4.1 Viscous air, viscous liquid

When the reduced Reynolds number in each fluid is small, the air pressure scale is given by (3.1) and the appropriate horizontal length scale is given by (3.2). Much of the following analysis follows that for viscous fibres, as described by Howell (Reference Howell1994), van de Fliert et al. (Reference van de Fliert, Howell and Ockendon1995) and Howell (Reference Howell1996). To derive the most general possible model, we shall assume that $\mathit{Re}_{l}$ and $\mathit{Re}_{g}$ are of order unity. We expand all of the independent variables in powers of $\unicode[STIX]{x1D700}_{l}^{2}$ – this is without loss of generality, since we have assumed that $\unicode[STIX]{x1D702}=O(1)$ .

In the jet, the leading-order form of the Navier–Stokes equations (2.1)–(2.3) subject to the stress conditions on the upper and lower free surfaces (2.14), (2.15), (2.18) and (2.19) allows us to deduce that, to leading order, the horizontal component of velocity and the pressure do not vary across the jet, since it is thin. Thus, $u=u(x,t)$ and $p=-(2/\mathit{Re}_{l})u_{x}$ at leading order. Utilisation of (2.3) along with the leading-order versions of the kinematic conditions (2.11) and (2.12) allows us to deduce the usual conservation of mass equation

(4.1) $$\begin{eqnarray}h_{t}+(uh)_{x}=0.\end{eqnarray}$$

In the air layer, the leading-order form of the Navier–Stokes equations (2.4)–(2.6) subject to the no-slip, no-flux condition on the substrate (2.9), the kinematic condition (2.10) and the continuity of velocity condition (2.13a ) leads to the lubrication equation,

(4.2) $$\begin{eqnarray}H_{t}+\left(\frac{uH}{2}-\frac{H^{3}P_{x}}{12}\right)_{x}=0,\end{eqnarray}$$

where, to leading order, $P=P(x,t)$ . We can clearly see both the Couette and the Poiseuille parts of the flow in (4.2).

We can now integrate the $O(\unicode[STIX]{x1D700}_{l}^{2})$ version of the horizontal momentum equation (2.1) across the jet and apply the shear stress conditions (2.18) and (2.19) to deduce that

(4.3) $$\begin{eqnarray}h(u_{t}+uu_{x})=\frac{4}{\mathit{Re}_{l}}(hu_{x})_{x}.\end{eqnarray}$$

The term on the right-hand side is the Trouton term seen in extensional flow models for viscous fibres. To close the system, we integrate the $O(\unicode[STIX]{x1D700}_{l}^{2})$ version of the vertical momentum equation (2.2) across the jet and apply the normal stress conditions (2.14) and (2.15). The algebraic details are presented in appendix B, yielding

(4.4) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D702}h\left[\left(u^{2}-\frac{4u_{x}}{\mathit{Re}_{l}}-\frac{\unicode[STIX]{x1D6FE}}{h}\right)H_{xx}+2uH_{xt}+H_{tt}\right] & = & \displaystyle P-h{\mathcal{F}}+\frac{\unicode[STIX]{x1D6FE}}{2}h_{xx}-h^{2}u_{x}^{2}\nonumber\\ \displaystyle & & \displaystyle +\,\frac{2}{\mathit{Re}_{l}}(2h(h_{x}u_{x})_{x}+h^{2}u_{xxx}).\qquad\end{eqnarray}$$

We note that by writing $\unicode[STIX]{x1D702}H=H_{c}-h/2$ , where $H_{c}$ is the centreline of the jet, we can reduce (4.4) to the inertial centreline equation for the stretching of a two-dimensional viscous sheet, as given by (94) in Howell (Reference Howell1996), with the addition of the air pressure term, $P(x,t)$ . We shall continue to use $H$ and $h$ in our analysis, since the thickness of the air layer is most frequently what we are interested in.

In summary, in the viscous jet/viscous air regime, the leading-order model for the horizontal component of the jet velocity, $u(x,t)$ , the jet thickness, $h(x,t)$ , the air layer thickness, $H(x,t)$ , and the air pressure, $P(x,t)$ , is given by the extensional flow equations (4.1) and (4.3) in the jet coupled with the lubrication equation in the air layer, (4.2), with the centreline deflection determined from the pressure-jump equation, (4.4). The length scale over which we see such a deflection is given by the deflection length scale for a viscous air layer, (3.2).

4.2 Viscous air, inviscid liquid

In the limit in which the reduced Reynolds number is large in the jet and small in the air, we can make progress under the additional assumption that the flow is irrotational in the jet, as in Wagner theory. The pressure scale in the air is still given by (3.1), so that the appropriate length scale is again given by (3.2). On expanding the variables in asymptotic series in powers of $\unicode[STIX]{x1D700}_{l}^{2}$ , a similar analysis to § 4.1 reveals that, at leading order, $u(x,t)$ , $h(x,t)$ , $H(x,t)$ and $P(x,t)$ , satisfy

(4.5) $$\begin{eqnarray}\displaystyle & h_{t}+(uh)_{x}=0, & \displaystyle\end{eqnarray}$$

(4.6) $$\begin{eqnarray}\displaystyle & u_{t}+uu_{x}=0, & \displaystyle\end{eqnarray}$$

(4.7) $$\begin{eqnarray}\displaystyle & \displaystyle H_{t}+\left(\frac{uH}{2}-\frac{H^{3}P_{x}}{12}\right)_{x}=0, & \displaystyle\end{eqnarray}$$

(4.8) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D702}h\left[\left(u^{2}-\frac{\unicode[STIX]{x1D6FE}}{h}\right)H_{xx}+2uH_{xt}+H_{tt}\right]=P-h{\mathcal{F}}+\frac{\unicode[STIX]{x1D6FE}}{2}h_{xx}-h^{2}u_{x}^{2}. & \displaystyle\end{eqnarray}$$

Here, we note that (4.5) and (4.6) are the usual Wagner jet equations; see, for example, Howison et al. (Reference Howison, Ockendon and Wilson1991) and Riboux & Gordillo (Reference Riboux and Gordillo2015).

4.3 Inviscid air, viscous liquid

If the reduced Reynolds number is small in the jet, but large in the air, we can make progress under the additional assumption that the flow in the air is irrotational (thus assuming that there is no separation of viscous boundary layers on the time scale under consideration). The air pressure scale is now given by (3.3), so that the appropriate horizontal length scale is defined by the deflection length scale for an inviscid air layer, equation (3.4). Proceeding as in § 4.1, the leading-order problem for $u(x,t)$ , $h(x,t)$ , $H(x,t)$ , $P(x,t)$ and the horizontal component of the air velocity, $U(x,t)$ , is found to be given by

(4.9) $$\begin{eqnarray}\displaystyle & h_{t}+(uh)_{x}=0, & \displaystyle\end{eqnarray}$$

(4.10) $$\begin{eqnarray}\displaystyle & \displaystyle h(u_{t}+uu_{x})=\frac{4}{\mathit{Re}_{l}}(hu_{x})_{x}, & \displaystyle\end{eqnarray}$$

(4.11) $$\begin{eqnarray}\displaystyle & U_{t}+UU_{x}=-P_{x}, & \displaystyle\end{eqnarray}$$

(4.12) $$\begin{eqnarray}\displaystyle & H_{t}+(UH)_{x}=0, & \displaystyle\end{eqnarray}$$

(4.13) $$\begin{eqnarray}\displaystyle & & \displaystyle \displaystyle \unicode[STIX]{x1D702}h\left[\left(u^{2}-\frac{4u_{x}}{\mathit{Re}_{l}}-\frac{\unicode[STIX]{x1D6FE}}{h}\right)H_{xx}+2uH_{xt}+H_{tt}\right]\nonumber\\ \displaystyle & & \displaystyle \quad =P-h{\mathcal{F}}+\frac{\unicode[STIX]{x1D6FE}}{2}h_{xx}-h^{2}u_{x}^{2}+\frac{2}{\mathit{Re}_{l}}(2h(h_{x}u_{x})_{x}+h^{2}u_{xxx}).\end{eqnarray}$$

