2 Regimes and transitions in the breakup phenomenon

This article focuses on the breakup of incompressible Newtonian liquids with constant viscosity ${\it\mu}$ and density ${\it\rho}$ , which have a constant surface tension ${\it\sigma}$ at their liquid–gas interface and are surrounded by a dynamically passive gas. The breakup is considered axisymmetric with the axis of the thread along the $z$ -axis of a cylindrical coordinate system ( $r,z$ ), as shown in figure 1 for a liquid bridge geometry, with characteristic length scale $R$ . In general, the flow is governed by the Navier–Stokes equations, but when inertial forces are weak the dominant balance of viscous and capillary forces gives a characteristic speed of breakup of $U={\it\sigma}/{\it\mu}$ so that the capillary number $Ca={\it\mu}U/{\it\sigma}=1$ . Then the appropriate dimensionless number characterising breakup is the Ohnesorge number $Oh$ , which is obtained by substituting $U$ into the Reynolds number $Re={\it\rho}UR/{\it\mu}={\it\rho}{\it\sigma}R/{\it\mu}^{2}=Oh^{-2}$ . The Ohnesorge number is the square root of the ratio of a viscous length scale $\ell _{{\it\mu}}={\it\mu}^{2}/({\it\rho}{\it\sigma})$ to the characteristic scale of the system $R$ . Then $Oh^{-1}=0$ is Stokes flow and $Oh=0$ gives inviscid flow.

Figure 1. Illustration of breakup in the liquid bridge geometry using the (dimensionless) coordinate system $(r,z)$ , with the black lines showing the computational domain. The minimum radius of the thread is $r_{min}$ , which can change vertical position as the breakup proceeds if satellite drops are formed (b), and the inward normal to the free surface is $\boldsymbol{n}$ .

The scales used so far are based only on global quantities. To analyse the breakup process locally it is useful to introduce a time-dependent local Reynolds number $Re_{local}(t)={\it\rho}U_{z}L_{z}/{\it\mu}$ based on instantaneous axial scales $U_{z}(t)$ and $L_{z}(t)$ (which will be larger than, or comparable to, radial scales). The practicalities of calculating this quantity are discussed in § 5.1. This local parameter will indicate the dominant forces during a breakup event and hence should identify the flow regime.

2.1 Regimes of breakup

Near breakup much research has focused on finding similarity solutions for the different regimes, all of which are categorised in § 4 of Eggers & Villermaux (Reference Eggers and Villermaux2008). A summary of the key results is provided here, with particular attention paid to the dependence of the minimum thread radius $r_{min}$ , the maximum axial velocity $w_{max}$ and the local Reynolds number $Re_{local}$ on the time from breakup ${\it\tau}=t_{b}-t$ , where $t_{b}$ is the breakup time. Quantities have been made dimensionless with characteristic scales for lengths, velocities and time of $R$ , ${\it\sigma}/{\it\mu}$ and ${\it\mu}R/{\it\sigma}$ .

2.1.1 The inertial regime

In the ‘inertial regime’, henceforth the I-regime, where there is a balance between inertial and capillary forces, dimensional analysis (Keller & Miksis Reference Keller and Miksis1983; Brenner et al. Reference Brenner, Lister, Joseph, Nagel and Shi1997) of inviscid flow ( $Oh=0$ ) gives as ${\it\tau}\rightarrow 0$ that

(2.1a-c ) $$\begin{eqnarray}\displaystyle r_{min}=A_{I}(Oh\,{\it\tau})^{2/3},\quad Re_{local}\sim ({\it\tau}/Oh^{2})^{1/3},\quad w_{max}\sim (Oh^{2}/{\it\tau})^{1/3}, & & \displaystyle\end{eqnarray}$$

where $A_{I}$ is a constant of proportionality, given in Eggers & Villermaux (Reference Eggers and Villermaux2008) as $A_{I}\approx 0.7$ . Notably, computations performed in the inertial regime have never listed $A_{I}$ and so it will be of interest to determine this value so that (2.1) is precisely determined and becomes a predictive tool. It will be useful to note, in terms of $r_{min}$ , that $Re_{local}\sim Oh^{-1}\,r_{min}^{1/2}$ and $w_{max}\sim Oh\,r_{min}^{-1/2}$ .

A feature of the inertial regime is the ‘overturning’ of the free surface, observed in both experiments (Chen, Notz & Basaran Reference Chen, Notz and Basaran2002) and simulations in this regime (Schulkes Reference Schulkes1994), which invalidates attempts at a slender description of the thread. In Day, Hinch & Lister (Reference Day, Hinch and Lister1998) it was predicted that at breakup the free surface forms a double-cone shape with angles of $18.1^{\circ }$ and $112.8^{\circ }$ with the $z$ -axis. This phenomenon has been observed experimentally in Castrejón-Pita et al. (Reference Castrejón-Pita, Castrejón-Pita, Hinch, Lister and Hutchings2012) and computationally in Wilkes, Phillips & Basaran (Reference Wilkes, Phillips and Basaran1999).

2.1.2 The viscous regime

The solutions derived for both the viscous and viscous–inertial regimes, henceforth referred to as the V-regime and the VI-regime, were derived in a one-dimensional approximation of the Navier–Stokes equations which relies on the thread remaining ‘slender’ as the breakup is approached. For a free surface represented by $r=r(z,t)$ this means that the gradient of the free surface must remain small, $\partial r/\partial z\ll 1$ .

For the V-regime, the similarity solution for Stokes flow ( $Oh^{-1}=0$ ) was derived in Papageorgiou (Reference Papageorgiou1995a ,Reference Papageorgiou b ) and is given by

(2.2a-c ) $$\begin{eqnarray}\displaystyle r_{min}=0.0709{\it\tau},\quad Re_{local}\sim {\it\tau}^{2{\it\beta}-1}/Oh^{2},\quad w_{max}\sim {\it\tau}^{{\it\beta}-1}, & & \displaystyle\end{eqnarray}$$

where ${\it\beta}\approx 0.175$ . Recently, Eggers (Reference Eggers2012) showed that this is the only similarity solution of infinitely many possible ones which remains stable to small perturbations. The free-surface profile associated with this similarity solution remains symmetric about the pinch point and the local profile depends on an external length scale (i.e. it is not ‘universal’), so that it is a self-similar problem of the second kind (Barenblatt Reference Barenblatt1996). The persistence of the V-regime for high-viscosity liquids has been observed experimentally in McKinley & Tripathi (Reference McKinley and Tripathi2000).

