, , ,

# The Effect of Pressure Fluctuations on the Shapes of Thinning Liquid
Curtains

Bridget M. Torsey  School of Mathematical Sciences, Rochester Institute of
Technology, Rochester, NY 14623, USA  Steven J. Weinstein  School of
Mathematical Sciences, Rochester Institute of Technology, Rochester, NY 14623,
USA  Department of Chemical Engineering, Rochester Institute of Technology,
Rochester, NY 14623, USA  David S. Ross  School of Mathematical Sciences,
Rochester Institute of Technology, Rochester, NY 14623, USA  Nathaniel S.
Barlow  School of Mathematical Sciences, Rochester Institute of Technology,
Rochester, NY 14623, USA

###### Abstract

We consider the time-dependent response of a gravitationally-thinning inviscid
liquid sheet (a coating curtain) leaving a vertical slot to sinusoidal ambient
pressure disturbances. The theoretical investigation employs the hyperbolic
partial differential equation developed by [27]. The response of the curtain
is characterized by the slot Weber number, $W_{e_{0}}=\rho qV/2\sigma$, where
$V$ is the speed of the curtain at the slot, $q$ is the volumetric flow rate
per unit width, $\sigma$ is the surface tension, and $\rho$ is the fluid
density. Flow disturbances travel along characteristics with speeds relative
to the curtain of $\pm\sqrt{uV/W_{e_{0}}}$, where $u=\sqrt{V^{2}+2gx}$ is the
curtain speed at a distance $x$ downstream from the slot. When the flow is
subcritical ($W_{e_{0}}<1$), upstream traveling disturbances near the slot
affect the curtain centerline, and the slope of the curtain centerline at the
slot oscillates with an amplitude that is a function of $W_{e_{0}}$. In
contrast, all disturbances travel downstream in supercritical curtains
($W_{e_{0}}>1$) and the slope of the curtain at the slot is vertical. Here, we
specifically examine the curtain response under supercritical and subcritical
flow conditions near $W_{e_{0}}=1$ to deduce whether there is a substantial
change in the overall shape and magnitude of the curtain responses. Despite
the local differences in the curtain solution near the slot, we find that
subcritical and supercritical curtains have similar responses for all imposed
sinusoidal frequencies.

##  I Introduction

Curtain coating is a common industrial process that uses wide, thin planar
liquid sheets (curtains) to deposit uniform thin films on moving substrates
[28]. In one of its simplest configurations, a curtain leaves an inverted slot
die and thins under the influence of gravity as it falls, see figure 1.
Curtains are subjected to ambient disturbances that deflect them; these
deflections can cause nonuniform liquid coatings and imperfections in final
dried products. Flow disturbances are often examined using linear theory
because coated product quality is sensitive to even small thickness
variations. Curtains have significant surface area in contact with the
surrounding air along their long and wide faces, so understanding how they
respond to pressure disturbances is of practical importance. In this study we
focus on the effects of sinusoidal time-varying pressure disturbances on the
shapes and deflections of curtains.

The response of liquid curtains to pressure disturbances has been well-studied
both experimentally and theoretically. Much of that work has focused on the
related problem of water bells, i.e. radially-symmetric liquid sheets (among
the many references, see [14]; [16]; [20], [4], [19]). In aggregate, this work
shows that agreement with experiment is obtained when models use the
approximation of inviscid flow with small variations in thickness about the
curved centerline of the water bell; that is, the curtain is gradually
thinning in the direction of flow. These studies also demonstrate that small
pressure differences across the surface of a liquid sheet can have a large
impact on the shape of water bells.

Curtains have been similarly studied with respect to ambient pressure
disturbances. In previous work, [27] justify a potential flow approximation to
the flow in a curtain, and derive an equation that governs deflections of a
curtain’s centerline, $y=F(x)$ shown in figure 1, given as:

$\left(\frac{\partial}{\partial t}+g\frac{\partial}{\partial
u}\right)^{2}F-\frac{2\sigma g^{2}}{\rho q}\frac{\partial}{\partial
u}\left(\frac{1}{u}\frac{\partial F}{\partial
u}\right)=\frac{(M_{B}-M_{A})u}{\rho q}$ (1)

where:

$u=\sqrt{V^{2}+2gx}$ (2)

and the local thickness of the curtain, $h$, is expressed as:

$h(x)=\frac{q}{u}.$ (3)