4.4 Inviscid air, inviscid liquid

Finally, in the limit in which the reduced Reynolds number is large in both the air and the jet, we assume that both flows are irrotational and proceed as in the previous sections. The pressure scale and appropriate horizontal length scale are given by (3.3) and (3.4) respectively. The leading-order problem for $u(x,t)$ , $h(x,t)$ , $H(x,t)$ , $U(x,t)$ and $P(x,t)$ is given by

(4.14) $$\begin{eqnarray}\displaystyle & h_{t}+(uh)_{x}=0, & \displaystyle\end{eqnarray}$$

(4.15) $$\begin{eqnarray}\displaystyle & u_{t}+uu_{x}=0, & \displaystyle\end{eqnarray}$$

(4.16) $$\begin{eqnarray}\displaystyle & U_{t}+UU_{x}=-P_{x}, & \displaystyle\end{eqnarray}$$

(4.17) $$\begin{eqnarray}\displaystyle & H_{t}+(UH)_{x}=0, & \displaystyle\end{eqnarray}$$

(4.18) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D702}h\left[\left(u^{2}-\frac{\unicode[STIX]{x1D6FE}}{h}\right)H_{xx}+2uH_{xt}+H_{tt}\right]=P-h{\mathcal{F}}+\frac{\unicode[STIX]{x1D6FE}}{2}h_{xx}-h^{2}u_{x}^{2}. & \displaystyle\end{eqnarray}$$

4.5 Summary

We have outlined the derivation of the equations governing the motion of a thin jet above a thin air layer in four different asymptotic regimes. Although we have derived these equations in the context of an infinite jet, they are equally relevant along the bulk of a long but finite jet, provided that the length of the jet is comparable to the appropriate deflection length scales given in (3.2) and (3.4). In these latter instances, appropriate boundary and initial conditions will need to be specified, which is non-trivial since the equations are highly nonlinear. We shall discuss this in more detail in § 6.1.

It is perhaps worth reiterating that we have considered the particular distinguished limit in which the air pressure jump, surface tension and gravity influence the centreline deflection, but have no effect on the jet thickness or horizontal component of velocity at leading order. Thus, in each of §§ 4.1–4.4, the jet flow equations decouple from the air flow equations and centreline equation. Hence, in order to put our model in context, we discuss how our analysis could be used to extend the model for a detached lamella in Riboux & Gordillo (Reference Riboux and Gordillo2015). First, we could find the horizontal component of the jet velocity and the jet thickness by solving (4.1), (4.3) or (4.5), (4.6) subject to appropriate jet-root boundary conditions, for example the Wagner jet-root conditions modified to account for local viscous effects, as in Riboux & Gordillo (Reference Riboux and Gordillo2015). In principle, we could then use $u$ and $h$ to solve (4.2), (4.4) or (4.7), (4.8) subject to appropriate boundary conditions for the motion of the jet centreline under the action of the air-cushioning layer; the effect of the latter is not considered by Riboux & Gordillo (Reference Riboux and Gordillo2015). We discuss the relevance of our assumptions on the importance of surface tension and gravity by considering experiments in several studies in § 7.1. It is certainly possible to consider other distinguished limits, for example regimes where surface tension plays a more important role, which we discuss in appendix A.

5.1 Viscous air layer

When $u=1$ and $h=1$ , the Reynolds lubrication equation (4.2) and the centreline deflection equation (4.4) reduce to

(5.3a,b ) $$\begin{eqnarray}H_{t}+\left(\frac{H}{2}-\frac{H^{3}P_{x}}{12}\right)_{x}=0,\quad \unicode[STIX]{x1D702}[(1-\unicode[STIX]{x1D6FE})H_{xx}+2H_{xt}+H_{tt}]=P-{\mathcal{F}}.\end{eqnarray}$$

There is a spatially uniform solution of (5.3) given by

(5.4a,b ) $$\begin{eqnarray}H=1,\quad P={\mathcal{F}},\end{eqnarray}$$

where we take the constant solution for the thickness of the air layer to be unity without loss of generality. We seek a dispersion relation between the frequency, $\unicode[STIX]{x1D714}$ , and the wavenumber, $k$ , by expanding in the usual way via

(5.5) $$\begin{eqnarray}(H,P)=(1,{\mathcal{F}})+\unicode[STIX]{x1D6FF}(H_{1},P_{1})\text{e}^{\text{i}(kx-\unicode[STIX]{x1D714}t)},\end{eqnarray}$$

where $\unicode[STIX]{x1D6FF}\ll 1$ . On substituting (5.5) into (5.3), we deduce from the linearised problem (at $O(\unicode[STIX]{x1D6FF})$ ) that $\unicode[STIX]{x1D714}$ must satisfy the quadratic

(5.6) $$\begin{eqnarray}-\text{i}\unicode[STIX]{x1D714}+\frac{\text{i}k}{2}-\unicode[STIX]{x1D6E4}((1-\unicode[STIX]{x1D6FE})k^{4}-2k^{3}\unicode[STIX]{x1D714}+\unicode[STIX]{x1D714}^{2}k^{2})=0,\end{eqnarray}$$

where $\unicode[STIX]{x1D6E4}=\unicode[STIX]{x1D702}/12$ . We find that the two roots of this dispersion relation are given by

(5.7) $$\begin{eqnarray}\unicode[STIX]{x1D714}_{\pm }=k-\frac{\text{i}}{2\unicode[STIX]{x1D6E4}k^{2}}\left(-1\pm \sqrt{1-4(a-b\text{i})}\right),\quad \text{where}~a=\unicode[STIX]{x1D6E4}^{2}\unicode[STIX]{x1D6FE}k^{6},b=\frac{\unicode[STIX]{x1D6E4}k^{3}}{2}\end{eqnarray}$$

and the principal branch of the square root is taken. It is trivial to note that $a,b>0$ for all $k>0$ .

The imaginary part of (5.7) dictates the growth rate of the infinitesimal disturbances: we find that

(5.8) $$\begin{eqnarray}\text{Im}(\unicode[STIX]{x1D714}_{\pm })=\frac{1}{2\unicode[STIX]{x1D6E4}k^{2}}\left(-1\pm \left(\frac{(1-4a)+\sqrt{(1-4a)^{2}+16b^{2}}}{2}\right)^{1/2}\right).\end{eqnarray}$$

For an instability in either branch, we require $\text{Im}(\unicode[STIX]{x1D714}_{\pm })>0$ . Since the term $(1-4a)^{2}+16b^{2}$ is always positive, $\text{Im}(\unicode[STIX]{x1D714}_{-})<0$ for all $k$ , so that we can only get instability if $\text{Im}(\unicode[STIX]{x1D714}_{+})>0$ . This only occurs when $b^{2}>a$ or

(5.9) $$\begin{eqnarray}\unicode[STIX]{x1D6FE}=\frac{2}{\unicode[STIX]{x1D700}_{l}\,\mathit{We}}<\frac{1}{4}.\end{eqnarray}$$

It is interesting to note that this requirement is independent of $k$ ; hence, if $\unicode[STIX]{x1D6FE}$ satisfies (5.9), then the positive branch of the dispersion relation is unstable for all wavenumbers $k>0$ . We plot $\text{Im}(\unicode[STIX]{x1D714}_{\pm })$ with $\unicode[STIX]{x1D6E4}=1$ for various values of $\unicode[STIX]{x1D6FE}$ in figure 6(a,b), which illustrates that, as we decrease the Weber number – increasing $\unicode[STIX]{x1D6FE}$ – we see three different behaviours in $\text{Im}(\unicode[STIX]{x1D714}_{+})$ , while there is no qualitative change in the behaviour of $\text{Im}(\unicode[STIX]{x1D714}_{-})$ . From (5.9), we know that the critical value at which we see a change in stability is given by $\unicode[STIX]{x1D6FE}=1/4$ . For $\unicode[STIX]{x1D6FE}<1/4$ , that is for sufficiently small surface tension, perturbations are linearly unstable to all wavenumbers $k>0$ , with the growth rate for short wavelengths decaying like

(5.10) $$\begin{eqnarray}\text{Im}(\unicode[STIX]{x1D714}_{+})\sim \frac{1}{2\unicode[STIX]{x1D6E4}}\left(-1+\sqrt{\frac{1}{4\unicode[STIX]{x1D6FE}}}\right)\frac{1}{k^{2}}\quad \text{as}~k\rightarrow \infty .\end{eqnarray}$$

When $\unicode[STIX]{x1D6FE}=1/4$ , $\text{Im}(\unicode[STIX]{x1D714}_{+})$ is identically zero for all $k>0$ , and hence this branch is neutrally stable. For $\unicode[STIX]{x1D6FE}>1/4$ , we note that $\text{Im}(\unicode[STIX]{x1D714}_{+})<0$ for all $k>0$ , so that we have linear stability for all wavenumbers.

