3.2.1. Existence of a static solution

The static solution describing the morphology of a thin drop driven by centrifugal forces in spin coating or by a jet of impinging air was reported by McKinley et al. (Reference McKinley, Wilson and Duffy1999), McKinley & Wilson (Reference McKinley and Wilson2002) and Schwartz & Roy (Reference Schwartz and Roy2004). In terms of present dimensionless variables, the solution without gravity can be written as

(3.1) \begin{equation} h=\frac {\left (a^{2}_{\infty }-r^{2}\right )\left [16-\beta a_{\infty }\left (a^{2}_{\infty }-r^{2}\right )\right ]}{32a_{\infty }}, \end{equation}

which can also be derived from the lubrication equation (2.3) together with the boundary conditions (2.5) and (2.6). Here, $a_\infty$ is the radius of the contact line, which is also the limit of the contact-line radius as $t$ goes to infinity in the low-speed regime. According to the volume conservation condition (2.9), $a_\infty$ can be determined as

(3.2) \begin{equation} a_{\infty }=\left [\frac {4}{\beta }\left (3\pm \sqrt {9-12\beta }\right )\right ]^{1/3}. \end{equation}

Clearly, the solution exists only when $\beta \leqslant \beta _c=3/4$ .

The relationship between $a_\infty$ and $\beta$ is plotted in figure 3(a). It is seen that the solution bifurcates at $\beta =\beta _c$ , giving rise to two branches of the curve. The lower and upper branches of the solution are expected to be stable and unstable to axisymmetric disturbances, respectively. Therefore, it is only the lower branch of the solution that can be realized in our axisymmetric calculations. Representative drop profiles of the static solution are presented in figure 3(b). As $\beta$ increases along the lower branch, the drop height is significantly reduced, while the contact-line radius increases by a relatively small amount. At the threshold, the contact-line radius for the lower branch attains its maximum $a_{c,\infty }=2^{4/3}$ , and the corresponding drop profile is given by the red curve in figure 3(b). There is another critical point, $\beta =2/3$ , beyond which the drop is concave around the axis. When $\beta \lt 2/3$ , the solution of the upper branch exhibits negative film thickness and hence is unphysical.

Figure 3.

(a) Bifurcation curves of the static solution in terms of $a_\infty$ as functions of $\beta$ . The solid and dashed lines correspond to the negative and positive signs in (3.2), respectively. (b) Static drop profiles given by (3.1) for representative values of $\beta$ . The solid and dashed lines correspond to the lower and upper branches in (a), respectively.

The static solution is compared with the numerical results of the unsteady lubrication equation in figure 4 for $\beta =0.5$ and 0.7. In both cases, the numerical solution converges to an equilibrium state. For sufficiently large times, e.g. $t=100$ , the interface profiles show perfect agreement with the static solution denoted by the dashed lines. This comparison in turn demonstrates the accuracy of our numerical algorithm.

Figure 4.

Interface evolution of an axisymmetric drop at $t = 0, 0.1, 1, 10, 100$ for (a) $\beta =0.5$ and (b) $\beta =0.7$ in the low-speed regime. Solid lines represent the numerical solution of (2.3) at different time $t$ . Dashed lines represent the static solution (3.1).

3.2.2. Asymptotic solution of low-speed drop spreading

When $\beta \leqslant \beta _c$ , the drop is expected to spread at a rather slow rate, i.e. $\dot {a}\ll 1$ . An asymptotic theory can be developed in the limit $\lambda \rightarrow 0$ following the procedure proposed by Hocking (Reference Hocking1983), who considered the spreading of a thin drop in the absence or presence of gravity. For the present spin coating problem, the structure of the drop consists of an outer region and an inner region near the contact line. In the outer region, the drop exhibits a quasi-static behaviour and the slip effect is insignificant. The inner region is characterized by a length of $O(\lambda )$ , over which a significant variation of the surface slope occurs due to the large viscous force therein. A matching of the outer and inner solutions typically yields a relation between the speed and position of moving contact lines.

