3.1. Initial conditions due to impact of individual printed droplets

The presence of contact-angle hysteresis means that the final resting state depends on the initial conditions, which are chosen to approximate the successive printing of individual droplets. The initial conditions that are relevant for our models are the shape and location of the new wetted area, the increase in total fluid volume, and for Model II, the distribution of the fluid within the wetted area.

Visser et al. (Reference Visser, Tagawa, Sun and Lohse2012) describe several stages of impact of a single drop on a dry substrate. Prior to impact, the air layer between falling drop and substrate becomes strongly squeezed. The drop spreads over this air layer, reaching a maximum spreading radius before making contact with the substrate at its outer edge. Splashing may occur as the drop spreads over the layer, but is typically not observed in microscale experiments (Visser et al. Reference Visser, Tagawa, Sun and Lohse2012). During spreading over the air layer, the fluid is not in contact with the substrate. Hence, the maximum spreading radius can be limited either by viscosity or surface tension, but is only weakly dependent on the substrate properties such as contact angle.

In our experiments, the Weber and Reynolds numbers at impact have values of $\mathit{We}=2{\it\rho}\,R_{f}U_{i}^{2}/{\it\sigma}=2.4$ and $\mathit{Re}=2{\it\rho}\,R_{f}U_{i}/{\it\mu}=7.7$ . The results compiled by Visser et al. (Reference Visser, Tagawa, Sun and Lohse2012) from a variety of experimental studies indicate that our impacts are towards the lower end of the range of both $\mathit{We}$ and $\mathit{Re}$ , and thus may be interpreted as slow. Within this limit, the measurements of the maximum spreading radius presented by Visser et al. saturate at approximately 1.3 times the radius of the falling drop. In our experiments, the falling radius is $R_{f}=10.3\ {\rm\mu}\text{m}$ so that the maximum spreading radius achieved during impact would be $R_{i}=1.3\,R_{f}=13.4\ {\rm\mu}\text{m}$ based on this estimate. The radius $R_{i}$ is smaller than the contact radius $R_{c}=18.0\ {\rm\mu}\text{m}$ of a spherical cap with contact angle ${\it\theta}={\it\theta}_{A}$ . This implies that a single drop has a contact angle larger than ${\it\theta}_{A}$ at impact, and so will spread axisymmetrically until it reaches radius $R_{c}$ , at which point the contact line stops moving.

The first drop in each printed line lands on dry substrate, but the impacts of the second and subsequent drops are likely to be affected by interactions with the preexisting fluid layer. Full resolution of the interactions would require significant computation and will depend on modelling assumptions used to describe the incompletely understood physics. Instead, we approximate the impact process by choosing initial conditions for spreading according to a simple scheme that imitates the printing of circular, offset drops. When a new drop is printed, its volume $V_{0}$ is distributed over a circular region of radius $R_{p}$ offset in the $x$ direction by a distance ${\rm\Delta}x$ from the previous drop. The footprint ${\rm\Omega}$ is then the union of all wetted regions and we increment the total volume accordingly. This scheme is illustrated in figure 5. Effectively, we lump the effects of drop impact into a single measurable parameter that sets the size $R_{p}$ of the circular region that each drop is spread over. In principle, the parameter could be determined from full impact simulations, but experimental measurement avoids the problem of modelling-assumption dependence.

Figure 5.

An illustration of our scheme for ‘printing’ individual droplets. The centres of the drops are located at the filled black circles, each separated by a distance ${\rm\Delta}x=2{\it\delta}R_{c}$ from the previous drop. To determine the initial conditions for evolution just after drop impact, we spread the volume of each drop over a radius $R_{p}={\it\phi}R_{c}$ . We take the wetted area to be the union of the shaded regions. We calculate the dimensionless spreading parameter ${\it\phi}$ from our side-view experiments by measuring the distance between the dashed lines, i.e. the horizontal distance between the maximum extent of the fluid and the centre of the just-impacted drop. The values of ${\it\phi}$ measured in this way from our experiments are shown in figure 6.