Figure 6.

(a,b) Dispersion relation branches with $\unicode[STIX]{x1D6E4}=1$ for various values of the inverse Weber number: $\unicode[STIX]{x1D6FE}=0$ (solid line), $\unicode[STIX]{x1D6FE}=1/10$ (filled circles), $\unicode[STIX]{x1D6FE}=1/4$ (circles), $\unicode[STIX]{x1D6FE}=3/10$ (crosses) and $\unicode[STIX]{x1D6FE}=1/2$ (squares). (c) The wavenumber of the most unstable mode, $k_{m}$ , as a function of $\unicode[STIX]{x1D6FE}$ with $\unicode[STIX]{x1D6E4}=1$ .

To facilitate comparisons with experiments and simulations, we wish to discern the most unstable mode; that is, the wavenumber $k$ that maximises $\text{Im}(\unicode[STIX]{x1D714}_{+})$ . We find that the dimensionless wavenumber of the most unstable mode, $k_{m}$ , is given by

(5.11) $$\begin{eqnarray}k_{m}=\frac{1}{\unicode[STIX]{x1D6E4}^{1/3}}(v^{2}(v^{2}-1))^{1/6},\end{eqnarray}$$

where $v$ is the unique positive root of the sextic

(5.12) $$\begin{eqnarray}0=4\unicode[STIX]{x1D6FE}v^{4}(3v^{2}+3v-2)+v^{2}-3v-2.\end{eqnarray}$$

If we fix $\unicode[STIX]{x1D6E4}=1$ , we can investigate the effect of $\unicode[STIX]{x1D6FE}$ on $k_{m}$ , and we plot the result in figure 6(c). As $\unicode[STIX]{x1D6FE}$ increases, so too does the wavelength of the most unstable mode.

If we fix $\unicode[STIX]{x1D6FE}$ , $k_{m}$ is inversely proportional to the cubic root of $\unicode[STIX]{x1D6E4}$ . Therefore, increasing $\unicode[STIX]{x1D6E4}$ increases the wavelength of the most unstable mode. Since

(5.13) $$\begin{eqnarray}\unicode[STIX]{x1D6E4}=\frac{\unicode[STIX]{x1D702}}{12}=\frac{\ell _{g}}{12\ell _{l}},\end{eqnarray}$$

if we decrease the thickness of the liquid jet or increase the thickness of the air layer, we increase the wavelength of the most unstable mode. We note that as $k_{m}$ decreases, $\text{Im}(\unicode[STIX]{x1D714}_{+}(k_{m}))$ decreases, so that the growth rate is reduced.

5.2 Inviscid air layer

In the inviscid air regime, we perform a similar linear stability analysis of the centreline deflection equations (4.16)–(4.18) with $u=h=1$ to that in § 5.1 with the additional assumption that the air has a constant tangential speed in the base state, which we denote by $U_{0}$ . By perturbing about the spatially uniform steady-state solution by setting

(5.14) $$\begin{eqnarray}(H,U,P)=(1,U_{0},{\mathcal{F}})+\unicode[STIX]{x1D6FF}(H_{1},U_{1},P_{1})\text{e}^{\text{i}(kx-\unicode[STIX]{x1D714}t)},\end{eqnarray}$$

where $\unicode[STIX]{x1D6FF}\ll 1$ , we find that the dispersion relation in the inviscid air regime is given by

(5.15) $$\begin{eqnarray}0=(\unicode[STIX]{x1D702}k^{2}+1)\unicode[STIX]{x1D714}^{2}-2k(\unicode[STIX]{x1D702}k^{2}+U_{0})\unicode[STIX]{x1D714}+\unicode[STIX]{x1D702}(1-\unicode[STIX]{x1D6FE})k^{4}+U_{0}^{2}k^{2},\end{eqnarray}$$

giving the two roots

(5.16) $$\begin{eqnarray}\unicode[STIX]{x1D714}_{\pm }=\frac{k}{1+\unicode[STIX]{x1D702}k^{2}}(U_{0}+\unicode[STIX]{x1D702}k^{2}\pm k\sqrt{\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D702}(1+\unicode[STIX]{x1D702}k^{2})-\unicode[STIX]{x1D702}(U_{0}-1)^{2}}).\end{eqnarray}$$

Again, we require $\text{Im}(\unicode[STIX]{x1D714}_{\pm })>0$ for instability.

If $U_{0}=1$ , that is if the unperturbed air velocity is the same as the jet velocity, the disturbance is neutrally stable, which is in agreement with Tammisola et al. (Reference Tammisola, Sasaki, Lundell, Matsubara and Söderberg2011), who show that a co-flowing gas can stabilise a liquid jet provided that the difference between the mean gas and jet speeds is sufficiently small.

If $U_{0}\neq 1$ , we can find a range of unstable wavenumbers provided that the surface-tension parameter, $\unicode[STIX]{x1D6FE}$ , is sufficiently small. Considering (5.16), if $0<\unicode[STIX]{x1D6FE}<(U_{0}-1)^{2}$ , the range of unstable wavenumbers is

(5.17) $$\begin{eqnarray}0<k<\sqrt{\frac{1}{\unicode[STIX]{x1D702}\unicode[STIX]{x1D6FE}}((U_{0}-1)^{2}-\unicode[STIX]{x1D6FE})}.\end{eqnarray}$$

If $\unicode[STIX]{x1D6FE}>(U_{0}-1)^{2}$ , we have neutral stability. Thus, we identify $\unicode[STIX]{x1D6FE}_{c}=(U_{0}-1)^{2}$ as the critical value of the surface-tension parameter. We plot $\text{Im}(\unicode[STIX]{x1D714}_{\pm })$ for $\unicode[STIX]{x1D702}=1$ and $U_{0}=1/2$ for various values of $\unicode[STIX]{x1D6FE}$ in figure 7. We find that, when $0<\unicode[STIX]{x1D6FE}<\unicode[STIX]{x1D6FE}_{c}$ , the most unstable wavenumber is given by

(5.18) $$\begin{eqnarray}k_{m}=\left(\frac{1}{2\unicode[STIX]{x1D702}}\left(-3+\sqrt{1+\frac{8}{\unicode[STIX]{x1D6FE}}(U_{0}-1)^{2}}\right)\right)^{1/2}.\end{eqnarray}$$

It should be noted that $k_{m}\rightarrow 0$ as $\unicode[STIX]{x1D6FE}\rightarrow \unicode[STIX]{x1D6FE}_{c}$ . Furthermore, it should be noted that as $\unicode[STIX]{x1D6FE}\rightarrow 0$ , the range of unstable wavenumbers gets larger; indeed, if $\unicode[STIX]{x1D6FE}=0$ , the jet is unstable to disturbances of all wavelengths. Nevertheless, the growth rate is bounded for arbitrarily short wavelengths in the zero-surface-tension case, with

(5.19) $$\begin{eqnarray}\text{Im}(\unicode[STIX]{x1D714}_{+})\sim \sqrt{\frac{1}{\unicode[STIX]{x1D702}}}(U_{0}-1)-\left(\frac{1}{\unicode[STIX]{x1D702}}\right)^{3/2}\frac{U_{0}-1}{k^{2}}+\cdots \quad \text{as}~k\rightarrow \infty .\end{eqnarray}$$

Figure 7.