For the outer solution, it is instructive to replace $r$ with the new variable $y=r/a$ . Accordingly, the problem is transformed onto a fixed domain $0\leqslant y\leqslant 1$ . Then the lubrication equation of $h=h(y,t)$ can be written as

(3.3) \begin{equation} h_{t}-\frac {\dot {a}}{a}yh_y+\frac {1}{a^4y}\left \{yh^{3}\left [\left (\frac {\left (yh_{y}\right )_{y}}{y}\right )_{y}+\beta a^4y\right ]\right \}_{y}=0, \end{equation}

where the slip effect has been neglected. The supplementary conditions include

(3.4) \begin{equation} h_y=0,\ h_{yyy}=0 \quad \mbox {at}\ y=0, \end{equation}

(3.5) \begin{equation} h=0 \quad \mbox {at}\ y=1, \end{equation}

and the condition of the mass conservation

(3.6) \begin{equation} \int _{0}^1 yh \,\mathrm {d}y=a^{-2}. \end{equation}

Note that it is unnecessary for the outer solution to satisfy the contact-angle condition.

When the contact-line velocity $\dot {a}$ is small, it turns out that $h$ can be expanded as

(3.7) \begin{equation} h(y,t )=h_{0} (y,a )+\dot {a}\,h_{1} (y,a )+O(\dot {a}^{2}), \end{equation}

where $h_0$ and $h_1$ represent the hydrostatic profile and the first-order correction by the flow, respectively. Accordingly, the time derivative in (3.3) can be expressed as

(3.8) \begin{equation} h_{t}=\dot {a} h_{0a}+O(\dot {a}^{2}). \end{equation}

Substituting (3.7) and (3.8) into (3.3), we obtain equations for each power of $\dot {a}$ . The leading-order equation, after one integration, reads

(3.9) \begin{equation} \left [\frac {\left (yh_{0y}\right )_{y}}{y}\right ]_{y}+\beta a^{4}y=0. \end{equation}

The solution is given by

(3.10) \begin{equation} h_{0}=\frac {\left (1-y^{2}\right )\left [4-2b\left (1-3y^2\right )\right ]}{a^{2}}, \end{equation}

where $b=\beta a^{6}/{192}$ . This quasi-static solution satisfies boundary conditions in the form of (3.4)–(3.6) with $h$ replaced by $h_0$ . One can easily verify that (3.10) reduces to the equilibrium solution (3.1) when $a=a_\infty$ given by (3.2). The first-order correction is determined by the equation

(3.11) \begin{equation} h_{0a}-\frac {y}{a}h_{0y}+\frac {1}{a^{4}y}\left \{yh_{0}^{3}\left [\frac {\left (yh_{1y}\right )_{y}}{y}\right ]_{y}\right \}_y=0. \end{equation}

The details of solving (3.11) for $h_{1}$ are given in Appendix B. Of present interest is the slope close to the contact line. Correspondingly, we have

(3.12) \begin{equation} h_{1y}\left (y\right )\sim -\frac {a^7}{64\left (1+b\right )^{2}}\left \{\ln \left [\mathrm {e}^2\,\frac {2-b}{1+b}\left (1-y\right )\right ]-\frac {3b}{4-2b}\right \}\quad \mbox {as}\ y\rightarrow 1. \end{equation}

It is convenient to calculate the cube of the surface slope, which can be directly matched by the inner solution without invoking an intermediate region (Eggers Reference Eggers2005; Savva & Kalliadasis Reference Savva and Kalliadasis2009; Sibley et al. Reference Sibley, Nold and Kalliadasis2015; Savva et al. Reference Savva, Rezgdnikov and Colinet2017). Combining (3.12) and the derivative of (3.10), we obtain

(3.13) \begin{eqnarray} h_{y}^{3} & =& h_{0y}^{3}+3\dot {a}h_{0y}^{2}h_{1y}+O (\dot {a}^{2} ) \nonumber \\ & \sim & -\left [\frac {8(1+b)}{a^{2}}\right ]^{3}-3a^{3}\dot {a}\left \{\ln \left [\mathrm {e}^{2}\,\frac {2-b}{1+b}\left (1-y\right )\right ]-\frac {3b}{4-2b}\right \}\quad \mbox {as}\ y\rightarrow 1. \end{eqnarray}

Since this solution does not satisfy the contact-angle condition, there must be an inner region, where the slip effect should be taken into account.