In (1) - (3), $t$ is time, $u$ is the curtain speed at position $x$, $V$ is
the curtain speed at the slot, $g$ is the acceleration of gravity, $q$ is the
volumetric flow rate, $\sigma$ is the surface tension (which we take to be
constant), $\rho$ is the liquid density, and $M_{A}$ and $M_{B}$ are the
respective pressures on the front and back faces of the curtain as shown in
figure 1. [27] derive the equation (1.1) as follows. The equation for the
velocity field and thickness of the undisturbed time-independent curtain
(centerline $y=0$ for all $x$) is determined via an asymptotic expansion in
the small parameter $\epsilon=gq/V^{3}$. The limit $\epsilon\rightarrow 0$
corresponds physically to a gradually-thinning liquid curtain. The resulting
lowest-order flow in the curtain is plug at each location $x$ according to
equation (1.2), where velocity variations across the curtain thickness (i.e.
the $y$-direction in figure 1) are of $\mathcal{O}(\epsilon^{2})$. The time-
dependent potential flow equations are then linearized for small perturbations
about the asymptotic steady-state equations (the base flow). Since the base
flow is approximate, terms to $\mathcal{O}(\epsilon^{2})$ are required to
preserve accuracy in the linearization. The resulting time-dependent equation
(1.1) is valid for curtain deflections such that $F<<h$. In the limit taken,
pressure disturbances do not affect the local thickness of the curtain to
leading order, so the resulting curtain response is captured by the deflection
of its centerline [27].

When examined under conditions of small deflection, the equation of [10] is
identical to (1.1) at steady state. Equations (2) and (3) are valid regardless
of the magnitude of deflection provided that the curtain is long and thin.
[27] shows via dominant balance that the inviscid result, equation (1.2), is
consistent with Taylor’s equation (derived in the Appendix of [3]) for large
$x$, even when viscosity is included. Predictions of (1) and (1.2) agree well
with experiments under steady-state ([10]) and transient ([5]) conditions. In
the latter study, [5] demonstrate that predictions agree with experiments when
the initial velocity is adjusted to account for the effect of an entrance
region through extrapolation to the slot location at $x=0$. This result agrees
with the empirical equation of [3] for an undisturbed vertical curtain. Note
that equations (1.1) and (1.2) are strictly valid in a region displaced
downstream from the slot because the loss in viscous traction from the slot
leads to a flow rearrangement not captured in these equations ([6], [26];
[23]; [11]). Other studies have used the slender curtain/inviscid approach to
examine the effect of nonlinear dynamics ([22]) and stationary wave formation
([8]).

Closely related to the current work is the nappe oscillation configuration,
which is used to model observed perturbations in liquid sheets formed as water
flows over waterfalls, dams, and weirs. In such a configuration, pressure
disturbances are affected by the curtain motion via an enclosure that includes
the curtain as one of its long and wide sides; the other side of the curtain
is maintained at atmospheric pressure. As a result, the motion of the curtain
affects the volume of the enclosed region, which affects the pressure in that
region and provides a restoring force. There have been many experimental
studies of this phenomena (see for example, [2]; [24]; [18]), and observed
oscillation frequencies measured in the curtain correlate with those measured
in the enclosure itself. Theoretical analyses follow the modeling approach
used to obtain equations (1.1) to (1.3) and predict the natural frequencies of
the system in the absence of surface tension ([25], [9]); these provide a more
precise model of the curtain dynamics than the simplified theory provided by
[2]. The more recent analysis of [12] incorporates surface tension and
predicts natural frequencies that agree favorably with the cited experiments
of [2] and [24].

The governing equation, (1.1), is a second-order hyperbolic partial
differential equation (PDE). The two sets of characteristics associated with
the PDE carry information that moves relative to the curtain at speeds of
$\pm\sqrt{uV/W_{e_{0}}}$. It is established in hyperbolic PDE theory that the
number of constraints specified along a boundary must be equal to the number
of characteristics that emanate from that boundary at each point [17]. The
directions of the characteristics, and therefore the associated boundary
conditions, are determined by the Weber number at the slot, given by
$W_{e_{0}}=\rho qV/2\sigma$. In supercritical curtains ($W_{e_{0}}>1$), both
sets of characteristics leave the slot and are oriented downstream. Thus, two
conditions must be specified along this boundary. This constraint placement
corresponds to the physics of supercritical flows, in which the momentum flux
is greater than the surface tension at every location, so any disturbances are
washed downstream. [10] establish experimentally that supercritical curtains
leave the slot vertically under steady conditions, confirming that points
downstream do not influence upstream locations. In a subcritical curtain
($W_{e_{0}}<1$), there is a region in which one set of characteristics is
oriented upstream and the other set is oriented downstream. This region begins
at the slot and ends at the critical point, the point in the curtain at which
the local Weber number, $W_{e}=\rho qu/2\sigma$, equals 1. Downstream of the
critical point, both sets of characteristics are oriented downstream.
Therefore, if $W_{e_{0}}<1$, only one set of characteristics leaves the slot,
so only one condition is applied along this boundary. Disturbances in this
region of the curtain move upstream and downstream along the characteristics.
[10] demonstrate experimentally that when a constant pressure drop is applied
across a curtain containing such a region, the curtain takes on an angle at
the slot. They show theoretically that this angle can be predicted by removing
a singularity in the governing equation that arises at the location where
$W_{e}=1$, the precise location where the direction of characteristics
changes. As demonstrated in the appendix of [10], a subcritical water bell
issuing from an annular slit and having an applied pressure difference takes
on an angle at the slit. Their analytical solution, obtained under conditions
of negligible gravity, accounts for the earlier qualitative experimental
observations of [1]. [21] confirms this conclusion in a later examination of
this problem. [4] examine water bells with applied pressure differences that
undergo a subcritical-to-supercritical transition, both experimentally and
theoretically. They predict water bell shapes that agree well with experiment.
As noted previously, the water bell equations use the same thin-film-modeling
assumptions that [10] used to study the planar configuration, so these water
bell results also may be used to confirm the gradually-thinning and inviscid
assumptions used in the development of equations (1.1) to (1.3).

