavelength at different rotation rates with $M=0.74$ and (a) $Bo=0.73$ and $Ma=0$ and (b) $Bo=1.0$ and $Ma=0.35$ . Red dashed lines and blue solid lines indicate the most dangerous wavelength obtained from linear stability analysis.

We have also measured the average spacing between the free-surface bulges as soon as they become visible over a range of rotation rates (figure 15). This spacing should correspond the most dangerous wavelength, and is shown in non-dimensional form (scaled with $R$ ) in figure 15. Each point in figure 15 represents a mean of up to four measurements. For the surfactant-free solution, the largest deviation from the mean value of the most dangerous wavelength was $0.3\unicode[STIX]{x03C0}$ , whereas for the surfactant-free solution, the largest deviation from the mean was much smaller, $0.05\unicode[STIX]{x03C0}$ , presumably due to the reduced shifting and merging of rings and droplets. Also shown in figure 15 are predictions of the most dangerous wavelength from LSA for the RT and RP instabilities (see supplementary material and § 4), yielding plots similar to those for the cutoff wavelength shown in figure 4.

The most dangerous wavelengths of the RT and RP instabilities in the surfactant-free case are (Evans et al. Reference Evans, Schwartz and Roy2005)

(7.1a,b ) $$\begin{eqnarray}\text{RT instability:}\quad \unicode[STIX]{x1D706}_{drop}^{\star }=\frac{2\sqrt{2}\unicode[STIX]{x03C0}}{\sqrt{Bo}},\quad \text{RP instability:}\quad \unicode[STIX]{x1D706}_{ring}^{\star }=\frac{2\sqrt{2}\unicode[STIX]{x03C0}}{1+S},\end{eqnarray}$$

where $S=WBo^{2}$ is the Weber number defined in § 3. When $Ma\neq 0$ , analytical expressions for these quantities are not available, but calculations show that surfactant has only a very minor influence (see supplementary material). We thus use the expressions given in (7.1a,b ) even when $Ma\neq 0$ , similar to figure 4.

For the surfactant-free solution (figure 15 a), the average spacing oscillates about $\unicode[STIX]{x1D706}_{drop}^{\star }$ (red dotted line) at low rotation rates, and is always larger than $\unicode[STIX]{x1D706}_{ring}^{\star }$ (blue solid line) at high rotation rates. The discrepancy at high rotation rates is consistent with the merging of the rings (figure 14 b), which lowers the number of rings along cylinder axis and causes the average spacing to increase. No stable speed window is observed, perhaps due to the large value of $M$ and a relatively thick and non-uniform initial film.

For the surfactant-laden solution, (figure 15 b), the average spacing is generally smaller compared to the surfactant-free case. This is consistent with the observation that surfactant lowers the mean surface tension, $\unicode[STIX]{x1D70E}_{m}$ (larger $Bo$ ), which gives rise to a larger value of $Bo$ and a smaller value of the most dangerous wavelength (7.1a,b ). Again, no stable speed window is observed over the range of rotation rates considered.

When $W=0.028$ , the average spacing measured in our experiments is $\unicode[STIX]{x1D706}/\unicode[STIX]{x03C0}=2.0$ ( ${\sim}0.4$ cm) in the surfactant-laden solution, which is much smaller than $\unicode[STIX]{x1D706}_{drop}^{\star }/\unicode[STIX]{x03C0}=2.8$ calculated using LSA (red dotted line in figure 15 b). With an increase in $W$ , the average spacing increases monotonically in the RT-instability regime and reaches its maximum at $W=0.12$ . The discrepancy between the experimental values and the LSA predictions is consistent with our simulation results, which show that Marangoni stresses lead to the formation of extra droplets along the cylinder axis (figure 10), which would cause the average spacing to be smaller than that predicted by the simple RT LSA result. An increase in the rotation rate weakens the effects of gravity, and as a result, the coating becomes more stable to the RT instability. This explains the increase in the average spacing with increasing $W$ in the regime of the RT instability.

A further increase in $W$ brings the system into the regime of the RP instability. Here, the average spacing decreases as the rotation rate increases, and there is much better agreement between the experimental observations and the LSA predictions relative to the RT regime (blue solid line in figure 15 b). This can be rationalized by recalling that Marangoni stresses suppress the growth rate of the RP instability but do not modify the spacing between rings (§ 6.2). The behaviour of the most dangerous wavelength shown in figure 15(b) is analogous to the behaviour of the cutoff wavelength shown in figure 4(b), where the cutoff wavelength from the simulations is smaller than the LSA prediction at low rotation rates, but in much better agreement at high rotation rates.

We have also attempted to perform some nonlinear simulations to predict the most dangerous wavelength using an initial film thickness estimated from (1.1) and experimental observations of $\unicode[STIX]{x1D6FA}_{c}$ . However, since the initial film thickness is much larger than that used for the simulations in §§ 4–6, large free-surface deformations quickly develop that we cannot accurately resolve, and as a consequence, we are unable to obtain predictions of the most dangerous wavelength from the simulations. The same issue arises if we try to use a non-axisymmetric initial condition similar to that in the experiments.

Nevertheless, the results of these flow visualization experiments appear to be qualitatively consistent with predictions from LSA and other behaviour observed in our simulations. We note that while the simulations and LSA are for the case of an insoluble surfactant, the surfactant in the experiments is soluble. Nevertheless, it appears that much of what is observed in the experiments can largely be understood by a model that assumes surfactant insolubility, suggesting that surfactant solubility plays a secondary role in these experiments.