2.1. Governing equations

Consider an incompressible ideal fluid of density $\rho$ ejected from an infinitely long horizontal slot (outlet) of fixed width. Let the flow be homogeneous in the along-the-outlet direction – i.e. depend on a single horizontal coordinate $x$ and the vertical coordinate $z$ (see figure 1).

Figure 1. The setting: a two-dimensional liquid curtain ejected from an outlet of width $2H$. Here, $(x,z)$ are the Cartesian coordinates and $(l,\tau )$ are the curvilinear coordinates associated with the curtain's centreline (corresponding to $\tau = 0$). The important quantities not shown in the figure include: the local angle $\alpha (l,t)$ between the centreline and the horizontal, its near-outlet value $\alpha _{0}$ (determined by the outlet's orientation – hence, time independent) and the near-outlet curvature $\alpha _{0}^{\prime }(t)=(\partial \alpha /\partial l) _{l=0}$.

Unless the liquid is ejected vertically, the curtain's trajectory is curved due to gravity, making the Cartesian coordinates awkward to use. It is more convenient to employ curvilinear coordinates associated with the curtain's centreline (Entov & Yarin Reference Entov and Yarin1984; Wallwork Reference Wallwork2001; Wallwork et al. Reference Wallwork, Decent, King and Schulkes2002; Shikhmurzaev & Sisoev Reference Shikhmurzaev and Sisoev2017; Decent et al. Reference Decent, Părău, Simmons and Uddin2018; Benilov Reference Benilov2019).

Consider curvilinear coordinates $( l,\tau )$ related to their Cartesian counterparts by

(2.1a,b)\begin{equation} x=x(l,\tau ,t),\quad z=z(l,\tau ,t), \end{equation}

where $t$ is the time. Let the set $( l,\tau )$ be orthogonal with a unit Jacobian,

(2.2a,b)\begin{equation} \frac{\partial x}{\partial l}\frac{\partial x}{\partial \tau }+\frac{ \partial z}{\partial l}\frac{\partial z}{\partial \tau }=0,\quad \frac{ \partial x}{\partial l}\frac{\partial z}{\partial \tau }-\frac{\partial x}{ \partial \tau }\frac{\partial z}{\partial l}=1. \end{equation}

Since the solution of (2.2a,b) is not constrained by boundary conditions, the relationship between $(x,z)$ and $( l,\tau )$ is not unique, leaving one an opportunity to choose the set $( l,\tau )$ that makes the forthcoming calculations simpler.

In what follows, the so-called Lamé coefficients

(2.3a,b)\begin{equation} h_{l}=\sqrt{\left( \frac{\partial x}{\partial l}\right) ^{2}+\left( \frac{ \partial z}{\partial l}\right) ^{2}},\quad h_{\tau }=\sqrt{\left( \frac{ \partial x}{\partial \tau }\right) ^{2}+\left( \frac{\partial z}{\partial \tau }\right) ^{2}} \end{equation}

will be needed. It follows from (2.2a,b) that $h_{l}h_{\tau }=1$, i.e. the transformation $(x,z) \rightarrow ( l,\tau )$ preserves the elemental area.

Let the flow be characterised by the velocity components $( u_{l},u_{\tau })$ and the pressure $p$. Representing the gravitational force by $-\rho g\boldsymbol {\nabla }z$ ($g$ is the acceleration due to gravity), one can write the Euler equations in the form

(2.4)\begin{align} &h_{l}\frac{\partial u_{l}}{\partial t}+\left[ u_{l}-\frac{1}{h_{l}}\left( \frac{\partial x}{\partial l}\frac{\partial x}{\partial t}+\frac{\partial z}{\partial l}\frac{\partial z}{\partial t}\right) \right] \left( \frac{ \partial u_{l}}{\partial l}+\frac{u_{\tau }}{h_{\tau }}\dfrac{\partial h_{l} }{\partial \tau }\right) \nonumber\\ &\quad +\frac{h_{l}}{h_{\tau }}\left[ u_{\tau }-\frac{1}{h_{\tau }}\left( \frac{ \partial x}{\partial \tau }\frac{\partial x}{\partial t}+\frac{\partial z}{ \partial \tau }\frac{\partial z}{\partial t}\right) \right] \frac{\partial u_{l}}{\partial \tau } \nonumber\\ &\quad +\frac{u_{\tau }}{h_{\tau }}\left[ \dfrac{\partial x}{\partial l}\frac{ \partial ^{2}x}{\partial t\,\partial \tau }+\dfrac{\partial z}{\partial l} \frac{\partial ^{2}z}{\partial t\,\partial \tau }-\frac{1}{h_{\tau }^{2}} \left( \frac{\partial x}{\partial \tau }\frac{\partial x}{\partial t}+\frac{ \partial z}{\partial \tau }\frac{\partial z}{\partial t}\right) \left( \dfrac{\partial x}{\partial l}\frac{\partial ^{2}x}{\partial \tau ^{2}}+ \dfrac{\partial z}{\partial l}\frac{\partial ^{2}z}{\partial \tau ^{2}} \right) -u_{\tau }\frac{\partial h_{\tau }}{\partial l}\right] \nonumber\\ &\quad +\dfrac{1}{\rho }\dfrac{\partial p}{\partial l}={-}g\dfrac{\partial z}{ \partial l}, \end{align}

(2.5)\begin{align} & h_{\tau }\frac{\partial u_{\tau }}{\partial t}+\frac{h_{\tau }}{h_{l}}\left[ u_{l}-\frac{1}{h_{l}}\left( \frac{\partial x}{\partial l}\frac{\partial x}{\partial t}+\frac{\partial z}{\partial l}\frac{\partial z}{\partial t} \right) \right] \frac{\partial u_{\tau }}{\partial l} \nonumber\\ &\quad +\left[ u_{\tau }-\frac{1}{h_{\tau }}\left( \frac{\partial x}{\partial \tau } \frac{\partial x}{\partial t}+\frac{\partial z}{\partial \tau }\frac{ \partial z}{\partial t}\right) \right] \left( \frac{\partial u_{\tau }}{ \partial \tau }+\frac{u_{l}}{h_{l}}\frac{\partial h_{\tau }}{\partial l} \right) \nonumber\\ &\quad+\frac{u_{l}}{h_{l}}\left[ \dfrac{\partial x}{\partial \tau }\frac{\partial ^{2}x}{\partial t\,\partial l}+\dfrac{\partial z}{\partial \tau }\frac{ \partial ^{2}z}{\partial t\,\partial l}-\frac{1}{h_{l}^{2}}\left( \frac{ \partial x}{\partial l}\frac{\partial x}{\partial t}+\frac{\partial z}{ \partial l}\frac{\partial z}{\partial t}\right) \left( \dfrac{\partial x}{ \partial \tau }\frac{\partial ^{2}x}{\partial l^{2}}+\dfrac{\partial z}{ \partial \tau }\frac{\partial ^{2}z}{\partial l^{2}}\right) -u_{l}\frac{ \partial h_{l}}{\partial \tau }\right] \nonumber\\ &\quad +\dfrac{1}{\rho }\dfrac{\partial p}{\partial \tau }={-}g\dfrac{\partial z}{ \partial \tau }, \end{align}

(2.6)\begin{align} & \hspace{11pc}\frac{\partial \left( u_{l}h_{\tau }\right) }{\partial l}+\frac{\partial \left( u_{\tau }h_{l}\right) }{\partial \tau }=0. \end{align}

For a stationary coordinate system (such that $\partial x/\partial l=\partial x/\partial t=0$), (2.4)–(2.6) can be found in most textbooks (e.g. Kochin, Kibel & Roze Reference Kochin, Kibel and Roze1964), with the general case considered in B19.