Figure 6.

Measurements of ${\it\phi}$ from each impact in our experiments, with (a–d) corresponding to figure 3. The error bars show the average contact-line position in the first five frames after each impact. The centre of the falling drop is at ${\it\phi}=0$ . The shaded region indicates the extent of the wetted region just before each impact; the centre of the falling drop is above the wetted region for (a,b), very close to the edge in (c), and above dry substrate in (d). The horizontal line indicates the value ${\it\phi}=0.7$ which we use in all our simulations except for the results for ${\it\phi}=1$ shown in figures 8 and 16.

We define the printing radius $R_{p}={\it\phi}R_{c}$ as the contact radius of the drop after initial impact, but before significant interaction with the sessile fluid. Detailed studies of impact dynamics when $N=2$ have shown that the spreading parameter ${\it\phi}$ can take a wide range of values (Graham et al. Reference Graham, Farhangi and Dolatabadi2012; Lee et al. Reference Lee, Kim, Chandra and Yoon2013). As our printed lines are substantially longer than in these detailed studies, we choose ${\it\phi}$ based on measurements from our experimental videos. To do so, we measure $R_{p}$ as the distance in the $x$ direction between the centre of mass of the impacting drop and the contact-line extent just after impact (these two locations are marked by dashed lines in figure 5). Figure 6 shows the resulting values for ${\it\phi}$ for each impact in the experimental sequences shown in figure 3. After the first few drops for figure 6(a,b), and for all impacts in (c), the results are clustered around ${\it\phi}=0.7$ , but in (d), the average value of ${\it\phi}$ is slightly lower. Within the margin of error in our measurements of ${\it\phi}$ , we note that the value ${\it\phi}=0.7$ is equal to the value we would obtain if we set $R_{p}=R_{i}=1.3\,R_{f}=0.74R_{c}$ based on Visser et al. (Reference Visser, Tagawa, Sun and Lohse2012). We also note that $2^{1/3}\,R_{f}\simeq 0.72R_{c}$ ; $2^{1/3}$ is the radius ratio between a hemispherical drop and a spherical drop of identical volumes. Thus, the value ${\it\phi}=0.7$ can also be interpreted as each impacting drop being reconfigured into a hemisphere immediately post-impact.

The first drop will spread axisymmetrically until it reaches the advancing contact angle, so for the first drop, we set $R_{p}=R_{c}$ , the resting radius of a single drop. For subsequent drops, however, we set $R_{p}={\it\phi}R_{c}$ . As ${\it\phi}$ is largely independent of $N$ , we choose to use ${\it\phi}=0.7$ throughout the simulations, but we will also show selected results for ${\it\phi}=1$ . We would expect ${\it\phi}$ to increase for larger values of ${\it\delta}$ because the fluid will spread until either it comes into contact with the resting fluid, giving ${\it\phi}\geqslant {\it\delta}$ , or the advancing contact angle is reached at ${\it\phi}=1$ .

3.2. Model I: Strong surface tension limit

In Model I, we assume that there is no viscous resistance to movement of fluid within the footprint ${\rm\Omega}(t)$ . The fluid arranges itself so that pressure is spatially constant. The pressure within the fluid is linked to the interface curvature ${\it\kappa}(t)$ by the Young–Laplace equation, so that $p={\it\sigma}{\it\kappa}$ , and in the thin-film approximation ${\it\kappa}=-{\rm\nabla}^{2}h$ , where ${\rm\nabla}^{2}=\partial _{xx}+\partial _{yy}$ . If both ${\rm\Omega}$ and total fluid volume $V$ are known, we can determine $h(x,y,t)$ and $p(t)$ by solving the linear equations

with boundary condition $h=0$ on the contact line.