Branches of the dispersion relation for fixed $\unicode[STIX]{x1D706}=1$ , $\unicode[STIX]{x1D702}=1$ and $U_{0}=1/2$ for various values of the inverse Weber number: $\unicode[STIX]{x1D6FE}=0$ (solid line), $\unicode[STIX]{x1D6FE}=0.1$ (dashed line), $\unicode[STIX]{x1D6FE}=0.2$ (circles), $\unicode[STIX]{x1D6FE}=0.25$ (squares). It should be noted that $\unicode[STIX]{x1D6FE}_{c}=1/4$ for this particular example.

6.1 Modelling assumptions

We therefore seek to adapt our model to consider scenarios in which the liquid jet is finite, and return to the configuration depicted in figure 5. A full model has the potential to be extremely complicated, as discussed in detail in Moore (Reference Moore2014). We present a simple first model for a growing jet that applies the analysis of § 4 along with suitable boundary conditions to model the roles played by the jet root and tip. Since it is the simplest regime, we shall look in detail at the case where the reduced Reynolds number in the air is small and that in the jet is large.

In this section, we solve (4.5)–(4.8) with $\unicode[STIX]{x1D6FE}=0$ . For (4.5) and (4.6), we prescribe inlet data $u=1$ , $h=1$ at $x=0$ for $t>0$ , with the jet having zero length at $t=0$ (subject to the aforementioned caveat about neglecting the stage of jet formation in this paper). In the absence of air, the solution must be $u=1$ , $h=1$ for $0<x<t$ ; that is, the jet tip does not retract according to the zero-gravity shallow-water equations which do not take account of the effects of surface tension. In appendix C, we show that even in the presence of air, retraction of the jet tip occurs on a much longer time scale provided that $0<\unicode[STIX]{x1D6FE}\ll 1$ . We note that in the particular problem of droplet impact, appropriate inlet data for $u$ and $h$ would vary with time; see, for example, Oliver (Reference Oliver2002) or Riboux & Gordillo (Reference Riboux and Gordillo2015). We have chosen the conditions above simply for illustrative purposes, as in § 5.

We therefore propose the model for the evolution of the jet centreline given by (4.7) and (4.8), with $u=1$ , $h=1$ and $\unicode[STIX]{x1D6FE}=0$ , i.e.

(6.1a,b ) $$\begin{eqnarray}H_{t}+\left(\frac{H}{2}-\frac{1}{12}(H^{3}P_{x})\right)_{x}=0,\quad H_{xx}+2H_{xt}+H_{tt}=P\quad \text{for}~0<x<t.\end{eqnarray}$$

We must now prescribe appropriate boundary data for (6.1). The simplest way to determine what boundary conditions are sufficient to solve the problem is to rewrite (6.1) in terms of the travelling-wave variable moving with the jet, $\unicode[STIX]{x1D709}=x-t$ , so that the jet tip is fixed at $\unicode[STIX]{x1D709}=0$ . Under this change of variables, (6.1) becomes

(6.2a,b ) $$\begin{eqnarray}H_{t}+\left(-\frac{H}{2}-\frac{1}{12}H^{3}P_{\unicode[STIX]{x1D709}}\right)_{\unicode[STIX]{x1D709}}=0,\quad H_{tt}=P\end{eqnarray}$$

for $-t<\unicode[STIX]{x1D709}<0$ . We note that, in this particular limit, the jet motion is exactly the same as that of a string above a thin air layer, with the tension of the string being much smaller than its mass. The model also bears similarities to models for flapping flags with small bending rigidity; see, for example, Connell & Yue (Reference Connell and Yue2007) and Shelley & Zhang (Reference Shelley and Zhang2011). Let us view (6.2b ) as an equation for $H$ given $P$ . The characteristics of (6.2b ) are simply $\unicode[STIX]{x1D709}=\text{const.}$ , and (6.2b ) is effectively an ordinary differential equation for $H$ given $P$ . Thus, we need two boundary conditions at $\unicode[STIX]{x1D709}=-t$ . It is also clear from (6.2a ) that we require two further boundary conditions: a third condition at the nozzle and one that is found by noting that the air pressure scale in the tip region is an order of magnitude smaller than that in the main body of the air layer, as described in appendix C. Motivated by the configuration illustrated in figure 5 at the nozzle, we prescribe the jet height, $H_{J}(t)$ , the jet angle, $G_{J}(t)$ , and zero flux of air back under the nozzle. Thus, returning to the $(x,t)$ frame, we solve (6.1) for $t>0$ subject to

(6.3a-d ) $$\begin{eqnarray}H(0,t)=H_{J}(t),\quad H_{x}(0,t)=G_{J}(t),\quad \left.\left(\frac{H}{2}-\frac{1}{12}(H^{3}P_{x})\right)\right|_{(0,t)}=0,\quad P(t,t)=0.\end{eqnarray}$$

In general, we must solve (6.1) numerically subject to (6.3). We do not require initial conditions because the jet has zero length at $t=0$ ; therefore, to initiate the numerical scheme, we use the small-time solution of (6.1) and (6.3), which we obtain by performing a regular perturbation analysis as $t\rightarrow 0^{+}$ . We find that

(6.4) $$\begin{eqnarray}\displaystyle & \displaystyle P\sim \frac{6}{H_{J}(0)^{2}}(x-t)+\frac{6}{H_{J}(0)^{2}}(x-t)((x-t){\dot{H}}_{J}(0)-(x+t)G_{J}(0)), & \displaystyle\end{eqnarray}$$

(6.5) $$\begin{eqnarray}\displaystyle H & {\sim} & \displaystyle H_{J}(0)+{\dot{H}}_{J}(0)t+G_{J}(0)x-\frac{1}{2}(x-t)((x+t)\ddot{H}_{J}(0)+2x{\dot{G}}_{J}(0))\nonumber\\ \displaystyle & & \displaystyle +\,\frac{3t^{2}}{H_{J}(0)^{2}}(x-t)+t(x-t)^{2}\left(\frac{6}{H_{J}(0)^{2}}+\frac{\ddot{G}_{J}(0)}{2}+\frac{\dddot{H}_{J}(0)}{2}\right)\nonumber\\ \displaystyle & & \displaystyle +\,(x-t)^{3}\left(\frac{3}{H_{J}(0)^{2}}+\frac{\ddot{G}_{J}(0)}{2}+\frac{\dddot{H}_{J}(0)}{3}\right)\end{eqnarray}$$

for $0<x<t$ as $t\rightarrow 0^{+}$ . When $x,t=O(\unicode[STIX]{x1D700}_{l})$ , the leading-order air and liquid flows are more complicated and we do not consider them here, although the early stage of jet formation is an important problem in its own right.

Here, we note that, under the assumptions we made at the start of this section, the jet tip moves ballistically: if the jet tip is defined by $N(t)=H(t,t)$ , we have, by (6.1b ) and (6.3a,b,d ),

(6.6) $$\begin{eqnarray}\ddot{N}(t)=(H_{xx}+2H_{xt}+H_{tt})|_{(t,t)}=P(t,t)=0,\end{eqnarray}$$

so that

(6.7) $$\begin{eqnarray}N(t)=(G_{J}(0)+{\dot{H}}_{J}(0))t+H_{J}(0).\end{eqnarray}$$

Thus, if we were to shoot the jet down at an angle towards the substrate, we would expect it to touch down on the substrate at a finite time $t=-H_{J}(0)/(G_{J}(0)+{\dot{H}}_{J}(0))$ , provided that the denominator is negative. In general, however, we are interested in whether the jet can be induced to touch down purely by the action of the pressure jump across it. We shall therefore concentrate on conditions that do not lead to tip touchdown.