Figure 1: Side view schematic of a liquid exiting a slot of height $h_{0}$ and
falling under the influence of gravity while subjected to ambient gas
pressures $M_{A}$ and $M_{B}$ on its sides. The curtain is assumed to be
infinite and invariant in the $z$ direction, oriented out of the figure. The
centerline of the curtain and its local thickness are denoted as $y=F(x)$ and
$h(x)$, respectively. The governing equations (1.1)-(1.3) are valid for
curtain deflections $F<<h$, so dimensions in the figure are chosen for clarity
and are not to drawn to scale.

With this background, we now return to the nappe configuration analysis of
[12] cited above. There, the equation governing the curtain motion caused by
the applied pressure is the same as that of equation (1.1) although the
pressure disturbances $M_{A}-M_{B}$ are expressed in terms motion of the
curtain shape itself through the enclosure volume. Despite this mathematical
difference, the characterization of subcritical and supercritical flows
discussed above in terms of $W_{e_{0}}$ applies. [12] examine curtain flows
with slot Weber numbers near one. As we discussed above, when the flow is
subcritical ($W_{e_{0}}<1$), there exists a location in the curtain at which
$W_{e}=1$, below which the flow becomes supercritical. As in the steady
problem of [10], the time-dependent equation is singular at the point at which
$W_{e}=1$. Because (1.1) is of second order in the spatial dimension, the
general solution of the initial-boundary value problem has two degrees of
freedom that allow us to satisfy spatial constraints. In supercritical cases
we use those degrees of freedom to specify two conditions: that the curtain
centerline is not displaced at the slot exit, and that the curtain leaves the
slot with its centerline aligned with that of the slot. In subcritical cases
we still use one degree of freedom to enforce the condition that the curtain
centerline is not displaced at the slot exit, but we use the second degree of
freedom to enforce the condition that the solution be smooth at the singular
point. This switch in the application of the second degree of freedom
corresponds to the physical facts that subcritical curtains leave the slot at
non-zero, and sometimes varying, angles, and that the transition from
subcritical to supercritical flow in such curtains is smooth. In this paper we
refer to subcritical curtains as those for which $W_{e_{0}}<1$ and for which
there exists a location where the flow transitions from subcritical to
supercritical. This definition is necessary, as [10] show experimentally that
it is possible for curtains to remain subcritical over their entire lengths.
This observation is also consistent with experimental findings of [7], who
show that long subcritical curtains can persist without rupture. We define
supercritical curtains as those for which $W_{e_{0}}>1$ and thus the flow is
supercritical over the entire domain. For supercritical curtains, [12] and
[10] both find that the slope of the curtain at the slot is vertical.

[12] report a distinct increase in dominant oscillation frequency as the slot
Weber number is decreased from supercritical to subcritical. It is not clear
from the investigation whether this effect is a result of an inherent
susceptibility of the curtain to pressure disturbances because of the
oscillation of the curtain centerline slope at the slot exit, or is a result
of the pressure coupling present in a nappe configuration. In this paper we
study the imposition of a pressure drop across the curtain that is sinusoidal
in time and is not coupled to the curtain motion. This configuration is itself
practically relevant to curtain coating processes. We specifically examine the
response of the curtain as the flow is reduced from supercritical to
subcritical, and we determine whether the corresponding change in the boundary
conditions at the slot exit leads to a substantial change in the overall shape
and magnitude of the curtain response. This provides a comparison of the
capacities of subcritical and supercritical curtains to resist pressure
disturbances.