We calculate the contact angle ${\it\theta}$ from $\tan {\it\theta}=-\boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}h$ , where $\boldsymbol{n}$ is the two-dimensional (2D) unit normal directed out of ${\rm\Omega}$ , and $h(x,y)$ is known from the solution of (3.2). Under the small-angle approximation, we use the linear expression ${\it\theta}=-\boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}h$ . We allow the contact line to spread outwards with normal velocity controlled by the spreading law (3.1). Specifically, if a material point on the contact line is at $(x,y)=\boldsymbol{R}(t)$ , then

At each time step, we use (3.3) to update the footprint ${\rm\Omega}$ , and then recalculate $h$ and ${\it\theta}$ . Andrieu et al. (Reference Andrieu, Beysens, Nikolayev and Pomeau2002) proposed a similar model for relaxation of near-spherical droplets towards equilibrium, but performed calculations only for circular contact lines. We solve the equations with a second-order accurate implicit scheme using the finite-element library oomph-lib (Heil & Hazel Reference Heil and Hazel2006). The domain is discretised via a triangular mesh that deforms as the ink spreads, and so the mesh boundary remains fitted to the contact line. We can therefore obtain results for arbitrary contact-line shapes.

Top- and side-view results from Model I are shown in figure 7. For each value of ${\it\delta}$ , the sequence features a growing primary head that connects to a long, shallow rivulet extending to the deposition point. The general behaviour of the primary head is similar to the experimental images, with the head being most pronounced for small  ${\it\delta}$ . In contrast to the experiments, however, there is no secondary bulge formation for any value of ${\it\delta}$ .

Figure 7.

Superimposed contact-line positions (solid lines) and final side-view profile (shaded) for Model I, with (a) ${\it\delta}=0.125$ , (b) ${\it\delta}=0.25$ , (c) ${\it\delta}=0.375$ and (d)  ${\it\delta}=0.5$ , matching figure 3. The dashed line indicates the contact line position at $N=N_{1}$ , where the side profile is no longer convex. For this calculation, we have used ${\it\phi}=0.7$ , infinite time between printed drops, and a receding angle of zero. Scalloping with the same wavelength as the drop spacing is visible along the edges of the rivulet in (c,d) and is more pronounced for larger ${\it\delta}$ . Similar scalloping is seen in the experimental top views shown by Soltman & Subramanian (Reference Soltman and Subramanian2008) for ${\it\delta}\approx 0.83$ and by Dalili et al. (Reference Dalili, Chandra, Mostaghimi, Fan and Simmer2014) for ${\it\delta}$ in the range 0.5–1, again more pronounced for larger ${\it\delta}$ . In this figure and also in figures 8, 11, 13 and 16, all lengths are scaled on $R_{c}$ .

For the results of Model I shown in figure 7, sufficient time is allowed between the printing of successive drops so that the contact line always comes to rest before the next drop is deposited, which renders the results independent of $U_{0}$ . More generally, in order to construct a timescale for contact-line spreading from $U_{0}$ , we use the distance $R_{c}$ and typical contact angle ${\it\theta}_{A}$ , to reach the dimensional timescale $T_{c}=R_{c}/({\it\theta}_{A}U_{0})$ . The competition between this spreading timescale and the period between successive drops $T_{p}$ is quantified by the dimensionless time parameter ${\it\tau}_{p}=T_{p}/T_{c}$ , which is approximately 1 if $U_{0}=0.01\ \text{m}\ \text{s}^{-1}\ \text{rad}^{-1}$ . Results with infinite ${\it\tau}_{p}$ and with ${\it\tau}_{p}=1$ , shown in figure 8(a,b), respectively, are virtually indistinguishable. In particular, the value of $U_{0}=0.3\ \text{m}\ \text{s}^{-1}\ \text{rad}^{-1}$ provided by the Cox–Voinov law leads to a relatively large dimensionless time ${\it\tau}_{p}=30$ between drops, which tends towards the infinite time results shown in figures 7 and 8(a). Furthermore, as shown in figure 8(c), increasing the value of ${\it\phi}$ from ${\it\phi}=0.7$ to ${\it\phi}=1.0$ leads to the formation of a wider rivulet because the pressure difference between bulk and drop is smaller and, hence, less fluid is driven into the bulk. Thus, the height difference between the primary bulge and the rivulet is reduced, but the qualitative features of the fluid footprint remain unchanged.