Finally, if we had relaxed our choice of $\unicode[STIX]{x1D706}=1$ to $\unicode[STIX]{x1D706}=O(1)$ when choosing  $L$ , the pressure term on the right-hand side of (6.2b ) would have had a prefactor $\unicode[STIX]{x1D706}$ . Then, the limit in which $\unicode[STIX]{x1D706}\rightarrow 0^{+}$ is equivalent to neglecting the air pressure on the length scale $L$ (air effects being confined to shorter horizontal length scales). In this small- $\unicode[STIX]{x1D706}$ limit, to leading order, the jet particles simply travel ballistically at the height and angle at which they are ejected from the nozzle, so that

(6.8) $$\begin{eqnarray}H=H_{J}(t-x)+x({\dot{H}}_{J}(t-x)+G_{J}(t-x)),\end{eqnarray}$$

where a dot indicates differentiation with respect to argument. We shall refer back to (6.8) during our numerical analysis to highlight the influence of the cushioning layer.

6.2 Numerical solution

We have developed two separate codes to solve (6.1) and (6.3) numerically. The first is an implicit Newtonian solver on the growing domain which we describe in detail in § D.1. The second uses the finite element method having mapped to a fixed domain, as discussed in § D.2. The former has the benefit that it is relatively straightforward to implement, while the finite element code is advantageous in that it is readily refined about a given location, which makes investigation of touchdown possible. We validated the two codes, as discussed in § D.1 and Moore (Reference Moore2014), and they give good agreement with each other.

Various aspects of the solution are shown in figure 8 for a jet shot out at a constant height, parallel to the substrate, so that $H_{J}(t)=1$ and $G_{J}(t)=0$ . By (6.8), if there were no air, the jet would continue from the nozzle in a straight line, so the effect of the cushioning layer is significant. We have plotted on the physical domain $0<x<t$ , with the code initiated using the small-time solution (6.4), (6.5) evaluated at $t=t_{0}=0.01$ . In figure 8(a), it is clear that the numerical solution agrees well with the small-time solution (6.4), (6.5) for $t>t_{0}$ , with the maximum difference between the asymptotic and numerical solutions at, for example, $t=0.02$ being approximately $3.3\times 10^{-8}$ , which is excellent considering that (6.4) and (6.5) are only accurate up to  $O(t^{3})$ .

Figure 8.

Numerical solution of (6.1) and (6.3), with $H_{J}(t)=1$ and $G_{J}(t)=0$ . The code is initiated using (6.4) and (6.5) at $t=0.01$ . (a) Comparison of the small-time solution (blue thick dashed) and numerical solution (red thin solid) at, from left to right, $t=0.011$ , 0.015 and 0.02. The initial jet shape is given by the bold line. (b) Plots of the jet profile for $t=0.51,1.01,1.51,\ldots ,3.51,4.01$ . The tip is denoted by a filled circle in each plot, while the bold line corresponds to $t=2.51$ . (c) The pressure profile at $t=2.51$ . (d) The jet shape and the air streamlines at $t=2.51$ .

The lower free surface of the jet is shown at later times in figure 8(b). As time increases, the jet is greatly influenced by the air layer away from the tip. The small trough seen in the small-time solution in figure 8(a) grows and moves upstream, driven by a region of backflow in the air, which is a result of the no-flux condition below the nozzle, as illustrated in figure 8(d). As the free surface approaches the substrate, the air pressure increases. Eventually, we see in figure 8(b) that, at later times, the jet bends further from the nozzle, creating new crests and troughs. The emergence of new troughs may be explained as follows. It should be recalled that in $(x,t)$ coordinates the vertical velocity in the air on the free surface is given by

(6.9) $$\begin{eqnarray}V=H_{t}+H_{x},\end{eqnarray}$$

since $U=1$ on the jet surface at leading order. Upstream of a trough, we have $H_{x}<0$ , and, since the air is pulling the jet down (at least for small times), $H_{t}<0$ as well. Therefore, $V<0$ just upstream of a trough, and the jet continues to move towards the substrate. On the other hand, downstream of the trough, $H_{x}>0$ , so that eventually $H_{x}$ is larger than $H_{t}$ and the vertical velocity changes sign, causing the jet to move away from the substrate. This explains the undulations seen at later times in figure 8(b).

The large pressure variations in the air due to these undulations are depicted at $t=2.51$ in figure 8(c). Where the jet is closest to the substrate, the deflection has greatly increased resistance to the air flow, leading to very large pressure peaks. It should be noted that the largest pressure peak in figure 8(c) is relatively thin compared with the jet length, which is why the pressure solution converges more slowly than the free surface solution, as illustrated in figure 13 in § D.1.

We plot the streamlines in the air at $t=2.51$ in figure 8(d). Since we have imposed no flux of air back through the inlet at $x=0$ , we see some recirculation of air upstream of the first jet trough. Downstream of this trough, the air is flowing out towards the tip of the jet, with variations to the velocity profile caused by the jet shape.

6.3 Touchdown

Of particular interest to our applications is whether the jet touches down on the solid substrate in finite time. As can be seen in figure 8(b), the original trough approaches the substrate quite rapidly, but eventually slows down, with undulations forming further downstream. Other minima develop downstream of the first trough, eventually superseding the first trough in size. The numerical simulations fail to converge once the global minimum of the lower jet surface is predicted to be negative; for the example in figure 8, the code stops running at $t=t_{\ast }=4.5975$ , with the jet profile at this time depicted in figure 9. We suggest that this breakdown of the numerical simulation is due to the touchdown of the jet on the solid substrate. As is shown in figure 9, the profile touches down at a station downstream of the original trough that we saw in figure 8(a). It is unclear why the first trough does not touch down and this later one does.

Figure 9.

Free surface profile and corresponding air streamlines when the code stops running due to the free surface touching down at $t=t_{\ast }$ .

In other two-fluid problems, for example droplet impact onto a deep pool in Smith et al. (Reference Smith, Li and Wu2003) and Purvis & Smith (Reference Purvis and Smith2004) or droplet impact onto a solid in Duchemin & Josserand (Reference Duchemin and Josserand2011), the free surface exhibits similarity behaviour close to touchdown. We have investigated whether this is present here by looking at the location of the free surface minimum at times close to touchdown. We utilise the finite element solver, since it is possible to refine the grid close to the minimum. In order to look at such behaviour, we require an accurate prediction of both the time, $t_{c}$ , and location, $x_{c}$ , of touchdown. It is somewhat unreliable to use the time at which the code first predicts a negative value of $H$ as the touchdown time, since the large pressures and small values of $H$ are likely to make the numerics less accurate. Therefore, we isolate the global minimum value of $H$ at times close to $t_{\ast }$ and use Richardson extrapolation to predict $t_{c}$ and $x_{c}$ . We depict the behaviour of the minimum air layer thickness and the location of this minimum close to touchdown in figure 10. We have plotted against the time-to-touchdown variable $\unicode[STIX]{x1D70F}=t_{c}-t$ . We appear to have the minimum air layer thickness decaying like $\unicode[STIX]{x1D70F}^{3/2}$ and the location of the minimum decaying linearly in $\unicode[STIX]{x1D70F}$ , although as $\unicode[STIX]{x1D70F}$ gets small the resolution of the numerical scheme becomes more unreliable. These scales bear a resemblance to those in Duchemin & Josserand (Reference Duchemin and Josserand2011), although in that paper the lubrication equation in the air layer is coupled to a potential problem in the droplet. Clearly, additional numerical investigation is required to investigate touchdown further, although our preliminary numerical simulations suggest that the interface touches down in finite time, with the pressure becoming singular at the touchdown point.

Figure 10.

Log–log plots of the free surface minimum and the location of the minimum close to touchdown as a function of $\unicode[STIX]{x1D70F}=t_{c}-t$ , where $t_{c}$ is the estimated touchdown time using Richardson extrapolation. The straight lines correspond to $\unicode[STIX]{x1D70F}^{3/2}$ and $-\unicode[STIX]{x1D70F}$ respectively.