For the inner solution, we consider a small- $\dot {a}$ expansion again, just as for the outer region:

(3.14) \begin{equation} H (\xi ,t )=H_{0} (\xi ,a )+\dot {a}\,H_{1} (\xi ,a )+O(\dot {a}^{2}), \end{equation}

where $H=h/(3\lambda )$ and $\xi = (a-r )/{(3\lambda )}$ . The inner solution has been solved in many papers (Hocking Reference Hocking1983; Eggers Reference Eggers2005; Sibley et al. Reference Sibley, Nold and Kalliadasis2015). The asymptotic behaviour of the solution for large $\xi$ is

(3.15) \begin{equation} H_{\xi }=1+\dot {a}\ln (\mathrm {e}\xi )+O (\dot {a}^{2}). \end{equation}

Furthermore, the cube of $H_\xi$ has the form

(3.16) \begin{equation} H^{3}_{\xi }=1+3\dot {a}\ln (\mathrm {e}\xi )+O (\dot {a}^{2}), \end{equation}

which can be matched with the outer solution.

Rewriting the inner solution (3.16) using the outer variables $h$ and $y$ , and matching to the outer solution (3.13), we obtain a relation between $a$ and $\dot {a}$ :

(3.17) \begin{equation} \left [\frac {8(1+b)}{a^{3}}\right ]^{3}-1=3\dot {a}\left [\ln \left (\frac {1+b}{2-b}\,\frac {a}{3\mathrm {e}\lambda }\right )+\frac {3b}{4-2b}\right ]. \end{equation}

Note that $b\leqslant 1$ is satisfied in the low-speed regime such that the argument of the logarithmic term remains positive. The left-hand side of (3.17) represents the cube difference between the apparent and equilibrium contact angles. To validate (3.17), we explore two special cases. First, when $\dot {a}=0$ , it is trivial to show that the contact-line position of the static solution, i.e. (3.2), is recovered. Second, considering the case $b=0$ , (3.17) reduces to

(3.18) \begin{equation} \left ( \frac {2}{a}\right )^{9}-1=3\dot {a}\ln \left (\frac {a}{6\mathrm {e} \lambda }\right ), \end{equation}

which is exactly the relation presented by Hocking (Reference Hocking1983) for drop spreading without centrifugal and gravitational forces.

Figure 5.

Variation of (a) the contact-line radius $a$ and (b) the contact-line velocity $\dot {a}$ as functions of time. The solid and dashed curves represent the numerical solutions of (2.3) and the asymptotic solution predicted by (3.17), respectively.

The asymptotic solution of $a$ and $\dot {a}$ can be calculated conveniently by solving (3.17), starting from the initial value $a=2$ . The results for representative values of $\beta$ are shown in figure 5, where the numerical solutions of the lubrication equation (2.3) are also plotted for comparison. It can be seen that the asymptotic theory is in perfect agreement with the numerical results when $\beta \lt 3/4$ , since the underlying assumption of the asymptotic analysis, i.e. $\dot {a}\ll 1$ , does hold according to these results. We also observe that drops at larger values of $\beta$ move faster, but take longer to reach equilibrium.

Furthermore, it is noteworthy that the derivation of (3.17) does not rely on the condition $\beta \leqslant \beta _c$ . When $\beta$ slightly exceeds $3/4$ , the early stage of spreading is also quasi-static and (3.17) remains valid, as confirmed in figure 5 for $\beta =0.8$ and 1. The predictions of the quasi-static analysis deviate from the exact solutions for large $t$ due to the formation of thin films around the axis, as discussed later.