Figure 8.

The effect on Model I results for ${\it\delta}=0.25$ of varying the time allowed between printed drops, and of setting ${\it\phi}=1$ rather than our standard value of ${\it\phi}=0.7$ . The dimensionless printing interval ${\it\tau}_{p}$ is defined as ${\it\tau}_{p}=T_{p}{\it\theta}_{A}U_{0}/R_{c}$ .

Our final investigation of Model I is to explore the effect of a non-zero receding contact angle ${\it\theta}_{R}$ , governed by the dimensionless parameter ${\it\epsilon}\equiv {\it\theta}_{R}/{\it\theta}_{A}$ . If ${\it\epsilon}=1$ , the only equilibrium configurations are circular islands. For ${\it\epsilon}=0$ , we obtain the non-receding system, in which a very long, very shallow rivulet can form. In figure 9, we show results for ${\it\epsilon}=0.5$ , calculated using the spreading law

for the case ${\it\delta}=0.5$ , ${\it\phi}=0.7$ . We find that the deposited ink forms a sequence of identical, non-circular fluid islands, each consisting of exactly five fluid drops. These islands form because a non-zero receding angle allows very elongated rivulets to retract towards the primary head. Sufficient retraction prevents new drops from overlapping the existing fluid, resulting in the formation of a new line. If ${\it\delta}$ and ${\it\phi}$ are fixed and ${\it\epsilon}$ decreases, increasingly elongated rivulets can be sustained without retraction, so that the islands contain an increasing number of drops. It is possible that the retraction of elongated rivulets, rather than the breakup of a very long uniform line, may cause the regular disconnected islands of fluid observed at large drop spacing by Duineveld (Reference Duineveld2003) and also in our experiments, as shown in the second row of figure 2.

Figure 9.

Model I simulations with and without a receding contact line. Here ${\it\delta}=0.5$ , ${\it\phi}=0.7$ , and there are 20 drops in each row. We define ${\it\epsilon}={\it\theta}_{R}/{\it\theta}_{A}$ , and show results for (a) ${\it\epsilon}=0$ and (b) ${\it\epsilon}=0.5$ . The shape of the contact line is exactly the same in both cases for the first three drops. The fourth contour shows small differences. The contact line after drop five is shown in black in each row; these differ significantly. In the case with ${\it\epsilon}=0.5$ , the extended rivulet has retracted sufficiently after the fifth drop to prevent overlap with the next printed drop. Instead, the next printed drop forms the start of a new line. Continuing to print equally spaced drops leads to an exactly repeated sequence of islands each containing five drops.

3.3. Model II: Viscosity regulates fluid redistribution within the footprint

Secondary bulge formation requires viscous resistance to fluid redistribution within the footprint, which is the new effect incorporated in Model II. Our formulation uses a simplified description of the rivulet geometry, characterised by its width $W(x,t)$ and centreline height $H(x,t)$ , in order to avoid introducing additional slip mechanisms required for a full 2D viscous lubrication simulation, and to minimise computational cost. We use Cartesian coordinates, where the $x$ -axis is aligned with the direction of printing, the footprint lies in the $(x,y)$ plane and $z$ is directed out of the plane, as shown in figure 10. We will assume that the film is shallow and slowly varying, or equivalently that $L\gg W_{0}\gg H_{0}$ , where $L_{0}$ , $W_{0}$ and $H_{0}$ are typical lengthscales of the fluid island in the $x$ , $y$ and $z$ directions, respectively.

Figure 10.

A 3D interface represented by the parabolic height profile (3.6), characterised by its width $W(x,t)$ and centreline height $H(x,t)$ .

Under the thin film assumptions (i.e.  $H_{0}\ll W_{0}$ and $H_{0}\ll L$ ), the evolution of the film height is governed by the free-surface version of the lubrication equations:

noting that the factor of three arises because we apply boundary conditions representing no slip on the solid substrate at $z=0$ , and no tangential stress on the free surface at $z=h$ .