Let the liquid be bounded by two free surfaces described by equations $\tau =\tau _{-}(l,t)$ and $\tau =\tau _{+}(l,t)$, where the functions $\tau _{\pm }(l,\tau )$ satisfy the free-boundary kinematic condition. If written in terms of time-dependent curvilinear coordinates, it takes the form

(2.7)\begin{align} &\frac{\partial \tau _{{\pm} }}{\partial t}+\frac{1}{h_{l}}\left[ u_{l}-\dfrac{1 }{h_{l}}\left( \frac{\partial x}{\partial l}\frac{\partial x}{\partial t}+ \frac{\partial z}{\partial l}\frac{\partial z}{\partial t}\right) \right]\dfrac{\partial \tau _{{\pm} }}{\partial l} \nonumber\\ &\quad -\frac{1}{h_{\tau }}\left[ u_{\tau }-\dfrac{1}{h_{\tau }}\left( \frac{ \partial x}{\partial \tau }\frac{\partial x}{\partial t}+\frac{\partial z}{ \partial \tau }\frac{\partial z}{\partial t}\right) \right] =0\quad \text{if}\ \tau =\tau _{{\pm} }. \end{align}

The dynamic boundary condition, in turn, assumes that the pressure at a free boundary is determined by its curvature and surface tension $\sigma$,

(2.8)\begin{equation} p={\mp} \sigma \left\{ \frac{\partial }{\partial l}\left[ \frac{\dfrac{h_{\tau }^{2}}{h_{l}}\dfrac{\partial \tau _{{\pm} }}{\partial l}}{\sqrt{1+\left( \dfrac{h_{\tau }}{h_{l}}\dfrac{\partial \tau _{{\pm} }}{\partial l}\right) ^{2} }}\right] -\frac{\partial }{\partial \tau }\left[ \frac{h_{l}}{\sqrt{ 1+\left( \dfrac{h_{\tau }}{h_{l}}\dfrac{\partial \tau _{{\pm} }}{\partial l} \right) ^{2}}}\right] \right\} \quad \text{if}\ \tau =\tau _{{\pm} } \end{equation}

(for more details on conditions (2.7)–(2.8), see B19).

Let the outlet's width be $2H$. In this paper, the simplest particular case is examined, where the streamwise component of the velocity at the outlet is not sheared (i.e. is independent of $\tau$), while the cross-stream component is identically zero,

(2.9)\begin{equation} u_{l}=u_{0},\quad u_{\tau }=0\quad \text{if}\ l=0,\quad \tau \in \left({-}H,H\right) , \end{equation}

where $u_{0}$ may depend on $t$. The outlet conditions for the curtain's boundaries are, obviously,

(2.10)\begin{equation} \tau _{{\pm} }={\pm} H\quad \text{if}\ l=0. \end{equation}

Given an initial condition and a specific solution of the coordinate equations (2.2a,b), the boundary-value problem (2.3a,b)–(2.10) (presumably uniquely) determines $p$, $u_{l}$, $u_{\tau }$ and $\tau _{\pm }$.

It is convenient to identify the curtain's centreline with the curve $\tau =0$. To do so, we require that

(2.11)\begin{equation} \tau _{{\pm} }={\pm} W, \end{equation}

where $W(l,t)$ is the curtain's half-width.

2.2. The scaling

There are two non-dimensional parameters in the problem at hand. The Froude number

(2.12)\begin{equation} Fr=\frac{u_{0}^{2}}{gH}, \end{equation}

reflects the balance of inertia and gravity, and the Weber number

(2.13)\begin{equation} We=\frac{\rho Hu_{0}^{2}}{\sigma } \end{equation}

that of inertia and surface tension. This paper is concerned the limit $We\approx 1$, which implies that the velocity should be scaled by

(2.14)\begin{equation} U=\left( \frac{\sigma }{\rho H}\right) ^{1/2}. \end{equation}

In B19, the curtain's spatial scale $L$ was derived from the assumption that the centrifugal force (due to the curtain's curvature), surface tension, and gravity are in balance – which implies that $L=H\,Fr$, so that the slender-curtain approximation is valid if $Fr\gg 1$.

If, however, $We\rightarrow 1$, the leading-order surface tension and centrifugal force tend to cancel each other. As a result, gravity remains unopposed and makes the curtain bend much more steeply (see figure 2 in B19).

To determine $L$ for the case $We\approx 1$, one should balance gravity with higher-order corrections to centrifugal force and surface tension – which have to be calculated first, of course – which is, however, impossible without a conjecture as to what $L$ actually is. This task is exacerbated even more by the fact that the next-to-leading-order corrections do contribute to the eventual asymptotic equation, forcing one to delve into yet another order.

Thus, $L$ was effectively determined through a trial-and-error approach, and it has turned out that a consistent asymptotic theory can be derived if

(2.15)\begin{equation} L=\frac{H}{\epsilon }, \end{equation}

where $\epsilon$ is the cube root of the Bond number,

(2.16)\begin{equation} \epsilon =\left( \frac{\rho gH^{2}}{\sigma }\right) ^{1/3}, \end{equation}

or, equivalently, $\epsilon =( We/Fr) ^{1/3}\approx Fr^{-1/3}$.

In the present paper, $\epsilon$ is assumed small – hence, $L\gg H$. The larger scale, $L$, will be used to non-dimensionalise the Cartesian coordinates $(x,z)$ and the streamwise variable $l$, whereas the cross-stream variable $\tau$ and the curtain's half-width $W$ will be non-dimensionalised by $H$.

The following non-dimensional variables will be used:

(2.17)\begin{equation} \left.\begin{gathered} l_{nd}=\dfrac{l}{L},\quad \tau _{nd}=\dfrac{\tau }{H},\quad t_{nd}=\dfrac{t }{T},\quad x_{nd}=\dfrac{x}{L}, \quad z_{nd}=\dfrac{z}{L},\\ \left( u_{l}\right) _{nd}=\dfrac{u_{l}}{U},\quad \left( u_{\tau }\right) _{nd}=\dfrac{u_{\tau }}{\epsilon ^{2}U},\quad p_{nd}=\dfrac{p}{P},\quad W_{nd}=\dfrac{W}{H}. \end{gathered}\right\} \end{equation}

The pressure scale $P$ is such that the centrifugal force is, to leading order, balanced by the capillary-pressure gradient, which implies

(2.18)\begin{equation} P=\frac{\rho gH}{\epsilon ^{2}}. \end{equation}

The time scale $T$ is determined by the balance of the Coriolis force and gravity – i.e. the time derivatives of $x$ and $z$ in (2.5) and its right-hand side, respectively – which amounts to

(2.19)\begin{equation} T=\frac{H}{\epsilon ^{3}U}. \end{equation}

Rewriting (2.2a,b)–(2.8) in terms of variables (2.17)–(2.19), taking into account (2.14)–(2.16) and omitting the subscript $_{nd}$, one obtains

(2.20a,b)\begin{gather} \frac{\partial x}{\partial l}\frac{\partial x}{\partial \tau }+\frac{ \partial z}{\partial l}\frac{\partial z}{\partial \tau }=0,\quad \frac{1}{ \epsilon }\left( \frac{\partial x}{\partial l}\frac{\partial z}{\partial \tau }-\frac{\partial x}{\partial \tau }\frac{\partial z}{\partial l}\right) =1, \end{gather}

(2.21a,b)\begin{gather} h_{l}=\sqrt{\left( \frac{\partial x}{\partial l}\right) ^{2}+\left( \frac{ \partial z}{\partial l}\right) ^{2}},\quad h_{\tau }=\frac{1}{\epsilon } \sqrt{\left( \frac{\partial x}{\partial \tau }\right) ^{2}+\left( \frac{ \partial z}{\partial \tau }\right) ^{2}}, \end{gather}

(2.22)\begin{align} &\epsilon h_{l}\frac{\partial u_{l}}{\partial t}+\left[ u_{l}-\frac{\epsilon ^{2}}{h_{l}}\left( \frac{\partial x}{\partial l}\frac{\partial x}{\partial t} +\frac{\partial z}{\partial l}\frac{\partial z}{\partial t}\right) \right] \left( \frac{1}{\epsilon }\frac{\partial u_{l}}{\partial l}+\frac{u_{\tau }}{ h_{\tau }}\dfrac{\partial h_{l}}{\partial \tau }\right)\nonumber\\ &\quad +\frac{h_{l}}{h_{\tau }}\left[ u_{\tau }-\frac{1}{\epsilon h_{\tau }}\left( \frac{\partial x}{\partial \tau }\frac{\partial x}{\partial t}+\frac{ \partial z}{\partial \tau }\frac{\partial z}{\partial t}\right) \right] \frac{\partial u_{l}}{\partial \tau } \nonumber\\ &\quad +\frac{\epsilon ^{2}u_{\tau }}{h_{\tau }}\left[ \dfrac{\partial x}{\partial l }\frac{\partial ^{2}x}{\partial t\,\partial \tau }+\dfrac{\partial z}{ \partial l}\frac{\partial ^{2}z}{\partial t\,\partial \tau }-\frac{1}{ \epsilon ^{2}h_{\tau }^{2}}\left( \frac{\partial x}{\partial \tau }\frac{ \partial x}{\partial t}+\frac{\partial z}{\partial \tau }\frac{\partial z}{ \partial t}\right) \left( \dfrac{\partial x}{\partial l}\frac{\partial ^{2}x }{\partial \tau ^{2}}+\dfrac{\partial z}{\partial l}\frac{\partial ^{2}z}{ \partial \tau ^{2}}\right) \right.\nonumber\\ &\quad \left.-\epsilon u_{\tau }\frac{\partial h_{\tau }}{ \partial l}\right] +\frac{\partial p}{\partial l}={-}\epsilon \dfrac{\partial z}{\partial l}, \end{align}