2.1.3 The viscous–inertial regime

As ${\it\tau}\rightarrow 0$ , from (2.1) and (2.2) we find in the I-regime that $Re_{local}\rightarrow 0$ and in the V-regime that $Re_{local}\rightarrow \infty$ . This contradicts the initial assumptions behind their derivations (Lister & Stone Reference Lister and Stone1998). Therefore, neither the I- nor the V-regime is valid right up to breakup, and a regime where viscous and inertial effects are in balance, so that $Re_{local}\sim 1$ , must be considered. In this VI-regime, the Navier–Stokes equations are required. It was shown in Eggers (Reference Eggers1993) that for the VI-regime a ‘universal’ set of exponents exist for which

(2.3a-c ) $$\begin{eqnarray}\displaystyle r_{min}=0.0304{\it\tau},\quad Re_{local}\sim 1,\quad w_{max}=1.8Oh\,{\it\tau}^{-1/2}, & & \displaystyle\end{eqnarray}$$

with a highly asymmetric free-surface shape joining a thin thread to a much steeper ‘drop-like’ profile. It was later demonstrated in Brenner, Lister & Stone (Reference Brenner, Lister and Stone1996) that this solution is the most favourable of a countably infinite set of similarity solutions. Notably $w_{max}=\max (w)$ is the maximum velocity out of the thin thread ( $w>0$ in what follows), rather than $\text{max}|w|$ , which scales in the same way but has a pre-factor $3.1$ (Eggers Reference Eggers1993). In terms of $r_{min}$ this gives $w_{max}=0.3Oh\,r_{min}^{-1/2}$ .

2.2 Transitions between regimes

It has been established that whichever regime the breakup process starts in, it will eventually end up in the VI-regime. However, it is possible that the transition from either the V- or the I-regime will occur at such small scales that this regime is irrelevant, at least from a practical perspective. For example, experiments in Burton, Rutledge & Taborek (Reference Burton, Rutledge and Taborek2004) for millimetre-sized mercury drops show that the similarity solution in the I-regime adequately describes the breakup down to the nanoscale. Therefore, determining the transitions between the regimes is an important part of understanding the breakup process.

Phase diagrams which identify the regions of $(Oh,r_{min})$ space associated with different regimes have been constructed for the related problem of drop coalescence in Paulsen et al. (Reference Paulsen, Burton, Nagel, Appathurai, Harris and Basaran2012), Paulsen (Reference Paulsen2013) and Sprittles & Shikhmurzaev (Reference Sprittles and Shikhmurzaev2014b ) – though, intriguingly, these publications disagree on the number of regimes – whereas for the breakup phenomenon an equivalent diagram is currently lacking. Most progress in this direction has been made in Castrejón-Pita et al. (Reference Castrejón-Pita, Castrejón-Pita, Thete, Sambath, Hutchings, Hinch, Lister and Basaran2015), where sketches of two characteristic phase space trajectories were presented (in their figure 1E). Here, we provide a systematic exploration of $(Oh,r_{min})$ space in order to construct a phase diagram for breakup which is equivalent to those obtained for coalescence. To do so, the scalings proposed for the transitions between the various regimes as $r_{min},{\it\tau}\rightarrow 0$ are considered.

2.2.1 Viscous to viscous–inertial transition (V $\rightarrow$ VI)

When $Oh\gg 1$ , the V-regime adequately describes the initial stages of breakup. However, the inertial term in the Navier–Stokes equations does not remain negligible as $r_{min}\rightarrow 0$ and so a transition to the VI-regime occurs at a bridge radius $r_{min}^{V\rightarrow VI}$ . In Basaran (Reference Basaran2002) and Eggers (Reference Eggers2005), by balancing the solutions in the V- and VI-regimes, i.e. by taking $Re_{local}\sim 1$ in (2.2), it was shown that this should occur when

(2.4) $$\begin{eqnarray}\displaystyle r_{min}^{V\rightarrow VI}\sim Oh^{2/(2{\it\beta}-1)}\sim Oh^{-3.1}, & & \displaystyle\end{eqnarray}$$

However, experimental evidence in Rothert, Richter & Rehberg (Reference Rothert, Richter and Rehberg2003) appears to contradict this result, finding instead that $r_{min}^{V\rightarrow VI}$ is constant, i.e. independent of $Oh$ . Investigating these regimes computationally should provide new insight into the transition.

2.2.2 Inertial to viscous–inertial transition (I $\rightarrow$ VI)

When $Oh\ll 1$ the Euler equations accurately capture the initial stages of breakup until the local Reynolds number in (2.1) drops to $Re_{local}\sim 1$ , which occurs when ${\it\tau}\sim Oh^{2}$ so that

(2.5) $$\begin{eqnarray}\displaystyle r_{min}^{I\rightarrow VI}\sim Oh^{2}. & & \displaystyle\end{eqnarray}$$

This cross-over was first confirmed computationally in Notz, Chen & Basaran (Reference Notz, Chen and Basaran2001) and experimentally in Chen et al. (Reference Chen, Notz and Basaran2002) for the case of a water–glycerol mixture with $Oh=0.16$ , but has never been studied systematically across a range of $Oh$ .

2.2.3 The discovery of multiple regime transitions in Castrejón-Pita et al. (Reference Castrejón-Pita, Castrejón-Pita, Thete, Sambath, Hutchings, Hinch, Lister and Basaran2015)

From the predicted transitions of (2.4) and (2.5) the phase diagram may be expected to look qualitatively like figure 2(a). In this figure, to see which regimes are encountered during breakup at a given $Oh$ , one follows a vertical line (at the given $Oh$ ) from the top axis downwards (through decreasing $r_{min}$ ), e.g. see paths 1 and 2. For small $Oh$ (path 1) one has an I-regime crossing into a VI-regime when $r_{min}\sim Oh^{2}$ , whilst for large $Oh$ (path 2) one has a V-regime crossing into the VI-regime at $r_{min}\sim Oh^{-3.1}$ .

However, a recent publication by Castrejón-Pita et al. (Reference Castrejón-Pita, Castrejón-Pita, Thete, Sambath, Hutchings, Hinch, Lister and Basaran2015) shows that the picture can be more complex: the breakup can pass transiently through multiple different regimes. For example, Castrejón-Pita et al. (Reference Castrejón-Pita, Castrejón-Pita, Thete, Sambath, Hutchings, Hinch, Lister and Basaran2015) show that, counter-intuitively, at $Oh=0.23$ there is no I $\rightarrow$ VI transition but rather an I $\rightarrow$ V $\rightarrow$ VI transition, so that a new unexpected low- $Oh$ V-regime is encountered. Similarly, a high- $Oh$ I-regime is observed. This behaviour is sketched out qualitatively in a phase diagram (their figure 1E) showing how $Re_{local}$ varies as breakup is approached for different values of $Oh$ .

