2.1 Boundary element method (BEM)

The Stokes equation (2.1) and the continuity equation (2.2) can be formulated in the form of the boundary integral equations; a method which has been used extensively to study many interfacial flow problems (Pozrikidis Reference Pozrikidis1992). In this approach the velocity $\boldsymbol{u}(\boldsymbol{s}_{0})$ at any point $\boldsymbol{s}_{0}$ can be written in terms of integrals involving the stress $\boldsymbol{f}$ and the velocity on the boundary. For the axisymmetric Stokes flow problem we study in this article, the boundary integral equations (Pozrikidis Reference Pozrikidis1992) read

(2.9) $$\begin{eqnarray}u_{\unicode[STIX]{x1D6FC}}(\boldsymbol{s}_{0})=-\frac{A}{4\unicode[STIX]{x03C0}\unicode[STIX]{x1D702}}\int _{c}\bar{\unicode[STIX]{x1D60E}}_{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}}(\boldsymbol{s}_{0},\boldsymbol{s})f_{\unicode[STIX]{x1D6FD}}(\boldsymbol{s})\,\text{d}l(\boldsymbol{s})+\frac{A}{4\unicode[STIX]{x03C0}}\int _{c}\bar{\unicode[STIX]{x1D61B}}_{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D701}}(\boldsymbol{s}_{0},\boldsymbol{s})u_{\unicode[STIX]{x1D6FD}}(\boldsymbol{s})n_{\unicode[STIX]{x1D701}}(\boldsymbol{s})\,\text{d}l(\boldsymbol{s}),\end{eqnarray}$$

where the subscripts $\unicode[STIX]{x1D6FC}$ , $\unicode[STIX]{x1D6FD}$ and $\unicode[STIX]{x1D701}$ represent either the radial ( $r$ ) or the vertical ( $z$ ) components in cylindrical coordinates, and $c$ is the contour line (boundary) over which the integration takes place. The repeated Greek indices conform to the Einstein summation convention, that is they are summed over the radial and the vertical components. For the expression of the tensor components $\bar{\unicode[STIX]{x1D60E}}_{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}}$ and $\bar{\unicode[STIX]{x1D61B}}_{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D701}}$ , we refer to § A.1. The value of $A$ depends on the position  $\boldsymbol{s}_{0}$ :

(2.10) $$\begin{eqnarray}A=\left\{\begin{array}{@{}ll@{}}1/2 & \text{for}~\boldsymbol{s}_{0}~\text{inside the liquid domain enclosed by the boundary},\\ 1 & \text{for}~\boldsymbol{s}_{0}~\text{on the closed boundary}.\end{array}\right.\end{eqnarray}$$

We note that $\bar{\unicode[STIX]{x1D60E}}_{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}}$ and $\bar{\unicode[STIX]{x1D61B}}_{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D701}}$ are singular at $\boldsymbol{s}=\boldsymbol{s}_{0}$ ; the integral over the singular point is thus computed analytically by expanding the tensor components in series about $\boldsymbol{s}=\boldsymbol{s}_{0}$ (Lee & Leal Reference Lee and Leal1982; van Lengerich & Steen Reference van Lengerich and Steen2012).

The main advantage of the boundary element method is that the velocity field is explicitly written in terms of the velocities and the stresses on the boundary. No discretization of elements inside the droplet is required to solve for the flow fields. As given by the boundary conditions, not all the velocities and stresses at the boundary of the liquid domain are known. For example, the velocities at the free interface are unknowns, yet the unknown quantities can be found by solving (2.9) for the case that $\boldsymbol{s}_{0}$ lies on the boundary. For a numerical treatment of the problem, the contour is discretized into small linear elements. The velocity and the stress are taken to be constant within the same element. A system of linear equations is then obtained from (2.9), and the unknown quantities can be computed. Note that the normal stress for the numerical element of the liquid/air interface containing the contact line is computed with the boundary condition of the imposed equilibrium contact angle  $\unicode[STIX]{x1D703}_{e}$ . Once the velocities at the free surface have been computed, one can determine the profile evolution using the kinematic condition (2.7).