##  II Theory

The dimensionless form of (1) is

$\frac{\partial^{2}\bar{F}}{\partial\bar{t}^{2}}+2\bar{u}\frac{\partial^{2}\bar{F}}{\partial\bar{x}\partial\bar{t}}+\bar{u}\frac{\partial}{\partial\bar{x}}\left[\left(\bar{u}-\frac{1}{W_{e_{0}}}\right)\frac{\partial\bar{F}}{\partial\bar{x}}\right]=\bar{u}(\bar{P}_{2}-\bar{P}_{1}).$
(4)

Here, $\bar{P}_{2}-\bar{P}_{1}$ is the pressure difference across the curtain,
whose displacement from its centerline is $\bar{F}(\bar{x},\bar{t})$ [27]. The
dimensionless version of (1.2) is

$\bar{u}=\sqrt{1+2\bar{x}},$ (5)

where dimensionless variables are defined as:

$\bar{F}=\frac{F}{h_{0}}\qquad\bar{u}=\frac{u}{V}\qquad\bar{x}=\frac{xg}{V^{2}}\qquad\bar{t}=\frac{tg}{V}$
$None$
$\bar{P}_{2}-\bar{P}_{1}=\frac{M_{B}-M_{A}}{\rho\epsilon^{2}V^{2}}\qquad\epsilon=\frac{gq}{V^{3}}\qquad
W_{e_{0}}=\frac{\rho qV}{2\sigma}.$ $None$

In (2.3a), $h_{0}$ is the slot height, and other variables have been defined
in section 1 and shown schematically in figure 1. We then convert the
coordinate system from Eulerian to Lagrangian by making the following variable
change:

$\xi=\bar{u}-1.$ (8)

In order to study the response of a curtain, we consider an applied sinusoidal
pressure disturbance of the form $\bar{P}_{2}-\bar{P}_{1}=\beta
e^{i\bar{\omega}\bar{t}}$ such that (2.1) is written, using coordinate
transformation (2.5), as:

$\frac{\partial^{2}\bar{F}}{\partial\bar{t}^{2}}+2\frac{\partial^{2}\bar{F}}{\partial\xi\partial\bar{t}}+\frac{\partial}{\partial\xi}\left(\left(1-\frac{1}{(\xi+1)W_{e_{0}}}\right)\frac{\partial\bar{F}}{\partial\xi}\right)=(\xi+1)\beta
e^{i\bar{\omega}\bar{t}}$ (9)

where $\bar{\omega}$ and $\beta$ are given by

$\bar{\omega}=\frac{\omega V}{g}\qquad\beta=\frac{\alpha
e^{i\theta_{R}}}{\rho\epsilon^{2}V^{2}}.$ $None$

Here, $\omega$ is the angular frequency and $\alpha$ and $\theta_{R}$ are the
magnitude and reference phase of the pressure disturbance, respectively. The
reference phase, $\theta_{R}$, is adjusted to provide clarity in the
presentation of results to follow (see discussion in Section 3). It is
understood that only the real part of $\bar{F}$ is taken as the actual
solution. The periodic solution for $\bar{F}$ is of the form

$\bar{F}(\xi,\bar{t})=\beta\bar{H}(\xi)e^{i\bar{\omega}\bar{t}}$ (11)

where $\bar{H}(\xi)$ is to be determined. We obtain a second-order ordinary
differential equation (ODE) by substituting (11) into (9):

$\left((\xi+1)^{2}-\frac{(\xi+1)}{W_{e_{0}}}\right)\frac{d^{2}\bar{H}}{d\xi^{2}}+\left(2i(\xi+1)^{2}\bar{\omega}+\frac{1}{W_{e_{0}}}\right)\frac{d\bar{H}}{d\xi}-(\xi+1)^{2}\bar{\omega}^{2}\bar{H}=(\xi+1)^{3}.$
(12)

A second change of coordinates is implemented to simplify calculations, given
by:

$z=\xi-c\qquad c=\frac{1}{W_{e_{0}}}-1.$ $None$

In these coordinates, the critical point is $z=0$ and (12) becomes

$z(z+c+1)\frac{d^{2}\bar{H}}{dz^{2}}+(2i(z+c+1)^{2}\bar{\omega}+c+1)\frac{d\bar{H}}{dz}-(z+c+1)^{2}\bar{\omega}^{2}\bar{H}=(z+c+1)^{3}.$
(14)

As discussed in section 1, there are two characteristics that are oriented
downstream from the slot in supercritical curtains ($W_{e_{0}}>1$), and the
appropriate constraints to apply here are

$\bar{H}(-c)=0$ (15)

$\frac{d\bar{H}}{dz}(-c)=0.$ (16)

Note that the subcritical case ($W_{e_{0}}<1$) corresponds to $c>0$. Condition
(15) sets the curtain centerline to be that of the slot, and (16) specifies
that the curtain be vertical there. We can see that for supercritical
curtains, (14), along with the conditions stated in (15) and (16), constitute
a well-posed initial value problem [15]. Supercritical solutions for this
system may be obtained over the entire domain using a fourth-order accurate
Runge-Kutta marching scheme.