3.3.1. Flat central film

Figures 2(b,c) demonstrate that the film in the central zone flattens rapidly upon the application of rotation. Given the near-uniform thickness of the film, the curvature and hence the capillary force can be safely neglected in this region. Consequently, the flow is primarily governed by the balance between viscous and centrifugal forces. Eliminating the surface tension term and the slip effect, the lubrication equation (2.3) becomes

(3.19) \begin{equation} h_{\tau }+\frac {1}{r}\left (r^{2}h^{3}\right )_{r}=0. \end{equation}

We have introduced a rescaled time $\tau =\beta t$ , which scales out the surface tension according to (2.2) and (2.4). Emslie et al. (Reference Emslie, Bonner and Peck1958) and Schwartz & Roy (Reference Schwartz and Roy2004) solved (3.19) under initial conditions of a general form and a uniform film, respectively. In particular, given an initially uniform film of thickness $h_p$ , the solution is

(3.20) \begin{equation} h_{f}=\frac {h_p}{\sqrt {1+4h_p^2\tau }}, \end{equation}

which describes the thinning of a flat film with thickness $h_f$ . The long-time asymptotics of these solutions has the form

(3.21) \begin{equation} h_{f}=\tfrac {1}{2}\tau ^{-{1}/{2}}, \end{equation}

and has also been reported by Fraysse & Homsy (Reference Fraysse and Homsy1994). It is trivial to verify that (3.21) is a self-similar solution of (3.19). Clearly, as initial and boundary conditions are not involved, this solution holds for long-time evolution of the film and is independent of the film structure near the contact line. The comparison of the numerical results and the theoretical solution (3.21) is depicted in figure 6. It can be seen that the numerical results for all values of $\beta$ collapse onto the theoretical curve at large $\tau$ . According to (3.20), the range of $\tau$ over which the data approach the similarity solution depends on $h_p$ rather than $\beta$ . However, for the initial condition of a drop profile, the timescale for the collapse depends on $\beta$ ; as shown in the inset of figure 6, it takes a longer time to approach the theoretical curve for smaller $\beta$ . We emphasize that surface tension still plays a role in the early surface evolution due to the finite curvature, and can only be disregarded as the liquid film is well flattened. As shown in figure 6, deviations of the numerical results from the theoretical curve can be observed at the early stage due to the effect of surface tension, especially for small values of $\beta$ .

Figure 6.

Variation of the thickness of the central film $h_f$ as a function of the rescaled time $\tau$ . The red solid curve represents (3.21). Other curves represent the numerical solutions of (2.3) measured at the axis of symmetry. The inset shows a log–log plot.

3.3.2. Capillary ridge

An annular ridge occurs in the vicinity of the contact line such that capillary force becomes locally important. The ridge is connected to the LLD-type film (figure 2 b) or the thin flat film zone (figure 2 c), depending on the spinning rate and time stage. We first discuss the case without the LLD-type film. Figure 7(a) shows the film profiles at $\beta =2$ and four time instants when the flat film and the ridge are well developed. As the film spreads, both the thickness of the flat region and the height and width of the ridge decrease. The relative height of the ridge with respect to the thickness of the flat region actually grows with time, as shown in the inset of figure 7(a). The influence of $\beta$ on the ridge size is depicted in figure 7(b), which plots the film profiles at a given $\tau$ , corresponding to an approximately identical value of $h_f$ . As demonstrated later, the shape of the ridge is determined by the balance between capillary and centrifugal forces. With an increase in $\beta$ , the centrifugal force becomes more pronounced. Therefore, the ridge size is reduced such that the capillary force increases to balance the centrifugal force.

Figure 7.

(a) Instantaneous film profiles at $\beta =2$ . The inset shows the local structures of the capillary ridge with $h$ rescaled by $\tau ^{-1/2}$ or equivalently the thickness of the flat region. (b) The ridge profiles for different $\beta$ at $\tau =600$ .