Initially, the droplet has a spherical-cap shape with a contact angle $\unicode[STIX]{x1D703}_{i}$ . Due to the small molecular relaxation time scale at the contact line, the contact angle quickly reaches the equilibrium contact angle $\unicode[STIX]{x1D703}_{e}$ microscopically (McGraw et al. Reference McGraw, Chan, Maurer, Salez, Benzaquen, Raphaël, Brinkmann and Jacobs2016). To approximate this initial microscopic contact angle in our numerical computations, we assume that at $t=0$ , there is a kink in the interface profile at the contact line position. The line connecting the first numerical marker point and the contact line makes an angle $\unicode[STIX]{x1D703}_{e}$ with the substrate. Due to this kink, the magnitude of the approximated interfacial curvature near the contact line is larger than that on the rest of the interface, thus the Laplace pressure is unbalanced and the pressure gradient initiates a flow. Hence, the contact line starts to move towards the centre. The kink then quickly relaxes to a smooth shape due to surface tension. This type of initial profile is a natural choice as it smoothly connects the boundary condition at the contact line and the initial spherical-cap shape of the droplet. Note that our model is not able to describe the physics happening in the very vicinity of the contact line at earlier times. A different approach from hydrodynamics is required to solve that specific problem, such as molecular dynamics simulations.

We non-dimensionalize the problem as follows: all lengths are rescaled by the initial maximum height of the droplet $h_{0}$ and all the times by the viscous capillary time scale $h_{0}\unicode[STIX]{x1D702}/\unicode[STIX]{x1D6FE}$ . All these dimensionless variables are denoted with a tilde. We are thus left with three independent dimensionless parameters. In the following, we consider the initial contact angle $\unicode[STIX]{x1D703}_{i}$ , the equilibrium contact angle $\unicode[STIX]{x1D703}_{e}$ and the rescaled slip length $\tilde{b}\equiv b/h_{0}$ as the control parameters. For all our numerical computations, 300 marker points are used to describe the interface profile of the droplet. The vertical separation between two marker points is approximately 0.003. For smaller separations, the profile evolution becomes unstable. We then set the smallest rescaled slip length to $\tilde{b}=0.023$ , which is about ten times the marker separation. Hence for all our computations, the rescaled slip length is varied in a range $\tilde{b}>0.023$ . We also check the conservation of volume in our numerics. The difference between the initial and the final volumes of the droplets is less than 1 %. Further demonstrations of the numerical precision of our BEM, and a comparison to analytical results, are shown in § A.2.

Figure 1.

Evolution of the droplet profiles. The initial shape of the droplet is a flat spherical cap with contact angle $\unicode[STIX]{x1D703}_{i}=10^{\circ }$ . The final equilibrium droplet has contact angle $\unicode[STIX]{x1D703}_{e}=62^{\circ }$ . One of the profiles in each figure is plotted with a dashed line to emphasize the difference between the transient shapes. (a)  $\tilde{b}=0.46$ . The time interval $\unicode[STIX]{x1D6FF}\tilde{t}$ between two neighbouring curves is 3.48. (b)  $\tilde{b}=23.2$ and $\unicode[STIX]{x1D6FF}\tilde{t}=1.28$ . Insets: the contact line position $\tilde{R}(\tilde{t})$ as a function of time  $\tilde{t}$ .

3.1 Interfacial profile evolution and non-sphericity of the profiles

Here we briefly consider the droplet geometries studied in McGraw et al. (Reference McGraw, Chan, Maurer, Salez, Benzaquen, Raphaël, Brinkmann and Jacobs2016), namely an initial spherical cap with $\unicode[STIX]{x1D703}_{i}=10^{\circ }$ and an equilibrium contact angle $\unicode[STIX]{x1D703}_{e}=62^{\circ }$ . As discussed in McGraw et al. (Reference McGraw, Chan, Maurer, Salez, Benzaquen, Raphaël, Brinkmann and Jacobs2016), the main feature of the profile evolution is the appearance, or absence, of a transient ridge, defined as the fluid region in between the contact line and the outermost inflection point of the droplet profile (i.e. $\unicode[STIX]{x2202}^{2}\tilde{h}/\unicode[STIX]{x2202}\tilde{r}^{2}|_{\tilde{r}=\tilde{r}_{inf}}=0$ ). The ridge may develop to a global bump, characterized by a maximum in the height profile at $\tilde{r}\neq 0$ . The properties of the global bump will be discussed in § 3.4. We first look at two different rescaled slip lengths, $\tilde{b}=0.46$ and 23.2, which respectively demonstrate the formation or absence of a transient ridge. The evolution of the free interface profiles $\tilde{h}(\tilde{r},\tilde{t})$ is shown in figure 1(a) for $\tilde{b}=0.46$ and in figure 1(b) for $\tilde{b}=23.2$ . The main difference between the two cases is that, for $\tilde{b}=0.46$ , the profile around the centre does not change appreciably at early times. The fluid accumulates in a rim as the contact line moves towards the centre of the droplet and forms a transient ridge. In contrast, for $\tilde{b}=23.2$ , the height of the interface profile at the centre of the droplet increases at early times due to elongational flow (McGraw et al. Reference McGraw, Chan, Maurer, Salez, Benzaquen, Raphaël, Brinkmann and Jacobs2016). No ridge is developed in this case.