For subcritical curtains ($W_{e_{0}}<1$), only one characteristic is oriented
downstream from the slot, so we only apply (15). Furthermore, the coefficient
of the highest order term in (14) goes to zero at the critical point ($z=0$),
and therefore the ODE is singular ([13]). An implication of this singularity
is that any marching scheme will become infinitely stiff as $z$ approaches 0.
Thus, we use a power series centered at $z=0$ to determine the solution from
the slot ($z=-c$) to a point downstream from the transition point ($z=c$).
Then, taking the first coefficients of the power series solution as initial
conditions, we use the Runge-Kutta scheme mentioned above to complete the
solution from $z=c$ to the end of the curtain ($z=z_{L}$); the flow is
supercritical below the critical point, allowing us to apply a marching scheme
in this way. We present the derivation of the power series in appendix A.1. In
the power series, enough terms are used to achieve machine precision.

##  III Results and Discussion

Figures 2-9 provide results for subcritical and supercritical curtains
corresponding to the configuration shown in figure 1. Note that these figures
have been rotated counter-clockwise 90 degrees, such that the gravitational
force points right, along the $\bar{x}$-axis. In all of the figures, the
location of the unperturbed curtain centerline, $\bar{y}=0$, is shown as a
dashed line and the shapes that the curtains adopt at 1/4 increments of the
periods of motion are represented by the solid lines. For clarity in the
presentation of results, the reference phase of the pressure disturbance,
$\theta_{R}$ in (2.7b), is chosen to be

$\theta_{R}=\arctan\left(\frac{\text{Im}(\bar{H}(\bar{x}_{L}))}{\text{Re}(\bar{H}(\bar{x}_{L}))}\right),$
(17)

where Im and Re denote the imaginary and real parts of the complex argument.
Note that reference phase merely adjusts the time at which the curtain adopts
a shape, but does not affect the sequence of shapes predicted through its
period of oscillation. Equation (3.1) assures that the bottom of the curtain,
located at $\bar{x}=\bar{x_{L}}$, sweeps through the same locations as it
moves forward and backward throughout an oscillation cycle–this makes it
easier to interpret the curtain response.

Figures 2 and 3 provide curtain responses for $W_{e_{0}}=1.1$ (supercritical)
and $W_{e_{0}}=0.9$ (subcritical) conditions both with $\bar{\omega}=0.1$. The
insets in these figures show the magnifications of the curtain shapes near the
slot exit. As described earlier, the centerline slope of supercritical
curtains is always zero at the slot exit, while the slope of supercritical
curtains changes throughout the oscillation cycle. Despite this change in
behavior, figures 2 and 3 show that the overall shape of the curtain is quite
similar for subcritical and supercritical curtains, with the magnitude of the
subcritical responses comparable. Figures 4 and 5 ($\bar{\omega}=0.5$) and
figures 6 and 7 ($\bar{\omega}=2$) compare $W_{e_{0}}=1.1$ and $W_{e_{0}}=0.9$
curtain responses. Again, the magnitude and shapes of the curtain responses
are similar for supercritical and subcritical flows. Figures 8 and 9 provide
additional results for $W_{e_{0}}=1.3$ and $W_{e_{0}}=0.7$, respectively, both
with $\bar{\omega}=2$. Even though the range of Weber numbers has extended
further into the supercritical and subcritical regimes compared with figures 6
and 7, there is no marked change in curtain shape or magnitude. Additionally,
we have solved Equation 2.11 for a selection of subcritical and supercritical
slot Weber numbers ($W_{e_{0}}$) sweeping over a wide range of frequencies.
The trends shown in Figures 2-7 are maintained over all frequencies and slot
Weber numbers surveyed. That is, the number of spatial oscillations within a
given curtain length increases, and the maximum magnitude of curtain responses
decreases with increasing frequency.