In this work, we will use computations to build the first fully computed and quantitatively constructed phase diagram for the breakup so that the nature of the transitions can be established. To orientate the reader over the forthcoming sections, figure 2(b) gives a preview of the computed phase diagram, in a form that is only asymptotically accurate as $r_{min}\rightarrow 0$ but serves to neatly illustrate the computed flow transitions. The result shows clearly how the new phase diagram differs from the expected behaviour (figure 2 a) due to the appearance of the low- $Oh$ V-regime sandwiched between the I- and VI-regimes, so that path 1 now first encounters an I $\rightarrow$ V transition. In contrast to Castrejón-Pita et al. (Reference Castrejón-Pita, Castrejón-Pita, Thete, Sambath, Hutchings, Hinch, Lister and Basaran2015), no evidence for the high- $Oh$ I-regime can be seen. These features will be considered in further detail over the forthcoming sections.

In §§ 7–9 we will show how the phase diagram in figure 2(b) was actually constructed, but before doing so we must specify the problem considered.

Figure 2. Phase diagrams of Ohnesorge number $Oh$ against minimum bridge radius $r_{min}$ : (a) expected phase diagram; (b) computed phase diagram. The paths of typical breakup events are given by arrowed lines 1 and 2, which show how different regimes are encountered as $r_{min}$ decreases. At higher $Oh$ (path 2) a V $\rightarrow$ VI transition is both expected and computed, whilst at lower $Oh$ (path 1), a single I $\rightarrow$ VI transition was expected but a more complex behaviour was computed, with a low- $Oh$ V-regime encountered before the VI-regime, so that I $\rightarrow$ V $\rightarrow$ VI transitions are found, as discovered in Castrejón-Pita et al. (Reference Castrejón-Pita, Castrejón-Pita, Thete, Sambath, Hutchings, Hinch, Lister and Basaran2015). The dashed lines show the scaling of the transitions between the different regimes as $r_{min}\rightarrow 0$ , with I $\rightarrow$ V and V $\rightarrow$ VI transitions scaling as $r_{min}\sim Oh^{2}$ and V $\rightarrow$ VI occurring when $r_{min}\sim Oh^{-3.1}$ .

3 Problem formulation

A liquid bridge geometry (figure 1) is used with the liquid trapped between two stationary solid discs of (dimensional) radius $R$ a distance $2R$ apart and surrounded by a dynamically passive gas. This set-up allows us to isolate the breakup dynamics, limit the elongation of the domain (e.g. in contrast to dripping phenomena) and retain an experimentally realisable set-up. A future work will consider the effects of geometry through elongation by allowing the plates to move apart. The contact line where the liquid–gas free surface meets the solid remains pinned at the disc’s edge throughout. Assuming gravitational effects are negligible allows us to consider a plane of symmetry at $z=0$ of a cylindrical polar coordinate system $(r,z,{\it\theta})$ so that, using $R$ as a characteristic scale for lengths, the solid is located at dimensionless position $z=1$ and the free surface is pinned at $(r,z)=(1,1)$ . The extension to include gravity is not a difficult one, but it does not add significant value to a study focused on the small-scale breakup.

Using $U={\it\sigma}/{\it\mu}$ as a scale for velocity, $T=R/U$ as a scale for time and ${\it\mu}U/R$ for pressure, the (dimensionless) Navier–Stokes equations are

(3.1a,b ) $$\begin{eqnarray}\displaystyle \boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\boldsymbol{u}=0,\quad \frac{\partial \boldsymbol{u}}{\partial t}+\boldsymbol{u}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}\boldsymbol{u}=Oh^{2}\,\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\unicode[STIX]{x1D64B}, & & \displaystyle\end{eqnarray}$$

where the stress tensor is

(3.2) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D64B}=-p\unicode[STIX]{x1D644}+[\boldsymbol{{\rm\nabla}}\boldsymbol{u}+(\boldsymbol{{\rm\nabla}}\boldsymbol{u})^{\text{T}}]. & & \displaystyle\end{eqnarray}$$

Here, $\boldsymbol{u}$ and $p$ are, respectively, the velocity and pressure in the liquid, and the Ohnesorge number is $Oh={\it\mu}/\sqrt{{\it\rho}{\it\sigma}R}$ .

On the free surface, whose location $f(r,z,t)=0$ must be obtained as part of the solution, the kinematic equation

(3.3) $$\begin{eqnarray}\displaystyle \frac{\partial f}{\partial t}+\boldsymbol{u}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}f=0 & & \displaystyle\end{eqnarray}$$

is applied alongside the usual balance of fluid stresses with capillarity in the directions tangential and normal to the free surface:

(3.4a,b ) $$\begin{eqnarray}\displaystyle \boldsymbol{n}\boldsymbol{\cdot }\unicode[STIX]{x1D64B}\boldsymbol{\cdot }(\unicode[STIX]{x1D644}-\boldsymbol{n}\boldsymbol{n})=\mathbf{0},\quad \boldsymbol{n}\boldsymbol{\cdot }\unicode[STIX]{x1D64B}\boldsymbol{\cdot }\boldsymbol{n}=\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\boldsymbol{n}, & & \displaystyle\end{eqnarray}$$

where $\boldsymbol{n}$ is the inward normal, $\unicode[STIX]{x1D644}$ is the metric tensor of the coordinate system and $p$ is taken relative to the (constant) gas pressure.

At the liquid–solid boundary, no-slip and impermeability conditions are applied, $\boldsymbol{u}=0$ and the free surface is pinned at the contact line $f(1,1,t)=0$ .

4 Multiscale finite element computations

The mathematical model formulated in § 3 is for an unsteady free-surface flow influenced by the forces of inertia, viscosity and capillarity. To study this system over the entire range of $Oh$ requires computational methods.

5 Quantitatively identifying regimes

Computations are unable to resolve down to $r_{min}=0$ , i.e. the point at which the topological change occurs, so that the precise time $t=t_{b}$ of pinch-off is unknown. This can be problematic when comparing to scaling laws of the form $r_{min}=A{\it\tau}^{B}+C$ , where ${\it\tau}=t_{b}-t$ requires the time of breakup, as small changes in $t_{b}$ can have a large effect on a curve of $r_{min}$ versus ${\it\tau}$ , which is then used to determine which regime a breakup process is in. Similar conclusions were reached for coalescence in Thoroddsen et al. (Reference Thoroddsen, Takehara and Etoh2005). In the final regime one has $C=0$ , otherwise for ‘transitional’ or ‘transient’ regimes $C$ must be determined as well. In general, there is no easy method to overcome these issues without assuming a particular functional form for the breakup.

However, in the V- and VI-regimes it is possible to exploit $r_{min}$ being a linear function of ${\it\tau}$ ( $B=1$ ) so that the speed at which the minimum thread radius evolves is constant, i.e. independent of both $t_{b}$ and $C$ . Therefore, to identify these regimes the ‘speed of breakup’ given by ${\dot{r}}_{min}=\text{d}r_{min}/\text{d}t$ can be compared to the predictions of (2.2) that ${\dot{r}}_{min}=-0.071$ and (2.3) that ${\dot{r}}_{min}=-0.030$ as a function of $r_{min}$ without needing to know $t_{b}$ . This has the further advantage that $C$ does not have to be fitted when a regime is observed transiently, as in Castrejón-Pita et al. (Reference Castrejón-Pita, Castrejón-Pita, Thete, Sambath, Hutchings, Hinch, Lister and Basaran2015).