It is believed that the capillary ridge in spin coating evolves in a quasi-steady way, similar to the two-dimensional ridge encountered in the front of a draining film on a wall (Troian et al. Reference Troian, Herbolzheimer, Safran and Joanny1989; Fraysse & Homsy Reference Fraysse and Homsy1994). The present calculations allow for more quantitative validation of this analogy. When the film is well spread out, the width of the capillary ridge $W$ is much smaller than the contact-line radius $a$ . Therefore, the local film around the ridge can be treated effectively as a two-dimensional region, in which the spatial variation of the driving centrifugal force can be neglected. Moreover, as the contact-line speed is small (see figure 10b), the evolution of the ridge is quasi-steady in a reference frame moving with the contact line. Accordingly, introducing $x=r-a$ , one gets an equation governing the film thickness in the ridge zone:

(3.22) \begin{equation} \frac {1}{\beta }h_{xxx}=\frac {a_{\tau }}{h\left (h+3\lambda \right )}-a, \end{equation}

where $a_{\tau }=\mathrm {d}a/\mathrm {d}\tau =\dot {a}/\beta$ is the rescaled contact-line velocity, and the three terms represent capillary, viscous and centrifugal forces, respectively. For $\lambda \ll 1$ , (3.22) possesses the solution of uniform thickness

(3.23) \begin{equation} h=\sqrt {\frac {a_{\tau }}{a}}. \end{equation}

It is convenient to rescale this equation by introducing

(3.24) \begin{equation} x=\left (\frac {a_{\tau }}{\beta ^{2}a^{3}}\right )^{1/6}X, \quad h=\sqrt {\frac {a_{\tau }}{a}}\,\mathcal {H},\quad \lambda =\sqrt {\frac {a_{\tau }}{a}}\,\Lambda , \end{equation}

which gives rise to

(3.25) \begin{equation} \mathcal {H}_{XXX}=\frac {1}{\mathcal {H}\left (\mathcal {H}+3\Lambda \right )}-1. \end{equation}

The boundary conditions are $\mathcal {H}(0)=0$ , $\mathcal {H}_X(0)=-1/\sqrt [3]{\beta a_{\tau }}$ and $\mathcal {H}\rightarrow 1$ as $X\rightarrow -\infty$ for $\Lambda \ll 1$ by ignoring higher-order $O(\Lambda )$ terms. Equations of similar forms have also been investigated by Tuck & Schwartz (Reference Tuck and Schwartz1990) and Spaid & Homsy (Reference Spaid and Homsy1996). A shooting method is employed to solve (3.25), with $a_\tau$ provided by the numerical solution of the full lubrication equation (2.3). As illustrated in figure 8, the quasi-steady theory agrees well with the numerical results. In particular, the asymptotically flat film of the ridge coincides with the central region and adapts to its decreasing thickness. The rescaled slip length $\Lambda$ plays only a weak role in the height of the ridge. As the film spreads, the increase of the maximum height of the ridge relative to the flat film, shown in the inset of figure 7(a), is caused by the slowing down of the contact-line speed, which results in an increase of the rescaled contact angle $1/\sqrt [3]{\beta a_{\tau }}$ .

Figure 8.

Comparison of the ridge profiles for $\beta =6$ and $\tau =600,1200,1800$ . The solid and dashed curves represent the solutions of the full lubrication equation (2.3) and the quasi-steady equation (3.25), respectively.

Figure 9.

(a) Evolution of the ridge volume for different $\beta$ . (b) Variation of the ridge volume as a function of $\beta$ at $\tau =5000$ , with the solid curve from (3.30). The inset illustrates a comparison of the ridge profiles, with the solid and dashed curves denoting the numerical solution and (3.28) at $\beta =6$ , respectively.

Furthermore, the quasi-steady model of the capillary ridge allows us to develop an approximate prediction of the macroscopic morphology of the ridge and the amount of liquid accumulated in it. At small spreading rates, the viscous flow is important only in the region of small film thicknesses, i.e. the thin flat film and the region near the contact line. Alternatively, the macroscopic shape of the ridge is determined mainly by the balance between the capillary and centrifugal forces. By neglecting the viscous term in (3.22), we have

(3.26) \begin{equation} h_{xxx}=-\beta a. \end{equation}

The thickness of the capillary ridge should vanish at the contact line. On the other side, the ridge connects to a flat film with an asymptotically vanishing thickness. Correspondingly, the boundary conditions of (3.26) have the form

