2.1 Non-dimensionalisation

We non-dimensionalise distances by $R_{+}$ , velocities by $V_{+}$ , time by $R_{+}/V_{+}$ and pressure by $\unicode[STIX]{x1D70C}_{l}V_{+}^{2}$ . The key dimensionless parameters are the Reynolds number in the liquid, the Weber number and the Froude number, defined by

(2.7a-c ) $$\begin{eqnarray}\displaystyle Re=\frac{\unicode[STIX]{x1D70C}_{l}R_{+}V_{+}}{\unicode[STIX]{x1D707}_{l}},\quad We=\frac{\unicode[STIX]{x1D70C}_{l}R_{+}V_{+}^{2}}{\unicode[STIX]{x1D70E}},\quad Fr^{2}=\frac{V_{+}^{2}}{gR_{+}}. & & \displaystyle\end{eqnarray}$$

Moreover, the density, viscosity, radius of curvature and impact speed ratios are defined by

(2.8a-d ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D70C}=\frac{\unicode[STIX]{x1D70C}_{g}}{\unicode[STIX]{x1D70C}_{l}},\quad \unicode[STIX]{x1D707}=\frac{\unicode[STIX]{x1D707}_{g}}{\unicode[STIX]{x1D707}_{l}},\quad R=\frac{R_{-}}{R_{+}},\quad V=\frac{V_{-}}{V_{+}}. & & \displaystyle\end{eqnarray}$$

2.2 Modelling assumptions

For the purposes of our analysis in § 3, we shall make the assumption that $\text{Re}$ , $We$ and $Fr^{2}$ are large, so that viscosity, surface tension and gravity are negligible, with the motion dominated by inertia. For the impact of a water droplet with radius of curvature $R_{+}=10^{-3}$ m at speed $V_{+}=10~\text{m}~\text{s}^{-1}$ with air as the surrounding gas, these numbers are

(2.9a-c ) $$\begin{eqnarray}\displaystyle Re\approx 10^{4},\quad We\approx 10^{3},\quad Fr^{2}\approx 10^{2}, & & \displaystyle\end{eqnarray}$$

so that this assumption is not unreasonable at early time, at least in the bulk of the droplet. However, since the free surface is highly curved at the root of the splash jet, it may be that viscosity and surface tension are more relevant there, see for example Moore et al. (Reference Moore, Ockendon, Ockendon and Oliver2014). We shall discuss this further in § 5. We also note here that the viscosities we consider in this paper (for water) are in general at the lower limit of those considered in Thoroddsen (Reference Thoroddsen2002) and Josserand & Zaleski (Reference Josserand and Zaleski2003). Moreover, for the example above, the typical time scale is $R_{+}/V_{+}\sim 10^{-4}$  s, which is significantly longer than the time scale $t_{v}=2R_{+}/(V_{+}Re)\sim 10^{-8}$  s, over which viscous effects dominate the flow at the root of the jet, discussed by Josserand & Zaleski (Reference Josserand and Zaleski2003). Hence, we expect the inviscid regime to be appropriate.

In addition to these assumptions, since, in general, the gas-liquid viscosity and density ratios are small – $\unicode[STIX]{x1D70C}\approx 10^{-3},~\unicode[STIX]{x1D707}\approx 10^{-2}$ for air–water systems – we shall also neglect the role of the gas layer post-impact in this analysis. Thus, for our analytical model, we will consider a one-fluid inviscid impact model known as the Wagner model.

3.1 Asymptotic structure

Figure 2. The Wagner asymptotic structure based on the small time scale $\unicode[STIX]{x1D6FF}$ . The fluid in each droplet is coloured with different shades of grey for ease of viewing. The horizontal coordinates of the upper and lower turnover points are denoted by $\pm d_{\pm }(\hat{t})$ respectively. Note that the problem is symmetric about the $y$ -axis.

Figure 3. A close-up of the right-hand jet root to highlight the main quantities of interest local to the turnover region. The fluid in each droplet is coloured with different shades of grey for ease of viewing. The turnover points $\boldsymbol{x}_{j\pm }$ as defined in (3.10) are taken to be the points at which the curvature of the free surface is locally maximised. The thickness of the jet is taken to be the distance between the turnover points and is denoted by $J(\hat{t})$ . The angle $\unicode[STIX]{x1D703}_{jet}(\hat{t})$ is a measure of the local angle the jet makes to the horizontal. Specifically, we take $\unicode[STIX]{x1D703}_{jet}(\hat{t})$ to be the angle formed when we take the local tangent to the interface between the fluids at the point at which the line joining the turnover points intersects the interface.

The impact breaks down into three distinct regimes based on the time scale $\unicode[STIX]{x1D6FF}$ , which are depicted in figure 2. In the outer region, where lengths scale with $\unicode[STIX]{x1D6FF}^{1/2}$ , the thin jet can be neglected and the free surface and interface boundary conditions linearise on to the undisturbed waterline, leading to a mixed boundary value problem. Local to the jet roots, the outer velocity and pressure are singular, which is rectified by matching to an inner region of size $O(\unicode[STIX]{x1D6FF}^{3/2})$ centred around the root, that is two orders of magnitude smaller than the outer region. In the inner region, the free surface turns over into the splash jet. The final regions are the thin, fast-moving jets, whose lengths are comparable to the outer region and whose thicknesses are comparable to the inner.

Since some of the key results that we aim to extract from the model and compare with our numerical simulations are the location of the jet roots, the thickness of the jet at its root and the angle the jet makes to the horizontal, we briefly clarify what we mean by these quantities here. Without loss of generality we shall describe these properties for the right-hand jet. The upper and lower jet-root coordinates are taken to be $\boldsymbol{x}_{j\pm }=(d_{\pm }(\hat{t}),h_{\pm }(d_{\pm }(\hat{t}),\hat{t}))$ , and, as seen in figure 3, these are taken to be the points at which the curvatures of the free surfaces are maximised locally. Note that this requires the jet to have formed: when the jet initially emerges, this criterion is not necessarily a sensible definition, particularly with the presence of a tip, but recall that we are not considering the initial emergence of the jet in our analysis. Furthermore, this definition may also break down in the event that the free surface becomes unstable to small perturbations (perhaps driven by viscosity or surface tension forces) local to the jet root. In this purely inviscid theory, we do not see such instabilities, but it is something that must be considered later in the numerical study. As seen in § 5.1, however, over the time scales we consider and with the fluids we study, we do not see such issues in this analysis. We shall refer to these upper and lower jet locations as the turnover points of the droplet free surfaces throughout our analysis. In summary, for the whole impact, there are four turnover points, two for each of the jets, with the right-hand turnover points located at $\boldsymbol{x}_{j\pm }^{\ast }$ , where

(3.10) $$\begin{eqnarray}\displaystyle \boldsymbol{x}_{j\pm }^{\ast }=R_{+}\boldsymbol{x}_{j\pm }=R_{+}(d_{\pm }(\hat{t}),h_{\pm }(d_{\pm }(\hat{t}),\hat{t})), & & \displaystyle\end{eqnarray}$$

using the notation of figure 3. The left-hand turnover points are easily deduced from the symmetry of the problem.

Integral to the inviscid theory is that the distance between the turnover points (scaled by $R_{+}$ ), defined by the thickness $J(\hat{t})$ in figure 3, is of $O(\unicode[STIX]{x1D6FF}^{3/2})$ . We note that this is an order of magnitude smaller than the $O(\unicode[STIX]{x1D6FF})$ stated in Josserand & Zaleski (Reference Josserand and Zaleski2003) for an inviscid approximation of the local jet thickness, but it is well established in Wagner theory – see, for example, Howison et al. (Reference Howison, Ockendon and Wilson1991) – that the only consistent asymptotic structure for a purely inviscid model requires the local jet thickness at its root to be two orders of magnitude smaller than the size of the outer region, which is of $O(\unicode[STIX]{x1D6FF}^{1/2})$ , as stated above. The veracity of the above assumption is realised by both the consistency of the matched asymptotic solution and the excellent comparisons between the predictions of the location of the turnover points and the numerical results in § 5.