Figure 2.

(a) The non-sphericity, $\unicode[STIX]{x0394}V/V$ , versus the rescaled contact line displacement $(\tilde{R}(0)-\tilde{R}(\tilde{t}))/(\tilde{R}(0)-\tilde{R}(\infty ))$ for $\unicode[STIX]{x1D703}_{e}=62^{\circ }$ and for three different slip lengths. The maxima are indicated by solid symbols. (b) The maximum of $\unicode[STIX]{x0394}V/V$ shown in (a), $\unicode[STIX]{x0394}V_{m}/V$ , as a function of slip length $\tilde{b}$ for different equilibrium contact angles $\unicode[STIX]{x1D703}_{e}$ . For both (a) and (b), the initial contact angle is $\unicode[STIX]{x1D703}_{i}=10^{\circ }$ .

The transient profiles of the droplets in both cases shown in figure 1 deviate significantly from the shape of a spherical cap. To quantify the non-sphericity of the droplet, we introduce an observable $\unicode[STIX]{x0394}V$ in the following way. For each time $\tilde{t}$ , we determine the spherical cap of profile $\tilde{z}=\tilde{S}(\tilde{r},\tilde{t};\tilde{\unicode[STIX]{x1D70C}},\tilde{S}_{0})$ that best fits the profile of the droplet; $\tilde{S}$ is given implicitly by $\tilde{\unicode[STIX]{x1D70C}}^{2}=\tilde{r}^{2}+(\tilde{S}-\tilde{S}_{0})^{2}$ , where $\tilde{S}_{0}$ is the vertical shift of the sphere centre while $\tilde{\unicode[STIX]{x1D70C}}$ is its radius of curvature. We introduce $\unicode[STIX]{x0394}V$ as the total volume of the non-overlapped region between the droplet and the corresponding spherical cap. The spherical cap is selected under the condition that the total non-overlapped volume is minimized. More precisely, we define

(3.1) $$\begin{eqnarray}\unicode[STIX]{x0394}V=\min _{\tilde{\unicode[STIX]{x1D70C}},\tilde{S}_{0}}\left(\int _{0}^{\infty }\,\text{d}\tilde{r}2\unicode[STIX]{x03C0}\tilde{r}|\tilde{h}(\tilde{r},\tilde{t})-\tilde{S}(\tilde{r},\tilde{t};\tilde{\unicode[STIX]{x1D70C}},\tilde{S}_{0})|\right),\end{eqnarray}$$

under the constraint of identical total volumes:

(3.2) $$\begin{eqnarray}V=\int _{0}^{\infty }\,\text{d}\tilde{r}2\unicode[STIX]{x03C0}\tilde{r}\tilde{h}(\tilde{r},\tilde{t})=\int _{0}^{\infty }\,\text{d}\tilde{r}2\unicode[STIX]{x03C0}\tilde{r}\tilde{S}(\tilde{r},\tilde{t};\tilde{\unicode[STIX]{x1D70C}},\tilde{S}_{0}).\end{eqnarray}$$

Note that

(3.3) $$\begin{eqnarray}\left.\begin{array}{@{}ll@{}}\tilde{h}(\tilde{r},\tilde{t})=0 & \text{for}~\tilde{r}>\tilde{R}(\tilde{t}),\\ \tilde{S}(\tilde{r},\tilde{t};\tilde{\unicode[STIX]{x1D70C}},\tilde{S}_{0})=0 & \text{for}~\tilde{r}>\tilde{R}_{cap}(\tilde{t};\tilde{\unicode[STIX]{x1D70C}},\tilde{S}_{0}),\end{array}\right\}\end{eqnarray}$$

where $\tilde{R}(\tilde{t})$ and $\tilde{R}_{cap}(\tilde{t};\tilde{\unicode[STIX]{x1D70C}},\tilde{S}_{0})$ are the contact line radius of the droplet and the spherical cap respectively.

