# Stretching and breakup of two-dimensional falling Newtonian sheets

Kindness Isukwem, Liam Fève, Ramy Nemer, Edith Peuvrel-Disdier, Elie Hachem, Anselmo Soeiro Pereira

► To cite this version:

Kindness Isukwem, Liam Fève, Ramy Nemer, Edith Peuvrel-Disdier, Elie Hachem, et al. Stretching and breakup of two-dimensional falling Newtonian sheets. Physics of Fluids, 2025, 37 (6), pp.063125. 10.1063/5.0270377. hal-05212149

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Submitted on 30 Nov 2025

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Distributed under a Creative Commons Attribution 4.0 International License
Stretching and breakup of two-dimensional falling Newtonian sheets

Kindness Isukwem,\* Liam Fève,† Ramy Nemer,‡ Edith Peuvrel-Disdier,§ Elie Hachem,¶ and Anselmo Pereira\*\*

Mines Paris, PSL University, Centre for material forming (CEMEF),
UMR CNRS 7635, rue Claude Daunesse, 06904 Sophia-Antipolis, France

(Dated: May 8, 2025)

This theoretical and numerical study investigates the stretching and the subsequent breakup process in a Newtonian sheet which falls from a slot in ambient air under gravity. We perform two-dimensional numerical simulations based on a variational multi-scale approach devoted to multiphase fluid flows. The results are analyzed by considering the sheet stretching dynamics, energy transfers and scaling laws. Three flow regimes associated with the early times of the stretching process are highlighted: gravito-inertial (free fall); gravito-viscous; and mixed gravito-inertio-viscous. In late times, the sheet becomes balanced by a competition between inertial and capillary stresses. The findings are summarized in two-dimensional diagrams that correlate the sheet breakup time and the generation of satellite objects after the liquid fragmentation with different flow regimes using the important dimensionless parameters of the problem.

**Keywords:** falling Newtonian sheet; thinning process; stretching and breakup; multiphase flow; 2D numerical simulation; scaling laws.

I. INTRODUCTION

The stretching and subsequent breakup of liquids is a cardinal matter in fluid mechanics. It is directly related to critical environmental and industrial multiscale situations, such as the formation and fragmentation of large-scale cosmic sheets and filaments in galaxy clusters [1], the optimisation of pesticide deposition via liquid atomisation [2], the optimisation of liquid-propulsion systems [3], coating [4], microfluidics [5], inkjet printing [6], and 3D bioprinting [7, 8].

In this work, we focus on the stretching and breakup of two-dimensional Newtonian sheets falling from a slot in ambient air under gravity, a fundamental multiphase problem intrinsically linked with the fragmentation of magma sheets during volcanic eruptions [9] and water-falls [10], the destabilization of saliva films during speech [11], the manufacturing of thin glass sheets [12], paper fabrication [13], the control of container filling processes from the liquid injection to its rupture [14], and the stabilization of thin and uniform fluid films (or curtains) during their deposition on a moving solid substrate [15].

Following the seminal works of Savart [16–18], Taylor [19–22], Squire [23], Hagerty & Shea [24], and Brown [25], numerous investigations have addressed the growth of small amplitude sinuous (antisymmetric) waves and dilatational (symmetric) perturbations within the falling sheet, from which its rupture would emanate [26–32]. Despite the lack of consensus on the physical mechanics

driving the sheet destabilization, the mentioned studies have pointed out that aerodynamic forces arising from liquid-air interactions would be the primary source of instability amplification. In such a context, small transverse perturbations in the falling sheet would be naturally amplified due to the shear with the surrounding air (e.g., differences between the liquid and the air velocities). The latter would then induce the growth of unstable periodic oscillations that oppose stabilizing surface tension and viscous effects [33–35]. These oscillations would be advected by the flow (Kelvin-Helmholtz convective instabilities) through the downstream part of the sheet where the inertial stresses overpass the capillary stresses, ultimately inducing fragmentation [31, 34–37].

Another relevant and unelucidated question concerns the decay of the sheet thickness during the stretching process. Such a lack of comprehension emerges from a singularity exhibited by the equations governing the time evolution of small perturbations, which appears at a critical part of the sheet where the surface stress becomes equal to the inertial one. Although the nature of the singularity has been deeply analyzed by numerous researchers [10, 36, 38–42], the solution of the time-dependent equations driving the falling sheet remains a matter of discussion. Additionally, more details regarding the formation of satellite objects after the sheet rupture are needed. Hence, physically consistent attempts to provide a deeper understanding of the thinning process in a falling sheet and the eventual formation of secondary objects following its rupture would be highly welcome.