(2.23)\begin{align} &\epsilon ^{4}h_{\tau }\frac{\partial u_{\tau }}{\partial t}+\frac{\epsilon ^{2}h_{\tau }}{h_{l}}\left[ u_{l}-\frac{\epsilon ^{2}}{h_{l}}\left( \frac{ \partial x}{\partial l}\frac{\partial x}{\partial t}+\frac{\partial z}{ \partial l}\frac{\partial z}{\partial t}\right) \right] \frac{\partial u_{\tau }}{\partial l} \nonumber\\ &\quad +\epsilon ^{3}\left[ u_{\tau }-\frac{1}{\epsilon h_{\tau }}\left( \frac{ \partial x}{\partial \tau }\frac{\partial x}{\partial t}+\frac{\partial z}{ \partial \tau }\frac{\partial z}{\partial t}\right) \right] \left( \frac{ \partial u_{\tau }}{\partial \tau }+\frac{u_{l}}{\epsilon h_{l}}\frac{ \partial h_{\tau }}{\partial l}\right) \nonumber\\ &\quad +\frac{u_{l}}{h_{l}}\left[ \epsilon \left( \dfrac{\partial x}{\partial \tau } \frac{\partial ^{2}x}{\partial t\,\partial l}+\dfrac{\partial z}{\partial \tau }\frac{\partial ^{2}z}{\partial t\,\partial l}\right) -\frac{\epsilon }{ h_{l}^{2}}\left( \frac{\partial x}{\partial l}\frac{\partial x}{\partial t}+ \frac{\partial z}{\partial l}\frac{\partial z}{\partial t}\right) \left( \dfrac{\partial x}{\partial \tau }\frac{\partial ^{2}x}{\partial l^{2}}+ \dfrac{\partial z}{\partial \tau }\frac{\partial ^{2}z}{\partial l^{2}} \right) \right.\nonumber\\ &\quad \left.-\frac{u_{l}}{\epsilon }\frac{\partial h_{l}}{\partial \tau }\right]+\frac{\partial p}{\partial \tau }={-}\epsilon \dfrac{\partial z}{\partial \tau }, \end{align}

(2.24)\begin{equation} \frac{1}{\epsilon }\frac{\partial \left( u_{l}h_{\tau }\right) }{\partial l}+ \frac{\partial \left( u_{\tau }h_{l}\right) }{\partial \tau }=0, \end{equation}

(2.25)\begin{align} &\epsilon \frac{\partial W}{\partial t}+\dfrac{1}{\epsilon h_{l}}\left[ u_{l}- \dfrac{\epsilon ^{2}}{h_{l}}\left( \frac{\partial x}{\partial l}\frac{ \partial x}{\partial t}+\frac{\partial z}{\partial l}\frac{\partial z}{ \partial t}\right) \right] \dfrac{\partial W}{\partial l} \nonumber\\ &\quad \mp \frac{1}{h_{\tau }}\left[ u_{\tau }-\dfrac{1}{\epsilon h_{\tau }}\left( \frac{\partial x}{\partial \tau }\frac{\partial x}{\partial t}+\frac{ \partial z}{\partial \tau }\frac{\partial z}{\partial t}\right) \right] =0\quad \text{if}\ \tau ={\pm} W, \end{align}

(2.26)\begin{equation} p={-}\epsilon \frac{\partial }{\partial l}\left[ \frac{\dfrac{h_{\tau }^{2}}{ h_{l}}\dfrac{\partial W}{\partial l}}{\sqrt{1+\left( \dfrac{\epsilon h_{\tau }}{h_{l}}\dfrac{\partial W}{\partial l}\right) ^{2}}}\right] \pm \frac{1}{ \epsilon }\frac{\partial }{\partial \tau }\left[ \frac{h_{l}}{\sqrt{1+\left( \dfrac{\epsilon h_{\tau }}{h_{l}}\dfrac{\partial W}{\partial l}\right) ^{2}}} \right] \quad \text{if}\ \tau ={\pm} W. \end{equation}

When non-dimensionalising the boundary conditions (2.9) and (2.10), it is convenient to introduce the non-dimensional ‘excess injection velocity’ $v_{0}$ such that

(2.27)\begin{equation} u_{0}=U(1+\epsilon ^{2}v_{0}). \end{equation}

Then, (2.9) becomes

(2.28)\begin{equation} u_{l}=1+\epsilon ^{2}v_{0},\quad u_{\tau }=0\quad \text{if}\ l=0,\quad \tau \in \left({-}1,1\right) . \end{equation}

Finally, the non-dimensional version of the boundary condition (2.10) and (2.11), is

(2.29)\begin{equation} W=1\quad \text{if}\ l=0. \end{equation}

2.3. How should the curvilinear coordinates be chosen?

As stated before, the curvilinear coordinates $( l,\tau )$ are associated with the curtain's centreline – but this association has not been reflected in the general equations (2.20a,b) relating $(l,\tau )$ to the Cartesian coordinates.

To single out the desired solution of (2.20a,b), introduce the centreline's Cartesian coordinates $x=\bar {x}(l,t)$ and $z=\bar {z}(l,t)$, and the local angle $\alpha (l,t)$ between the centreline and the horizontal. Let $l$ be the centreline's arclength, which implies

(2.30a,b)\begin{gather} \frac{\partial \bar{x}}{\partial l}=\cos \alpha ,\quad \frac{\partial \bar{z }}{\partial l}=\sin \alpha , \end{gather}

(2.31)\begin{gather} \bar{x}=0,\quad \bar{z}=0\quad \text{if}\ l=0. \end{gather}

For a given $\alpha (l,t)$, (2.30a,b) and (2.31) uniquely determine $\bar {x}(l,t)$ and $\bar {z}(l,t)$.

Now, seek a solution (2.20a,b) in the form of a series in powers of $( \epsilon \tau )$, with the zero-order terms in the expansions of $x$ and $z$ being $\bar {x}(l)$ and $\bar {z}(l)$, respectively. After straightforward algebra (for more detail, see B19), one obtains

(2.32)\begin{gather} x=\bar{x}-\epsilon \tau \sin \alpha -\frac{(\epsilon \tau )^{2}}{2}\frac{ \partial \alpha }{\partial l}\sin \alpha -(\epsilon \tau )^{3}\left[ \frac{1 }{6}\frac{\partial ^{2}\alpha }{\partial l^{2}}\cos \alpha +\frac{1}{2} \left( \frac{\partial \alpha }{\partial l}\right) ^{2}\sin \alpha \right] + {O}(\epsilon ^{4}), \end{gather}

(2.33)\begin{gather}z=\bar{z}+\epsilon \tau \cos \alpha +\frac{(\epsilon \tau )^{2}}{2}\frac{ \partial \alpha }{\partial l}\cos \alpha -(\epsilon \tau )^{3}\left[ \frac{1 }{6}\frac{\partial ^{2}\alpha }{\partial l^{2}}\sin \alpha -\frac{1}{2} \left( \frac{\partial \alpha }{\partial l}\right) ^{2}\cos \alpha \right] + {O}(\epsilon ^{4}). \end{gather}

Substitution of these expressions into (2.21a,b) yields the following expressions for the Lamé coefficients:

