3.3.1. Zone 1: thin film

From figure 3 we see that the thickness of the film around the north pole only varies slightly such that capillary forces can be neglected, which will be validated later, and (2.10) can be written as

(3.2)\begin{equation} \frac{\partial H}{\partial \tau}-\frac{\partial}{\partial z}\left[H^3\left(1-z^2\right)\right]=0. \end{equation}

This equation has a self-similar solution

(3.3)\begin{equation} H(z,\tau)=\eta(z)\tau^{-1/2}, \end{equation}

where $\eta (z)$ satisfies the following ordinary differential equation:

(3.4)\begin{equation} \tfrac{1}{2}\eta+\left[\eta^3\left(1-z^2\right)\right]'=0, \end{equation}

where a prime denotes the derivative with respect to $z$.

Furthermore, the exact solution of (3.4) under the constraint $|\eta '(1)|<+\infty$ is

(3.5a,b)\begin{equation} \eta(z)=\frac{[\varphi(1)-\varphi(z)]^{1/2}}{\left(1-z^2\right)^{1/3}},\quad \varphi(z)\equiv\frac{z}{3}{_2}F_1\left(\frac{1}{3},\frac{1}{2};\frac{3}{2};z^2\right), \end{equation}

where ${_2}F_1$ is the hypergeometric function. This solution is monotonic in $z$ (dashed line in figure 6). The monotonic behaviour is a manifestation of the fact that the film drained by gravity becomes thicker in the flow direction. Similar behaviours occur in falling films along a vertical wall (de Gennes, Brochard-Wyart & Quéré Reference de Gennes, Brochard-Wyart and Quéré2004). The film configuration was previously described by Lee et al. (Reference Lee, Brun, Marthelot, Balestra, Gallaire and Reis2016) and Balestra et al. (Reference Balestra, Nguyen and Gallaire2018) as

(3.6)\begin{equation} \eta(z)=\tfrac{1}{2}+\tfrac{1}{10}(1-z)+O((1-z)^2), \end{equation}

which is a truncated version of (3.5a,b) and is valid for small $1-z$, while (3.5a,b) remains precise throughout zone 1.

Figure 6.

Rescaled film thickness for $Bo=100$ at $\tau =10,100,1000$ (solid lines). The dashed and dotted lines represent (3.5a,b) and (3.6), respectively.

In particular, we have $\eta (1)=1/2$ at the north pole, corresponding to

(3.7)\begin{equation} H|_{z=1}=\tfrac{1}{2}\tau^{-1/2},\quad \tau\to+\infty. \end{equation}

This result has been previously obtained by Takagi & Huppert (Reference Takagi and Huppert2010), and is the same as the late-stage result of a drop impacting on a sphere (Bakshi et al. Reference Bakshi, Roisman and Tropea2007).

The statement that capillarity can be neglected is validated by checking the flow rate (2.12). The capillary flow rate decays as

(3.8)\begin{equation} Bo^{-1}H^3\frac{\partial}{\partial z}\left[(1-z^2)^2\frac{\partial^2 H}{\partial z^2}\right]\propto\tau^{-2}, \end{equation}

faster than the gravitational flow rate which is $H^3(1-z^2)\propto \tau ^{-3/2}$. This means that capillarity can be neglected in zone 1.

In figure 6, our theory is compared with numerical solution at three different times at the late stage. These shapes collapse well throughout this region, confirming the similarity solution (3.3). As shown in the figure, our prediction of the film configuration (3.5a,b) is more precise than that in Lee et al. (Reference Lee, Brun, Marthelot, Balestra, Gallaire and Reis2016) and Balestra et al. (Reference Balestra, Nguyen and Gallaire2018). The profile at lower position of the sphere cannot be described by the capillarity-free theory, as it serves as a reflection of capillarity to be discussed below.

3.3.2. Zone 4: pendant drop

At late times of the evolution, the film on the lower body of the sphere is quasi-static, forming a pendant drop. This zone is essentially the same as the solution of steady states without rotation, as reported by Kang et al. (Reference Kang, Nadim and Chugunova2016), but with coefficients not provided explicitly in their formula. Here we present a fully analytical solution of the pendant drop. In this region, capillary force is balanced with gravity. Since the flow within the static pendant drop can be ignored, we set $Q=0$ in (2.12) and obtain a third-order ordinary equation for $H(z)$:

(3.9)\begin{equation} Bo^{-1}\frac{\mathrm{d}}{\mathrm{d}z}\left[\left(1-z^2\right)^2\frac{\mathrm{d}^2H}{\mathrm{d}z^2}\right]-\left(1-z^2\right)=0. \end{equation}