The same approach cannot be used to identify the inertial regime where $B=2/3$ , so that the speed of breakup is no longer constant. Therefore, we try a similar idea to that above and determine an $Oh$ -independent quantity which should be linear in the inertial regime. This relies on $C\approx 0$ , so that this regime must be close to breakup. Then, given that $r_{min}=A_{I}(Oh\,{\it\tau})^{2/3}$ , we introduce $l_{min}=Oh^{-1}r_{min}^{3/2}$ , which will satisfy $l_{min}=A_{I}^{3/2}{\it\tau}$ if the expected scaling holds. Differentiating this quantity with respect to time gives $\dot{l}_{min}=\text{d}l_{min}/\text{d}t=-A_{I}^{3/2}$ , i.e. a time-independent constant.

A key aspect of this work is to develop techniques to quantify when the breakup dynamics is in a certain regime (Sprittles & Shikhmurzaev Reference Sprittles and Shikhmurzaev2014b ). To do so, we must define what it means to be ‘in a regime’. We choose to define a breakup process to be in the V- or VI-regime when the speed of breakup ${\dot{r}}_{min}$ is within $0.015$ of that predicted by (2.2) or (2.3), respectively. In other words, speeds in the range ${\dot{r}}_{min}=-0.071\pm 0.015$ indicate V-regime dynamics whilst ${\dot{r}}_{min}=-0.030\pm 0.015$ for the VI-regime. In a similar way, the inertial regime is defined by points at which $\dot{l}_{min}$ is within 0.15 of $A_{I}^{3/2}$ , where we will later see that $A_{I}^{3/2}=0.5$ . Choosing smaller (larger) margins to define each regime will shrink (expand) the area of a given regime in the phase diagram but should not alter the qualitative picture. The particular values chosen avoided regimes overlapping and enabled us to satisfactorily map the phase diagram. The method for identifying regimes is illustrated in the Appendix.

Rather than assuming that the regimes fill all of phase space, we will be careful to split the entrance to and exit from a particular regime. For example, the exit from the V-regime will be labelled as $r_{min}^{V\rightarrow }$ and the entrance to the VI-regime as $r_{min}^{\rightarrow VI}$ , rather than assuming a priori that these coincide at $r_{min}^{V\rightarrow VI}$ .

6 Overview

In §§ 7–9, respectively, the dynamics of breakup for large ( $Oh>1$ ), intermediate ( $0.1<Oh<1$ ) and small ( $Oh<0.1$ ) Ohnesorge numbers are analysed separately and used to establish the limits of applicability of the different similarity solutions described in § 2.1. This will allow us to define the regimes of breakup on a phase diagram.

In each section, the initial focus will be on a breakup event which highlights the features of the prominent regime, namely the V-regime ( $Oh=10$ ), VI-regime ( $Oh=0.16$ ) and I-regime ( $Oh=10^{-3}$ ), with a movie of each breakup in the supplementary material. In each case, snapshots of the free-surface shape, axial velocity and pressure distributions in the breakup region will be presented. Evidence that the breakup is in a given regime will be provided by comparing the computational results with the similarity solution’s predictions for the evolution of the free-surface shape, minimum bridge radius $r_{min}$ , maximum axial velocity $w_{max}$ and scaling of the local Reynolds number $Re_{local}$ .

Figure 3. For $Oh=10$ , characteristic of the V-regime, plots show (a) evolution of the entire free surface, (b) a close-up of the breakup region, (c) the axial velocity at the free surface, $w_{fs}$ , and (d) the pressure at the free surface, $p_{fs}$ , at $r_{min}=10^{-1}$ (1, blue), $r_{min}=10^{-2}$ (2, red), $r_{min}=10^{-3}$ (3, green) and $r_{min}=10^{-4}$ (4, brown).

Having confirmed the existence of a regime, we then identify its boundaries on the phase diagram to determine the regime transitions. As explained in § 5, these boundaries are specified by calculating the values of ${\dot{r}}_{min}$ and $\dot{l}_{min}$ as a function of $r_{min}$ . Having discovered the transitional behaviour, the local measures $Re_{local}$ and slenderness $L_{r}/L_{z}$ (introduced in § 5.1) are studied in an attempt to rationalise these findings.

The analysis in §§ 7–9 will not only allow us to construct a phase diagram (cf. figure 18) for the process but also open up new questions about the breakup process; to avoid distracting from the main focus in §§ 7–9, these aspects will be considered in more detail in § 11.

7 Viscous-dominated flow (large $Oh$ )

For $Oh=10$ free-surface profiles in figure 3(a,b) show the development of a thin thread of liquid whose smallest minimum radius remains at $z=0$ throughout, so that no satellite drops form. The axial velocity (figure 3 c) at the free surface $w_{fs}$ , which is approximately $r$ -independent, shows a rapid drainage from the thread driven by a pressure gradient from $z=0$ (figure 3 d).

Evidence that this breakup is in the V-regime can be seen in figure 4(b), where the free-surface profile obtained at $r_{min}=10^{-4}$ is almost indistinguishable from the similarity solution for the V-regime, and figure 5(a), in which the computed axial velocity (curve 3) follows the predicted (singular) form $w_{max}\sim r_{min}^{-0.825}$ as $r_{min}\rightarrow 0$ .

Figure 4. A comparison of the computed free-surface profiles with the similarity solutions for the (a) VI-regime and (b) V-regime. Computed profiles are at the minimum radius $r_{min}=10^{-4}$ for (a) $Oh=0.16$ and (b) $Oh=10$ , typical of breakup in the VI- and V-regimes, respectively. These profiles compare well to the similarity solutions (dashed lines) in (a) with ( ${\it\xi}_{VI},{\it\phi}_{VI}$ ) provided in Eggers (Reference Eggers1993), giving $(r,z)=({\it\tau}{\it\phi}_{VI},z_{0}+Oh\,{\it\tau}^{1/2}{\it\xi}_{VI})$ with no free parameters, and in (b) with ( ${\it\xi}_{V},{\it\phi}_{V}$ ) from Papageorgiou (Reference Papageorgiou1995b ) and Eggers (Reference Eggers1997), so that $(r,z)=({\it\tau}{\it\phi}_{V},Oh^{2-2{\it\beta}}\,{\it\tau}^{{\it\beta}}\ell _{z}{\it\xi}_{V})$ with $\ell _{z}=1.85\times 10^{-3}$ . In (a), the height of the pinch point $z_{0}=0.268$ for the VI-regime is calculated using the expression in Eggers (Reference Eggers1993) for the drift of this point $z(r=r_{min})=z_{0}-1.6Oh\,{\it\tau}^{-1/2}$ (shown in figure 11 as the dashed line).