(3.27) \begin{equation} h|_{x=-W}=0, \quad \frac {\partial h}{\partial x}\bigg |_{x=-W}=0, \quad h|_{x=0}=0, \quad \frac {\partial h}{\partial x}\bigg |_{x=0}=-1, \end{equation}

where the ridge width $W$ is defined as the distance from the dimple (see figure 8) to the contact line, and remains to be determined. Note that we simply impose $h_x=-1$ at $x=0$ , which is used as an additional condition to determine $W$ . This condition is based on the fact that the contact speed is small in the late stage, hence the apparent contact angle of the ridge remains close to the equilibrium contact angle, as confirmed by our numerical calculations. The solution to problem (3.26)–(3.27) can be readily obtained as

(3.28) \begin{equation} h=-\frac {\beta a}{6}x\left (x+W\right )^{2}, \quad W =\sqrt {\frac {6}{\beta a}}. \end{equation}

Correspondingly, the maximum height of the ridge is

(3.29) \begin{equation} h_{ridge} = \frac {4}{9}\sqrt {\frac {2}{3\beta a}}, \end{equation}

and the volume of the ridge can thus be calculated as

(3.30) \begin{equation} V_{ridge}=\int _{a-W}^a r h\,\mathrm {d}r\approx \int _{-W}^0 a h\,\mathrm {d}x=\frac {1}{2\beta }. \end{equation}

Interestingly, $V_{ridge}$ approaches a finite constant instead of vanishing after a long time. This scenario is confirmed by the numerical results shown in figure 9(a). Here, $V_{ridge}$ is computed by integrating from the local minimum between the flat region and the ridge to the contact line. It is seen that $V_{ridge}$ decreases with time, and varies very little in the late stages. The volume of the ridge at a large time instant $\tau =5000$ is plotted in figure 9(b) as a function of $\beta$ , and compares satisfactorily with the theoretical prediction (3.30). The agreement is due to a reasonable description of the ridge profile provided by (3.28), as illustrated in the inset of figure 9(b). Note that the capillary ridge can accommodate a significant amount of liquid at small $\beta$ .

Previous works on spin coating usually neglect the presence of the capillary ridge and assume a uniformly flat film to predict the growth of the contact radius (Emslie et al. Reference Emslie, Bonner and Peck1958; Fraysse & Homsy Reference Fraysse and Homsy1994). As a result, the volume conservation condition can be expressed as $a^2h_f=1$ , which, together with (3.21), yields the long-time asymptotics of the contact line:

(3.31) \begin{equation} a=2\tau ^{1/4},\quad a_{\tau }=\tfrac {1}{2}\tau ^{-3/4}. \end{equation}

In figure 10, we present a comparison of the theoretical and numerical solutions for different $\beta$ . The observed kinks in the curves for $\beta =0.8$ and $\beta =1$ are attributed to the appearance of LLD-type film, which will be discussed in § 3.3.3. The numerical results for $a$ are significantly below the values predicted by (3.31), especially for small values of $\beta$ . The deviation from the theoretical curve for the contact-line speed can also be observed in figure 10(b). The over-prediction of the theory was also observed in the work of Fraysse & Homsy (Reference Fraysse and Homsy1994), who reported an improved solution based on (3.20); however, there is still a remarkable deviation from their experimental results (see also Spaid & Homsy Reference Spaid and Homsy1997; Holloway et al. Reference Holloway, Habdas, Semsarillar, Burfitt and de Bruyn2007). These expressions can be improved by reconsidering the volume conservation condition, but with the volume of the capillary ridge subtracted. It turns out that the resulting radius and speed of the contact line differ from (3.31) only by a factor, and read

(3.32) \begin{equation} a=2\tau ^{1/4}\sqrt {1-\frac {1}{2\beta }},\quad a_{\tau }=\frac {1}{2}\tau ^{-3/4}\sqrt {1-\frac {1}{2\beta }}. \end{equation}

The numerical results are rescaled according to (3.32) and plotted in figure 11, which shows much better agreement between theory and calculations. The remaining difference at $\beta =0.8$ and $\beta =1$ is due to the occurrence of the LLD-type film, which is described later.

Figure 10.

(a) Position and (b) velocity of the contact line as functions of time $\tau$ for different $\beta$ . The red solid curve indicates (3.31); the other curves are numerical solutions of (2.3).