(2.34)\begin{gather} h_{l}=1-\epsilon \tau \frac{\partial \alpha }{\partial l}-\frac{(\epsilon \tau )^{2}}{2}\left( \frac{\partial \alpha }{\partial l}\right) ^{2}-(\epsilon \tau )^{3}\left[ \frac{1}{6}\frac{\partial ^{3}\alpha }{ \partial l^{3}}+\frac{1}{2}\left( \frac{\partial \alpha }{\partial l}\right) ^{3}\right] +{O}(\epsilon ^{4}), \end{gather}

(2.35)\begin{gather}h_{\tau }=1+\epsilon \tau \frac{\partial \alpha }{\partial l}+\frac{ 3(\epsilon \tau )^{2}}{2}\left( \frac{\partial \alpha }{\partial l}\right) ^{2}+(\epsilon \tau )^{3}\left[ \frac{1}{6}\frac{\partial ^{3}\alpha }{ \partial l^{3}}+\frac{5}{2}\left( \frac{\partial \alpha }{\partial l}\right) ^{3}\right] +{O}(\epsilon ^{4}). \end{gather}

3.1. Summary

All characteristics of the flow can be related to the curtain's coordinates $\bar {x}(l,t)$ and $\bar {z}(l,t)$, and the local angle $\alpha (l,t)$ between its centreline and the horizontal. The former satisfy (2.30a,b) and (2.31), and the latter is governed by the following asymptotic equation:

(3.1)\begin{equation} \frac{\partial \alpha }{\partial t}+\frac{\partial \alpha }{\partial l}\left[ v_{0}-\frac{1}{2}\bar{z}+\frac{1}{12}\left( \frac{\partial \alpha }{\partial l}\right) ^{2}-\frac{1}{6}\alpha _{0}^{\prime 2}\right] +\frac{1}{6}\frac{ \partial ^{3}\alpha }{\partial l^{3}}={-}\frac{1}{2}\cos \alpha , \end{equation}

where $v_{0}$ is the excess injection velocity (see (2.28)) and

(3.2)\begin{equation} \alpha _{0}^{\prime }=\left( \frac{\partial \alpha }{\partial l}\right) _{l=0} \end{equation}

is the curtain's curvature near the outlet. This characteristic plays an important role in the curtain's global dynamics.

Equation (3.1) requires two boundary conditions at the outlet, with one of these prescribing the ejection angle,

(3.3)\begin{equation} \alpha =\alpha _{0}\quad \text{if}\ l=0, \end{equation}

where $\alpha _{0}$ may vary with $t$. An additional boundary condition follows from the analysis of a near-outlet boundary layer: the solution there matches the global solution only if the latter satisfies

(3.4)\begin{equation} \frac{\partial ^{2}\alpha }{\partial l^{2}}=0\quad \text{if}\ l=0. \end{equation}

The set comprising (3.1)–(3.4) and (2.30a,b)–(2.31) is the desired asymptotic model for liquid curtains with a large Froude number and a near-unity Weber number.

The following comments should be helpful to understand the physical meaning of the asymptotic model.

1. (i) Equation (3.1) is, essentially, the cross-stream momentum equation integrated across the curtain, and so each of its terms can be interpreted physically.

   1. (a) The time derivative represents the Coriolis force – which is a cross product of the fluid velocity by the angular velocity of the coordinate frame. The former is, to leading order, unity and $\partial \alpha /\partial t$ is the latter.

   2. (b) The term $v_{0}-\frac {1}{2}\bar {z}$ is an approximate expression for the excess velocity affected (through the energy conservation) by the local height. Physically, it is the phase velocity of upstream-propagating capillary waves.

   3. (c) The right-hand side of (3.1) represents the force of gravity.

   4. (d) The remaining terms represent the centrifugal force and capillary-pressure gradient (since they both depend on the curtain's curvature, it is impossible to match each of the corresponding terms to a single effect).

2. (ii) The presence of $v_{0}$ and $\alpha _{0}^{\prime }$ among the coefficients of equation (3.1) makes the curtain dynamics non-local, as the effect of what happens near the outlet is immediately sensed downstream. The non-locality is a result of the difference between the (slow) time scale of the curtain evolution and the (fast) time scale of fluid particles passing through the region where (3.1) applies.

3. (iii) The derived model governs sinuous oscillations of curtains, whereas varicose oscillations (which are also present in the exact model – see Benilov, Barros & O'Brien Reference Benilov, Barros and O'Brien2016) have been scaled out. The latter are still generated in a boundary layer near the outlet (as shown in Appendix A), but their small amplitude and short wavelength make their impact on the global dynamics negligible.

4. (iv) It can be demonstrated that

   (3.5)\begin{equation} We=(1+\epsilon ^{2}v_{0}) ^{2}, \end{equation}
   i.e. the deviation of $We$ from unity is controlled by the excess injection velocity $v_{0}$.

   Curtains with negative (positive) $v_{0}$ will be referred to as subcritical (supercritical).

3.2. Derivation of (3.1)

The solution of set (2.22)–(2.29), (2.32)–(2.35) will be sought in the form

(3.6)\begin{equation} \left.\begin{gathered} u_{l}=1+\epsilon u_{l}^{(1)}+\epsilon ^{2}u_{l}^{(2)}\ldots,\quad u_{\tau }=u_{l}^{(0)}+\epsilon u_{l}^{(1)}\ldots, \quad p=p^{(0)}+\epsilon p^{(1)}\ldots,\\ W=1+\epsilon W^{(1)}+\epsilon ^{2}W^{(2)}\ldots\,. \end{gathered}\right\} \end{equation}

The fact that the leading-order $u_{l}$ and $W$ are constant suggests that the described dynamics mainly occurs near the outlet, so that the change in the potential energy is too small to alter the fluid velocity and, consequently, the curtain's width. The same fact also implies that the curtain's evolution mainly consists in the centreline changing its shape.

Unlike the physical variables, those associated with the coordinate system do not have to be expanded. Higher-order corrections for $\bar {x}$, $\bar {z}$ and $\alpha$ need to be introduced only if one intends to derive asymptotic equations for them, and I do not.

To leading order, (2.22)–(2.29) and (2.32)–(2.35) yield

(3.7)\begin{gather} \frac{\partial u_{l}^{(1)}}{\partial l}+\frac{\partial p^{(0)}}{\partial l} =0, \end{gather}

(3.8)\begin{gather}\frac{\partial \alpha }{\partial l}+\frac{\partial p^{(0)}}{\partial \tau } =0, \end{gather}

(3.9)\begin{gather}\frac{\partial }{\partial l}\left( u_{l}^{(1)}+\tau \frac{\partial \alpha }{ \partial l}\right) +\frac{\partial u_{\tau }^{(0)}}{\partial \tau }=0, \end{gather}

(3.10)\begin{gather}\dfrac{\partial W^{(1)}}{\partial l}\mp \left( u_{\tau }^{(0)}+\frac{ \partial \bar{x}}{\partial t}\sin \alpha -\frac{\partial \bar{z}}{\partial t} \cos \alpha \right) =0\quad \text{if}\ \tau ={\pm} 1, \end{gather}

(3.11)\begin{gather}p^{(0)}={\mp} \frac{\partial \alpha }{\partial l}\quad \text{if}\ \tau ={\pm} 1. \end{gather}

(3.12)\begin{gather}u_{l}^{(1)}=0,\quad \text{if}\ l=0,\quad \tau \in \left({-}1,1\right) , \end{gather}

(3.13)\begin{gather}u_{\tau }^{(0)}=0\quad \text{if}\ l=0,\quad \tau \in \left({-}1,1\right) , \end{gather}

(3.14)\begin{gather}W^{(1)}=0\quad \text{if}\ l=0. \end{gather}

Treating $\alpha$ as if it were given, one can deduce from (3.8) and (3.11) that

(3.15)\begin{equation} p^{(0)}={-}\tau \frac{\partial \alpha }{\partial l}. \end{equation}

Substituting $p^{(0)}$ into (3.7) and taking into account (3.12), one obtains

(3.16)\begin{equation} u_{l}^{(1)}=\tau \left( \frac{\partial \alpha }{\partial l}-\alpha _{0}^{\prime }\right) , \end{equation}

where $\alpha _{0}^{\prime }$ is defined by (3.2). Next, it follows from (3.9)–(3.10) and (3.14) that