Figure 5. Curves are for 1: $Oh=10^{-3}$ , 2: $Oh=0.16$ and 3: $Oh=10$ . (a) Evolution of the maximum axial velocity $w_{max}$ as the minimum radius $r_{min}$ decreases, showing excellent agreement with the similarity solutions (dashed lines) from the I-regime, $w_{max}\sim r_{min}^{-0.5}$ , the VI-regime, $w_{max}=0.3Oh\,r_{min}^{-0.5}$ , and the V-regime, $w_{max}\sim r_{min}^{-0.825}$ . (b) The minimum bridge radius $r_{min}$ against time from breakup ${\it\tau}$ . Computations converge to the similarity solutions for the I-, VI- and V-regimes, respectively, with the lower dashed line being the similarity solution for the I-regime ((2.1) with $A_{I}=0.63$ ), the middle one for the VI-regime (2.3) and the highest one for the V-regime (2.2).

Assuming that the breakup remains in the V-regime, the data can be extrapolated to obtain the breakup time $t_{b}$ and, as a result, plot $r_{min}$ against ${\it\tau}$ in figure 5(b) (curve 3), where there is excellent agreement with the similarity solution of (2.2) for $r_{min}<10^{-2}$ .

7.1 Identification of the V-regime

When $Oh\gg 1$ the breakup is expected to start in the V-regime before transitioning into the VI-regime. Figure 6(a) shows that when $Oh=10$ (curve 5), the breakup speed ${\dot{r}}_{min}$ reaches the value predicted in the V-regime, $-0.071$ , at approximately $r_{min}=10^{-2}$ and remains there until $r_{min}=10^{-4}$ , i.e. until the end of the computation. For higher $Oh$ , curves are graphically indistinguishable from the case of $Oh=10$ , so this is the Stokes flow solution. Therefore, for $r_{min}\geqslant 10^{-4}$ the VI-regime is not observed at large $Oh$ .

Figure 6. (a) Evolution of the breakup speed ${\dot{r}}_{min}$ for 1: $Oh=1$ , 2: $Oh=1.5$ , 3: $Oh=2.5$ , 4: $Oh=5$ and 5: $Oh=10$ . The similarity solution in the V-regime ( ${\dot{r}}_{min}=-0.071$ ) is shown as a dashed line. At higher $Oh$ (e.g. curve 5) the speed remains at $-0.071$ , whilst at lower values (e.g. curve 1) there is a transition towards the VI-regime where ${\dot{r}}_{min}=-0.030$ . (b) Variation of the local Reynolds number for 1: $Oh=0.25$ , 2:  $Oh=0.5$ , 3: $Oh=1$ , 4: $Oh=2.5$ and 5: $Oh=10$ . Curves show that in the V-regime $Re_{local}$ gradually increases until a critical value is achieved, at which point the V-regime is exited. The increase of $Re_{local}$ in the V-regime is shown to follow the predicted scaling from (2.2) of $Re_{local}\sim r_{min}^{-0.65}$ (lower dashed line). The transition out of the V-regime ( $r_{min}=r_{min}^{V\rightarrow }$ marked as circles) is found to occur when $Re_{local}\approx 0.85$ (horizontal dashed line).

Figure 7 shows the region of $Oh$ – $r_{min}$ phase space where V-regime dynamics is encountered, with the circles marking the computationally determined boundaries of this regime. The procedure for identifying this regime is shown in the Appendix and, as discussed in § 5, it is based on the speed of breakup ${\dot{r}}_{min}$ remaining between $-0.071\pm 0.015$ . Next, the changes in behaviour triggering the entrance and exit from this V-regime are considered.

7.1.1 Entrance into the V-regime

The V-regime is entered ( $r_{min}^{\rightarrow V}$ ) at a constant $Oh$ -independent value $r_{min}\approx 10^{-2}$ , which suggests that this transition may occur when the thread can be considered slender, so that the assumptions behind the similarity solution (2.2) become valid. To determine whether the geometry near the breakup point is ‘slender’ or not, we plot $L_{r}/L_{z}$ in figure 8. For $Oh=10$ (curve 5), and generally for larger $Oh$ , initially $L_{r}/L_{z}\sim 1$ (no scale separation), but as $r_{min}$ shrinks the breakup region becomes slender. Defining the region to be slender when $L_{r}/L_{z}<0.1$ , it is found that for all $Oh>1$ slenderness occurs at $r_{min}\approx 10^{-2}$ , in agreement with our result for $r_{min}^{\rightarrow V}$ . Therefore, the transition into the V-regime appears to be dictated by the geometry of the thread.

Figure 7. Phase diagram showing the existence of a V-regime where there is a balance between viscous and capillary forces. The transition $r_{min}^{\rightarrow V}$ is where the breakup enters this regime and $r_{min}^{V\rightarrow }$ is where it leaves. Computational results (circles) show that the entrance is at a constant $Oh$ -independent $r_{min}^{\rightarrow V}=0.014$ (horizontal dashed line) whilst the exit follows $r_{min}^{V\rightarrow }=5.5\times 10^{-4}Oh^{-3.1}$ (lower dashed line).

Figure 8. Evolution of the slenderness of the free surface in the breakup region for 1: $Oh=10^{-3}$ , 2: $Oh=10^{-2}$ , 3: $Oh=0.14$ , 4: $Oh=0.16$ , 5: $Oh=10$ . This quantity, which must be small for the thread to be slender (as is required by the V- and VI-regimes), is defined using $L_{r}=r_{min}$ and $L_{z}=|z(r=r_{min})-z(w=w_{max})|$ .

7.1.2 Exit from the V-regime

The V-regime is left when the speed of breakup ${\dot{r}}_{min}$ diverges from $-0.071\pm 0.015$ (e.g. curve 1 in figure 6 a). Figure 7 shows that this transition follows the proposed scaling for V $\rightarrow$ VI that $r_{min}=A\,Oh^{-3.1}$ , with computations finding $A=5.5\times 10^{-4}$ .

The transition out of the V-regime ( $r_{min}^{V\rightarrow }$ ) occurs when inertial effects become non-negligible, so that (2.2) is no longer valid. Figure 6(b) shows that the increase in $Re_{local}$ follows remarkably well the scaling predicted by the similarity solution for the V-regime that $Re_{local}\sim r_{min}^{-0.65}$ . This also serves to validate our definition of $Re_{local}$ . Circles placed at $r_{min}^{V\rightarrow }$ on curves in figure 6(b) show that the transition out of the V-regime occurs when $Re_{local}\approx 0.85$ . One may expect that at this point, where $Re_{local}\sim 1$ , the VI-regime will be encountered, and this will be the focus of the next section.