The latter observation is relevant as it mathematically eliminates the
possibility of forcing the curtain via a pressure disturbance in such a way
that natural curtain frequencies are excited. If resonance were to arise,
there would be a large amplification in the vicinity of a given imposed
frequency. At the precise resonant frequency, the assumed form of the forced
solution (2.8) would become invalid, necessitating a secular time dependence
(such as $te^{\bar{\omega}t}$). The monotonic decrease in curtain deflection
with increasing frequency (as seen in figures 2 through 9) demonstrates that
such behavior does not occur. This mathematical result has a physical basis.
In supercritical curtains, disturbances are washed downstream, so the
reinforcement of repeated disturbances that characterizes resonance is not
possible. In curtains that are subcritical near the slot but turn
supercritical downstream, one might expect that some reinforcement is
possible, because disturbances in the subcritical region propagate upstream to
the slot. However these reflect at the slot and are washed downstream.

Figure 10 provides the maximum curtain angle at the slot as a function of
$W_{e_{0}}$ for various frequencies. Note that the curtain takes on a range of
angles at the slot as it cycles through its motion. The maximum angle of the
response at the slot immediately drops to zero as the slot Weber number
increases past $W_{e_{0}}=1$, in accordance with a subcritical to
supercritical transition.

Despite the marked difference in curtain motion at the slot quantified by
figure 10, the subcritical and supercritical curtain responses shown in
figures 2-9 are not appreciably different at a given frequency. These results
indicate that there is not a significant change in a curtain’s sensitivity to
pressure disturbances in the subcritical to supercritical transition. The
abrupt frequency shift observed by [12] in this transition is thus attributed
to the coupling between the pressure disturbances and curtain motion that
arises in the nappe configuration.

##  IV Conclusion

We consider the time-dependent response of a gravitationally-thinning liquid
curtain that is subjected to sinusoidal ambient pressure disturbances under
subcritical and supercritical conditions. Consistent with previous studies
under both steady state and transient conditions, we find that the centerline
slope of the curtain is not the same as that of the slot under subcritical
conditions. This is a direct consequence of the flow of information
propagating along characteristics in the governing hyperbolic PDE. We examine
whether the increased susceptibility of the curtain to pressure disturbances
arises because of the oscillation in curtain centerline slope at the slot exit
for subcritical curtains. Our investigation shows there are no abrupt
differences in the responses of subcritical and supercritical curtains for
Weber numbers near 1 for all imposed sinusoidal frequencies. Experimental
confirmation of these results is needed in future studies.

Figure 2: The location of the curtain centerline, $\text{Re}(\bar{F})$, as a
function of distance down the curtain, $\bar{x}$, when it is subjected to a
sinusoidal pressure disturbance. The inset is a magnification of the region
near the slot. Here, the flow is supercritical ($W_{e_{0}}=1.1$) and
$\bar{\omega}=0.1$, $\beta=1$, and $\theta_{R}=-0.4048$ radians (see (3.1)).
The dashed line denotes the centerline of the unperturbed curtain. Figure 3:
The location of the curtain centerline, $\text{Re}(\bar{F})$, as a function of
distance down the curtain, $\bar{x}$, when it is subjected to a sinusoidal
pressure disturbance. The inset is a magnification of the region near the
slot. Here, the flow is subcritical ($W_{e_{0}}=0.9$) and $\bar{\omega}=0.1$,
$\beta=1$, and $\theta_{R}=-0.4366$ radians (see (3.1)). The dashed line
denotes the centerline of the unperturbed curtain. Figure 4: The location of
the curtain centerline, $\text{Re}(\bar{F})$, as a function of distance down
the curtain, $\bar{x}$, when it is subjected to a sinusoidal pressure
disturbance. The inset is a magnification of the region near the slot. Here,
the flow is supercritical ($W_{e_{0}}=1.1$) and $\bar{\omega}=0.5$, $\beta=1$,
and $\theta_{R}=-1.7794$ radians (see (3.1)). The dashed line denotes the
centerline of the unperturbed curtain. Figure 5: The location of the curtain
centerline, $\text{Re}(\bar{F})$, as a function of distance down the curtain,
$\bar{x}$, when it is subjected to a sinusoidal pressure disturbance. The
inset is a magnification of the region near the slot. Here, the flow is
subcritical ($W_{e_{0}}=0.9$) and $\bar{\omega}=0.5$, $\beta=1$, and
$\theta_{R}=-1.7924$ radians (see (3.1)). The dashed line denotes the
centerline of the unperturbed curtain. Figure 6: The location of the curtain
centerline, $\text{Re}(\bar{F})$, as a function of distance down the curtain,
$\bar{x}$, when it is subjected to a sinusoidal pressure disturbance. The
inset is a magnification of the region near the slot. Here, the flow is
supercritical ($W_{e_{0}}=1.1$) and $\bar{\omega}=2$, $\beta=1$, and
$\theta_{R}=-2.8491$ radians (see (3.1)). The dashed line denotes the
centerline of the unperturbed curtain. Figure 7: The location of the curtain
centerline, $\text{Re}(\bar{F})$, as a function of distance down the curtain,
$\bar{x}$, when it is subjected to a sinusoidal pressure disturbance. The
inset is a magnification of the region near the slot. Here, the flow is
subcritical ($W_{e_{0}}=0.9$) and $\bar{\omega}=2$, $\beta=1$, and
$\theta_{R}=-2.9888$ radians (see (3.1)). The dashed line denotes the
centerline of the unperturbed curtain. Figure 8: The location of the curtain
centerline, $\text{Re}(\bar{F})$, as a function of distance down the curtain,
$\bar{x}$, when it is subjected to a sinusoidal pressure disturbance. The
inset is a magnification of the region near the slot. Here, the flow is
supercritical ($W_{e_{0}}=1.3$) and $\bar{\omega}=2$, $\beta=1$, and
$\theta_{R}=-2.7377$ radians (see (3.1)). The dashed line denotes the
centerline of the unperturbed curtain. Figure 9: The location of the curtain
centerline, $\text{Re}(\bar{F})$, as a function of distance down the curtain,
$\bar{x}$, when it is subjected to a sinusoidal pressure disturbance. The
inset is a magnification of the region near the slot. Here, the flow is
subcritical ($W_{e_{0}}=0.7$) and $\bar{\omega}=2$, $\beta=1$, and
$\theta_{R}=3.1252$ radians (see (3.1)). The dashed line denotes the
centerline of the unperturbed curtain. Figure 10: The maximum angle at the
slot, $\theta_{0,\text{max}}=\underaccent{\forall
t}{\text{max}}[\arctan(\text{Re}(\partial\bar{F}/\partial x))]_{\bar{x}=0}$,
as a function of slot Weber number, $W_{e_{0}}$, for a range of dimensionless
disturbance frequencies, $\bar{\omega}$, obtained from the solution of (14)
with $\beta=1$. The maximum angle at the slot, $\theta_{0,max}$, is taken here
to correspond with the dimensionless curtains shown in figures 2-9 (as well as
additional cases).