Aiming to shed light on the questions underlined above, we present here a theoretical and numerical study focused on the physical mechanisms governing the stretching and the subsequent breakup process in a Newtonian sheet which falls from a slot in ambient air under gravity. Specifically, we build upon previous studies by exploring the thinning process in a gravity-elongated liquid sheet. Our approach utilizes two-dimensional (2D)

<footnote>\* kindness-chinwendu.isukwem@minesparis.psl.eu</footnote>

<footnote>† liam.feve@minesparis.psl.eu</footnote>

<footnote>‡ ramy.nemer@minesparis.psl.eu</footnote>

<footnote>§ edith.peuvrel-disdier@minesparis.psl.eu</footnote>

<footnote>¶ elie.hachem@minesparis.psl.eu</footnote>

<footnote>\*\* anselmo.soeiro.pereira@minesparis.psl.eu (corresponding author)</footnote>
<figure><img src="image_2.png" /><figcaption>FIG. 1. (a) Multiphase 2D numerical simulation: typical stretch and breakup of a Newtonian sheet (shown in red) with initial height $h_0$, initial thickness $d_0$, density $\rho$, viscosity $\eta$ and surface tension $\sigma$ falling from a slot (yellow lines). The initial sheet profile (first snapshot) combines a rectangle of dimensions $d_0 \times h_0 - 0.5d_0$ and a semicircle of diameter $d_0$. The sheet is surrounded by air (turquoise part), characterised by a density $\rho_{air}$ and viscosity $\eta_{air}$. It is stretched by gravity until it breaks apart at time $t_b$ (breakup instant). The mesh, illustrated with black lines on the right side of each snapshot, consists of approximately 300000 elements per unit height [m⁻¹], with the smallest element size being 1µm. A movie showing a typical numerical simulation is available online youtube.com/watch?v=hTLT648O6Xs (in this movie $h_0/d_0 = 2$, $\rho = 1000kg/m³$, $\eta = 0.001Pa·s$, $\sigma = 0.072N/m$, $\rho_{air} = 1kg/m³$, and $\eta_{air} = 0.00001Pa·s$). | (b) Fall criterion (each circle represents a numerical simulation): the sheet falls when $\rho gh_0 > 4\sigma/d_0$ (blue closed circles) and retracts when $\rho gh_0 \le 4\sigma/d_0$ (open grey circles). A movie showing these two situations is available on youtube.com/watch?v=FPfXwdHsQN0 in which $\rho gh_0 < 4\sigma/d_0$ for the left-hand side case (retraction) and $\rho gh_0 > 4\sigma d_0$ for the right-hand side case (fall).</figcaption></figure>

numerical simulations based on a variational multi-scale framework devoted to multiphase fluid flows. The results are depicted through stretching dynamics analyses, energy transfers, and scaling laws. The findings are summarized in two-dimensional diagrams that correlate the sheet breakup time and the generation of satellite objects after the liquid rupture with different flow regimes using the important dimensionless parameters of the problem.

The organization of the paper is as follows. A detailed description of the physical formulation and the numerical method is presented in section II. The important parameters of the problem are equally stressed. Numerical results are discussed in Section III. Finally, conclusions and perspectives are drawn in the closing section.

## II. PHYSICAL FORMULATION, NUMERICAL METHOD, DIMENSIONLESS NUMBERS, AND FALL CRITERION

As mentioned in the previous section, this study presents a theoretical and numerical analysis of the stretching and subsequent breakup in falling Newtonian sheets, as depicted in Fig. 1(a). The sheets, characterized by an initial length $h_0$, initial thickness $d_0$, density $\rho$, viscosity $\eta$ and surface tension $\sigma$, are surrounded by

air (turquoise part). They fall from a slot of thickness $d_0$ (yellow lines) in ambient air under gravity $g$, being stretched and subsequently breaking apart at instant $t_b$. After breakup ($t > t_b$), the part of the sheet which remains attached to the slot retracts by surface effects, while its bottom part continues to fall. The surrounding air is characterised by a density $\rho_{air}$ and viscosity $\eta_{air}$.

Our computational method is based on a massively parallel finite element library [CIMLIB-CFD; 43, 44] devoted to multiphase flows [45–54]. More specifically, we apply the momentum conservation equation presented below to the solenoidal flow ($\nabla \cdot u = 0$) described earlier (see Fig. 1a):

$$ \rho \left( \frac{\partial u}{\partial t} + u \cdot \nabla u - g \right) = -\nabla p + \nabla \cdot \tau + f_{st}, \quad (1) $$

in which $u$, $p$, $\tau$, $\nabla$, $g$, $\nabla \cdot$ and $f_{st}$ are, respectively, the velocity vector $u = \{u_x, u_y\}$, the pressure, the extra-stress tensor, the gradient operator, the gravity vector, the divergence operator, and a capillary term related to the surface tension force. The latter is defined as $f_{st} = -\sigma\kappa\Phi n$, where $\sigma$, $\kappa$, $\Phi$, and $n$ are the sheet surface tension with the surrounding air, the curvature of the sheet-air interface, the Dirac function locating the
sheet-air interface, and its normal vector, respectively. In addition, the extra-stress tensor is given by $\tau = \eta\dot{\gamma}$, in which $\dot{\gamma}$ represents the rate-of-strain tensor defined as $\dot{\gamma} = (\nabla u + \nabla u^T)$. The norm of $\dot{\gamma}$ is called deformation rate, being defined as $|\dot{\gamma}| = (\frac{1}{2}\dot{\gamma} : \dot{\gamma})^{\frac{1}{2}}$.