We write the contact-line position as $y=\pm W(x,t)$ . If $W$ varies slowly with $x$ , the parameter ${\it\lambda}=W_{0}/L$ is small. Expanding (3.5) in powers of ${\it\lambda}$ , we find at leading order that any pressure gradients in the $y$ direction will be quickly driven to equilibrium. Pressure gradients in the $x$ direction decay over a timescale larger by a factor of ${\it\lambda}^{-2}$ . On this slow timescale, the pressure is effectively independent of $y$ , and the leading-order height profile is a solution to $h_{yyy}=0$ with boundary condition $h=0$ on $y=\pm W$ . We solve these equations to yield the parabolic profile

The fluid line is thus characterised by its width $W(x,t)$ and centreline height $H(x,t)$ . An example of a 3D free surface described by (3.6) is shown in figure 10.

Figure 11.

Model I results recalculated within the geometrical description of § 3.2, which exploits a slowly varying rivulet to describe the 3D geometry by 1D equations. The values of ${\it\delta}$ in (a–d) again match those in figure 3. These results should be compared with figure 7 for a less-restrictive geometrical description with the same physical mechanisms as here. Figure 13 uses the same geometrical description, but with the addition of viscous resistance to fluid motion in the $x$ -direction; we find that viscous resistance leads to secondary bulge development.

Any pressure gradients in the $x$ direction will drive a net flow resisted by viscosity. We obtain a one-dimensional (1D) viscous transport equation to describe this process by integrating (3.5) across a line of constant $x$ , thus relating the cross-sectional area $A$ and the net volume flux $Q$ :

We now suppose that pressure is independent of $y$ , $p=P(x,t)$ , and use the parabolic height profile (3.6) to write $A$ and $Q$ in terms of $H$ and $W$ , which yields

In order to calculate $P(x,t)$ , we use the pressure along the fluid centreline:

We will allow $W$ to evolve according to the same contact-line spreading scheme as used in Model I. In terms of $W$ , at $y=\pm W$ , we have $\boldsymbol{R}=(x,\pm W(x,t))$ and $\boldsymbol{n}=(-W_{x},\pm 1)/(1+W_{x}^{2})^{1/2}$ . We substitute these expressions for $\boldsymbol{R}$ and $\boldsymbol{n}$ , along with the height profile (3.6), into (3.3) to obtain

We retain quasi-static contact-line dynamics as the spreading of $W$ is perpendicular to any pressure gradient, which would be along the length of the line. For simplicity, we do not allow spreading at the two ends of the domain, instead pinning $H=0$ and $W=0$ there. In order to allow uninhibited competition between drop arrival, viscous transport and contact-line spreading, we no longer wait for the contact line to come to rest before printing the next drop, but instead fix the printing time interval at $T_{p}=1/f$ .

Figure 12.

The predicted maximum bulge width for Model I, calculated according to the full 2D equations (solid line, full results shown in figure 7), and the 1D equations derived in § 3.3 (dashed line, full results shown in figure 11), with parameters in (a–d) matching figure 3. In both cases, we neglect viscous resistance, set ${\it\phi}=0.7$ , and allow sufficient time between successive drops for the contact line to come to rest.

The most important difference between Model II and a standard 1D lubrication model for fluid redistribution is the $2H/W^{2}$ term in (3.9), representing the curvature of the interface in the $y{-}z$ plane. Hernández-Sánchez et al. (Reference Hernández-Sánchez, Lubbers, Eddi and Snoeijer2012) used the standard 1D lubrication model to study flow in a connecting liquid bridge during coalescence of millimetre-sized droplets on a substrate, and found that the equations captured the correct similarity form. However, the $2H/W^{2}$ term is not negligible for slowly varying rivulets, as this term is typically much larger than $H_{xx}$ , and it supplies the only coupling between centreline height and contact line shape in the constant-pressure equilibrium state reached long after printing. The evolution equation (3.8) is also related to the equations for flow in a bounded rivulet on an inclined plane (Duffy & Moffatt Reference Duffy and Moffatt1995), although our system is driven by surface tension rather than gravity.