Three boundary conditions are needed. If we use $z_{d}$ to represent the position of the dimple ring (to be determined in the following), the extremely thin film here is seen as an apparent contact line, and thus leads to

(3.10)\begin{equation} H(z_{d})=0. \end{equation}

Since the film here is connected to the film upwards, the apparent contact line should be accompanied with a vanishing apparent contact angle:

(3.11)\begin{equation} H'(z_{d})=0. \end{equation}

At the bottom of the pendant drop, we have the third condition from (2.6), which in terms of $z$ is

(3.12)\begin{equation} \left.\sqrt{1-z^2}H'(z)\right|_{z=-1}=0. \end{equation}

The solution to the problem (3.9)–(3.12) is

(3.13)\begin{equation} H=\frac{Bo}{3}\left(z_{d}\frac{z_{d}-z}{1-z_{d}}+z\ln\frac{1-z_{d}}{1-z}\right). \end{equation}

An extra condition is needed to determine $z_{d}$. Since the film thickness in zones 1, 2 and 3 tends to vanish as $\tau \to +\infty$, zone 4 will finally contain the entire liquid, and the volume conservation (2.11) becomes

(3.14)\begin{equation} \int_{-1}^{z_{d}} H\,\mathrm{d}z=2, \end{equation}

which yields that $Bo$ and $z_{d}$ (or $\theta _{d}$) are dependent on each other:

(3.15)\begin{equation} Bo=\frac{24(1-z_{d})}{(1+z_{d})^3}=\frac{24(1-\cos\theta_{d})}{(1+\cos\theta_{d})^3}. \end{equation}

It is trivial to confirm that $\mathrm {d}^2H/\mathrm {d}\theta ^2$ is positive at $z=z_{d}$ and negative at $z=-1$. Therefore, there must be an inflection point within this region.

When time approaches infinity, our numerical result in zone 4 has a limit which is exactly given by (3.13) (figure 7). Correspondingly, (3.15) predicts the final position of the dimple ring. Figure 8 shows that this prediction is fairly precise. As we see, the larger the value of $Bo$, the lower is the dimple ring. In particular, when $Bo=24$, we have $z_{d}=0$, which means the dimple ring is located on the equator of the sphere.

Figure 7.

Interface evolution for $Bo=24$. Solid lines represent solutions at $\tau =0,1,10,100,1000$. The dashed line displays our analytical solution (3.13).

Figure 8.

(a) Position of dimple (the first minimum from the bottom of the sphere) at $\tau =100$. Circles: numerical solution of (2.4); solid line: (3.15). (b) Effect of $Bo$ on the profile of the pendant drop.

3.3.3. Zone 3: dimple ring

Unlike zone 1 and zone 4, this zone is dominated by capillary and viscous forces, and gravity is neglected (validated later), similar to the ‘pinch region’ proposed by Lamstaes & Eggers (Reference Lamstaes and Eggers2017). The width across the dimple ring is much smaller than the circumference of the sphere, expressed as $|z-z_{d}|\ll 1$. Not only the film thickness but also the width of the zone is decreasing, and this decreasing is believed to be self-similar since there is no characteristic length within this region. For our problem, this decreasing can be measured by flow current. Integrating (2.10) throughout zone 1, i.e. from $z=1$ to $z=z_{d}$, and using (3.3), we obtain

(3.16)\begin{equation} Q(\tau,z_{d})=-\int_1^{z_{d}}\frac{\partial H}{\partial\tau}\,\mathrm{d}z=\frac{1}{2}\tau^{-3/2}\int_1^{z_{d}}\eta(z)\,\mathrm{d}z. \end{equation}

Here we have neglected the flow rate variation in zone 2, which will be estimated as $O(\tau ^{-8/5})$ based on (3.35a,b) and decays faster than (3.16). On the other hand, within zone 3, the function $Q(\tau ,z)$ would be

(3.17)\begin{equation} Q(\tau,z)=Bo^{-1}H^3\frac{\partial}{\partial z}\left[\left(1-z^2\right)^2\frac{\partial^2H}{\partial z^2}\right], \end{equation}

where gravity has been neglected. Since zone 3 is narrow, i.e. $z\approx z_{d}$, we have

(3.18)\begin{equation} Q(\tau,z_{d})\approx Bo^{-1}\left(1-z_{d}^2\right)^2H^3\frac{\partial^3H}{\partial z^3}, \end{equation}

where the term $\partial ^2H/\partial z^2$ has been neglected since $|z-z_{d}|\ll 1$ leads to $|\partial ^2H/\partial z^2|\ll |\partial ^3H/\partial z^3|$.