(3.17)\begin{gather} u_{\tau }^{(0)}=(1-\tau ^{2}) \frac{\partial ^{2}\alpha }{ \partial l^{2}}-\frac{\partial \bar{x}}{\partial t}\sin \alpha +\frac{ \partial \bar{z}}{\partial t}\cos \alpha , \end{gather}

(3.18)\begin{gather}W^{(1)}=0. \end{gather}

Treating the next-to-leading order in a similar fashion, one obtains

(3.19)\begin{gather} p^{(1)}=( 1-2\tau ^{2}) \left( \frac{\partial \alpha }{\partial l} \right) ^{2}+( \tau ^{2}-1) \alpha _{0}^{\prime }\frac{\partial \alpha }{\partial l}, \end{gather}

(3.20)\begin{gather} \hspace{-12pc} u_{l}^{(2)}=u_{0}-z+\frac{\partial \bar{x}}{\partial t}\cos \alpha +\frac{ \partial \bar{z}}{\partial t}\sin \alpha \nonumber\\ \hspace{1pc} -\left( 1-\frac{3\tau ^{2}}{2}\right) \left( \frac{\partial \alpha }{ \partial l}\right) ^{2}+(2-\tau ^{2}) \alpha _{0}^{\prime }\frac{ \partial \alpha }{\partial l}-\left( 1+\frac{\tau ^{2}}{2}\right) \alpha _{0}^{\prime 2}, \end{gather}

(3.21)\begin{gather} u_{\tau }^{(1)}=\left( 3\tau -\frac{11}{3}\tau ^{3}\right) \frac{\partial \alpha }{\partial l}\frac{\partial ^{2}\alpha }{\partial l^{2}}-2\left( \tau -\frac{\tau ^{3}}{3}\right) \alpha _{0}^{\prime }\frac{\partial ^{2}\alpha }{ \partial l^{2}}+\tau \sin \alpha , \end{gather}

(3.22)\begin{gather} W^{(2)}=\bar{z}-\frac{1}{3}\left( \frac{\partial \alpha }{\partial l}\right) ^{2}-\frac{4}{3}\alpha _{0}^{\prime }\frac{\partial \alpha }{\partial l}+ \frac{5}{3}\alpha _{0}^{\prime 2}. \end{gather}

In the next order, one only needs the cross-stream equation (2.23) and the capillary-pressure condition (2.26), which have the form

(3.23)\begin{align} &\frac{\partial u_{\tau }^{(0)}}{\partial l}+\frac{\partial \alpha }{\partial t}-\left( \frac{\partial x}{\partial t}\cos \alpha +\frac{\partial z}{ \partial t}\sin \alpha \right) \frac{\partial \alpha }{\partial l}+(2u_{l}^{(2)}+u_{l}^{(1)2})\frac{\partial \alpha }{\partial l} \nonumber\\ &\quad +4\tau u_{l}^{(1)}\left( \frac{\partial \alpha }{\partial l}\right) ^{2}+ \frac{\tau ^{2}}{2}\left[ 8\left( \frac{\partial \alpha }{\partial l}\right) ^{3}+\frac{\partial ^{3}\alpha }{\partial l^{3}}\right] +\frac{\partial p^{(2)}}{\partial \tau }={-}\cos \alpha , \end{align}

(3.24)\begin{equation} p^{(2)}\pm \frac{\partial p^{(1)}}{\partial \tau }W^{(1)}={\mp} \left[ \frac{3 }{2}\left( \frac{\partial \alpha }{\partial l}\right) ^{3}+\frac{1}{2}\frac{ \partial ^{3}\alpha }{\partial l^{3}}\right] \quad \text{if}\ \tau ={\pm} 1. \end{equation}

Observe that (3.23) and (3.24) involve only one unknown, $p^{(2)}$ – which can be actually eliminated. Integrating (3.23) from $\tau =-1$ to $\tau =1$ and taking into account (3.24), one obtains

(3.25)\begin{align} &\int_{{-}1}^{{-}1}\frac{\partial u_{\tau }^{(0)}}{\partial l}\mathrm{d}\tau +2 \frac{\partial \alpha }{\partial t}-2\left( \frac{\partial x}{\partial t} \cos \alpha +\frac{\partial z}{\partial t}\sin \alpha \right) \frac{\partial \alpha }{\partial l} \nonumber\\ &\quad +\frac{\partial \alpha }{\partial l}\int_{{-}1}^{{-}1}\left( 2u_{l}^{(2)}+u_{l}^{(1)2}\right) \mathrm{d}\tau +4\left( \frac{\partial \alpha }{\partial l}\right) ^{2}\int_{{-}1}^{{-}1}\tau u_{l}^{(1)}\,\mathrm{d}\tau +\frac{1}{2}\left[ 8\left( \frac{\partial \alpha }{\partial l}\right) ^{3}+ \frac{\partial ^{3}\alpha }{\partial l^{3}}\right] \nonumber\\ &\quad -3\left( \frac{\partial \alpha }{\partial l}\right) ^{3}-\frac{\partial ^{3}\alpha }{\partial l^{3}}-\left[ \left( \frac{\partial p^{(1)}}{\partial \tau }\right) _{\tau ={-}1}+\left( \frac{\partial p^{(1)}}{\partial \tau } \right) _{\tau =1}\right] W^{(1)}={-}2\cos \alpha . \end{align}

Substituting in this equality expressions (3.15)–(3.24) for the lower-order unknowns and evaluating the integrals involved, one obtains (3.1) as required.

3.3. Derivation of condition (3.4)

An attentive reader may have noticed that the boundary condition (3.13) has not been used in the above calculation of the leading-order cross-stream velocity $u_{\tau }^{(0)}$. As a result, $u_{\tau }^{(0)}$ does not assume the prescribed value at the outlet: as follows from (3.17) and (2.30a,b),

(3.26)\begin{equation} u_{\tau }^{(0)}\rightarrow \left( 1-\tau ^{2}\right) \alpha _{0}^{\prime \prime }\quad \text{as}\ l\rightarrow 0, \end{equation}

where

(3.27)\begin{equation} \alpha _{0}^{\prime \prime }=\left( \frac{\partial ^{2}\alpha }{\partial l^{2}}\right) _{l=0}. \end{equation}

Thus, condition (3.13) holds only if $\partial ^{2}\alpha /\partial l^{2}$ happens to vanish at $l=0$. This discrepancy arises due to the fact that none of the leading-order equations (3.7)–(3.14) includes $\partial u_{\tau }^{(0)}/\partial l$ – hence, the solution cannot satisfy a requirement imposed at a fixed $l$.

This discrepancy suggests that a boundary layer exists, with a solution satisfying the boundary conditions at the outlet and matching the outer solution (described by expansions (3.6)) far from the outlet.

To derive the equations describing the boundary layer, one needs to rescale the streamwise coordinate: instead of $l_{nd}=l/L$, introduce

(3.28)\begin{equation} \left( l_{b}\right) _{nd}=\frac{l}{H}. \end{equation}

It can be safely assumed that, near the outlet, the curtain does not experience sharp turns – hence, in the boundary layer, $\alpha$ can be treated as a given function determined by the outer region. Expanding $\alpha$ in powers of the long-scale coordinate $l$,

(3.29)\begin{equation} \alpha =\alpha _{0}+l\alpha _{0}^{\prime }+\frac{l^{2}}{2}\alpha _{0}^{\prime \prime }+\frac{l^{3}}{6}\alpha _{0}^{\prime \prime \prime }+ {O}(l^{4})\quad \text{as}\ l\rightarrow 0 \end{equation}

and rewriting the series in terms of $( l_{b}) _{nd}$, one obtains (the subscript $_{nd}$ omitted)

(3.30)\begin{equation} \alpha =\alpha _{0}+\epsilon l_{b}\alpha _{0}^{\prime }+\frac{\epsilon ^{2}l_{b}^{2}}{2}\alpha _{0}^{\prime \prime }+\frac{\epsilon ^{3}l_{b}^{3}}{6 }\alpha _{0}^{\prime \prime \prime }+{O}(\epsilon ^{4}). \end{equation}

Expressions (2.34)–(2.35) for the Lamé coefficients should be treated in a similar manner,