It should be noted that, when the air layer becomes very thin, our assumptions in the derivation of the jet model break down. In particular, the jet is now much thicker than the air layer locally. Furthermore, rapid changes in the centreline curvature may mean that surface tension becomes important close to the minima, which may inhibit or slow touchdown; see Purvis & Smith (Reference Purvis and Smith2004) and Duchemin & Josserand (Reference Duchemin and Josserand2011). Three-dimensional effects may also become important.

6.4 Varying jet angle

Our first example was for a jet shot parallel to a substrate at a fixed height. However, in reality, the inlet conditions are not likely to be so simple in many impact scenarios. Indeed, when considering the examples of droplet impact onto a deep pool as considered in Thoroddsen et al. (Reference Thoroddsen, Thoraval, Takehara and Etoh2011) and Thoraval et al. (Reference Thoraval, Takehara, Etoh, Popinet, Ray, Josserand, Zaleski and Thoroddsen2012), the ejecta is initially emitted at a shallow angle to the undisturbed pool water line, but this angle increases and the ejecta is eventually emitted almost parallel to the droplet side. In these experiments, it should be noted that the thinnest air layer is that between the ejecta and the drop, so this is the cushioning layer most relevant to our models. In order to model this situation, we now consider the following boundary conditions:

(6.10a,b ) $$\begin{eqnarray}H_{J}(t)=1,\quad G_{J}(t)=\text{e}^{-5t}.\end{eqnarray}$$

In (6.10), while the nozzle height is fixed, the angle varies such that the jet is initially aimed away from the substrate, but as $t$ increases, it approaches the horizontal.

We plot our numerical results in figure 11. Following (6.7), the jet tip increases linearly in time, moving further away from the substrate. However, due to the action of the air layer, the jet eventually begins to bend towards the substrate, eventually touching down at $x_{c}\approx 2.7285$ at time $t_{c}\approx 5.2919$ . In figure 11(b), we also plot a dashed line corresponding to the solution when the air is neglected, (6.8). From this, one can clearly see the influence of the cushioning layer in inducing the ‘S’-like shapes seen in the evolution of ejecta in, for example, the frames at $t=80$ and $130~\unicode[STIX]{x03BC}\text{s}$ in figure 2 in Thoraval et al. (Reference Thoraval, Takehara, Etoh, Popinet, Ray, Josserand, Zaleski and Thoroddsen2012). While our results can only be qualitative due to the fact that our model has a substrate rather than a liquid bulk, our reduced model predicts that an air-cushioning layer can cause a thin liquid sheet to bend and touch down, even if the jet is initially shot away from the substrate.

Figure 11.

Numerical results with inlet conditions given in (6.10). The code is again initiated at $t=0.01$ . (a) Comparison of the small-time solution (blue dashed) and numerical solution (red) at, from left to right, $t=0.011$ , 0.015 and 0.02. The initial jet shape is given by the bold black line. (b) Plots of the jet profile at time intervals of 0.5 from 0.51 to 5.01. The bold line corresponds to $t=2.51$ while the dashed line indicates the no-air solution given by (6.8) at $t=2.51$ . (c) The pressure profile at $t=2.51$ . (d) The jet shape and the air streamlines at $t=2.51$ .

6.5 Summary

We have shown that the centreline of a thin liquid jet can bend due to its interaction with a thin air-cushioning layer. Furthermore, we have given evidence that these jets can be affected to such an extent that they can touch down on the solid substrate upstream of the jet tip. This is reminiscent of what is seen in the experiments of Thoroddsen et al. (Reference Thoroddsen, Thoraval, Takehara and Etoh2011) (cf. figure 2) and the simulations of Thoraval et al. (Reference Thoraval, Takehara, Etoh, Popinet, Ray, Josserand, Zaleski and Thoroddsen2012). The ejecta bends and eventually touches down on the large liquid bulk, with fluid ligaments then being flung away from the impacting drop as satellite droplets. As mentioned previously, in the context of the current analysis, the thin air-cushioning layer should be taken to be that trapped between the ejecta and the upper bulk of the droplet, i.e. the thinnest air layer. Noticeably in, for example, the frames at $t=80$ and $130~\unicode[STIX]{x03BC}\text{s}$ in figure 2 in Thoraval et al. (Reference Thoraval, Takehara, Etoh, Popinet, Ray, Josserand, Zaleski and Thoroddsen2012), we see that the ejecta bends towards the upper bulk of the drop, and indeed the presence of an air bubble at $130~\unicode[STIX]{x03BC}\text{s}$ indicates that a touchdown event has recently occurred. A similar bubble entrapment is seen in figure 13(c) of Thoraval et al. (Reference Thoraval, Takehara, Etoh and Thoroddsen2013). Here, we have presented, albeit in the context of a solid substrate as opposed to a large liquid bulk, preliminary numerical evidence for finite-time touchdown caused by the air pressure jump across a thin jet, although we do not speculate on what happens to the jet after it touches the substrate.

The main aim of this section has been to show that a thin air-cushioning layer beneath a thin jet provides a mechanism for bending the jet centreline, which may eventually lead to touchdown. Naturally, given the approximations involved in the jet-root and jet-tip regions, our analysis is only indicative, but we have derived a toolbox of models that can be readily adapted to help to analyse a range of impact scenarios. Clearly, future work needs to investigate the behaviour close to touchdown in more detail, in particular concentrating on what physical effects may come back into the model when the air layer becomes sufficiently thin or the jet curvature becomes sufficiently large. Furthermore, it would be beneficial to use more general jet-inlet boundary conditions to see how this affects the jet evolution; certainly (6.1a,b ) becomes much more complicated for non-constant jet thicknesses and speeds.

7.1 Applicability of the model

While we have already discussed the results of our modelling in comparison with experimental data in §§ 5.3 and 6.5, we now wish to consider explicitly the key length scales given by (3.2) and (3.4). Since the models we have derived in this paper are meant to provide a toolbox for adaptation to particular impact scenarios, it is vital to establish the applicability of (3.2) and (3.4).

To this end, we shall consider three examples from the literature and compare the available parameters with those in our model. At the outset, we highlight two points. First, it has been difficult to find information on the typical thickness of the air layers trapped beneath splash jets and droplet lamellae. This is both because these layers are very thin, and hence hard to measure, and also because they may not have been the main object of interest in the study. Second, Thoraval et al. (Reference Thoraval, Takehara, Etoh, Popinet, Ray, Josserand, Zaleski and Thoroddsen2012), which we use as one of our examples (specifically because it is possible to estimate $\ell _{g}$ in this paper), is concerned with droplet impact on liquid pools in which there may be thin lubricating layers both above and below the jet at early times, which means that our model can only give a qualitative prediction for jet growth when one layer dominates the effect of the other.

Notwithstanding these concerns, we present in table 2 a list of values of the experimental/numerical data from the examples, where $L_{J}$ is a typical jet/lamella length. Furthermore, we use these data to calculate the relevant model parameters, leaving $\ell _{j}$ , $j=l,g$ , as unknowns when they are not given. The model parameters in table 2 should be interpreted as follows: parameters with a subscript $v$ correspond to values calculated assuming that the air layer is viscous – i.e. $L$ calculated from (3.2) – while parameters with a subscript $i$ correspond to values calculated assuming that the air layer is inviscid – i.e. $L$ calculated from (3.4). For each example, we aim to decide which regime is more appropriate: recall that if we take $L$ from (3.2), we require that $\unicode[STIX]{x1D700}_{g}^{2}\,\mathit{Re}_{g}=O(1)$ , while if we take $L$ from (3.4), we must have $\unicode[STIX]{x1D700}_{g}^{2}\,\mathit{Re}_{g}\gg 1$ .

Table 2.

Experimental and numerical data from the three papers concerning impact problems and our model parameters calculated using the data. It should be noted that a subscript  $v$ (respectively  $i$ ) indicates estimates made using the deflection length scale for a viscous (respectively inviscid) air layer.