Combining (3.16) and (3.18), we obtain the well-known ‘current equation’ (valid only in zone 3):

(3.19)\begin{equation} H^3\frac{\partial^3H}{\partial z^3}=J(\tau), \end{equation}

where $J(\tau )=-C\tau ^{-3/2}$ and according to (3.15) and (3.16) $C$ is

(3.20)\begin{equation} C=\frac{12}{\left(1+z_{d}\right)^5\left(1-z_{d}\right)^2}\int_{z_{d}}^1\eta(z)\,\mathrm{d}z. \end{equation}

Different forms of $J(\tau )$ have emerged in other problems (Dupont et al. Reference Dupont, Goldstein, Kadanoff and Zhou1993; Lamstaes & Eggers Reference Lamstaes and Eggers2017; van Limbeek et al. Reference van Limbeek, Sobac, Rednikov, Colinet and Snoeijer2019).

From (3.20) we see that $C$ is always positive and only depends on $z_{d}$ (or $Bo$). As shown in figure 9, $C$ has a minimum $C_{min}\approx 1.449$ at $Bo\approx 1.644$. In addition, for small and large values of $Bo$ we have

(3.21)\begin{equation} \left. \begin{gathered} C\sim \frac{9}{16Bo},\quad Bo\to0^+,\\ C\sim 4.805\left(\frac{Bo}{48}\right)^{5/3},\quad Bo\to+\infty, \end{gathered} \right\} \end{equation}

after analysis.

Figure 9.

Relation between $C$ and $Bo$ (solid line). Dashed and dash-dotted lines represent the asymptotics (3.21).

Now we explore self-similarity of (3.19) by choosing variables

(3.22a,b)\begin{equation} \psi(\xi)=\frac{H}{\tau^{\alpha}},\quad \xi=\frac{z_{d}-z}{\tau^{\beta}}, \end{equation}

where we anticipate both $\alpha$ and $\beta$ to be negative for the local shrinking behaviour. In this way, the flow current will be

(3.23)\begin{equation} -C\tau^{-3/2}=J(\tau)=H^3\frac{\partial^3H}{\partial z^3}=-\psi^3\psi'''\tau^{-4\alpha+3\beta}. \end{equation}

The presence of similarity solution requires

(3.24)\begin{equation} -4\alpha+3\beta=-\tfrac{3}{2}, \end{equation}

and the self-similar equation becomes

(3.25)\begin{equation} \psi^3\psi'''=C. \end{equation}

The asymptotic analysis to (3.25) shows $\psi ''(\xi )$ approaches a non-negative constant when $|\xi |\to +\infty$ (Lamstaes & Eggers Reference Lamstaes and Eggers2017). Approaching zone 4 corresponds to taking the limit $\xi \to +\infty$. The slope angle is zero and the curvature is approximately constant when approaching the apparent contact line, i.e. $z\to z_{d}^-$. The Taylor expansion of (3.13) is

(3.26)\begin{equation} H|_{z\to z_{d}^-}=\frac{4(2-z_{d})}{(1+z_{d})^3(1-z_{d})}(z-z_{d})^2+O((z-z_{d})^3). \end{equation}

Matching between zone 3 and zone 4 requires

(3.27)\begin{equation} \left.\frac{\mathrm{d}^2\psi}{\mathrm{d}\xi^2}\right|_{\xi\to+\infty}\tau^{\alpha-2\beta}=\left.\frac{\mathrm{d}^2H}{\mathrm{d}z^2}\right|_{z\to z_{d}^-}. \end{equation}

This matching condition leads to the second relation between $\alpha$ and $\beta$:

(3.28)\begin{equation} \alpha-2\beta=0. \end{equation}

The result after combining (3.24) with (3.28) is

(3.29a,b)\begin{equation} \alpha=-3/5,\quad \beta=-3/10. \end{equation}

So under the definition of

(3.30a,b)\begin{equation} \psi=\frac{H}{\tau^{-3/5}},\quad \xi=\frac{z_{d}-z}{\tau^{-3/10}}, \end{equation}

the scaled interface profile $\psi (\xi )$ would be approximately unchanged through time when $\tau \to +\infty$ (figure 10). Note that the limit configuration is not thought to exist physically because a circle of contact line will finally emerge.

Figure 10.

Self-similar configuration of the dimple ring. Solid lines represent numerical solutions of (2.4) at $\tau =1,10,\ldots ,10^5$ for $Bo=24$, and the dashed line represents the numerical solution to (3.25) in the limit $\tau \to +\infty$. The arrow points in the direction of increasing $\tau$.