8 Viscous–inertial balance (intermediate $Oh$ )

At $Oh=0.16$ in figure 9(a,b) one can see the appearance of a satellite drop centred at $z=0$ . The close-up images show how this is driven by symmetry breaking about the pinch point after $r_{min}\approx 10^{-2}$ (curve 2), as observed experimentally in Rothert, Richter & Rehberg (Reference Rothert, Richter and Rehberg2001). Consequently, two pinch points occur at a finite distance either side of the symmetry plane ( $z=0$ ) with a large pressure acting to push fluid out of the thinnest region (figure 9 d). In contrast to profiles in the I-regime (cf. § 9), the geometry near the pinch point remains slender; see curve 4 in figure 9(b) and the slenderness data in figure 8 (curve 4).

Features that indicate the breakup is occurring in the VI-regime are as follows. Figure 4(a) shows that the free-surface profile at $r_{min}=10^{-4}$ compares well with the similarity solution (dashed line) derived in Eggers (Reference Eggers1993) with no free parameters. Below the pinch point ( $z<0.25$ ) in figure 4, deviations of the computed free-surface shape from the similarity solution occur as there is no ‘far field’ in our set-up (as the plane of symmetry is at $z=0$ ). This also affects the velocity profiles (see figure 9 c), which agree with the similarity profiles in Eggers (Reference Eggers1993) above the pinch point but not below – a point that will be discussed further in § 11.2.1. However, curve 2 in figure 5(a) shows that the maximum axial velocity follows $w_{max}=0.3Oh\,r_{min}^{-0.5}$ from (2.3), again with no adjustable parameters.

Figure 9. For $Oh=0.16$ plots show (a) the entire free surface, (b) a close-up of the breakup region, (c) the axial velocity at the free surface $w_{fs}$ (with $w_{max}$ when $r_{min}=10^{-3},~10^{-4}$ shown) and (d) the pressure at the free surface $p_{fs}$ at 1: $r_{min}=10^{-1}$ (blue), 2: $r_{min}=10^{-2}$ (red), 3: $r_{min}=10^{-3}$ (green) and 4: $r_{min}=10^{-4}$ (brown).

In figure 10(a), the breakup speed is shown for $Oh>0.15$ , values whose significance will become apparent. In this range, all curves tend towards the value of $-0.030$ , indicating that breakup enters the VI-regime.

Figure 10. (a) Evolution of the speed of breakup when $Oh>Oh_{c}$ . Curves are for 1: $Oh=0.16$ , 2: $Oh=0.25$ , 3: $Oh=0.5$ . At $Oh>Oh_{c}$ curves converge towards the similarity solution in the VI-regime given by (2.3), i.e.  ${\dot{r}}_{min}=-0.030$ . (b) Different dynamics are observed for the breakup speed in the VI- and V-regimes with 1: $Oh=0.16$ , 2: $Oh=10$ . For these values of $Oh$ , the computed solutions converge in an oscillatory (1) and a monotonic (2) manner towards similarity solutions in the VI- and V-regimes, given by (2.3) and (2.2), respectively, which are the lower and upper dashed lines.

Figure 11. Evolution of the vertical position of the pinch point, i.e. minimum thread radius $z(r=r_{min})$ , indicating the formation of satellite drops once the pinch point moves away from the centre line $z(r=r_{min})>0$ . Curves are for 1: $Oh=10^{-3}$ , 2: $Oh=0.16$ , 3: $Oh=0.5$ and 4: $Oh=1$ . The dashed line shows the expression from Eggers (Reference Eggers1993) at $Oh=0.16$ for the drift of the pinch point, $z(r=r_{min})=z_{0}-1.6Oh\,{\it\tau}^{-1/2}$ with $z_{0}=0.268$ .

What is immediately striking about the curves in figure 10(a) is that they appear to oscillate around the value of  $-0.030$ , converging towards it as $r_{min}\rightarrow 0$ . To highlight this behaviour, figure 10(b) compares curves for $Oh=10$ , which remains in the V-regime, and $Oh=0.16$ , where the VI-regime is most prominent. Whilst the value of  $-0.071$ is approached monotonically from below by curve 2, the approach of curve 1 towards $-0.030$ is entirely different. The motion in the VI-regime is clearly oscillatory in the radial direction, and on closer inspection of figure 5(a) and figure 11 oscillations in axial quantities $w_{max}$ (and hence $Re_{local}$ in figure 6 b) and $z(r=r_{min})$ can also be seen. Having identified this behaviour, it can also be recognised in the plot of $r_{min}$ against ${\it\tau}$ in curve 2 of figure 5(b), with the bridge radius oscillating around and converging towards the dashed line predicted by the similarity solution (2.3). As far as we are aware, this is the first time this oscillatory behaviour has been observed and it will be discussed further in § 11.2.

Figure 12(a) focuses on the narrow range $Oh=0.14-0.16$ to highlight a sharp transition in the dynamics of breakup which occurs at a critical $Oh_{c}\approx 0.15$ . For $Oh=0.16>Oh_{c}$ (curve 2), the transition into the VI-regime occurs at $r_{min}\approx 10^{-2}$ , whilst for $Oh=0.14<Oh_{c}$ (curve 1) the transition into this regime is delayed until $r_{min}\approx 4\times 10^{-4}$ . Remarkably, we will show that this is due to the existence of a ‘low- $Oh$ V-regime’, first discovered in Castrejón-Pita et al. (Reference Castrejón-Pita, Castrejón-Pita, Thete, Sambath, Hutchings, Hinch, Lister and Basaran2015), which precedes entry into the VI-regime for $Oh<Oh_{c}$ . In figure 12(a) for $Oh=0.14$ (curve 1), it can be seen that this recently discovered regime is entered when $r_{min}^{\rightarrow V}=3\times 10^{-3}$ and exited again when $r_{min}^{V\rightarrow }=6\times 10^{-4}$ , during which period the speed of breakup is in the V-regime range of $-0.071\pm 0.015$ .

Figure 12. Curves are for 1: $Oh=0.14$ and 2: $Oh=0.16$ . (a) Changes in the evolution of the breakup speed ${\dot{r}}_{min}$ , against $r_{min}$ , around the critical point $Oh_{c}=0.15$ . Curve 1 shows the appearance of the low- $Oh$ V-regime around $r_{min}\approx 10^{-3}$ where the breakup speed dips to ${\dot{r}}_{min}\approx -0.071$ , in contrast to curve 2 where the speed immediately tends towards the value from the VI-regime of $-0.030$ . (b) The local Reynolds number $Re_{local}$ for the same values of $Oh$ shows that the low- $Oh$ V-regime coincides with a drop in $Re_{local}$ . The dashed line $Re_{local}=0.85$ is the value below which V-regime dynamics is to be expected.