Finally, we need to define an angle $\unicode[STIX]{x1D703}_{jet}(\hat{t})$ that gives a representation of the angle the jet makes to the horizontal. The choice of such an angle is not unique. In this paper, we instantaneously draw a straight line between the upper and lower turnover points, this line intersects the interface between the fluids. The jet angle, $\unicode[STIX]{x1D703}_{jet}(\hat{t})$ , is taken to be the angle made by the local tangent to the interface to the horizontal at this point. Other definitions of the jet angle have also been postulated, for example the angle between the normal to the line drawn between $\boldsymbol{x}_{j\pm }^{\ast }$ and the horizontal (as used in, for example, Thoraval et al. (Reference Thoraval, Takehara, Etoh, Popinet, Ray, Josserand, Zaleski and Thoroddsen2012)), or the angle made to the horizontal by a line drawn from the jet root to the jet tip (which does not take into account jet bending due to the surrounding air, see Moore & Oliver (Reference Moore, Whiteley and Oliver2018) for example). We do not seek to discuss the relative merits of one definition over another here, but state that we have chosen our definition of the jet angle to tie most closely to the inviscid analysis we pursue in the following sections. At the outset, we state that this is the quantity that is most poorly predicted by the inviscid theory, and it appears that viscosity and surface tension may have crucial roles in its evolution.

3.2 Outer region

In the outer region, we scale

(3.11) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}(x,y)=\unicode[STIX]{x1D6FF}^{1/2}(\hat{x},{\hat{y}}),\quad \unicode[STIX]{x1D719}_{\pm }=\unicode[STIX]{x1D6FF}^{1/2}\hat{\unicode[STIX]{x1D719}}_{\pm },\quad p_{\pm }=\unicode[STIX]{x1D6FF}^{-1/2}\hat{p}_{\pm },\\ h_{\pm }=\unicode[STIX]{x1D6FF}{\hat{h}}_{\pm },\quad \unicode[STIX]{x1D702}=\unicode[STIX]{x1D6FF}\hat{\unicode[STIX]{x1D702}},\quad d_{\pm }=\unicode[STIX]{x1D6FF}^{1/2}\hat{d}_{\pm },\end{array}\right\} & & \displaystyle\end{eqnarray}$$

and expand the variables in asymptotic series in powers of delta, viz.:

(3.12) $$\begin{eqnarray}\displaystyle \hat{\unicode[STIX]{x1D719}}_{\pm }=\hat{\unicode[STIX]{x1D719}}_{\pm 0}+\unicode[STIX]{x1D6FF}^{1/2}\hat{\unicode[STIX]{x1D719}}_{\pm 1}+O(\unicode[STIX]{x1D6FF}) & & \displaystyle\end{eqnarray}$$

etc. For notational convenience – and subject to the assumption mentioned above – we shall denote the leading-order term in the expansions of $\hat{d}_{\pm }$ by $\hat{d}_{0}$ .

Figure 4. The leading-order outer problem for the velocity potentials $\hat{\unicode[STIX]{x1D719}}_{\pm 0}$ . We have integrated the dynamic boundary conditions with respect to time and applied the leading-order initial condition to obtain the boundary conditions $\hat{\unicode[STIX]{x1D719}}_{\pm 0}=0$ on the free surfaces and $\hat{\unicode[STIX]{x1D719}}_{+0}=\hat{\unicode[STIX]{x1D719}}_{-0}$ on the interface. The problem is supplemented by the far-field conditions that $\hat{\unicode[STIX]{x1D719}}_{+0}\sim -{\hat{y}}$ as $\hat{x}^{2}+{\hat{y}}^{2}\rightarrow \infty$ and $\hat{\unicode[STIX]{x1D719}}_{-0}\sim V{\hat{y}}$ as $\hat{x}^{2}+{\hat{y}}^{2}\rightarrow \infty$ . The free surfaces have initial profiles given by ${\hat{h}}_{+0}=\hat{x}^{2}/2$ and ${\hat{h}}_{-0}=-\hat{x}^{2}/(2R)$ respectively. This problem is co-dimension two in the sense that $\hat{d}_{0}(\hat{t})$ must be solved for as part of the problem.

The leading-order problem is depicted in figure 4. It is a Riemann–Hilbert boundary value problem of index $-1$ , since, as is confirmed when considering the inner region, the velocity potentials have square-root behaviour close to the turnover points, $\pm \hat{d}_{0}(\hat{t})$ . The solution for the complex velocities ${\hat{w}}_{\pm 0}=\hat{\unicode[STIX]{x1D719}}_{\pm 0}+\text{i}\hat{\unicode[STIX]{x1D713}}_{\pm 0}$ can readily be found in terms of the complex variable $\hat{z}=\hat{x}+\text{i}{\hat{y}}$ :

(3.13) $$\begin{eqnarray}\displaystyle {\hat{w}}_{\pm 0}(\hat{z},\hat{t})=\text{i}\left[\left(\frac{1-V}{2}\right)\hat{z}+\left(\frac{1+V}{2}\right)\sqrt{\hat{z}^{2}-\hat{d}_{0}(\hat{t})^{2}}\right], & & \displaystyle\end{eqnarray}$$

where the branch cut for the square root is taken along the real axis for $|\hat{x}|>\hat{d}_{0}$ . Evaluating (3.13) on the real axis and integrating allows us to determine the leading-order upper and lower free surface profiles, which take the form

(3.14) $$\begin{eqnarray}\displaystyle {\hat{h}}_{+0} & = & \displaystyle \frac{\hat{x}^{2}}{2}-\left(\frac{1-V}{2}\right)\hat{t}-\left(\frac{1+V}{2}\right)\int _{0}^{\hat{t}}\frac{|\hat{x}|}{\sqrt{\hat{x}^{2}-\hat{d}_{0}(s)^{2}}}\,\text{d}s,\end{eqnarray}$$

(3.15) $$\begin{eqnarray}\displaystyle {\hat{h}}_{-0} & = & \displaystyle -\frac{\hat{x}^{2}}{2R}-\left(\frac{1-V}{2}\right)\hat{t}+\left(\frac{1+V}{2}\right)\int _{0}^{\hat{t}}\frac{|\hat{x}|}{\sqrt{\hat{x}^{2}-\hat{d}_{0}(s)^{2}}}\,\text{d}s.\end{eqnarray}$$

In order to complete the leading-order solution, we require a condition for the leading-order turnover point location, $\hat{d}_{0}$ . For this we use the Wagner condition, which is a matching condition with the inner region, but in essence says that in the outer region, the leading-order free surfaces and the vortex sheet must coincide at the turnover point, so that

(3.16) $$\begin{eqnarray}\displaystyle {\hat{h}}_{+0}(\hat{d}_{0}(\hat{t}),\hat{t})={\hat{h}}_{-0}(\hat{d}_{0}(\hat{t}),\hat{t})=\unicode[STIX]{x1D702}_{0}(\hat{d}_{0}(\hat{t}),\hat{t}). & & \displaystyle\end{eqnarray}$$

Upon substituting (3.14)–(3.15) into (3.16), we obtain an integral equation that may be inverted subject to the initial condition that $\hat{d}_{0}(0)=0$ , giving