(3.31)\begin{align} h_{l}&=1-\tau \left( \epsilon \alpha _{0}^{\prime }+\epsilon ^{2}l_{b}\alpha _{0}^{\prime \prime }+\frac{\epsilon ^{3}l_{b}^{2}}{2}\alpha _{0}^{\prime \prime \prime }\right) \nonumber\\ &\quad -\frac{\tau ^{2}}{2}(\epsilon ^{2}\alpha _{0}^{\prime 2}+2\epsilon ^{3}l_{b}\alpha _{0}^{\prime }\alpha _{0}^{\prime \prime }) -\epsilon ^{3}\tau ^{3}\left( \frac{1}{6}\alpha _{0}^{\prime \prime \prime }+\frac{1}{2 }\alpha _{0}^{\prime 3}\right) +{O}(\epsilon ^{4}), \end{align}

(3.32)\begin{align} h_{\tau }&=1+\tau \left( \epsilon \alpha _{0}^{\prime }+\epsilon ^{2}l_{b}\alpha _{0}^{\prime \prime }+\frac{\epsilon ^{3}l_{b}^{2}}{2}\alpha _{0}^{\prime \prime \prime }\right) \nonumber\\ &\quad +\frac{3\tau ^{2}}{2}(\epsilon ^{2}\alpha _{0}^{\prime 2}+2\epsilon ^{3}l_{b}\alpha _{0}^{\prime }\alpha _{0}^{\prime \prime }) +\epsilon ^{3}\tau ^{3}\left( \frac{1}{6}\alpha _{0}^{\prime \prime \prime }+\frac{5}{2 }\alpha _{0}^{\prime 3}\right) +{O}(\epsilon ^{4}). \end{align}

With $\alpha$ being a given function, the curvilinear coordinates no longer evolve with the flow and, thus, the curve $\tau =0$ does not necessarily coincide with the centreline. As a result, $\tau _{+}$ and $\tau _{-}$ should be treated as independent functions, not inter-related by constraint (2.11).

Replacing in non-dimensionalisation (2.17) the outer coordinate $l_{nd}$ with its inner counterpart (3.28) and keeping in mind expansions (3.30)–(3.32), one can rewrite the original equations (2.22)–(2.28) in the form (the subscript $_{nd}$ omitted)

(3.33)\begin{align} &\hspace{5pc}\frac{u_{l}}{\epsilon ^{2}}\frac{\partial u_{l}}{\partial l_{b}}-\epsilon \alpha _{0}^{\prime }u_{\tau }+\frac{1}{\epsilon }\frac{\partial p}{\partial l_{b}}={-}\epsilon \sin \alpha _{0}+{O}(\epsilon ^{2}), \end{align}

(3.34)\begin{align} &\epsilon \left( 1+2\epsilon \tau \alpha _{0}^{\prime }\right) \frac{\partial u_{\tau }}{\partial l_{b}}+u_{l}^{2}\left\{ \alpha _{0}^{\prime }+\epsilon (2\tau \alpha _{0}^{\prime 2}+l_{b}\alpha _{0}^{\prime \prime }) \vphantom{\frac{\epsilon ^{2}}{2}}\right. \nonumber\\ &\quad +\left. \frac{\epsilon ^{2}}{2}\left[ \tau ^{2}(8\alpha _{0}^{\prime 3}+\tau ^{2}\alpha _{0}^{\prime \prime \prime }) +8\tau l_{b}\alpha _{0}^{\prime }\alpha _{0}^{\prime \prime }+l_{b}^{2}\alpha _{0}^{\prime \prime \prime }\right] \right\} +\frac{\partial p}{\partial \tau }={-}\epsilon ^{2}\cos \alpha _{0}+{O}(\epsilon ^{3}), \end{align}

(3.35)\begin{align} &\frac{1}{\epsilon ^{2}}\frac{\partial }{\partial l_{b}}\left\{ u_{l}\left[ 1+\epsilon \tau \alpha _{0}^{\prime }+\epsilon ^{2}\left( \frac{3\tau ^{2}}{2 }\alpha _{0}^{\prime 2}+\tau l_{b}\alpha _{0}^{\prime \prime }\right) \vphantom{\frac{l_{b}^{2}}{2}}\right. \right. \nonumber\\ &\quad+\left. \left. \epsilon ^{3}\left( \frac{\tau ^{3}}{6}\alpha _{0}^{\prime \prime \prime }+\frac{5\tau ^{3}}{2}\alpha _{0}^{\prime 3}+3\tau ^{2}l_{b}\alpha _{0}^{\prime }\alpha _{0}^{\prime \prime }+\frac{\tau l_{b}^{2}}{2}\alpha _{0}^{\prime \prime \prime }\right) \right] \right\}\nonumber\\ &\quad +\frac{\partial }{\partial \tau }\left[ u_{\tau }\left( 1-\epsilon \tau \alpha _{0}^{\prime }\right) \right] ={O}(\epsilon ^{2}), \end{align}

(3.36)\begin{align} &\hspace{5pc}\frac{1}{\epsilon ^{2}}\dfrac{\partial \tau _{{\pm} }}{\partial l_{b}}-\left( 1\mp 2\epsilon \alpha _{0}^{\prime }\right) u_{\tau }={O}(\epsilon ^{2})\quad \text{if}\ \tau ={\pm} 1, \end{align}

(3.37)\begin{align} p&=\frac{\mp 1-3\epsilon \alpha _{0}^{\prime }}{\epsilon }\dfrac{\partial ^{2}\tau _{{\pm} }}{\partial l_{b}^{2}}\mp \alpha _{0}^{\prime }-\epsilon (\alpha _{0}^{\prime 2}\pm l_{b}\alpha _{0}^{\prime \prime }) \nonumber\\ &\quad \mp \frac{\epsilon ^{2}}{2}(3\alpha _{0}^{\prime 3}+\alpha _{0}^{\prime \prime \prime }\pm 4l_{b}\alpha _{0}^{\prime }\alpha _{0}^{\prime \prime }+l_{b}^{2}\alpha _{0}^{\prime \prime \prime }) + {O}(\epsilon ^{3})\quad \text{if}\ \tau ={\pm} 1. \end{align}

These equations were obtained under the following conjectures:

(3.38a,b)\begin{equation} u_{l}=1+{O}(\epsilon ^{2}),\qquad \tau _{{\pm} }={\pm} 1+{O} (\epsilon ^{2}), \end{equation}

accordingly, the solution of the boundary-value problem (3.33)–(3.37) should be sought in the form

(3.39)\begin{equation} \left.\begin{gathered} u_{l}=1+\epsilon ^{2}u_{l}^{(2)}\ldots,\quad u_{\tau }=u_{\tau }^{(0)}+\epsilon u_{\tau }^{(1)}\ldots, \quad p={-}\alpha _{0}^{\prime }\tau +\epsilon p^{(1)}\ldots,\\ \tau _{{\pm} }={\pm} 1+\epsilon ^{2}\tau _{{\pm} }^{(2)}\ldots\,. \end{gathered}\right\} \end{equation}

To leading order, one obtains

(3.40)\begin{gather} \left.\begin{gathered} \dfrac{\partial u_{l}^{(2)}}{\partial l_{b}}+\dfrac{\partial p^{(1)}}{ \partial l_{b}}=0, \\ \dfrac{\partial u_{\tau }^{(0)}}{\partial l_{b}}+2\tau \alpha _{0}^{\prime 2}+l_{b}\alpha _{0}^{\prime \prime }+\dfrac{\partial p^{(1)}}{\partial \tau }=0, \\ \dfrac{\partial u_{l}^{(2)}}{\partial l_{b}}+\tau \alpha _{0}^{\prime \prime }+\dfrac{\partial u_{\tau }^{(0)}}{\partial \tau }=0, \end{gathered} \right\} \end{gather}

(3.41)\begin{gather} \dfrac{\partial \tau _{{\pm} }^{(2)}}{\partial l_{b}}-u_{\tau }^{(0)}=0,\quad p^{(1)}={\mp} \dfrac{\partial ^{2}\tau _{{\pm} }^{(2)}}{\partial l_{b}^{2}} -\alpha _{0}^{\prime 2}\mp l_{b}\alpha _{0}^{\prime \prime }\quad \text{if} \ \tau ={\pm} 1, \end{gather}

(3.42)\begin{gather} u_{l}^{(2)}=u_{0},\quad u_{\tau }^{(0)}=0,\quad \tau _{{\pm} }^{(2)}=0\quad \text{if}\ l_{b}=0. \end{gather}

Equations (3.40)–(3.42) can be reduced to a boundary-value problem for $u_{\tau }^{(0)}$, which can be solved using the Fourier transformation. Omitting the technicalities (which are similar to those examined in Appendix A), one obtains

(3.43)\begin{equation} u_{\tau }^{(0)}={-}\alpha _{0}^{\prime \prime }l_{b}^{2}+A^{(0)}\sinh k_{{\ast} }\tau \sin k_{{\ast} }l_{b}, \end{equation}

where $A^{(0)}$ is an undetermined constant and $k_{\ast }$ satisfies

(3.44)\begin{equation} \cosh k_{{\ast} }-k_{{\ast} }\sinh k_{{\ast} }=0. \end{equation}

If solved numerically, (3.44) yields $k_{\ast }\approx 1.1997$.