For our first example, we consider a simulation of droplet impact onto a deep pool given in Thoraval et al. (Reference Thoraval, Takehara, Etoh, Popinet, Ray, Josserand, Zaleski and Thoroddsen2012), where we have estimated $\ell _{l}$ , $\ell _{g}$ and $L_{J}$ from the $80~\unicode[STIX]{x03BC}\text{s}$ image in figure 2 in that paper. The reader should recall our comment in § 6.5 that the thin air layer here is the upper layer. Using these figures, we see from table 2 that the sensible choice is to model the air as viscous, so that $L$ is given by (3.4) and the reduced Reynolds number in the air is $\unicode[STIX]{x1D700}_{g}^{2}\mathit{Re}_{g}\sim 0.0382$ . In this regime, we find that $\unicode[STIX]{x1D700}_{l}^{2}\mathit{Re}_{l}\sim 17.716$ , so that the analysis for viscous air and inviscid jets in § 4.4 is most appropriate, although it is possible that this experiment falls into the regime in which the boundary layer equations hold in the jet. It should be noted that the surface-tension parameter, $\unicode[STIX]{x1D6FE}$ , is small in the experiments of Thoraval et al. (Reference Thoraval, Takehara, Etoh, Popinet, Ray, Josserand, Zaleski and Thoroddsen2012). This is encouraging as it suggests that we were correct to ignore surface-tension effects at leading order in the centreline equation (4.8); moreover, as discussed in appendix C, this also implies that tip retraction occurs on a longer time scale than jet growth.

Driscoll & Nagel (Reference Driscoll and Nagel2011) present a series of experiments in which a droplet spreads on a solid substrate and ejects a thin sheet from the front of the spreading lamella. Using measurements of the intensity of light emitted from an LED, the authors show that a layer of air persists between this sheet and the substrate. Here, we use the results associated with their figure 1(b). Although no estimates of the thickness of the air layer are given, the authors state that the air layer is thicker than the ejected sheet. No explicit estimate of $U_{s}$ is given in Driscoll & Nagel (Reference Driscoll and Nagel2011), but in Driscoll et al. (Reference Driscoll, Stevens and Nagel2010) they perform similar experiments for which the speed of the lamella at the moment when the thin sheet is ejected from the front of the lamella (cf. figure 3) is approximately $3~\text{m}~\text{s}^{-1}$ . Using the estimates in table 2, we see that for air layers that are much thicker than $\ell _{c}\sim 3.963\times 10^{-5}$ , the reduced Reynolds number in the air is large and $L$ is given by (3.4), which means that $\unicode[STIX]{x1D700}_{l}^{2}\,\mathit{Re}_{l}\ll 8.79\times 10^{-2}$ . Therefore, we would expect the analysis in the inviscid air and viscous jet regime given in § 4.2 to be the most relevant. On the other hand, for air layers much thinner than $\ell _{c}$ , the reduced Reynolds number in the air is small and $L$ is given by (3.2), so that $\unicode[STIX]{x1D700}_{l}^{2}\,\mathit{Re}_{l}\sim 8.79\times 10^{-2}$ . Therefore, the viscous air and viscous jet limit described in § 4.1 is the appropriate model. We note that regardless of the value of $\ell _{g}$ , the surface-tension parameter $\unicode[STIX]{x1D6FE}$ is $O(1)$ , which again supports our decision to neglect surface tension in the leading-order jet motion in §§ 3–4. However, in this example, surface tension will have a role in the leading-order evolution of the centreline and the retraction of the jet tip in the growing-jet model. We shall discuss this further in § 7.2.

Our third and final example concerns the spreading of a viscous droplet on a solid substrate as described in Thoroddsen et al. (Reference Thoroddsen, Takehara and Etoh2010). The authors perform several experiments and note that as the droplet spreads, air bubbles become entrained under the lamella. They show that after a certain time, the lamella detaches from the substrate and rides on a layer of air that is thinner than the lamella itself. The lower free surface of the lamella can touch down on the substrate. We take our parameter estimates from figure 2(d) in that paper, although, as shown in table 2, we do not have concrete values for either $\ell _{l}$ or $\ell _{g}$ in this example. However, given that the air layer is suggested to be much thinner than the lamella, we postulate that the viscous air regime might be most appropriate, and can therefore compare $L$ calculated using (3.2) with $L_{J}$ and check the size of the resulting reduced Reynolds number in the air. For the air to be viscous, we require

(7.1) $$\begin{eqnarray}L_{J}>L\approx (4.05\times 10^{2})(\ell _{l}\ell _{g})^{2/3}~\text{m}\end{eqnarray}$$

and

(7.2) $$\begin{eqnarray}1\gg \unicode[STIX]{x1D700}_{g}^{2}\,\mathit{Re}_{g}=(1.982\times 10^{2})\frac{\ell _{g}^{4/3}}{\ell _{l}^{2/3}}.\end{eqnarray}$$

If the lamella had thickness $\ell _{l}\sim 10^{-4}$ m, an air layer thickness of $\ell _{g}\sim 10^{-5}$ m would satisfy (7.1) and (7.2), which does not seem to be implausible. In such an example, the reduced Reynolds number in the jet would also be small, so that the model in § 4.1 would be the most appropriate regime.

Although it would be desirable to have more data to compare with, there is good evidence that the length scales given by (3.2) and (3.4) are suitable for a range of scenarios, and, in addition, each of the regimes presented in §§ 4.1–4.4 may be relevant in different splashing/impact problems.

7.2 Summary

In this paper, we have presented models that investigate the role of an air-cushioning layer on the evolution of a liquid jet. Inspired by the splash jets produced in impact problems, as depicted in figures 2–3, we considered a long thin jet of liquid moving over a solid substrate with a thin layer of air cushioning it from the solid. Above the jet is a larger unbounded air region. In each model, we considered the limit in which deflections to the jet centreline are driven by the jump in air pressure across the jet. We looked at various regimes depending on the size of the reduced Reynolds number in each phase. The appropriate horizontal length scale of such deflections is given by

(7.3) $$\begin{eqnarray}L=L_{v}=\left(\frac{\unicode[STIX]{x1D70C}_{l}U_{s}\ell _{l}^{2}\ell _{g}^{2}}{\unicode[STIX]{x1D707}_{g}}\right)^{1/3}\end{eqnarray}$$

when the reduced Reynolds number in the air is small, so that viscous forces dominate, and by

(7.4) $$\begin{eqnarray}L=L_{i}=\frac{\ell _{l}\unicode[STIX]{x1D70C}_{l}^{1/2}}{\unicode[STIX]{x1D70C}_{g}^{1/2}}\end{eqnarray}$$

when the reduced Reynolds number is large, so that inertial forces dominate.

We considered the stability to small perturbations of an infinitely long constant-speed, constant-thickness liquid jet travelling over a constant-thickness air layer. We showed that the centreline of such a jet is destabilised by the pressure jump across it provided that the surface tension is sufficiently small. This is true both when the air is viscous dominated and when it is inertia dominated, and in each case we derived the wavenumber of the fastest-growing mode, given by (5.11) and (5.12) for the viscous air case and by (5.18) for the inviscid case. While comparisons were purely qualitative, we demonstrated that the experiments in Thoraval et al. (Reference Thoraval, Takehara, Etoh and Thoroddsen2013) fell into the viscous gas regime, so that ejecta instability could potentially be a mechanism for air bubble entrapment.

In § 6, we moved on to consider a jet emitted from a nozzle, concentrating on the regime in which the reduced Reynolds number in the air layer is small, so that the jet length scale is given by (7.3). We considered a constant-speed, constant-thickness jet shooting from a fixed orifice over a thin air cushion. After discussing the relevant scalings in the tip region in appendix C, we considered the particular regime in which retraction of the jet tip due to the action of surface tension happens on a longer time scale than the jet growth. By making a change of variables, we showed how the resulting equations can be related to the motion of a string on top of a thin air layer, and used this to motivate the boundary and initial conditions required to close the problem. When the surface tension is small, we only require one boundary condition at the jet tip: we imposed atmospheric air pressure motivated by the scaling arguments in appendix C. At the nozzle, we imposed boundary conditions fixing the thickness of the air cushion and the angle at which the jet is fired, while allowing no flux of air under the nozzle. By solving the resulting problem numerically, we were able to show that the jet bends as it advances due to the action of the air layer. Furthermore, we gave evidence that the jet can eventually touch the solid substrate away from its tip, which is reminiscent of the onset of droplet sling-shotting seen in droplet impact experiments in Thoraval et al. (Reference Thoraval, Takehara, Etoh, Popinet, Ray, Josserand, Zaleski and Thoroddsen2012); cf. figure 2 (note that, in this figure, the thin layer is between the jet and the main droplet).