In figure 2(a), $\unicode[STIX]{x0394}V$ rescaled by the volume of the droplet is plotted as a function of the contact line displacement $\tilde{R}(0)-\tilde{R}(\tilde{t})$ normalized by the total displacement $\tilde{R}(0)-\tilde{R}(\infty )$ for $\tilde{b}=0.023$ , 0.20 and 23.2. For all three cases, $\unicode[STIX]{x0394}V/V$ is zero at $\tilde{t}=0$ and at equilibrium because of the spherical-cap shape of the droplets. During the evolution, the non-sphericity attains a maximum. This maximal non-sphericity, $\unicode[STIX]{x0394}V_{m}/V$ , occurs at smaller contact line displacements for smaller slip lengths.

A full investigation of $\unicode[STIX]{x0394}V_{m}/V$ as a function of $\tilde{b}$ is shown in figure 2(b) for various $\unicode[STIX]{x1D703}_{e}$ . We observe that this maximal non-sphericity of the droplet evolution is non-monotonic with $\tilde{b}$ for all $\unicode[STIX]{x1D703}_{e}$ investigated. We note furthermore the presence of a well-defined maximum at a slip length that we denote by $\tilde{b}_{m}(\unicode[STIX]{x1D703}_{e})$ . For $\tilde{b}>\tilde{b}_{m}$ , $\unicode[STIX]{x0394}V_{m}/V$ decreases with $\tilde{b}$ and asymptotically saturates to a finite $\unicode[STIX]{x0394}V_{m}(\infty )/V$ . For $\tilde{b}<\tilde{b}_{m}$ , $\unicode[STIX]{x0394}V_{m}/V$ decreases with decreasing $\tilde{b}$ . As expected, the non-sphericity becomes smaller as the equilibrium contact angle $\unicode[STIX]{x1D703}_{e}$ approaches the initial contact angle  $\unicode[STIX]{x1D703}_{i}$ .

Figure 3.

After two different rescalings, the data in figure 2(b) collapse onto a single function for two different regions, namely $\tilde{b}>\tilde{b}_{m}$ as shown in (a) and for $0.023<\tilde{b}<\tilde{b}_{m}$ as shown in (c). (a)  $V_{1}$ , defined as $(\unicode[STIX]{x0394}V_{m}/V-\unicode[STIX]{x0394}V_{m}(\infty )/V)/V_{d}$ , versus $k_{1}\tilde{b}$ . (b) The dependence of $V_{d}$ and $k_{1}$ on the equilibrium contact angle $\unicode[STIX]{x1D703}_{e}$ . (c)  $V_{2}$ , defined as $\unicode[STIX]{x0394}V_{m}/V$ rescaled by $\unicode[STIX]{x0394}V_{m}(\tilde{b}_{m})/V$ , versus $(\tilde{b}/\tilde{b}_{m})^{k_{2}}$ . (d)  $\tilde{b}_{m}$ and $k_{2}$ as a function of $\unicode[STIX]{x1D703}_{e}$ .

The similar features of $\unicode[STIX]{x0394}V_{m}/V$ as a function of $\tilde{b}$ for different equilibrium contact angles $\unicode[STIX]{x1D703}_{e}$ suggest possible collapse of the curves after certain rescalings. First, we shift $\unicode[STIX]{x0394}V_{m}/V$ by $\unicode[STIX]{x0394}V_{m}(\infty )/V$ such that all the curves have the same reference level in the full-slip limit. Then we rescale the shifted $\unicode[STIX]{x0394}V_{m}/V$ by $V_{d}\equiv \unicode[STIX]{x0394}V_{m}(\tilde{b}_{m})/V-\unicode[STIX]{x0394}V_{m}(\infty )/V$ . We hence introduce a rescaled quantity $V_{1}(\tilde{b})$ as the following:

(3.4) $$\begin{eqnarray}V_{1}(\tilde{b})\equiv \frac{\unicode[STIX]{x0394}V_{m}(\tilde{b})-\unicode[STIX]{x0394}V_{m}(\infty )}{\unicode[STIX]{x0394}V_{m}(\tilde{b}_{m})-\unicode[STIX]{x0394}V_{m}(\infty )}.\end{eqnarray}$$