Figure 11.

Rescaled (a) position and (b) velocity of the contact line as functions of time $\tau$ for different $\beta$ . The red solid curve indicates (3.32); the other curves are numerical solutions of (2.3).

Figure 12.

Evolution of the LLD-type film from $\tau =1000$ to $\tau=5000$ with time interval $1000$ for $\beta =0.8$ .

3.3.3. LLD-type film

As mentioned earlier, the flat central film is connected to the capillary ridge through an intermediate film zone when $\beta$ is slightly larger than $\beta _c$ . The thin film between the central region and the capillary ridge is further illustrated in figure 12 for $\beta =0.8$ . Compared with the central region, this intermediate film exhibits a couple of distinct features. First, its thickness is much smaller yet non-uniform. Second, a careful examination of figure 12 shows that the film thickness in the intermediate zone decreases with $r$ rather than time $\tau$ . The film profile varies from the flat central region to the thinner film through a smooth zone at $\tau =1000$ , which subsequently becomes steeper and evolves into a moving capillary shock. Meanwhile, the capillary ridge slows down. The thinner film eventually disappears at $\tau =5000$ due to the reconnecting of the flat film and the ridge, which is also responsible for the fluctuations of the curves in figures 10(b) and 11(b). Then the whole film evolves following the principles described earlier.

The presence of the thinner film and the capillary shock is reminiscent of the film structures in dip coating (Snoeijer et al. 2006; Gao et al. Reference Gao, Li, Feng, Ding and Lu2016; Xia et al. Reference Xia, Qin and Gao2020). A thin film can be entrained by withdrawing a solid plate away from a liquid bath, as first investigated by Landau & Levich (Reference Landau and Levich1942) and Derjaguin (Reference Derjaguin1943). We anticipate that the film in the intermediate region is entrained from the travelling capillary ridge following a similar mechanism, and hence refer to it as the LLD-type film. In order to confirm this idea and to develop a theoretical prediction of the film thickness, we examine the dynamic meniscus connecting the LLD-type film and the capillary ridge. The dynamic meniscus is assumed to move with the speed of the contact line. Employing the reference frame for analysing the capillary ridge, the film profile near $x=-W$ is governed by

(3.33) \begin{equation} h_{xxx} = \frac {\dot {a}}{h^2}\left ({1-\frac {h_L}{h}} \right ), \end{equation}

where $h_L$ is the thickness of the thin film with its weak non-uniformity neglected. The meniscus is governed by the balance between viscous and capillary forces, and the centrifugal force has also been neglected. Equation (3.33) is known as the Landau–Levich equation, and can be further simplified by appropriately rescaling the variables. As shown in many works (e.g.Landau & Levich Reference Landau and Levich1942; Wilson Reference Wilson1982; Du & Gao Reference Du and Gao2022), the solution for $h\gg h_L$ has an asymptotically constant curvature

(3.34) \begin{equation} h''_{in} = 0.643\frac {\dot {a}^{2/3}}{h_L}, \end{equation}

which should be matched to an outer solution corresponding to the macroscopic geometry of the capillary ridge. According to (3.28), the curvature of the capillary ridge as $x\rightarrow -W$  is

(3.35) \begin{equation} h''_{out} = \sqrt {\frac {2}{3}\beta a}. \end{equation}

The matching of the two curvatures yields

(3.36) \begin{equation} h_{L}=0.643\times \sqrt {\frac {3}{2\beta a}}\,\dot {a}^{2/3}, \end{equation}

which is confirmed by the numerical solution of the lubrication equation in figure 13. Here, the numerical results for $h_L$ are measured at the local minimum of the film. Due to the time-dependent contact-line velocity and radius, the LLD-type film becomes progressively thinner in thickness, distinguishing it from the classical LLD film.

Figure 13.

Comparison of LLD-type film thickness with the asmptotic theory (3.36) at $\beta =1$ .

Since the LLD-type film is much thinner, it is expected that the flow rate in this zone is significantly suppressed. Consequently, the flat central film and the capillary ridge are decoupled and evolve independently, in contrast to the situations in the absence of the LLD-type film.