The term involving $A^{(0)}$ in solution (3.43) describes a short-scale varicose capillary wave coming from infinity – bouncing off the outlet – going back to infinity. Since the ‘infinity’ here means the ‘outer region’ and since short waves have been scaled out from the outer solution, one sets

(3.45)\begin{equation} A^{(0)}=0. \end{equation}

Comparing the inner solution (3.43)–(3.45) with the inner limit of the outer solution (given by (3.26)), one can see that they match only if $\alpha _{0}^{\prime \prime }=0$. This requirement amounts to the boundary condition (3.4) for the outer solution, as required.

Note that condition (3.4) could be derived by forcing the outer $u_{\tau }^{(0)}$ to satisfy the boundary condition at the outlet, i.e. without considering the boundary layer. A similar problem, however, arises in the next order: letting $\alpha _{0}^{\prime \prime }=0$ in (3.21), one obtains

(3.46)\begin{equation} u_{\tau }^{(1)}\rightarrow \tau \sin \alpha _{0}\quad \text{as}\ l\rightarrow 0, \end{equation}

i.e. $u_{\tau }^{(1)}$ vanishes at the outlet only if the curtain is ejected horizontally. This discrepancy can only be resolved by examining the next order of the boundary-layer problem (3.33)–(3.37), as done in Appendix A.

It is also shown in Appendix A that the boundary-layer solution describes varicose capillary waves.

4.1. The difference between upward- and downward-bending curtains

Three parameters appear in the boundary-value problem (4.1)–(4.4): the ejection angle $\alpha _{0}$, the excess ejection velocity $v_{0}$ and the curtain's near-outlet curvature $\alpha _{0}^{\prime }$. The first two are controlled in an experiment – hence, should be treated as given. The third parameter, in turn, can not be set by the experimentalist – hence, the mathematician should either treat it as arbitrary (and find a solution for each value of $\alpha _{0}^{\prime }$ for which a solution exists) or determine it as part of the solution (the same way eigenvalues are determined together with the eigenfunctions).

It turns out that the former is the case for the usual, downward-bending (DB) curtains – and the latter, for upward-bending (UB) curtains.

To describe a DB curtain, require

(4.5)\begin{equation} \alpha \rightarrow -\frac{{\rm \pi} }{2}\quad \text{as}\ l\rightarrow \infty . \end{equation}

To examine how the solution of (4.1)–(4.4) approaches this limit, let

(4.6a,b)\begin{equation} \alpha ={-}\frac{{\rm \pi} }{2}+\tilde{\alpha},\quad z={-}l+\varDelta _{\infty }+\tilde{z }, \end{equation}

where

(4.7)\begin{equation} \varDelta _{\infty }=\lim_{l\rightarrow \infty }\left( z+l\right) , \end{equation}

so that $\tilde {z}\rightarrow 0$ as $l\rightarrow \infty$. Linearising (4.1), one obtains

(4.8)\begin{equation} \frac{\mathrm{d}\tilde{\alpha}}{\mathrm{d}l}\left[ v_{0}-\frac{1}{2}\left({-}l+\varDelta _{\infty }\right) -\frac{1}{6}\alpha _{0}^{\prime 2}\right] +\frac{ 1}{6}\frac{\mathrm{d}^{3}\tilde{\alpha}}{\mathrm{d}l^{3}}={-}\frac{1}{2}\tilde{ \alpha}, \end{equation}

whereas the equation for $\tilde {z}$ decouples from the above and is unimportant.

Let the general solution of (4.8) be

(4.9)\begin{equation} \tilde{\alpha}=C_{1}\,\tilde{\alpha}_{1}(l)+C_{2}\,\tilde{\alpha} _{2}(l)+C_{3}\,\tilde{\alpha}_{3}(l), \end{equation}

where $C_{1,2,3}$ are arbitrary constants. The linearly independent solutions $\tilde {\alpha }_{1,2,3}(l)$ can be fixed by their large-$l$ asymptotics,

(4.10)\begin{equation} \tilde{\alpha}_{1}\sim \tilde{l}^{{-}1},\quad \tilde{\alpha}_{2}\sim \tilde{l} ^{{-}1/4}\sin \frac{2\tilde{l}^{3/2}}{3^{1/2}},\quad \tilde{\alpha}_{3}\sim \tilde{l}^{{-}1/4}\cos \frac{2\tilde{l}^{3/2}}{3^{1/2}}\quad \text{as}\ l\rightarrow \infty, \end{equation}

where

(4.11)\begin{equation} \tilde{l}=l-\varDelta _{\infty }+2u_{0}-\tfrac{1}{3}\alpha _{0}^{\prime 2}. \end{equation}

Most importantly, $\tilde {\alpha }_{1,2,3}$ all decay as $l\rightarrow \infty$ – which means that the point $\alpha =-{\rm \pi} /2$ is an attractor. As a result, solutions are likely to exist for a range of $\alpha _{0}^{\prime }$ (not just a set of discrete values): one can ‘shoot’ a solution of (4.1)–(4.4) from the outlet with a value of $\alpha _{0}^{\prime }$ from the allowed range, and this solution will end up at the attractor. Thus, there is no need to seek new solutions by parametric continuation from the already-found ones.

UB curtains, in turn, imply that

(4.12)\begin{equation} \alpha \rightarrow \frac{{\rm \pi} }{2}\quad \text{as}\ l\rightarrow \infty , \end{equation}

and the large-$l$ analysis yields

(4.13)\begin{equation} z\sim l+\varDelta _{\infty },\quad \alpha \sim \frac{{\rm \pi} }{2}+C_{1}\,\tilde{\alpha}_{1}+C_{2}\,\tilde{\alpha}_{2}+C_{3}\,\tilde{\alpha}_{3}\quad \text{as}\ l\rightarrow \infty , \end{equation}

where

(4.14)\begin{gather} \tilde{\alpha}_{1}\sim \tilde{l}^{{-}1},\quad \tilde{\alpha}_{2}\sim \tilde{l} ^{{-}1/4}\exp \frac{2\tilde{l}^{3/2}}{3^{1/2}},\quad \tilde{\alpha}_{3}\sim \tilde{l}^{{-}1/4}\exp \left( -\frac{2\tilde{l}^{3/2}}{3^{1/2}}\right) \quad \text{as}\ l\rightarrow \infty , \end{gather}

(4.15)\begin{gather}\tilde{l}=l+z_{\infty }-2u_{0}+\tfrac{1}{3}\alpha _{0}^{\prime 2}. \end{gather}

Evidently, $\tilde {\alpha }_{2}$ grows as $l\rightarrow \infty$ – hence, the point $\alpha ={\rm \pi} /2$ is not an attractor. As a result, problem (4.1)–(4.4), (4.12) may have a solution only for discrete values of $\alpha _{0}^{\prime }$, such that $C_{2}$ in asymptotic (4.13) vanishes.

4.2. Numerical results

The boundary-value problem (4.1)–(4.4) was solved numerically. The following results have been obtained.

(i) The numerical method for the UB problem (comprising (4.1)–(4.4) and (4.12)) is described in Appendix B. The parameter plane $( \alpha _{0},v_{0})$ has been thoroughly trawled, and it has turned out that, for a given pair $(\alpha _{0},v_{0})$, a UB curtain exists only for a single value of $\alpha _{0}^{\prime }$. The dependence of $\alpha _{0}^{\prime }$ on $v$ and $\alpha _{0}$ is illustrated in figure 2, and examples of UB curtains are shown in figure 3.