The maximum of $V_{1}$ is unity for any equilibrium contact angles. When plotting $V_{1}$ versus $\tilde{b}$ multiplied by a scaling factor $k_{1}(\unicode[STIX]{x1D703}_{e})$ in figure 3(a), we observe that the curves for different $\unicode[STIX]{x1D703}_{e}$ collapse into a single function for $\tilde{b}>\tilde{b}_{m}$ . The dependence of $V_{d}$ and $k_{1}$ on $\unicode[STIX]{x1D703}_{e}$ is shown in figure 3(b). Note that $k_{1}$ is not unique. The effect of $k_{1}$ is shifting the curve horizontally. Multiplying $k_{1}$ by an arbitrary factor will still collapse all the curves. Here we take $k_{1}=1$ for $\unicode[STIX]{x1D703}_{e}=17^{\circ }$ , which is the smallest equilibrium contact angle considered here.

For $0.023<\tilde{b}<\tilde{b}_{m}$ , a different rescaling is required to reach a collapse of the curves. In figure 3(c), $V_{2}$ , defined as $\unicode[STIX]{x0394}V_{m}/V$ rescaled by $\unicode[STIX]{x0394}V_{m}(\tilde{b}_{m})/V$ , is plotted as a function of $(\tilde{b}/\tilde{b}_{m})^{k_{2}}$ ; a single curve is thus obtained for $0.023<\tilde{b}<\tilde{b}_{m}$ . This rescaling suggests a relation of the form $V_{2}/k_{2}\sim \log (\tilde{b}/\tilde{b}_{m})$ . Such a logarithmic relation is reminiscent of the weak-slip models for non-equilibrium droplets (de Gennes Reference de Gennes1985; Cox Reference Cox1986), in which the contact line dynamics and the interface profile also depend on the slip length logarithmically. The dependence of $\tilde{b}_{m}$ and $k_{2}$ on $\unicode[STIX]{x1D703}_{e}$ is shown in figure 3(d). The different rescalings for $\tilde{b}<\tilde{b}_{m}$ and $\tilde{b}>\tilde{b}_{m}$ indicate the existence of different regimes of the droplet retraction dynamics. The details will be discussed in the following sections.

3.2 The transient ridge and early time dynamics

Figure 4.

(a) Rescaled contact line displacement $(\tilde{R}_{0}-\tilde{R}(\tilde{t}))/(\tilde{R}_{0}-\tilde{R}_{\infty })$ as a function of $\tilde{t}$ in log–log scale for different $\tilde{b}$ and fixed $\unicode[STIX]{x1D703}_{e}=62^{\circ }$ . The early time data for intermediate slip lengths around $\tilde{b}_{m}$ can be described by a power law. A single power law becomes less pronounced for large and small slip lengths away from $\tilde{b}_{m}$ . (b–d) Rescaled profiles for $\unicode[STIX]{x1D703}_{e}=62^{\circ }$ in (b), $\unicode[STIX]{x1D703}_{e}=40^{\circ }$ in (c) and $\unicode[STIX]{x1D703}_{e}=23^{\circ }$ in (d); $\tilde{b}=0.46$ in all cases.

As the initial driving force is at the contact line, it is important to examine the growth of the rim once the contact line has started to move. We first look at the motion of the contact line. To resolve the contact line motion for early times, we investigate the rescaled contact line displacement ${\mathcal{R}}(\tilde{t})\equiv (\tilde{R}(0)-\tilde{R}(\tilde{t}))/(\tilde{R}(0)-\tilde{R}(\infty ))$ . For given $\unicode[STIX]{x1D703}_{i}=10^{\circ }$ and $\unicode[STIX]{x1D703}_{e}=62^{\circ }$ , ${\mathcal{R}}(\tilde{t})$ as a function of time is plotted in figure 4(a) in log–log scale for different $\tilde{b}$ . For large $\tilde{b}$ , the slope of the curves decreases with time. A power law is observed for intermediate slip lengths in the vicinity of $\tilde{b}_{m}$ corresponding to the maximal non-sphericity, $\unicode[STIX]{x0394}V_{m}$ . We recall that $\tilde{b}_{m}=0.21$ for $\unicode[STIX]{x1D703}_{e}=62^{\circ }$ . For example, for $\tilde{b}=0.46$ and $0.12<\tilde{t}<30$ , the relation, i.e. ${\mathcal{R}}\sim \tilde{t}^{\unicode[STIX]{x1D6FD}}$ , describes the data with $\unicode[STIX]{x1D6FD}=0.59$ . The power-law relation becomes less pronounced when decreasing $\tilde{b}$ for $\tilde{b}<\tilde{b}_{m}$ . For $\tilde{b}=0.023$ , the curve is seen to bend upward with time. Similar features are also observed for other equilibrium contact angles. The exponent $\unicode[STIX]{x1D6FD}$ is found to be 0.57 for $\unicode[STIX]{x1D703}_{e}=40^{\circ }$ and 0.54 for $\unicode[STIX]{x1D703}_{e}=23^{\circ }$ , when $\tilde{b}=0.46$ .