(3.17) $$\begin{eqnarray}\displaystyle \hat{d}_{0}(\hat{t})=2\sqrt{\frac{(1+V)R}{1+R}\hat{t}}. & & \displaystyle\end{eqnarray}$$

Note that in the symmetric regime where $R,V=1$ this reproduces the well-known formula for the turnover point location for the impact of a parabola onto a solid, $\hat{d}_{0}=2\sqrt{\hat{t}}$ , see for example Korobkin (Reference Korobkin2007). The free surface height at the turnover point is therefore given by

(3.18) $$\begin{eqnarray}\displaystyle {\hat{h}}_{+0}(\hat{d}_{0}(\hat{t}),\hat{t})=\frac{(3RV+R-V-3)}{2(1+R)}\hat{t}. & & \displaystyle\end{eqnarray}$$

Finally, after evaluating (3.13) for $|\hat{x}|<\hat{d}_{0}$ and carefully integrating with respect to time, we are able to determine the leading-order vortex sheet location:

(3.19) $$\begin{eqnarray}\displaystyle \hat{\unicode[STIX]{x1D702}}_{0}=\frac{\hat{x}^{2}}{2}-\left(\frac{1-V}{2}\right)\hat{t}-\left(\frac{1+V}{2}\right)|\hat{x}|\int _{0}^{\hat{\unicode[STIX]{x1D714}}_{0}}\frac{1}{\sqrt{\hat{x}^{2}-\hat{d}_{0}(s)^{2}}}\,\text{d}s, & & \displaystyle\end{eqnarray}$$

where we have used the Wagner condition (3.16) and where $\hat{t}=\hat{\unicode[STIX]{x1D714}}_{0}(\hat{x})$ is the location of the turnover curve – i.e. $\hat{\unicode[STIX]{x1D714}}_{0}=\hat{d}_{0}^{-1}$ , provided that the inversion is well defined. As discussed in Howison et al. (Reference Howison, Ockendon and Wilson1991), inversion is possible provided that $\dot{\hat{d}}_{0}(\hat{t})>0$ .

Figure 5. The leading-order jet-root curve for $\unicode[STIX]{x1D6FF}=0.1$ and various values of $R$ and $V$ . Note that $V=0$ corresponds to a stationary lower droplet and $V=1$ to symmetric impact speeds, while $1/R=1$ represents identical droplets and $1/R\rightarrow 0$ allows the lower droplet to approach a half-space.

Therefore recalling (3.10), and that to leading-order $\boldsymbol{x}_{j+}=\boldsymbol{x}_{j-}=\boldsymbol{x}_{j}$ , we have

(3.20) $$\begin{eqnarray}\displaystyle \boldsymbol{x}_{j}=(\unicode[STIX]{x1D6FF}^{1/2}\hat{d}_{0}(\hat{t}),\unicode[STIX]{x1D6FF}{\hat{h}}_{+0}(\hat{d}_{0}(\hat{t}),\hat{t})), & & \displaystyle\end{eqnarray}$$

where $\hat{d}_{0}(\hat{t})$ and ${\hat{h}}_{+0}(\hat{d}_{0}(\hat{t}),\hat{t})$ are given by (3.17), (3.18). Hence, after returning to dimensional variables, to leading order the right-hand jet root moves along the curve

(3.21) $$\begin{eqnarray}\displaystyle \boldsymbol{x}_{j}^{\ast }=(x_{j}^{\ast },y_{j}^{\ast })=\left(2\sqrt{\frac{(1+V)R}{1+R}}\sqrt{R_{+}V_{+}t^{\ast }},\left(\frac{3RV+R-V-3}{2(1+R)}\right)V_{+}t^{\ast }\right). & & \displaystyle\end{eqnarray}$$

We shall henceforth refer to this as the ‘jet-root curve’.

We plot curves of the locus mapped out by the jet-root region for various values of $R$ and $V$ in figure 5. It is clear that the shape of the jet-root curve depends strongly on $R$ and $V$ . In particular, in figure 5(a), when the lower droplet is stationary, the jet-root location moves towards the lower droplet when the radii of curvature are similar, but curves away from the lower droplet as it approaches a half-space. There is similar behaviour as we increase the velocity of the lower droplet, although the transition from a downward-moving to an upward-moving jet root happens earlier, as seen in figure 5(b). In figure 5(c), the impact speeds are identical, so that for $R=1$ we see that $y_{j}=0$ to leading order, as expected. However, note that even for $R,V\neq 1$ we can get $y_{j}=0$ to leading order provided the numerator for the ${\hat{y}}$ -coordinate in (3.21) is zero, that is, when

(3.22) $$\begin{eqnarray}\displaystyle V=\frac{R-3}{1-3R}. & & \displaystyle\end{eqnarray}$$

Finally, we note that after substituting for $\hat{d}_{0}$ into (3.19), the vortex sheet location is found to be

(3.23) $$\begin{eqnarray}\displaystyle \hat{\unicode[STIX]{x1D702}}_{0}=\left(\frac{R-1}{4R}\right)\hat{x}^{2}-\left(\frac{1-V}{2}\right)\hat{t}. & & \displaystyle\end{eqnarray}$$

This is simply the average location of the undisturbed parabolae (3.1); as expected the vortex sheet moves with the flow, see Saffman (Reference Saffman1992). Moreover, since the tangential velocity is continuous across $\hat{\unicode[STIX]{x1D702}}$ to leading order, the vortex sheet is completely passive in the leading-order theory.

3.3 Inner region

Local to the jet root (3.21), the droplet free surfaces turn over in a region that is two orders of magnitude smaller than the outer region. This displacement of the free surface forms a fast-moving, thin jet of fluid. In this section, we shall report the existing results in the literature that allow us to determine the thickness and velocity of the jets at their roots, which we can then compare to the full numerical simulations we perform in § 5. By symmetry, we need only consider the right-hand jet root here.

Before starting, we note it is natural that the jet-root region should be aligned with the slope of the vortex sheet at $x^{\ast }=d^{\ast }(t^{\ast })$ , which can be calculated from (3.23) to be at an angle

(3.24) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FC}=\unicode[STIX]{x1D6FF}^{1/2}\left((R-1)\sqrt{\frac{1+V}{R(1+R)}}\right)\sqrt{\hat{t}}+o(\unicode[STIX]{x1D6FF}^{1/2}) & & \displaystyle\end{eqnarray}$$

to the horizontal. However, since $\unicode[STIX]{x1D6FC}$ is small and we shall only pursue a leading-order solution in this paper, we shall neglect any rotation of the jet-root region in our analysis, since it will make the analysis significantly more complex while having no impact on the leading-order results. We shall, however, return in § 5 to discuss the angle (3.24) and how it compares to the angle $\unicode[STIX]{x1D703}_{jet}$ as defined in figure 3.