Both Model I and Model II involve descriptions of the 3D geometry of the air–fluid interface and its contact line. In Model I, we used 2D equations to describe the 3D system, with a contact line free to move in all directions. In Model II, we use 1D equations to describe the 3D system, and the contact line is pinned at the two end points of the fluid line. In order to clarify the effect of the different geometric descriptions, we recalculate Model I results using the 1D equations, but with the same physical ingredients as in § 3.2, so that the system evolves according to the spreading law (3.10), with spatially constant pressure computed according to (3.9), and the volume constraint calculated by integrating (3.6) over the length of the line. The recalculated Model I results are shown in figure 11, and a quantitative comparison of the bulge width as predicted by the two sets of equations is shown in figure 12. These results are in good agreement except that in our 1D description, we do not allow the end points of the domain to move and so the contact line can never spread behind the initial wetted region; this leads to some squashing of the first few drops in each line which widens the initial bulge. We note that the $(1+W_{x}^{2})^{1/2}$ terms in (3.10) were neglected by Duineveld (Reference Duineveld2003) in his analysis of bulge shapes. We find that retention of these terms leads to much better agreement with the fully 2D Model I results, particularly for the footprint shape of the primary bulge.

Figure 13 shows top- and side-view results from Model II, which includes viscous resistance to flow along the length of the rivulet. Comparing figures 11 and 13, we see that viscous resistance has very little effect on the contact-line shape for the first drops in each line, and does not affect the initial development of the primary bulge. However, in figure 13, a secondary bulge does eventually appear for large $N$ , and we find that the secondary bulge arises only in simulations with viscous resistance. The secondary bulge develops several drop lengths behind the rivulet tip, migrating to slightly larger $x$ as more drops are printed. The primary head continues to grow very slowly as the second bulge forms.

Drop-by-drop comparison of the experimentally observed and predicted side-view profiles, shown in figures 14 and 15, and also in the four supplementary movies, reveals that the numerical results for the full Model II, with ${\it\phi}=0.7$ and $U_{0}=0.01\ \text{m}\ \text{s}^{-1}\ \text{rad}^{-1}$ , are in good quantitative agreement with the experiments for both the primary head and secondary bulge. Figure 15 also includes results from a static calculation for the initial bulge height: a cube-root prediction obtained by gathering $N$ drops of volume $V_{0}$ into a spherical cap with contact angle ${\it\theta}_{A}=50^{\circ }$ . The Model II results are in better agreement with the experimental data than the static calculation except for small $N$ at the smallest drop separation, when the fixed end points leads to an artificially elevated drop in Model II.

Figure 13.

Superimposed contact-line positions (solid lines) and final side-view profile (shaded) for Model II, with parameters in (a–d) matching figure 3. Here ${\it\phi}=0.7$ and $U_{0}=0.01\ \text{m}\ \text{s}^{-1}\ \text{rad}^{-1}$ . These calculations include viscous resistance. As in figure 3, we indicate the contact-line position at $N_{1}$ (dashed black line), where the side profile is no longer convex and $N_{2}$ (solid black line), where the side profile first develops a second local maximum.

Figure 14.

Comparison of top- and side-view Model I (from figure 7) and Model II (from figure 13) predictions to the side-view experimental results, after the final drop in each line is printed. A drop-by-drop comparison of Model II predictions and the side-view experimental results can be seen in the supplementary movies.

Figure 15.

Maximum height of the primary head and second bulge, and minimum rivulet height, for experimental results from figure 3 (shown with error bars representing the height of a single pixel) and Model II results from figure 13 (solid lines). Parameters in (a–d) match figure 3. The (red online) dashed line shows a cube root height prediction for the primary bulge, based on arranging $N$ drops of volume $V_{0}$ into a spherical cap of angle ${\it\theta}_{A}=50^{\circ }$ .