Given these power-law relations for slip lengths around $\tilde{b}_{m}$ , it is instructive to investigate whether the interface profiles near the contact line can be described by a similarity solution. Assuming a similarity solution of the form $\tilde{h}=\tilde{t}^{\unicode[STIX]{x1D6FC}}f((\tilde{R}(\tilde{t})-\tilde{r})/\tilde{t}^{\unicode[STIX]{x1D6FC}})$ , we found that the rescaled profiles in the rim region collapse best for values of $\unicode[STIX]{x1D6FC}=0.64$ for $\unicode[STIX]{x1D703}_{e}=62^{\circ }$ , 0.59 for $\unicode[STIX]{x1D703}_{e}=40^{\circ }$ and 0.49 for $\unicode[STIX]{x1D703}_{e}=23^{\circ }$ . The corresponding rescaled profiles are shown in figure 4(b–d). Note that these exponents are slightly different from the exponent $\unicode[STIX]{x1D6FD}$ for the contact line. The exponent 0.49 for the case of $\unicode[STIX]{x1D703}_{e}=23^{\circ }$ , in which the interfacial slope is small, is close to the $\unicode[STIX]{x1D6FC}=1/2$ scaling predicted from the lubrication calculation for intermediate slip, namely when the dissipation is dominated by the friction at the substrate (McGraw et al. Reference McGraw, Chan, Maurer, Salez, Benzaquen, Raphaël, Brinkmann and Jacobs2016).

3.3 Physical explanation for the behaviour of the non-sphericity: flow structures and the spreading of the ridge

In this section, we rationalize the non-monotonic behaviour of the non-sphericity in terms of the flow structure and the spreading of the ridge.

We compute the velocity field inside the droplet using the boundary integral equation (2.9) with $A=1/2$ . The flow fields inside the droplet when the contact line position $\tilde{R}=10.82$ are shown in figure 5 for three different rescaled slip lengths $\tilde{b}=0.023$ , 0.21 and 23.2, and with the same equilibrium contact angle $\unicode[STIX]{x1D703}_{e}=62^{\circ }$ . For $\tilde{b}\gg \tilde{b}_{m}$ , low friction at the substrate promotes an elongational flow which affects the whole droplet in a very short time, see figure 5 for $\tilde{b}=23.2$ . Therefore, the central height of the droplet increases even at early times due to the upward flow in the centre. This prevents mass accumulation at the edge of the droplet. When decreasing $\tilde{b}$ , the elongational flow becomes less dominant. The flow is concentrated in the rim, see figure 5 for $\tilde{b}=0.023$ and 0.21. Mass is thus accumulated in the rim while the contact line is moving towards the droplet centre. As a consequence, a pronounced transient ridge is observed and the non-sphericity, $\unicode[STIX]{x0394}V_{m}/V$ , becomes larger when decreasing $\tilde{b}$ for $\tilde{b}>\tilde{b}_{m}$ .

Figure 5.

The flow fields for three different rescaled slip lengths $\tilde{b}=$ 0.023, 0.21 and 23.2. The initial contact angle is $\unicode[STIX]{x1D703}_{i}=10^{\circ }$ and the equilibrium contact angle is $\unicode[STIX]{x1D703}_{e}=62^{\circ }$ . The contact line position is $\tilde{R}=10.82$ .

For $\tilde{b}$ close to $\tilde{b}_{m}$ , the ridge profiles in the early times can be described by similarity solutions as shown in § 3.2. For small equilibrium contact angles, the similarity solutions can be obtained from the intermediate-slip lubrication model in which the dissipation by the friction at the substrate becomes dominant (McGraw et al. Reference McGraw, Chan, Maurer, Salez, Benzaquen, Raphaël, Brinkmann and Jacobs2016).