The appropriate scales for this region are derived in Howison et al. (Reference Howison, Ockendon and Wilson1991) for the related problem of water entry – in particular, their deadrise angle $\unicode[STIX]{x1D700}$ is equivalent to $\unicode[STIX]{x1D6FF}^{1/2}$ here – and they are given by

(3.25a,b ) $$\begin{eqnarray}\displaystyle x=\unicode[STIX]{x1D6FF}^{1/2}\hat{d}_{+}(\hat{t})+\unicode[STIX]{x1D6FF}^{3/2}X,\quad y=\unicode[STIX]{x1D6FF}{\hat{h}}_{+0}(\hat{d}_{0}(\hat{t}),\hat{t})+\unicode[STIX]{x1D6FF}^{3/2}Y, & & \displaystyle\end{eqnarray}$$

with the velocity potential and pressure in each fluid scaled by

(3.26a,b ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D719}_{\pm }=\unicode[STIX]{x1D6FF}(\dot{\hat{d}}_{+}(\hat{t})X+\unicode[STIX]{x1D6F7}_{\pm }),\quad p_{\pm }=\unicode[STIX]{x1D6FF}^{-1}P_{\pm }. & & \displaystyle\end{eqnarray}$$

In the inner region, the free surfaces are denoted by $Y=H_{\pm }$ and the vortex sheet is denoted by $Y=\unicode[STIX]{x1D6E5}$ , where

(3.27a,b ) $$\begin{eqnarray}\displaystyle h_{\pm }=\unicode[STIX]{x1D6FF}{\hat{h}}_{+0}(\hat{d}_{0}(\hat{t}),\hat{t})+\unicode[STIX]{x1D6FF}^{3/2}H_{\pm },\quad \unicode[STIX]{x1D702}=\unicode[STIX]{x1D6FF}{\hat{h}}_{+0}(\hat{d}_{0}(\hat{t}),\hat{t})+\unicode[STIX]{x1D6FF}^{3/2}\unicode[STIX]{x1D6E5}. & & \displaystyle\end{eqnarray}$$

Upon expanding the rescaled equations in asymptotic series of the form $\unicode[STIX]{x1D6F7}_{\pm }=\unicode[STIX]{x1D6F7}_{\pm 0}+o(1)$ etc., the leading-order problem is quasi-steady and is depicted in figure 6. There is a stagnation point on the vortex sheet where dividing streamlines in the upper and lower droplets meet. Fluid to the left of the dividing streamlines heads back into the bulk of the droplets, while fluid to the right of the dividing streamlines is forced into the jet. The quantity $H_{j}(\hat{t})$ denotes the leading-order downstream thickness of the jet in the inner region (as opposed to $J(\hat{t})$ , which is measured at the jet root, see figure 3).

Figure 6. The leading-order inner problem for the velocity potentials, $\unicode[STIX]{x1D6F7}_{\pm 0}$ , upper and lower free surfaces $H_{\pm 0}$ and vortex sheet $\unicode[STIX]{x1D6E5}_{0}$ . Dividing streamlines in each fluid are denoted by the dashed lines. They meet on the vortex sheet at a stagnation point. The far-field jet thickness is denoted by $H_{j}(\hat{t})$ .

Since the vortex sheet essentially plays a passive role in the leading-order outer problem, we are able to deduce that the leading-order inner problem must be symmetric about the sheet, which is thus simply given by $\unicode[STIX]{x1D6E5}_{0}(X,\hat{t})=0$ at leading order. We would need to proceed to second order in the outer problem to show this rigorously, but we omit the analysis here for brevity. We can use the symmetry and the complex variable $Z=X+\text{i}Y$ to derive the parametric solution for the leading-order complex potentials $W_{\pm 0}=\unicode[STIX]{x1D6F7}_{\pm 0}+\text{i}\unicode[STIX]{x1D6F9}_{\pm 0}$ as given in Oliver (Reference Oliver2002):

(3.28a,b ) $$\begin{eqnarray}\displaystyle W_{\pm 0}=C_{\pm }(\hat{t})+\frac{\dot{\hat{d}}_{0}H_{j}}{2\unicode[STIX]{x03C0}}(\unicode[STIX]{x1D701}-\log \unicode[STIX]{x1D701}),\quad Z=-\frac{H_{j}}{2\unicode[STIX]{x03C0}}(1+\unicode[STIX]{x1D701}+4\sqrt{\unicode[STIX]{x1D701}}+\log \unicode[STIX]{x1D701}), & & \displaystyle\end{eqnarray}$$

where the branch cuts are taken along $\text{Re}(\unicode[STIX]{x1D701})<0$ , $\text{Im}(\unicode[STIX]{x1D701})=0$ with $H_{\pm 0}$ corresponding to $\text{Re}(\unicode[STIX]{x1D701})<0,\text{Im}(\unicode[STIX]{x1D701})=\mp 0$ respectively.

Matching with the leading-order outer solution as in Oliver (Reference Oliver2002) gives the leading-order jet thickness to be

(3.29) $$\begin{eqnarray}\displaystyle H_{j}(\hat{t})=\frac{\unicode[STIX]{x03C0}}{16}\frac{(1+V)^{2}\hat{d}_{0}(\hat{t})}{\dot{\hat{d}}_{0}(\hat{t})^{2}}=\frac{\unicode[STIX]{x03C0}(1+V)^{3/2}}{8}\sqrt{\frac{1+R}{R}}\hat{t}^{3/2}, & & \displaystyle\end{eqnarray}$$

with the far-field jet velocity, $U_{j}(\hat{t})$ , given by

(3.30) $$\begin{eqnarray}\displaystyle U_{j}=2\dot{\hat{d}}_{0}=2\sqrt{\frac{(1+V)R}{1+R}}\hat{t}^{-1/2}, & & \displaystyle\end{eqnarray}$$

where the factor of two occurs due to the velocity of the moving frame.

Although we omit this here for brevity, by proceeding to second order in the outer region it is possible to show that the $O(\unicode[STIX]{x1D6FF})$ -correction to the horizontal turnover point locations $\hat{d}_{1}=0$ , as in solid–liquid parabola impacts, see Korobkin (Reference Korobkin2007) and Oliver (Reference Oliver2007). In order to calculate the $O(\unicode[STIX]{x1D6FF}^{3/2})$ -corrections to the horizontal jet-root coordinates, $\hat{d}_{\pm 2}$ , we would need to proceed to second order in the inner region as well, which is a non-straightforward task, particularly as the rotation of the inner region will be important at this order. Unfortunately, without $\hat{d}_{\pm 2}$ we are unable to get an exact leading-order form for $J(\hat{t})$ , although it is clear that $J(\hat{t})\propto \hat{t}^{3/2}$ : in particular, the vertical distance between the turnover points in the rotated frame is given by $(1+4/\unicode[STIX]{x03C0})H_{j}(t)\propto \hat{t}^{3/2}$ , as can be deduced by evaluating (3.28) on $\unicode[STIX]{x1D709}<0,\unicode[STIX]{x1D702}=\pm 0$ and finding the turnover points. We shall discuss the thickness at the jet root in more detail in § 5.2.

Thus, in summary, the right-hand jet root is located at $\boldsymbol{x}_{j}^{\ast }$ given by (3.21) and is aligned at a small angle to the horizontal, (3.24). In the inner region local to $\boldsymbol{x}_{j}^{\ast }$ , the free surfaces turn over and fluid is ejected into a jet with leading-order velocity $\unicode[STIX]{x1D6FF}^{-1/2}V_{+}U_{j}$ , with the leading-order jet thickness given by $\unicode[STIX]{x1D6FF}^{3/2}R_{+}H_{j}$ as we approach the jet root from downstream in the jet.

3.4 Jet region

The information we have obtained from the leading-order outer and leading-order inner regions indicates that the fast-moving jet is emitted almost horizontally from the jet root as it moves along the curve (3.21). Since the jet has a long, thin aspect ratio, we can still derive a model for the leading-order jet thickness and leading-order horizontal component of velocity using the standard Wagner jet scalings discussed in Howison et al. (Reference Howison, Ockendon and Wilson1991). The angle at which the jet is emitted affects the centreline of the jet, not its thickness, so, assuming that the jet centreline does not deviate significantly from the curve (3.21) we can neglect the angle here. Such an assumption seems reasonable in the early stages of impact, particularly in the absence of air effects, and we shall discuss its validity in § 5.4. Determining the centreline of the jet is an interesting problem in its own right, see Moore & Oliver (Reference Moore, Whiteley and Oliver2018), but not one we seek to analyse here.