It can be easily inferred from (3.30a,b) that $\partial ^3H/\partial z^3\propto \tau ^{3/10}$ and diverges as $\tau \to +\infty$, corresponding to a singularity of the capillary pressure gradient; $\partial H/\partial z$ and $\partial ^2H/\partial z^2$ remain regular. In particular, we have $\partial H/\partial z\propto \tau ^{-3/10}$, indicating that the flow within the dimple can be treated as a parallel shear flow and lubrication approximation still holds. Furthermore, according to (3.30a,b), the neglected part of the flow rate generated by gravity in (2.12) is approximately $H^3(1-z_{d}^2)\propto \tau ^{-9/5}$, which is small compared to the capillary part $Bo^{-1}(1-z_{d}^2)^2H^3(\partial ^3H/\partial z^3)\propto \tau ^{-3/2}$, justifying the neglect of gravity at the beginning of this subsection.

The final shape of the dimple can be obtained by solving (3.25), which is much easier to solve than the original lubrication equation (2.4). The solution can be determined by imposing the following constraints:

(3.31)\begin{equation} \left. \begin{gathered} \psi''(+\infty)=\frac{8(2-z_{d})}{(1+z_{d})^3(1-z_{d})},\\ \psi''(-\infty)=0,\\ \psi'(0)=0. \end{gathered} \right\} \end{equation}

The first condition is due to (3.27), and the second condition will be checked at the end of § 3.3.4. The proposal of these two conditions is similar to that of earlier researchers (Lamstaes & Eggers Reference Lamstaes and Eggers2017; van Limbeek et al. Reference van Limbeek, Sobac, Rednikov, Colinet and Snoeijer2019). The remaining condition defines the position of the local minimum. Under these constraints, the solution to (3.25) for $Bo=24$ is obtained numerically and displayed as the dashed line in figure 10, where the shape of the dimple ring approaches the dashed line as $\tau$ increases.

3.3.4. Zone 2: ridge ring

Although the matching of the flow current between zone 1 and zone 3 works, we point out that the interface configuration does not match because of the different signs of $\partial H/\partial z$ in zone 1 and at the upper part of zone 3, which leads to the existence of zone 2, the ridge ring. In this narrow transition zone, all three forces are believed to be important. The position of the ridge ring moves down approaching zone 3, and the local configuration of the ridge ring keeps narrowing. According to our results, we have $z_{r}-z_{d}\to 0$ as $\tau \to +\infty$, where $z_{r}$ is the position of the peak of the ridge. Under the condition $|z-z_{d}|\ll 1$, we have

(3.32)\begin{equation} Bo^{-1}\left(1-z_{d}^2\right)^2\frac{\partial^3H}{\partial z^3}=\frac{Q(\tau;z_{d})}{H^3}+\left(1-z_{d}^2\right), \end{equation}

according to (2.12). It is confirmed that (3.32) can describe zone 1 by removing the term of capillarity (left-hand side) and zone 3 by eliminating the term of gravity (last term on the right-hand side). Similar to the ‘bump’ explored in van Limbeek et al. (Reference van Limbeek, Sobac, Rednikov, Colinet and Snoeijer2019), the slopes in zones 2 and 3 should match:

(3.33)\begin{equation} \frac{H_{r}}{z_{r}-z_{d}}\sim\left[\frac{\partial H}{\partial z}\right]_{{zone\ 3}}\propto\tau^{-3/10}, \end{equation}

where $H_{r}$ is the height of the peak of the ridge. The term $H_{r}/(z_{r}-z_{d})$ denotes the characteristic slope of the ridge, and $[\partial H/\partial z]_{{zone\ 3}}$ is the slope in zone 3 and can be estimated according to (3.30a,b).

On the other hand, all three terms in (3.32) should be of the same order in zone 2. The balance between the first and the third terms yields

(3.34)\begin{equation} \frac{H_{r}}{(z_{r}-z_{d})^3}\sim\left[\frac{\partial^3 H}{\partial z^3}\right]_{{zone\ 2}}\propto\tau^{0}. \end{equation}

Comparing (3.33) and (3.34), we obtain a self-similar scaling law of the ridge evolution (figure 11):

(3.35a,b)\begin{equation} z_{r}-z_{d}\propto\tau^{-3/20},\quad H_{r}\propto\tau^{-9/20}. \end{equation}

In this way, the curvature will be proportional to $\tau ^{-3/20}$ and diminish when time approaches infinity, which justifies the second condition in (3.31).

Figure 11.

Self-similar evolution of the ridge for $Bo=24$. Symbols represent the numerical solution of (2.4) and lines indicate the scaling law (3.35a,b).

According to (3.35a,b), the film thickness in zone 2 declines slower than in zone 1 and zone 3, which reconfirms the ridge ring emerging here. As a transition zone, the ridge ring makes it possible to match zone 1 and zone 3. The connection between the slowest and fastest declining rates in zone 2 and zone 3, respectively, forms an evident local fluctuation between the relatively smooth structures in zone 1 and zone 4.