When further decreasing $\tilde{b}$ from $\tilde{b}_{m}$ , the non-sphericity becomes less pronounced. For those small $\tilde{b}$ cases, the flow is more confined to the contact line region and presents a vertical parabolic profile associated with strong shear dissipation, see figure 5 for $\tilde{b}=0.023$ . In addition, similarity solutions cannot describe the early ridge profiles anymore. The question of how much mass is accumulated at the ridge depends on the contact line speed and how fast the mass is redistributed to the central part of the droplet by shear flow. This type of mass redistribution can be observed from the spreading of the ridge. One can imagine a situation when a contact line is pinned from a certain moment, the accumulated mass then has enough time to redistribute to the central part of the droplet and the development of a pronounced global ridge is avoided. Along this line of reasoning, we can understand the decrease of $\unicode[STIX]{x0394}V_{m}/V$ with decreasing $\tilde{b}$ . The characteristic speed of the contact line decreases logarithmically with decreasing $\tilde{b}$ for small $\tilde{b}$ (McGraw et al. Reference McGraw, Chan, Maurer, Salez, Benzaquen, Raphaël, Brinkmann and Jacobs2016), which means that the disturbance at the contact line will have more time to spread for smaller $\tilde{b}$ . This result is demonstrated in figure 6(a,b) for the case of $\unicode[STIX]{x1D703}_{e}=62^{\circ }$ . In figure 6(a), several interface profiles are shown at the same contact line position for $\tilde{b}=0.023$ and 0.14. For both cases, the slip lengths are smaller than $\tilde{b}_{m}$ , so the shear dissipation dominates over the elongational one. One clearly sees that the ridge spreads wider for the smaller slip length, namely $\tilde{b}=0.023$ .

Figure 6.

(a) Droplet profiles for $\tilde{b}=0.023$ and 0.14. The profiles are compared when they have the same contact line position. (b) The rescaled displacement of the inflection point $(\tilde{R}(0)-\tilde{r}_{inf}(\tilde{t}))/(\tilde{R}(0)-\tilde{R}(\infty ))$ versus the rescaled contact line displacement $(\tilde{R}(0)-\tilde{R}(\tilde{t}))/(\tilde{R}(0)-\tilde{R}(\infty ))$ .

From the profiles of figure 6(a), we observe an outermost inflection point where $\text{d}^{2}\tilde{h}/\text{d}\tilde{r}^{2}=0$ . The position of the inflection point $\tilde{r}_{inf}(\tilde{t})$ is used to characterize the extent of the ridge. The displacement of this inflection point $(\tilde{R}(0)-\tilde{r}_{inf}(\tilde{t}))$ normalized by $(\tilde{R}(0)-\tilde{R}(\infty ))$ is plotted as a function of the rescaled contact line displacement in figure 6(b). It is found that first, the inflection point moves faster than the contact line for both cases, and second, for the same contact line position, the inflection point displaces more for $\tilde{b}=0.023$ compared to $\tilde{b}=0.14$ . This result shows again that mass is redistributed over a wider extent for the smaller slip length, $\tilde{b}=0.023$ . Hence the non-sphericity decreases with decreasing $\tilde{b}$ . Although we are numerically limited to the smallest $\tilde{b}=0.023$ , from the trends shown in figure 2(b), we expect that $\unicode[STIX]{x0394}V_{m}/V$ diminishes in the limit of vanishing $\tilde{b}$ . Our study thus indicates a crossover from a non-quasistatic regime to a quasistatic regime when $\tilde{b}$ is small.

3.4 Characteristic of the global bump

Figure 7.

The global-bump height $h_{b}$ versus $\tilde{b}$ for $\unicode[STIX]{x1D703}_{i}=10^{\circ }$ . The equilibrium contact angle is $\unicode[STIX]{x1D703}_{e}=62^{\circ }$ in (a), $\unicode[STIX]{x1D703}_{e}=34.5^{\circ }$ or $36.2^{\circ }$ in (b). (c) Phase diagram showing the with-global-bump region and the without-global-bump region. The vertical dashed line indicates the initial contact angle $\unicode[STIX]{x1D703}_{i}=10^{\circ }$ . The horizontal dashed line indicates the smallest $\tilde{b}$ we have computed for, which is 0.023. Circles are $b^{\ast }$ as a function of $\unicode[STIX]{x1D703}_{e}$ . Squares are $\tilde{b}_{L}^{\ast }$ as a function of $\unicode[STIX]{x1D703}_{e}$ . (d) Zoom on the region where $\tilde{b}_{L}^{\ast }$ is computed. Note that $\tilde{b}$ ( $y$ -axis) is in log scale.