Both of these models indicate the regimes in which the jet motion may be heavily influenced by the surrounding air as it grows, and thus it cannot be ignored in any modelling or study of splash jets. In particular, the simultaneous small aspect ratios of the jet and the air layer, as well as the relevance of the substrate in the tip region, are key in the evolution of the jet centreline, and these have often been neglected in existing models. Furthermore, the growing-jet model we considered in § 6 showed qualitatively similar behaviour to splash jets seen in droplet impacts. However, any in-depth comparisons require several further details.

In terms of theory, there are plenty of future areas of interest. There are several simple extensions to cover other scenarios: for example, replacing the solid substrate with a deep liquid pool, which is more relevant in the experiments of Thoraval et al. (Reference Thoraval, Takehara, Etoh, Popinet, Ray, Josserand, Zaleski and Thoroddsen2012) and Agbaglah et al. (Reference Agbaglah, Thoraval, Thoroddsen, Zhang, Fezzaa and Deegan2015); considering limits in which the jet is much thinner than the cushioning layer or vice versa; or considering asymptotic regimes in which the reduced Reynolds numbers in either phase are of order unity.

The touchdown process should also be investigated in more detail. While our zero-surface-tension model predicts a finite-time touchdown, it is well known (see, e.g., Duchemin & Josserand Reference Duchemin and Josserand2011) that surface tension inhibits touchdown in related problems, so a more careful analysis is necessary. Furthermore, for sufficiently thin air layers, it is likely that non-continuum effects will play a role in the evolution through the Knudsen number. Such an analysis is beyond the scope of this paper, but non-continuum effects can be incorporated via an effective slip at the solid–gas and liquid–gas interfaces; see Marchand et al. (Reference Marchand, Chan, Snoeijer and Andreotti2012), Riboux & Gordillo (Reference Riboux and Gordillo2014) and Sprittles (Reference Sprittles2017). This perturbation to the boundary condition alters the local lubrication model in the air layer, which in turn will influence the centreline evolution.

Major areas for future work concern the growing jet: namely the small-time jet growth, the flow around the jet tip and the boundary conditions at the jet root. We shall now discuss these in slightly more detail.

In this paper, we have primarily focused on jets and lamellae that have attained the characteristic small aspect ratio seen in experiments and depicted in figures 2–3. It is therefore important to check whether the earlier dynamics, when the nascent jet or lamella has an order-unity aspect ratio, plays a significant role in the long-time evolution. It is particularly interesting in light of recent experiments by Zhang et al. (Reference Zhang, Toole, Fezzaa and Deegan2012a ,Reference Zhang, Toole, Fezzaa and Deegan b ) and Agbaglah et al. (Reference Agbaglah, Thoraval, Thoroddsen, Zhang, Fezzaa and Deegan2015) that multiple jets can form when a droplet impacts a deep pool. These jets may coalesce with one another, the droplet or the later-emerging lamella, but all are examples of liquid jets moving over a narrow air gap. Moreover, in this early stage of impact, vorticity shedding at the jet root may also be important, as discussed in Thoraval et al. (Reference Thoraval, Takehara, Etoh, Popinet, Ray, Josserand, Zaleski and Thoroddsen2012), Moore et al. (Reference Moore, Ockendon, Ockendon and Oliver2014) and Agbaglah et al. (Reference Agbaglah, Thoraval, Thoroddsen, Zhang, Fezzaa and Deegan2015).

The flow around the jet tip is perhaps the most important outstanding question. Our assumption in this paper has been that the amount of fluid in the jet tip is small compared with that in the bulk of the jet, and that the flow around the tip is well represented by applying an air pressure condition there. However, as discussed in appendix C, for sufficiently large surface tension, tip retraction occurs at leading order in the growing-jet problem considered in § 6, which makes the problem more complicated. In the outer region discussed in § 6.1, the approximation that the jet tip moves ballistically with the initial jet speed is no longer appropriate: the jet loses mass and momentum to the tip, so the tip location needs to be determined as part of the problem. Introduction of surface tension to (6.2) in the $(\unicode[STIX]{x1D709},t)$ frame gives

(7.5a,b ) $$\begin{eqnarray}H_{t}+\left(-\frac{H}{2}-\frac{1}{12}H^{3}P_{\unicode[STIX]{x1D709}}\right)_{\unicode[STIX]{x1D709}}=0,\quad -\unicode[STIX]{x1D6FE}H_{\unicode[STIX]{x1D709}\unicode[STIX]{x1D709}}+H_{tt}=P.\end{eqnarray}$$

To tackle (7.5), we would need to determine whether we need a further boundary condition at the tip. This depends on the speed of tip retraction (see Moore Reference Moore2014), and would involve coupling to an in-depth tip-flow model.

It is possible that some of the ideas from the related field of sheet retraction could be employed in modelling of the tip in such cases; see, for example, Erneux & Davis (Reference Erneux and Davis1993), Savva & Bush (Reference Savva and Bush2009), Gordillo et al. (Reference Gordillo, Agbaglah, Duchemin and Josserand2011) and the references therein. When a thin liquid sheet is punctured, it can either retract or work to close up the resulting hole. Typical models derive a thin-film approximation in the main part of the sheet, similar to our models in § 4. However, despite the lack of asymptotic robustness, use of the same thin-film approximation in the rim is shown to give surprisingly accurate results to full numerical simulations; see, for example, Gordillo et al. (Reference Gordillo, Agbaglah, Duchemin and Josserand2011). A similar ad hoc approach may bear fruit in our growing-jet model. Similarly, vertical and horizontal force balances akin to those in Riboux & Gordillo (Reference Riboux and Gordillo2015) could also be used as approximations to the tip dynamics, although the role of the solid substrate must be taken into account when the tip is sufficiently close to the substrate.

At the jet root, it is probable that we need to think more closely about the flow in both the liquid and the air. A possible approach would be to match the jet to a Wagner-type inner region, as discussed in Howison et al. (Reference Howison, Ockendon and Wilson1991) or Moore et al. (Reference Moore, Ockendon and Oliver2013) with surrounding air. It is also possible to account for the role of viscous forces in the jet-root region; see, for example, Riboux & Gordillo (Reference Riboux and Gordillo2015). Moreover, as mentioned previously, there is strong evidence that vorticity accumulates and can shed from the base of splash jets, so it is likely that this needs to be taken into account in any modelling.

At a more basic level, all of our analysis in this paper has been restricted to two dimensions, whereas three-dimensional effects may be important in describing some of the phenomena seen in the aforementioned experiments. Fortunately, it is relatively straightforward to extend our jet models to three dimensions, as reported in Moore (Reference Moore2014).

In this paper, we have presented several possible models to describe the evolution of liquid jets in the presence of a thin cushioning air layer. We have identified the governing physics of the problem, deriving the crucial length scales involved for different regimes. Moreover, we have shown that the resulting models are unstable to small perturbations of the jet centreline and the air pressure, and that it is possible to make the jet centreline bend purely due to the action of an air pressure jump. This indicates that the presence of a thin layer of air can be a mechanism for the bending and touchdown of splash jets, or the instability of a lamella that has become detached from a substrate. The models here are intended to be useful tools for future studies of impact problems. The equations of motion can be coupled with more sophisticated models of the flow behaviour near the jet root and jet tip (such as those in Riboux & Gordillo Reference Riboux and Gordillo2015), or adapted for different geometries. The key results are the systematic derivation of reduced models governing the evolution of the jet centreline when it is deflected by an air-cushioning layer and the fact that the resulting motion leads to potential new mechanisms for jet touchdown and air entrapment.