Figure 16 shows the effect of varying ${\it\phi}$ and $U_{0}$ on the Model II results. We find that bulge formation is not qualitatively affected by the spreading parameter ${\it\phi}$ , but decreasing ${\it\phi}$ reduces the wetted area, yields taller structures, and increases fluid mobility as a result of the increased contact angle. Figure 16 also shows Model II results for $U_{0}=0.3\ \text{m}\ \text{s}^{-1}\ \text{rad}^{-1}$ , which is the least well-established of the parameters in the model. This alternative value of $U_{0}$ is based on the Cox–Voinov law, and is 30 times larger than our experimentally measured value used for figure 13. However, for both values of $U_{0}$ with ${\it\phi}=0.7$ shown in figure 16, there is evidence of secondary bulge development, and so we conclude that the qualitative results are reasonably insensitive to $U_{0}$ , at least for the relatively short printed lines in our system.

Figure 16.

A comparison of Model II results for ${\it\delta}=0.25$ with and without viscous effects, and also the effect of varying ${\it\phi}$ and $U_{0}$ . We find that viscous resistance causes fluid to accumulate in the rivulet, which enables possible formation of a secondary bulge. In all cases with viscous resistance presented here, there is a second local maximum in the height profile. Variations in ${\it\phi}$ and $U_{0}$ do not prevent fluid accumulating, but do affect the shape of the second bulge. A pictorial comparison of the Model II results with ${\it\phi}=0.7$ and $U_{0}=0.01\ \text{m}\ \text{s}^{-1}$ with the experimental results is shown in figure 14, and demonstrates good agreement for the shape of both the primary and secondary bulge with the experimental side views.

The parameters in Model II can be used to construct three timescales, corresponding to viscous resistance, contact line spreading, and the enforced printing interval $T_{p}=1.9\times 10^{-3}\ \text{s}$ . Contact line spreading is again characterised by $T_{c}=R_{c}/({\it\theta}_{A}U_{0})=2.1\times 10^{-3}\ \text{s}$ ; the comparable values of $T_{p}$ and $T_{c}$ allow meaningful time for the contact line to spread before the next drop is deposited. Viscous resistance sets the timescale for fluid redistribution within the footprint, but depends strongly on the degree of elongation. If the rivulet varies in the axial direction over length $L$ , has typical height $H$ and width $W$ , we can use (3.8) and (3.9) to estimate the viscous timescale $T_{v}\sim {\rm\mu}L^{2}W^{2}/({\it\sigma}H^{3})$ . For a single drop with small contact angle, the appropriate lengthscales are $L\sim R_{c}$ , $W\sim R_{c}$ and $H\sim {\it\theta}_{A}R_{c}$ , and so the initial viscous timescale would be $T_{v}={\it\mu}R_{c}/({\it\sigma}{\it\theta}_{A}^{3})=4\times 10^{-6}\ \text{s}$ , which is much shorter than the timescale for contact-line spreading. However, after a large number of drops, Model I predicts that nearly all of the fluid is contained in the bulge, which is connected to a long shallow rivulet, of length $L=O({\it\delta}NR_{c})$ , height $H=O({\it\theta}_{A}R_{c}N^{-1/3})$ and width $W=O(R_{c})$ for large $N$ . This yields $T_{v}\sim {\it\mu}{\it\delta}^{2}N^{3}R_{c}/({\it\sigma}{\it\theta}_{A}^{3})$ , so that the viscous timescale increases strongly for large  $N$ .

A simple estimate for the onset of secondary bulge formation could be obtained by setting $T_{v}$ equal to $T_{p}$ , so that the secondary bulge grows when fluid is deposited in the rivulet more quickly than it can be removed. According to this criterion, decreasing the fluid viscosity or allowing a greater time interval between printed drops would allow more fluid to reach the primary bulge, and thus increase the number of drops required for the onset of secondary bulge formation. However, this estimate neglects the continual evolution of the system and the sensitivities of contact-line motion.