In this section we discuss the properties of the global bump, which reflects a global feature of the droplet profile. Understanding of this feature might be useful for droplet manipulations in micro- and nanofluidics. One can characterize the size of the global bump by measuring the difference between the maximum height of the profile and the central height of the droplet, which we refer to as the global-bump height. Like the non-sphericity $\unicode[STIX]{x0394}V_{m}/V$ , the global-bump height attains a maximum value, denoted as $h_{b}$ , throughout the profile evolution. A typical behaviour of $h_{b}$ as a function of slip length $\tilde{b}$ is shown in figure 7(a) for $\unicode[STIX]{x1D703}=62^{\circ }$ . The behaviours of $\unicode[STIX]{x0394}V_{m}/V$ and $h_{b}$ are similar. The maximum bump height $h_{b}$ is a non-monotonic function of the slip length and the maximum of $h_{b}$ occurs at almost the same $\tilde{b}$ as for $\unicode[STIX]{x0394}V_{m}/V$ . This indicates that the behaviour of both quantities have the same physical origin, namely the change of the flow structure when varying the slip length as discussed in § 3.3. For $\tilde{b}$ larger than the point of the maximum, we define the slip length at which $h_{b}$ goes to zero as $\tilde{b}\equiv \tilde{b}^{\ast }$ , which equals to 2.81 for the case of $\unicode[STIX]{x1D703}_{e}=62^{\circ }$ . No transient global bump is observed for $\tilde{b}>\tilde{b}^{\ast }$ .

Accessing more values of the equilibrium contact angle, we find an additional transition when $\tilde{b}$ is further decreased from $\tilde{b}^{\ast }$ . For example, for $\unicode[STIX]{x1D703}_{e}=34.5^{\circ }$ , $\tilde{b}^{\ast }=0.79$ , we observe a transition from ‘with global bump’ to ‘without global bump’ at a certain $\tilde{b}$ , which is denoted as $\tilde{b}_{L}^{\ast }$ here, and equals 0.050 in this case. This result means a transient global bump exists only when $\tilde{b}_{L}^{\ast }<\tilde{b}<\tilde{b}^{\ast }$ . This interesting behaviour can be observed clearly in figure 7(b) where the bump height $h_{b}$ is plotted as a function of  $\tilde{b}$ . To summarize the results, a phase diagram is plotted in figure 7(c,d) that indicates whether a global transient bump can be observed or not, for the specific case of $\unicode[STIX]{x1D703}_{i}=10^{\circ }$ . When $\unicode[STIX]{x1D703}_{e}$ is close to $\unicode[STIX]{x1D703}_{i}$ , namely $\unicode[STIX]{x1D703}_{e}<32.1^{\circ }$ , no global bump appears for any value of $\tilde{b}$ . In these cases, a ridge is observed at the early stage for small slip lengths. However, a global bump (with a profile maximum not at $\tilde{r}=0$ ) does not form because the initial and the final droplet shapes are too similar. In figure 7(d), one can observe the ‘with-global-bump’ region starts from $\unicode[STIX]{x1D703}_{e}=32.1^{\circ }$ . Although the second transition is not observed for $\unicode[STIX]{x1D703}_{e}>36.2^{\circ }$ due to numerical limitations, we expect the bump to diminish in magnitude also for very small slip lengths in this case; the decrease of $h_{b}$ in figure 7(a) for small $\tilde{b}$ supports this argument. Nevertheless, the slip length below which the global bump disappears is expected to be extremely small if the difference between the initial contact angle and the equilibrium contact angle is large. A recent study has demonstrated that a pronounced global bump exists in the dewetting of very flat droplets ( $h_{0}/R(0)\approx 0.02{-}0.1$ ) even though the slip length is very small ( $\tilde{b}\approx 10^{-5}$ ) (Edwards et al. Reference Edwards, Ledesma-Aguilar, Newton, Brown and McHale2016). In such cases, the transient global bump itself can be treated as quasistatic, as is the case in dewetting rims of thin liquid films (Redon et al. Reference Redon, Brochard-Wyart and Rondelez1991; Snoeijer & Eggers Reference Snoeijer and Eggers2010; Rivetti et al. Reference Rivetti, Salez, Benzaquen, Raphaël and Bäumchen2015).