In order to study the jet evolution it is sensible to move to a frame $(s,n)$ based on the curve (3.21), where we shall use $s$ to represent arc length along the curve and $n$ is in the normal direction to the curve. The curvature is denoted by $\unicode[STIX]{x1D705}(s)$ . Based on (3.21) and the known size of the jet root, we scale

(3.31a,b ) $$\begin{eqnarray}\displaystyle s=\unicode[STIX]{x1D6FF}^{1/2}\bar{s},\quad n=\unicode[STIX]{x1D6FF}^{3/2}\bar{n}. & & \displaystyle\end{eqnarray}$$

It is straightforward to show that $\unicode[STIX]{x1D705}=O(1)$ , so that the radius of curvature of the curve (3.21) is much larger than the length of the jet (essentially meaning we can replace $(\bar{s},\bar{n})$ by $(\hat{x},\unicode[STIX]{x1D6FF}{\hat{y}})$ ). Moreover, by the symmetry at leading order in the jet root, we can neglect the interface in the jet and simply treat the jet as a single fluid. Hence, introducing the leading-order tangential component of jet velocity $\bar{u}_{0}$ (scaled by $\unicode[STIX]{x1D6FF}^{-1/2}$ ) and the leading-order jet thickness $\bar{\unicode[STIX]{x1D712}}_{0}$ (scaled by $\unicode[STIX]{x1D6FF}^{3/2}$ ), we can derive, akin to the jet in solid–liquid impact, the zero-gravity shallow-water equations in $\bar{s}>\hat{d}_{0}(\hat{t})$ :

(3.32a,b ) $$\begin{eqnarray}\displaystyle \bar{u}_{0\hat{t}}+\bar{u}_{0}\bar{u}_{0\bar{s}}=0,\quad \bar{\unicode[STIX]{x1D712}}_{0\hat{t}}+(\bar{u}_{0}\bar{\unicode[STIX]{x1D712}}_{0})_{\bar{s}}=0, & & \displaystyle\end{eqnarray}$$

subject to the jet-root conditions

(3.33a,b ) $$\begin{eqnarray}\displaystyle \bar{u}_{0}(\hat{d}_{0}(\hat{t}),\hat{t})=U_{j}(\hat{t}),\quad \bar{\unicode[STIX]{x1D712}}_{0}(\hat{d}_{0}(\hat{t}),\hat{t})=H_{j}(\hat{t}). & & \displaystyle\end{eqnarray}$$

Using the method of characteristics, we can solve (3.32), (3.33) to find that

(3.34a,b ) $$\begin{eqnarray}\displaystyle \bar{u}_{0}=\frac{\bar{s}}{\hat{t}},\quad \bar{\unicode[STIX]{x1D712}}_{0}=\frac{4\unicode[STIX]{x03C0}(1+V)^{4}R^{2}}{(1+R)^{2}}\frac{\hat{t}^{4}}{\bar{s}^{5}}. & & \displaystyle\end{eqnarray}$$

Therefore, since $\bar{\unicode[STIX]{x1D712}}_{0}>0$ for all $\bar{s}>\hat{d}_{0}(\hat{t})$ , the jet is predicted to be infinitely long and to achieve this in an infinitesimally small amount of time (note that $\bar{u}_{0}(\hat{d}_{0}(\hat{t}),\hat{t})\sim \hat{t}^{-1/2}$ as $\hat{t}\rightarrow 0$ ). This is a well-known drawback of the Wagner jet model, with further physical effects likely to play a key role further down the jet. This problem is discussed in more detail for liquid–solid impacts after the jet has detached in Riboux & Gordillo (Reference Riboux and Gordillo2015), where the Wagner jet model given by (3.32), (3.33) is coupled to a model for the evolution of the tip (in their paper the lamella rim), which incorporates the role of surface tension and the surrounding gas. Note that in such a model, the location of the jet centreline is important, see Moore & Oliver (Reference Moore, Whiteley and Oliver2018), and this would require a more careful analysis of the second-order outer and second-order inner problems in the Wagner structure, as discussed in § 3.3.

5.1 Jet-root location

Of primary interest is the correct identification of the jet-root location, with the leading-order expressions given in (3.21). To accomplish this, a suitable large subset of the curves around the upper and lower turnover points are selected from the main liquid–gas interface. These are projected and interpolated onto regular grids, as many of the underlying data points are found to be in close proximity to one another due to the very fine grids used. We use a maximum curvature criterion in each local turnover region to then identify the precise location of the turnover points (cf. figure 3). Drawing a line between them, we then consider the intersection of this line with the passive interface as the jet-root location to be compared to (3.21). We note that the procedure of identifying the turnover points becomes non-trivial in view of the large parameter space and different aspect ratios and locations of the arising geometrical structures. Consequently, careful post-processing was required to ensure accurate measurements have been considered.

Figure 9. Comparison between analytical prediction (3.21) (lines) and numerical results (symbols) for two families of tests. (a) Relative impact velocity coincides ( $V=1$ ), while $R$ is varied from a case with identical drop size to the limiting case $R\rightarrow \infty$ , where a drop impacts a liquid pool. (b) The relative drop size is set to 1 and the relative impact velocity $V$ is varied from 0.0 (stationary drop) to 4.0.

The extracted results (symbols) are plotted alongside the analytically derived jet-root location curves in figure 9 for two different families of tests. Note that while the symbols are plotted relatively sparsely, the underlying datasets in each example have 200 points. Both sets of coordinates have been re-cast into dimensional variables (hence the $(\text{}^{\ast })$ superscripts) to facilitate the comparison. In figure 9(a), we keep the impact velocities fixed and equal to one another and vary the radius of the lower drop from a size equal to the upper drop to the limiting case in which the lower liquid region becomes a liquid pool. For $R=V=1$ , we find the anticipated horizontal location of the jet root, with the jet root moving away from the lower droplet/pool as $R$ is increased, as previously observed in figure 5(c). In figure 9(b), we fix $R=1$ and focus on the variation of the impact velocity, with $0\leqslant V\leqslant 4$ . The data are extracted based on a fixed time window, with the different curve lengths reflecting the jet-root coordinate dependence on parameters $R$ and $V$ . This is particularly noticeable in figure 9(b), in which the presence of $V$ solely in the numerator of the results summarised in (3.21) translates to an overall movement of the jet root proportional to its value. As alluded to previously in the discussion around figure 5, the non-trivial behaviour of the jet curving either towards or away from the lower drop is recovered depending on the value of $V$ . Excellent agreement is found across all examined test cases over a generous early time interval spanning the range of validity of the model. In the latter stages, the $x_{j}^{\ast }$ -coordinate of the jet-root location approaches 1 mm in size, which indicates that the jet-root location has travelled a distance of almost one full drop radius. At this stage, the leading-order analytical prediction underestimates the numerical results.

5.2 Jet-root thickness

We move on to investigate the evolution of the dimensional jet thickness, $J^{\ast }$ , in time, where the thickness is measured as the distance between the upper and lower turnover points as seen in figure 3). We recall that, as discussed in § 3.3, in Wagner theory the jet thickness must grow like $t^{\ast 3/2}$ . This is in contrast to growth proportional to $t^{\ast }$ in the inviscid regime suggested by Josserand & Zaleski (Reference Josserand and Zaleski2003) for axisymmetric droplet impact onto a thin film. In the same paper, the authors suggest that at early times when $t^{\ast }\ll t_{v}=(2R_{+}/V_{+})/\text{Re}$ , viscous effects play the dominant role in the jet thickness, with the appropriate growth rate being proportional to $t^{\ast 1/2}$ . However, we note that $t_{v}\approx 10^{-8}$ s here, so on the time scales we consider in the present paper, we expect the inviscid theory to dominate. In their study of planar droplet impact onto a thin layer, Coppola, Rocco & de Luca (Reference Coppola, Rocco and de Luca2011) show that the jet thickness is approximately linear at later times, although they also show that the horizontal projection of the jet thickness grows like $t^{\ast 3/2}$ .