The transition in flow behaviour at $Oh_{c}$ is also seen from the free-surface shapes in the vicinity of the breakup region. In figure 13(b) significant differences in the breakup geometry are shown for Ohnesorge numbers just below ( $Oh=0.14$ ) and above ( $Oh=0.16$ ) the critical value. At $r_{min}=10^{-4}$ , for $Oh=0.16$ (curve 1a) a long slender thread of length ${\approx}0.22$ connects the hemispherical volume above ( $z>0.27$ ) to a small satellite drop below ( $z<0.05$ ), whilst at $Oh=0.14$ (curve 1) the thread is much shorter at ${\approx}0.07$ . Consequently, the satellite drops formed in each case differ considerably, with $Oh=0.16$ forming a much fatter drop connected to a longer thin thread.

The free-surface profile in the low- $Oh$ V-regime (curve 2) does not have the slender symmetric shape in the breakup region that is characteristic of the conventional V-regime, which suggests that this regime is encountered in a weaker sense than the high- $Oh$ V-regime which satisfies all expected characteristics (see § 7). Using more stringent criteria for how the regimes are defined could result in this becoming a region of phase space where none of the similarity solutions are accurate; however, our definition is based on the breakup speed alone and this identifies it as a low- $Oh$ V-regime.

Figure 13. (a) Evolution of the free surface for $Oh=0.14$ , showing the development of a corner-like geometry (curve 3) while the flow is in the low- $Oh$ V-regime followed by the development of a thin thread once the VI-regime is entered (curve 4). (b) Comparison of free-surface shapes below the critical point, at $Oh=0.14$ (curves 1, 2), and above it, at $Oh=0.16$ (curves 1a, 2a), for 1, 1a: $r_{min}=10^{-4}$ and 2, 2a: $r_{min}=10^{-3}$ .

8.1 Identification of regimes

In figure 14 the VI- and low- $Oh$ V-regimes are shown on a phase diagram. The extreme change in behaviour at $Oh_{c}=0.15$ is clearly visible.

Figure 14. Phase diagram showing the low- $Oh$ V-regime and the VI-regime. The transition $r_{min}^{\rightarrow VI}$ follows the scaling $r_{min}=2.3\times 10^{-4}Oh^{-3.1}$ (curve 1) for $Oh>Oh_{c}=0.15$ . For $Oh<Oh_{c}$ the low- $Oh$ V-regime appears and is bounded by curves 2: $r_{min}=0.2Oh^{2}$ and 3: $r_{min}=10Oh^{2}$ as $r_{min}\rightarrow 0$ .

8.1.1 Entrance into the VI-regime for $Oh>Oh_{c}$

For $Oh>Oh_{c}$ , the transition into the VI-regime follows the expected scaling for V $\rightarrow$ VI of $r_{min}=B\,Oh^{-3.1}$ , with $B=2.3\times 10^{-4}$ . Curves 1–3 in figure 6(b) show that as expected $Re_{local}$ increases until it reaches ${\approx}1$ , a value characteristic of the VI-regime. The sharp changes in the gradients of these curves at the point where $Re_{local}$ is a maximum marks the beginning of the formation of satellite drops, i.e. the axial shift of the minimum bridge radius to $z(r=r_{min})>0$ , as can be seen from figure 11. This signals the appearance of satellite drops and entry into the VI-regime. Although $L_{r}/L_{z}$ increases at this point (figure 8), the thread remains slender. Once in the VI-regime, we do not see any evidence of an exit from this regime.

8.1.2 Entrance into the low- $Oh$ V-regime for $Oh<Oh_{c}$

When $Oh<Oh_{c}$ , appearance of the VI-regime is delayed by the presence of a low- $Oh$ V-regime. The entrance to this new regime follows a $r_{min}\sim Oh^{2}$ scaling as $r_{min}\rightarrow 0$ , which is characteristic of the I $\rightarrow$ VI transition, although here, as shown in § 9, this is an I $\rightarrow$ V transition.

Figure 12(b) corroborates the argument for a low- $Oh$ V-regime by showing, counter-intuitively, that once $Oh<Oh_{c}$ lower values of $Re_{local}$ are recovered. For example, for $Oh=0.16$ there is a dip to $Re_{local}=0.59$ , whilst for $Oh=0.14$ it falls as low as $Re_{local}=0.18$ . For $Oh=0.14$ , there is then a substantial period ( $4\times 10^{-4}<r_{min}<7\times 10^{-3}$ ) during which $Re_{local}$ remains at a value characteristic of the V-regime ( $Re_{local}<0.85$ ). Identifying the V-regime from the breakup speed in figure 12(a) results in a slightly smaller period ( $6\times 10^{-4}<r_{min}<3\times 10^{-3}$ ), a mismatch which appears to be caused by an initial lack of slenderness in the thread (figure 8), with $L_{r}/L_{z}<0.1$ only once $r_{min}<9\times 10^{-4}$ . This may explain why although ${\dot{r}}_{min}\approx -0.071$ in the new regime, there is no asymptotic approach to this value but rather a transient passing through speeds associated with a V-regime.

Entrance into the low- $Oh$ V-regime is triggered by the minimum radius moving away from $z=0$ as a satellite drop is formed. This is accompanied by other features that are characteristic of an I-regime, namely a corner-like free-surface shape (curve 3 in figure 13 a) and a decrease in the axial length scale $L_{z}$ due to the position of maximum velocity approaching the pinch point. However, in the newly created breakup region the axial velocity $U_{z}$ is still relatively small, so that $U_{z}L_{z}$ (i.e.  $Re_{local}$ ) is not large enough to trigger I-regime dynamics and instead a V-regime is encountered, a feature discovered in Castrejón-Pita et al. (Reference Castrejón-Pita, Castrejón-Pita, Thete, Sambath, Hutchings, Hinch, Lister and Basaran2015). What remains unclear is why this mechanism persists at smaller $Oh$ and prevents transitions from the I-regime directly into the VI-regime.

8.1.3 Entrance into the VI-regime for $Oh<Oh_{c}$

As can be seen from figure 14, the transition out of the low- $Oh$ V-regime is quickly followed by a transition into the VI-regime for $Oh<Oh_{c}$ . This boundary appears to follow a $r_{min}\sim Oh^{2}$ scaling as $r_{min}\rightarrow 0$ , although for $Oh>4\times 10^{-2}$ it is approximately constant. The ${\sim}Oh^{2}$ scaling was predicted for the I $\rightarrow$ VI transition and observed above for the entrance into the low- $Oh$ V-regime. However, why this scaling is followed for V $\rightarrow$ VI is unclear.