## Appendix A

###  A.1 Power Series

For subcritical cases ($W_{e_{0}}<1$), the power series
$\bar{H}(z)=\sum\limits_{j=0}^{\infty}(\lambda_{j}+i\Gamma_{j})z^{j}$,
centered at $z=0$ (the critical point), is a solution of (14) on the interval
$-b<z<b$, if the following conditions are met, where $b=c+1$ and
$Q_{j}=[\lambda_{j},\Gamma_{j}]^{T}$:

$Q_{1}=\left(A_{1}Q_{0}+\left[\begin{array}[]{cc}b^{4}\\\
-2\bar{\omega}b^{5}\\\ \end{array}\right]\right)s_{1}$ (18)
$Q_{2}=\left(A_{2}Q_{1}+B_{2}Q_{0}+\left[\begin{array}[]{cc}12b^{3}\\\
-12b^{4}\bar{\omega}\\\ \end{array}\right]\right)s_{2}$ (19)
$Q_{3}=\left(A_{3}Q_{2}+B_{3}Q_{1}+C_{3}Q_{0}+\left[\begin{array}[]{cc}27b^{2}\\\
-18b^{3}\bar{\omega}\\\ \end{array}\right]\right)s_{3}$ (20)
$Q_{4}=\left(A_{4}Q_{3}+B_{4}Q_{2}+C_{4}Q_{1}+\left[\begin{array}[]{cc}16b\\\
-8b^{2}\bar{\omega}\\\ \end{array}\right]\right)s_{4}$ (21)

And for all $j>3$

$Q_{j+1}=(A_{j+1}Q_{j}+B_{j+1}Q_{j-1}+C_{j+1}Q_{j-2})s_{j+1}$ (22)

where:

$A_{j}=\left[\begin{array}[]{cc}A_{11j}&A_{12j}\\\
A_{21j}&A_{22j}\end{array}\right]$ (23)

$A_{11j}=b(j+1)^{2}(b^{2}\bar{\omega}^{2}-j(j-1))-8b^{3}\bar{\omega}^{2}j(j+1)$
(24)

$A_{12j}=2\bar{\omega}b^{2}(2j(j+1)^{2}+(j+1)(b^{2}\bar{\omega}^{2}-j(j-1))$
(25)

$A_{21j}=-2\bar{\omega}b^{2}(2j(j+1)^{2}+(j+1)(b^{2}\bar{\omega}^{2}-j(j-1))$
(26)

$A_{22j}=b(j+1)^{2}(b^{2}\bar{\omega}^{2}-j(j-1))-8b^{3}\bar{\omega}^{2}j(j+1)$
(27)