We wish to investigate this in more detail for various values of $V$ , $R$ and we plot the results in figure 10. In the first plot, we consider the symmetric case $R=V=1$ , where we see $J^{\ast }$ exhibiting faster than linear growth, albeit significantly undershooting growth proportional to the prediction of leading-order Wagner theory. If the velocity ratio is increased, the thickness growth is now clearly faster than linear, although again there appears to be a deviation away from $t^{\ast 3/2}$ for larger times. In the final example, for droplet impact onto a liquid pool (with $R=\infty$ and $V=1$ in figure 10 c), the jet thickness clearly follows the Wagner prediction over several decades.

Overall, there is strong evidence that the jet thickness grows more rapidly than the linear prediction of Josserand & Zaleski (Reference Josserand and Zaleski2003), but in some cases more slowly than the predictions of Wagner theory. It may be that other physical effects play a role in thinning the jet: the highly curved free surfaces cause a build-up of vorticity in the jet root (see figure 8 d), so that viscous and capillary effects may play a role that cannot be picked up by the inviscid theory. However, the clear evidence in the third example of figure 10 that the jet thickness grows like $t^{\ast 3/2}$ suggests that there are regimes in which the Wagner prediction is valid.

5.3 Jet angle

Figure 10. Jet-root thickness $J^{\ast }$ evolution in time for three different cases in $R$ and $V$ . The symbols represent data from direct numerical simulations, while the underlying lines present evolutions proportional to $(t^{\ast })^{a}$ that best fit the early stages of the flow. The solid line corresponds to $a=3/2$ (the Wagner prediction), while the dashed line indicates linear growth in time as given by $a=1$ (the inviscid prediction of Josserand & Zaleski (Reference Josserand and Zaleski2003)).

Figure 11. Summary of instantaneous jet angle $\unicode[STIX]{x1D703}_{jet}$ evolution for two families of tests. (a) Relative impact velocity coincides ( $V=1$ ), while $R$ is varied from a case with identical drop sizes to the limiting case $R\rightarrow \infty$ , where a drop impacts a liquid pool. (b) The relative drop size is set to 1 and the relative impact velocity $V$ is varied from 0 (stationary drop) to 4.

The jet angle, $\unicode[STIX]{x1D703}_{jet}$ , was introduced in figure 3 as the instantaneous angle between the tangent to the interface between the two droplets and the horizontal at the point at which the line joining the upper and lower turnover points intersects the interface. In § 3.3, we discussed the difficulties in predicting this angle using Wagner theory, with the only sensible prediction of the jet angle being given by $\unicode[STIX]{x1D6FC}$ in (3.24), which is the slope of the vortex sheet in the outer region as it approaches the turnover point. In this section, we shall investigate the evolution of $\unicode[STIX]{x1D703}_{jet}$ for several different impact scenarios and we shall consider whether $\unicode[STIX]{x1D6FC}$ is a reasonable estimation of the jet angle.

To extract $\unicode[STIX]{x1D703}_{jet}$ numerically, at each instance in time we consider the point defining the root of the jet on the virtual interface between the two fluids. We then move ahead in a discrete sense, selecting another point further ahead along the jet and thus enabling us to define an angle with the horizontal axis. We have experimented with the choice in this reference point in order to study how the degree of locality affects the jet angle. In particular, we have varied our spatial window from one point ahead to 16 virtual points ahead, which for our resolution gives us a distance between $0.25~\unicode[STIX]{x03BC}\text{m}$ and approximately $4~\unicode[STIX]{x03BC}\text{m}$ . Figure 8(b–d) provides an indication of the bending of this jet and hints at the variability of the angle depending on the choice of reference points. To illustrate this aspect, we will refer to specific cases in more detail, however in terms of summarising the datasets we have selected the $16$ point heuristic as being sufficiently robust, with the outcomes presented in figures 11–13.

In figure 11, we plot the evolution of $\unicode[STIX]{x1D703}_{jet}$ as a function of the dimensional time for the first $17.5~\unicode[STIX]{x03BC}\text{s}$ of the impact. When we fix the ratio of the impact velocities at $V=1$ and vary $R$ in figure 11(a), we see a strong variation from the horizontal jet predicted when the impact is completely symmetric with $R=1$ (a prediction that is consistent with (3.24)). For $R>1$ , we see $\unicode[STIX]{x1D703}_{jet}\sim \sqrt{V_{+}t^{\ast }/R_{+}}$ , a growth rate predicted by Thoroddsen et al. (Reference Thoroddsen, Thoraval, Takehara and Etoh2011) – and indeed by (3.24) – using a simple geometric model. It is worth reiterating that the exact definition of the jet angle is important. Thoraval et al. (Reference Thoraval, Takehara, Etoh, Popinet, Ray, Josserand, Zaleski and Thoroddsen2012) take the angle to be that between the normal to the line between the turnover points and the horizontal in their study of droplet impact onto a deep pool. Over time scales similar to those we study here that angle grows linearly in time.

In contrast, in figure 11(b), we fix the ratio of the radii of curvature at $R=1$ and vary the impact speed ratio, $V$ . Perhaps surprisingly, we see that, even for starkly different impact speeds, when $R=1$ , the jet is essentially emitted horizontally, which is consistent with the Wagner prediction, (3.24). Therefore, even though vorticity will undoubtedly build up on the jet-root free surfaces, see for example figure 8(d), and will play a role in the jet angle evolution when $R\neq 1$ (as we discuss shortly), perhaps the largest contributing factor to the angle at which the jet is emitted is the ratio of the droplets’ radii of curvature.

Figure 12. Summary of the jet-root evolution for the case described by $R=1$ and $V=2$ . (a) The identified lower and upper turnover points along with the associated jet-root coordinates, with the symbols denoting the numerically obtained results and the continuous line illustrating the analytical prediction as given by (3.21). (b) The inviscid estimate of the jet angle, $\unicode[STIX]{x1D6FC}$ previously derived as expression (3.24). (c) Instantaneous jet angle $\unicode[STIX]{x1D703}_{jet}$ obtained numerically using distance criteria based on including either 1, 4 or 16 virtual interface points ahead of the jet root in order to calculate the angle the jet makes with the horizontal axis.

Figure 13. Summary of the jet-root evolution for the case described by $R=3$ and $V=0$ . (a) The identified lower and upper turnover points along with the associated jet-root coordinates, with the symbols denoting the numerically obtained results and the continuous line illustrating the analytical prediction as given by (3.21). (b) Jet-root region angle $\unicode[STIX]{x1D6FC}$ , with the analytical result previously derived as expression (3.24). (c) Instantaneous jet angle $\unicode[STIX]{x1D703}_{jet}$ obtained numerically using distance criteria based on including either 1, 4 or 16 virtual interface points ahead of the jet root in order to calculate the angle the jet makes with the horizontal axis.

We now move on to compare $\unicode[STIX]{x1D703}_{jet}$ with $\unicode[STIX]{x1D6FC}$ by carefully examining two particular cases in more detail, with the results displayed in figures 12 and 13. In these figures, we provide a comparison between the Wagner prediction for the leading-order jet angle as given by (3.24) and the numerically measured angle, $\unicode[STIX]{x1D703}_{jet}$ . At the same time, we also describe the methodology behind the results analysis in § 5.1 in more depth.