Figure 2. The near-outlet curvature $\alpha _{0}^{\prime }$ of UB curtains vs the excess ejection velocity $v_{0}$. The curves marked by ($\pm 1$) and ($0$) correspond to $\alpha _{0}=\pm {\rm \pi}/4$ and $\alpha _{0}=0$, respectively.

Figure 3. Examples of UB curtains, as described by the boundary-value problem (4.1)–(4.4), (4.12). The values of the ejection angle $\alpha _{0}$ are indicated in the corresponding panels, the values of $v_{0}$ mark the corresponding curves.

The mere fact of existence of gravity-defying flows is highly counter-intuitive, but this is not the only paradoxical feature of the solutions found. One would expect faster curtains (those with larger $v_{0}$) to be straighter than their small-$v_{0}$ counterparts – but, in reality, it is the other way around. This tendency is visible in the examples in figure 3, and is quantified in figure 2 (observe the growth of the near-outlet curvature $\alpha _{0}^{\prime }$ with increasing $v_{0}$). Furthermore, sufficiently fast curtains are so curved that they overshoot the vertical direction and come back to it after an inflection point (this pattern is not evident from figure 3, but it is clearly visible in figure 4). It should be emphasised, however, that solutions with large $\alpha _{0}^{\prime }$ may violate the slender-curtain approximation. Recalling how the problem was non-dimensionalised, one can show that the applicability condition of the solutions found is $\vert \alpha _{0}^{\prime }\vert \ll \epsilon ^{-1}$.

Figure 4. The local angle $\alpha$ between the curtain and the horizontal vs the centreline's arclength $l$, for the examples shown in figure 3(b) ($\alpha _{0}=0$, the values of $v_{0}$ mark the corresponding curves). Observe that curve 1 overshoots the vertical direction (shown by the dotted line) and comes back to it after an inflection point.

The dependence of UB curtains on the ejection angle $\alpha _{0}$ does not seem to be essential, as the trajectories of curtains with different values of $\alpha _{0}$ are qualitatively similar. Observe also that the curves with different $\alpha _{0}$ in figure 2 are close to each other. One can expect a different behaviour only in the limit $\alpha _{0}\rightarrow -{\rm \pi} /2$ (nearly vertical curtains), which is not considered here.

(ii) As mentioned before, the trajectories of DB curtains can be computed by simply choosing a value of $\alpha _{0}^{\prime }$ and shooting the solution from $l=0$ towards $l\rightarrow \infty$. Numerous examples have been computed via this approach, some of which are shown in figure 5.

Figure 5. Examples of DB curtains, as described by problem (4.1)–(4.4), (4.5) with $\alpha _{0}=0$. The values of the parameter $\alpha _{0}^{\prime }$ are indicated in the corresponding panels, the values of $v_{0}$ mark the corresponding curves. Panels (a,b) correspond to the two horizontal cross-sections of figure 6 labelled ‘Figure 5a’ and ‘Figure 5b’, respectively.

Three features of these examples catch one's eye. First, DB curtains are wavy (unlike the UB curtains examined above, as well as the DB curtains with $We\not \approx 1$ examined in B19). Second, curtains with a sufficiently negative $v_{0}$ become so wavy that they self-intersect (in figure 5, only the first intersection is shown; after that, the solution is meaningless physically). Third, some of the self-intersecting curtains bend initially upwards (but if their trajectories were extended beyond all intersections, one would see that they turn downwards eventually).

(iii) UB and DB curtains can be viewed as different members of the same family of solutions, as illustrated in figure 6.

Figure 6. Classification of curtains on the $( v_{0},\alpha _{0}^{\prime })$ plane, as described by problem (4.1)–(4.4) with $\alpha _{0}=0$. The thick solid curve corresponds to non-self-intersecting UB curtains (it is the same as curve 0 in figure 2). The dashed line bounds the region of self-intersecting curtains. The curtains depicted in figures 5 and 7 correspond to the circles connected by thin solid straight lines.

From now on, self-intersecting curtains will be classified as UB or DB depending on how they behave before the first intersection. Then, the curve corresponding to non-self-intersecting UB solutions should be viewed as a separatrix between self-intersecting UB and DB curtains.

Examples of near-separatrix curtains are shown in figure 7.

Figure 7. Examples of curtains with $\alpha _{0}=0$ and $v_{0}=1.5$. Curve ($u$) is the non-intersecting UB curtain ($\alpha _{0}^{\prime }\approx 3.3177$); curve ($i$) is an example of a self-intersecting UB curtain (with $\alpha _{0}^{\prime }=3.6467$); curve ($t$) is the self-touching curtain separating self-intersecting and non-self-intersecting DB curtains ($\alpha _{0}^{\prime }\approx 2.9799$). The solutions depicted correspond to the vertical cross-section of figure 6 labelled ‘Figure 7’.

4.3. Comparison with B19

(i) In B19, UB curtains could only rise to the height where the fluid velocity vanishes, and so all of the initial reserve of kinetic energy is used up. In the present model, on the other hand, the leading-order non-dimensional velocity is unity (see (3.6)) and, thus, cannot vanish – so the UB curtains formally rise infinitely high. However, it follows from (3.20) that

(4.16)\begin{equation} u^{(2)}\sim{-}z\quad \text{as}\ z\rightarrow \infty, \end{equation}

as a result, once $z$ becomes $O(\epsilon ^{-2})$, the whole expansion breaks down. This is the present model's equivalent of the limiting height of curtains in B19.

The growth of $u^{(2)}$ with growing $z$ invalidates the large-$z$ results for DB curtains too, but this occurs when they are almost vertical and their evolution is trivial.

(ii) One should keep in mind that B19 assumes the characteristic radius of curvature to be $L=H\,Fr$, whereas the present work considers a shorter scale, $L=H\,Fr^{1/3}$. In addition, the present work assumes $We\approx 1$.

Thus, the two sets of results should agree only if the limit $We\rightarrow 1$ is applied to the B19 solutions, whereas the present solutions are subject to

(4.17a,b)\begin{equation} \alpha _{0}^{\prime }\ll 1,\quad v_{0}\gg 1. \end{equation}

These two constraints guarantee that the curtain's non-dimensional curvature is small both near the outlet and globally (as follows from (4.1) with $v_{0}\gg 1$, the solution's global spatial scale is proportional to $v_{0}$).

(iii) According to the present results, UB curtains exist for all $v_{0}$, both positive and negative – whereas B19 found such solutions only in the subcritical case $v_{0}<0$.

The apparent contradiction can be resolved if one recalls that, for UB curtains, $\alpha _{0}^{\prime }$ is a function of $v_{0}$ and $\alpha _{0}$, such that $\alpha _{0}^{\prime }\rightarrow \infty$ as $v_{0}\rightarrow \infty$ (see figure 2). Clearly, such solutions are inconsistent with limit (4.17a,b), and so it comes as no surprise that supercritical UB curtains have been missed by the asymptotic model of B19.

(iv) There is another apparent discrepancy between B19 and the present work: in the former, a single DB curtain was found for given ejection velocity and angle, whereas the latter found a whole family of solutions, differing from each other by their values of $\alpha _{0}^{\prime }$.

To resolve the discrepancy, one should look at the curtains found in the present work under conditions (4.17a,b) – see figure 8. Observe that the medium-$v_{0}$ curtains with different $\alpha _{0}^{\prime }$ are located much closer together than their small-$v_{0}$ counterparts, and the curtains with the largest value of $v_{0}$ are hardly distinguishable.

Figure 8. Examples of DB curtains with $\alpha _{0}=0$. The three curves with $v_{0}=2$ are for $\alpha _{0}^{\prime }=\pm 2.5,0$, those with $v_{0}=5$ are for $\alpha _{0}^{\prime }=\pm 1,0$ and those with $v_{0}=12.5$ are for $\alpha _{0}^{\prime }=\pm 0.4,0$.

Thus, in limit (4.17a,b), curtains with different $\alpha _{0}^{\prime }$ collapse onto the same curve, i.e. the dependence on $\alpha _{0}^{\prime }$ becomes weak – which reconciles the present results with those of B19.

Figure 8 also illustrates the fact that limit (4.17a,b) makes curtains less wavy, just as they should be according to B19.