9 Inertia-dominated flow (small $Oh$ )

In figure 15 flow profiles for $Oh=10^{-3}$ are shown. As seen in § 8, a satellite drop is formed, but now the free surface forms a corner at the pinch point (figure 15 b). In a narrow region nearby a rapid increase in $w_{fs}$ and $p_{fs}$ (figure 15 c,d) can be seen. Notably, $w$ is no longer $r$ -independent. Below the pinch point the free surface forms an angle close to $18.1^{\circ }$ (predicted for the I-regime in Day et al. (Reference Day, Hinch and Lister1998)) with the $z$ -axis (dashed line), but there is no agreement with the predicted angle above ( $112.8^{\circ }$ ), which is found to be $78^{\circ }$ . Therefore, no ‘overturning’ of the free surface is observed, as discussed further in § 11.3.

Figure 15. For $Oh=10^{-3}$ plots are for (a) the entire free surface, (b) a close-up of the breakup region (with dashed line showing the angle $18.1^{\circ }$ from the $z$ -axis predicted in Day et al. (Reference Day, Hinch and Lister1998)), (c) the axial velocity at the free surface, $w_{fs}$ , and (d) the pressure at the free surface, $p_{fs}$ , at 1: $r_{min}=10^{-1}$ (blue), 2: $r_{min}=10^{-2}$ (red), 3: $r_{min}=10^{-3}$ (green) and 4: $r_{min}=10^{-4}$ (brown).

The maximum axial velocity is shown in figure 5(a) to scale as ${\sim}r_{min}^{-1/2}$ as predicted by the similarity solution (2.1). In contrast to the other two regimes, the radial velocity $-{\dot{r}}_{min}$ scales in the same way and is close to $w_{max}$ when multiplied by a factor of 5, showing that in the I-regime there is no separation of scales in the $r$ and $z$ directions. This is supported by observations that the thread is not slender as pinch-off is approached, as can be seen from curve 1 in figure 8.

Recalling that regions where $\dot{l}_{min}$ is approximately constant result in a $r_{min}\sim {\it\tau}^{2/3}$ scaling indicative of the I-regime, curve 1 in figure 16(a) shows the appearance of this regime for $Oh=10^{-3}$ around $10^{-3}<r_{min}<10^{-2}$ . Computations show that $\dot{l}_{min}=-A_{I}^{3/2}\approx -0.5$ so that $A_{I}=0.63$ , which can be used to extract the breakup time and plot $r_{min}$ against ${\it\tau}$ in curve 1 of figure 5(b). The value of $A_{I}$ is close to that given in Eggers & Villermaux (Reference Eggers and Villermaux2008) of $0.7$ .

Figure 16. Curves are for 1: $Oh=10^{-3}$ , 2: $Oh=4\times 10^{-3}$ and 3: $Oh=10^{-2}$ . (a) Identification of the I-regime, which is defined when $\dot{l}_{min}=-0.5\pm 0.15$ (dashed lines). (b) Evolution of the local Reynolds number: transition points out of the I-regime at $r_{min}=r_{min}^{I\rightarrow }$ are marked as circles and show that this transition occurs when $Re_{local}\approx 14$ (horizontal dashed line). The decrease of $Re_{local}$ in the I-regime follows the predicted scaling from the similarity solution (2.1) that $Re_{local}\sim r_{min}^{0.5}$ .

9.1 Identification of the I-regime

Using $\dot{l}_{min}$ to define the I-regime, and starting from the point at which the satellite drop is formed, i.e. $r_{min}<3\times 10^{-2}$ (figure 11), where the geometry of the I-regime starts to take shape, the boundaries of the I-regime are shown in figure 17.

Figure 17. Phase diagram showing the I-regime. The transition into this regime $r_{min}^{\rightarrow I}$ is constant, ${\approx}0.019$ , whilst the exit $r_{min}^{I\rightarrow }$ scales as $Oh^{2}$ with $r_{min}=45Oh^{2}$ .

9.1.1 Entrance into the I-regime

Similar to the V-regime, transitions into the I-regime $r_{min}^{\rightarrow I}$ are due to geometry, with a certain time required for the thread to form the corners which are characteristic of this regime. As one can see from curves 1 and 2 in figure 8, the breakup region is not slender and $L_{r}/L_{z}\approx 0.6$ remains approximately constant as $r_{min}\rightarrow 0$ .

9.1.2 Exit from the I-regime

The exit from the I-regime $r_{min}^{I\rightarrow }$ occurs when viscous effects become significant and the local Reynolds number drops below a critical value. Figure 16(b) shows that the reductions in $Re_{local}$ follow the ${\sim}r_{min}^{0.5}$ scaling predicted for the I-regime (2.1), which confirms that our definition of $Re_{local}$ is a good one. It is found that the I-regime is exited when $Re_{local}\approx 14$ . From § 8 we know that at this point the breakup will transition into the low- $Oh$ V-regime, and figure 17 shows that this entire boundary follows an ${\sim}Oh^{2}$ scaling.

10 A phase diagram for breakup

Having computed transitions into and out of the three regimes across parameter space, the results can be stitched together to produce the phase diagram shown in figure 18. Notably, once a regime is reached, which is around $r_{min}\approx 10^{-2}$ across $Oh$ , the transitions between regimes are relatively sharp, so that there are not vast regions of parameter space where none of the scaling laws are applicable. There are also no overlaps between the regimes, partially due to our choices of how to define being ‘in’ each regime. Consequently, rather than worry about transitions into and out of the different regimes, it is now sensible to talk about transitions between different regimes. This is how figure 18 was converted to produce the neat phase diagram figure 2(b) in § 2.2.3, where (i) transitions between the regimes are provided and (ii) only the asymptotic form of these transitions as $r_{min}\rightarrow 0$ are kept (i.e. using only the dashed lines in figure 18). It is of course possible that there is further complexity in the phase diagram below $r_{min}=10^{-4}$ . Here, analytical work is required as computational studies are inherently confined to a finite resolution below which the breakup behaviour can only be implied.

The most surprising results are the appearance of a low- $Oh$ V-regime which prevents I $\rightarrow$ VI transitions and the dramatic change in behaviour around a critical value $Oh_{c}=0.15$ . For $Oh>Oh_{c}$ the scaling for the V $\rightarrow$ VI transition follows that expected from theory ( $r_{min}^{V\rightarrow VI}\sim Oh^{-3.1}$ ), whilst both the transitions from the I-regime into the low- $Oh$ V-regime and the exit into the VI-regime follow the $r_{min}\sim Oh^{2}$ scaling predicted for the I $\rightarrow$ VI transition, but only as $r_{min}\rightarrow 0$ . Notably, it was shown that many of the transitions between regimes can be predicted from our measures $Re_{local}$ and $L_{r}/L_{z}$ .

Figure 18. Phase diagram for the breakup phenomenon which stitches together results from figure 7, figure 14 and figure 17, where specific expressions for the transitions between regimes can be found.

An experimentally verifiable prediction for the existence of a sharp transition at $Oh_{c}=0.15$ is that going from $Oh=0.16$ down to $Oh=0.14$ decreases the length of the thin liquid thread which connects the satellite drop to the main volume by a factor of three; see figure 13(b).