Firstly, guided by relation (3.24) and figure 11, we investigate a case which is predicted to have a horizontally evolving jet despite an asymmetric $R/V$ configuration, with $R=1$ and $V=2$ . In figure 12(a), we not only show the jet-root coordinate evolution alongside the analytical prediction as before, but also present the identified turnover points its detection is based on. In figure 12(b), we plot the analytic prediction of the jet angle as given by $\unicode[STIX]{x1D6FC}$ in (3.24), while in figure 12(c), we display the numerically measured angle $\unicode[STIX]{x1D703}_{jet}$ for each of the 1-, 4- and 16-point reference choices. For this case, where the jet is essentially emitted horizontally, there is very little difference between these choices, and it is clear that both the analytic and numerical jet angles agree.

In figure 13, however, we see somewhat different results. In this example, we have chosen $R=3,V=0$ – note that this leads to a horizontal jet-root coordinate of $y_{j}=0$ (cf. (3.22)). Firstly, we note that, despite the apparent noise in the data in figure 13(a), the jet-root coordinate varies within roughly $1{-}2~\unicode[STIX]{x03BC}\text{m}$ throughout its evolution, which is at the level of the grid cells used in the direct numerical simulations. Thus, given the sensitivity of this particular case in light of the fine balance between the impinging drop sizes and velocities, the result is actually very accurate. However, when comparing the vortex sheet angle $\unicode[STIX]{x1D6FC}$ (depicted in figure 13 b) and the measured instantaneous jet angle $\unicode[STIX]{x1D703}_{jet}$ seen in figure 13(c), we see that the Wagner prediction significantly underestimates the jet angle – by approximately a factor of two – although both figures capture the square-root growth of the angle in time. We postulate that this discrepancy is due to the role of vorticity (and, to a lesser extent, surface tension) in the inner region. Moore et al. (Reference Moore, Ockendon, Ockendon and Oliver2014) discuss the boundary layers induced by the highly curved free surface in the Wagner inner region, and it is well known from previous works in the area, for example Thoraval et al. (Reference Thoraval, Takehara, Etoh, Popinet, Ray, Josserand, Zaleski and Thoroddsen2012), Thoraval et al. (Reference Thoraval, Takehara, Etoh and Thoroddsen2013), that vorticity build-up on the free surface can even lead to vortex shedding and the formation vortex streets. Moreover, the vorticity build-up can readily be seen in figure 8(d), with the dimensionless vorticity $\unicode[STIX]{x1D714}$ varying from approximately $-500$ at the lower jet root to 500 at the upper jet root at $t=0.1$ . We believe that the angle of the jet is strongly influenced by this significant vorticity, with the jet angle increasing with time (so that the jet is emitted closer to the upper drop in this example) due to an imbalance in the magnitude of the vorticity at each jet root. This is also reported by Thoraval et al. (Reference Thoraval, Takehara, Etoh, Popinet, Ray, Josserand, Zaleski and Thoroddsen2012) who find the instantaneous maximum positive and negative vorticities in the impact of a droplet onto a deep pool. They find these extremal values occur near the upper and lower turnover points respectively, with a marked difference in the numerical values. In that example the jet angle increases sufficiently rapidly so as to impact on the side of the droplet (‘bumping’), causing bubble entrapment.

Nevertheless, when considering the examples in which $R=1$ , it appears that the ratio of the droplets’ radii of curvature also plays an important role the increase in $\unicode[STIX]{x1D703}_{jet}$ with time: when $R=1$ there are still large vorticities associated with the highly curved free surface local to the jet root, but the jet remains approximately horizontal for the duration of our simulations. Both of these assertions warrant further investigation, although we do not aim to pursue such an analysis here. What is clear however, is that Wagner theory cannot accurately predict the angle at which the jet is emitted, aside for cases in which the droplets have comparable radii of curvature.

We note that figure 13(c) also indicates how sensitive the measurement of $\unicode[STIX]{x1D703}_{jet}$ is to the choice of reference points: the 1- and 4-point based datasets resulting in lower angle values. Finally, in all cases the nature of the instantaneous jet angle calculation method results in strong variability in the early stages of jet formation, should the jet be so small that the necessary number of points is not available and hence the reference point is selected further along the interface and into the main body of the drop.

Figure 14. (a) Evolution of the jet for $R=2$ and $V=1$ from the time of its creation at $t^{\ast }=2.4\times 10^{-6}\,\text{s}$ to $t^{\ast }=1.24\times 10^{-5}\,\text{s}$ in increments of $1.0\times 10^{-6}\,\text{s}$ . The full shape up to the jet tip is extracted from the direct numerical simulations, while the coordinates of the jet root as obtained analytically are also presented. (b) Dimensional horizontal velocity $U^{\ast }(x^{\ast },t^{\ast })$ for each time step and comparison to the leading-order Wagner solution (3.34). Both a direct comparison (dotted line) and a simple correction to account for the presence of the entrapped gas bubble (dashed line) are illustrated.

5.4 Jet velocity and thickness

We now move on to consider the jet itself, in particular concentrating on the jet shape and velocity at various stages of the impact. We present our findings in figure 14 for the $R=2$ and $V=1$ case. In figure 14(a) we employ a different visualisation technique to overlay not only the leading-order jet-root locations, but also the full numerically obtained jet shapes up to when the liquid begins to thicken and form the tip. The more pronounced bend in the jet becomes visible as we advance towards this region, especially in the latter stages. The respective curves are constructed from the virtual interface given when considering the passive tracers in the flow. These have previously been shown as part of figure 8 (in light grey) and they may be used as an indication of the top and bottom liquid contributions into the jet. We note that in most asymmetric cases (in either $R$ or $V$ , or both) we anticipate this virtual interface not to be aligned with what can be defined as a geometrically central curve along the jet, an aspect which is not accounted for in the analytical model, which cannot predict non-symmetric contributions from the two droplets to the fluid in the jet at leading order.

Typical jet velocity profiles extracted at the coordinates given by such curves are shown in figure 14(b). Close to the jet root, there is a large initial increase in the velocity consistent with the large velocity scales in § 3.3. The velocity subsequently falls into a regime of linear growth, which compares favourably to the predicted leading-order tangential jet velocity profile given by (3.34), particularly for earlier times. With the reference impact velocity at $10~\text{m}~\text{s}^{-1}$ , we find velocities along the jet are larger by an order of magnitude in the tested conditions, with a steady overall decrease as time advances. We note that this type of linear dependence is also discernible in the potential flow calculations of Riboux & Gordillo (Reference Riboux and Gordillo2017), although in that paper, the authors have concentrated their attention on the characterisation of the boundary layer properties preceding the location of the jet root.

It should be noted that initially the results have not been modified or adjusted at the root of the jet to aid the comparison (represented through the dotted lines). The small gap between the curves for smaller times is believed to be associated with the existence of the trapped gas bubble, which is present in the full numerical set-up, but ignored in the analysis. We opted to illustrate the unaltered results to enable a clean contrast between the analytical predictions and the direct numerical simulation. Especially for the first few sets of curves, the slopes compare favourably. This is to be expected, as we are well within the regimes in which the asymptotic solution is valid. At later stages and sufficiently far into the jet, the agreement begins to deteriorate as the numerically obtained velocity starts to deflect away from the linear behaviour. This is an indication of further physical effects coming into play, for example the interaction of the jet with the surrounding gas. Having established the above, a corrective step similar to that undertaken by Philippi et al. (Reference Philippi, Lagrée and Antkowiak2016) consisting in a temporal and spatial adjustment accounting for the slight delay and shift in location of the coalescence is then conducted, using $t_{0}$ and $x_{0}$ to denote these offsets in figure 14(b). This results in significant improvement of the overall agreement and in particular for the early stages of the jet evolution, where any differences are essentially indistinguishable.