$B_{j}=\left[\begin{array}[]{cc}B_{11j}&B_{12j}\\\ B_{21j}&B_{22j}\\\
\end{array}\right]$ (28)

$B_{11j}=2b^{2}\bar{\omega}^{2}(j+1)(3-j)$ (29)

$B_{12j}=2b\bar{\omega}(j+1)(2b^{2}\bar{\omega}^{2}+(j+1)(j-1))$ (30)

$B_{21j}=-2b\bar{\omega}(j+1)(2b^{2}\bar{\omega}^{2}+(j+1)(j-1))$ (31)

$B_{22j}=2b^{2}\bar{\omega}^{2}(j+1)(3-j)$ (32)

$C_{j}=\left[\begin{array}[]{cc}C_{11j}&C_{12j}\\\ C_{21j}&C_{22j}\\\
\end{array}\right]$ (33)

$C_{11j}=b\bar{\omega}^{2}(j+1)^{2}$ (34)

$C_{12j}=2b^{2}\bar{\omega}^{3}(j+1)$ (35)

$C_{21j}=-2b^{2}\bar{\omega}^{3}(j+1)$ (36)

$C_{22j}=b\bar{\omega}^{2}(j+1)^{2}$ (37)

$s_{j}=\frac{1}{b^{2}j^{4}+4b^{4}\bar{\omega}^{2}j^{2}}\\\ $ (38)

If $\lambda_{0}$ and $\Gamma_{0}$ are known, the rest of the parameters are
determined. So we have a two-parameter family of solutions. In particular
cases the two parameters are determined by the boundary condition, which is
the Dirichlet condition given by Equation (2.13) for the cases we consider in
this paper.

###  A.2 Power Series Convergence Proof

Here we prove that the power series
$\bar{H}(z)=\sum\limits_{j=0}^{\infty}(\lambda_{j}+i\Gamma_{j})z^{j}$
converges for $|z|<b$ . Let $T_{j}=\|Q_{j}\|$. It follows from Equation A5 and
the triangle inequality that

$T_{j+1}\leq\|A_{j}s_{j}\|T_{j}+\|B_{j}s_{j}\|T_{j-1}+\|C_{j}s_{j}\|T_{j-2}$
(39)

The diagonal elements of $A_{j}$ are fourth-order polynomials in $j$ with
leading coefficient $b$, the off-diagonal elements are third-order polynomials
in $j$, and $s_{j}$ is a fourth-order polynomial with leading coefficient
$b^{2}$. Thus the matrix $A_{j}s_{j}$ approaches $\frac{1}{b}I$, where $I$ is
the identity matrix, as $j$ goes to infinity. All of the elements in the
matrices $B_{j}$ and $C_{j}$ are polynomials of order less than four, so
$B_{j}s_{j}$ and $C_{j}s_{j}$ approach the zero matrix as $j$ approaches
infinity. Thus for every $r>0$, there exists an integer $K$, such that for all
$j>K$:

$\|A_{j}s_{j}\|<\frac{1}{b}+r$ (40) $\|B_{j}s_{j}\|<\frac{r}{b}$ (41)
$\|C_{j}s_{j}\|<\frac{r}{b^{2}}$ (42)

If $\gamma>\left(\frac{1}{b}+3r\right)$, then:

$\gamma>\frac{1}{b}+r+\frac{r}{\gamma b}+\frac{r}{\gamma^{2}b^{2}}$ (43)

Let $q>0$ be such that $T_{j}<q\gamma^{j}$, for all $j<K+3$.

Then,

$q\gamma^{j}>q\gamma^{j-1}(\frac{1}{b}+r)+q\gamma^{j-2}\frac{r}{b}+q\gamma^{j-3}\frac{r}{b^{2}}>(\frac{1}{b}+r)T_{j-1}+\frac{r}{b}T_{j-2}+\frac{r}{b^{2}}T_{j-3}\geq
T_{j}$ (44)

Therefore by induction, $T_{j}<q\gamma^{j}$ for all $j$. So
$\sum\limits_{j=0}^{\infty}T_{j}|z^{j}|$ is dominated by the geometric series
$\sum\limits_{j=0}^{\infty}q\gamma^{j}|z^{j}|$ and this converges for
$|z|<\frac{1}{\gamma}$. Therefore, $\sum\limits_{j=0}^{\infty}Q_{j}z^{j}$
converges absolutely for $|z|<\frac{1}{\gamma}$. Because we have the derived
bound for every $r$, where $\gamma=r+\frac{1}{b}$, we can take $r$ as small as
we’d like to and establish that $b$ is the radius of convergence.

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