We found it significant that in this regime one of the classical definitions of the spreading radius as the point of maximal velocity in the flow becomes somewhat ambiguous. The increasing velocity inside the jet and the details around the tip of this structure dictate the outcome under this metric. This does not appear to be the case in lower speed impact scenarios, which select a point near the jet root as having this property, see for example Thoraval et al. (Reference Thoraval, Takehara, Etoh, Popinet, Ray, Josserand, Zaleski and Thoroddsen2012) and Josserand et al. (Reference Josserand, Ray and Zaleski2016).

Figure 15. Evolution of the jet thickness for the case described by $R=2$ and $V=1$ illustrated at five instances in time, from $t^{\ast }=6.7\times 10^{-6}$  s to $t^{\ast }=1.47\times 10^{-5}$  s in increments of $2.0\times 10^{-6}$  s. The grey symbols indicate results extracted from direct numerical simulations, while the solid lines represent the associated analytical predictions as given by expression (3.34). The leading-order jet thickness as given by analysing the inner region, previously derived as (3.29), is also presented in each case using white squares.

Finally we explore the thickness of the jet in detail and concentrate on the $R=2,V=1$ case in figure 15. In § 3.4 we derived an analytical result based on Wagner theory for the leading-order jet thickness $\bar{\unicode[STIX]{x1D712}}_{0}$ , (3.34), predicting a variation of $\bar{s}^{-5}$ along the jet where $\bar{s}$ is the arc length coordinate along the jet. The figure illustrates several such curves considered equidistantly in time, alongside the points delimiting the inner region from the jet region (in white squares). Alongside these we present the corresponding numerical results, which are calculated as follows. We start from the jet-root coordinate and the identified lower/upper turnover points used to calculate it. The distance between them forms the first natural thickness measurement. For the next thickness result along the jet, we move along the virtual interface by one discrete point and provide a subset of data points immediately ahead of the turnover regions from which we select the nearest distance candidates to the virtual interface point, both above and below. We then iterate on this procedure, marching forward along the jet whilst also sliding the candidate points above and below. The data extraction is halted once a significant jet thickening is observed, indicative of the presence of the tip. As may be anticipated, the specific parameter choices such as the size of the candidate points intervals does indeed affect the first few thickness measurements, however we found that beyond these first points along the jet, the algorithm becomes robust in the sense of the estimated thickness being insensitive to parametric changes. Consistently, the threshold point roughly corresponds to the identified limit of the inner region and start of the jet region.

As is clear from figure 15, the agreement between Wagner theory and the numerical simulations is excellent, which is perhaps surprising as the Wagner jet does not take any effects of the tip into account. Among others, Thoroddsen (Reference Thoroddsen2002), Josserand & Zaleski (Reference Josserand and Zaleski2003), Riboux & Gordillo (Reference Riboux and Gordillo2015) and Josserand et al. (Reference Josserand, Ray and Zaleski2016) discuss the motion of the jet tip, with strong indication that its motion is governed by not only the liquid viscosity and surface tension but also the action of the surrounding gas. Despite the present study considering flow regimes towards the inviscid limit of the parameter spaces in the respective investigations (either through the selection of water as opposed to high viscosity water–glycerine mixtures or through the consideration of comparatively larger impact velocities), it is likely that the decrease in accuracy of the inviscid jet model as time increases is a result of these effects on the jet tip, or indeed along the whole jet, as considered by Moore & Oliver (Reference Moore, Whiteley and Oliver2018).

5.5 Summary

In summary, the comparisons between the full numerical simulations and the leading-order analytical predictions are extremely encouraging. As discussed in Riboux & Gordillo (Reference Riboux and Gordillo2014), Riboux & Gordillo (Reference Riboux and Gordillo2015) and Philippi et al. (Reference Philippi, Lagrée and Antkowiak2016), it is well known that Wagner theory provides a good estimation of the size of the effective contact set in droplet impact onto a solid substrate. Here, we have shown that this is also true for the liquid–liquid regime for a range of droplet diameters and impact speeds, as evidenced in figure 9. Moreover, we have shown that Wagner theory also gives a reliable estimate of the elevation of the jet root and the shape of the curve mapped out by the jet root, as shown in figure 14(a). There is also good (subject to a consideration of the role of the entrapped central gas bubble present in the full model) agreement between the predicted leading-order jet velocity and the full numerical simulations in figure 14(b). Furthermore, there is also excellent agreement between the predicted leading-order jet thickness and the numerical solution, as seen in figure 15.

All of these comparisons indicate that in any investigation of the behaviour of the ejecta in droplet impacts, we can reliably use Wagner theory as a basis for prescribing initial conditions for the jet-root location and fluid velocity. This has the potential to make analysis of jet evolution much simpler, as we can neglect the flow in the bulk of the droplet and investigate interactions between the fast-moving jet and the surrounding gas in greater detail.

We do, however, need to be slightly more cautious when considering the jet thickness and angle. In terms of the jet thickness, we saw in figure 10 that although there was strong evidence of the jet thickness growing more rapidly than linearly with time, it was only for large values of $R$ (i.e. as we approach impact onto a deep pool) that the characteristic Wagner $t^{\ast 3/2}$ -growth was seen over a long range of times. It is possible that for smaller values of $R$ , other physical factors such as the role of viscosity or surface tension on the highly curved jet surfaces cause deviations from the inviscid prediction of the jet thickness.

We encounter similar problems when considering the jet angle. In cases where the ratio of the droplets’ radii of curvature is close to unity, both the theory and the numerical simulations predict that the jet is approximately horizontal, see figures 11 and 12(b,c). However, as the ratio increases, the difference in the vorticity building up on the highly curved jet-root free surfaces appears to lead the jet angle to increase much more quickly than Wagner theory predicts, as displayed in figure 13(b,c). Naturally this phenomenon cannot be captured by the purely inviscid theory and it appears a more careful analysis of the full Navier–Stokes equations is required to capture this behaviour, in a similar manner to Moore et al. (Reference Moore, Ockendon, Ockendon and Oliver2014), who consider the capillary and viscous boundary layers in the jet root region in liquid–solid impact. We do note, however, that although there is a noticeable influence of the vorticity build-up on the jet angle, there is at this stage no evidence of the vortex streets seen in similar computations, see for example Thoraval et al. (Reference Thoraval, Takehara, Etoh, Popinet, Ray, Josserand, Zaleski and Thoroddsen2012) (with the usual caveat that we are considering a two-dimensional problem here, rather than axisymmetric). Despite this, as we observe in figure 14(a), the jet is already curving downstream of the jet root, indicating that this deflection is likely not a product of vorticity shedding. It is an open question as to what causes jet bending, but it is probable that the surrounding gas plays a critical role, as described through a simple kinematic model and via comparison with low pressure experiments by Thoroddsen et al. (Reference Thoroddsen, Thoraval, Takehara and Etoh2011), as well as recently discussed in Moore & Oliver (Reference Moore, Whiteley and Oliver2018).

From a more general viewpoint, the validation has proven robust across a wide range of parameters and in various flow regions. Despite the differences in the two set-ups (presence of a trapped gas bubble, presence of surrounding gas, surface tension, gravity), the qualitative and, more importantly, quantitative validation of the asymptotically predicted structures has been successfully addressed. Furthermore, features beyond the analytically tractable arguments have been discussed through the use of full numerical results.