The numerical methods employed involve a Variational Multiscale Method (VMS) coupled with an anisotropic mesh adaptation technique [45–47]. The computational meshes consist of approximately 300000 elements per unit height [m⁻¹], with a minimum size of 1µm and a maximum size of d₀/6 (as illustrated by the black lines in Fig. 1a; the mesh sensitivity study is presented in the Appendix). A computation box of dimension 10d₀ × 100d₀ is used. The Courant–Friedrichs–Lewy (CFL) condition is lower than 0.1 for all numerical simulations considered here. The evolution of the interface of the sheet over time is tracked using a Level-Set function [44]. Fluid properties are interpolated through the Level-Set function whose value varies from positive within the sheet to negative in the air [and zero at the interfaces between the different fluid phases; supplemental details concerning the interface capturing and the Level-Set function are available in 52].

Concerning the boundary conditions, a fully developed Poiseuille (parabolic) profile is imposed as the liquid downwards vertical velocity at the slot inlet

$$u_{y,inlet} = \frac{3U_0}{2} \left[ 1 - \left( \frac{2|x|}{d_0} \right)^2 \right], \quad (2)$$

where $U_0$ denotes the liquid mean vertical velocity at the slot inlet and $|x|$ is the horizontal distance from the centerline (dashed line in Fig. 1a). Furthermore, a no-slip condition is imposed between the liquid and the walls of the slot (represented by the yellow lines). Lastly, zero normal stresses are imposed on the other walls of the computational domain. A movie showing a typical numerical simulation is available on youtube.com/watch?v=hTLT64806Xs in which $h_0/d_0 = 2$, $\rho = 1000\text{kg/m}^3$, $\eta = 0.001\text{Pa}\cdot\text{s}$, $\sigma = 0.072\text{N/m}$, $\rho_{\text{air}} = 1\text{kg/m}^3$, and $\eta_{\text{air}} = 0.00001\text{Pa}\cdot\text{s}$.

It is important to emphasize that our numerical multi-phase framework has been validated for a variety of flow scenarios, which includes the impact of drops [49–51], dam breaks [48, 53], the stretching and breakup of filaments [45], and the development of buckling instabilities [46, 47] using Newtonian and non-Newtonian fluids.

The dimensionless numbers governing the problem, derived from the Buckingham-Π theorem, are based on the dimensional variables $h_0$, $d_0$, $\rho$, $\eta$, $g$ (the y component of the gravity vector), $U_0$, $\rho_{\text{air}}$ and $\eta_{\text{air}}$, with the fundamental units of mass [kg], distance [m] and time [s]. These variables lead to six important dimensionless quantities:

$$\Pi_1 = \frac{h_0}{d_0}, \quad (3)$$

$$\Pi_2 = \frac{\sigma/h_0}{\rho g h_0}, \quad (4)$$

$$\Pi_3 = \frac{\eta \sqrt{g/h_0}}{\rho g h_0}, \quad (5)$$

$$\Pi_4 = \frac{\rho}{\rho_{\text{air}}}, \quad (6)$$

$$\Pi_5 = \frac{\eta_{\text{air}} \sqrt{g/h_0}}{\rho g h_0}, \quad (7)$$

$$\Pi_6 = \frac{\rho U_0^2}{\rho g h_0}. \quad (8)$$

$\Pi_2$ represents the inverse of the Bond number (Bo), $\Pi_3$ corresponds to the inverse Galileo number (Ga), $\Pi_5$ is the inverse air-based Galileo number ($Ga_{\text{air}}$), $\Pi_6$ is the Froude number based on the liquid’s mean velocity into the slot ($Fr_{\text{slot}}$) and $\sqrt{g/h_0}$ is a characteristic strain rate associated with the sheet stretching. In the present work, $1/Ga_{\text{air}} \lesssim 10^{-6}$, $\rho/\rho_{\text{air}}$ is fixed at 1000 and $Fr_{\text{slot}} \approx 10^{-10}$ ($U_0$ is much lower than the characteristic stretching velocity $\sqrt{g/h_0}$). Thus, the flow scenarios considered here varies with only three key dimensionless numbers:

$$\frac{h_0}{d_0} \quad (\text{aspect ratio}), \quad (9)$$

$$\frac{1}{\text{Bo}} = \frac{\sigma/h_0}{\rho g h_0} \quad (\text{inverse Bond number}), \quad (10)$$

$$\frac{1}{\text{Ga}} = \frac{\eta \sqrt{g/h_0}}{\rho g h_0} \quad (\text{inverse Galileo number}). \quad (11)$$

The aspect ratio $h_0/d_0$ ranges from 2 to 5, while $0.001 \le 1/\text{Bo} \le 3$ and $0.0001 \le 1/\text{Ga} \le 13$. The impact of these three dimensionless numbers on the stretching and rupture of Newtonian sheets is discussed in section III, where the results are presented in terms of these quantities.

It is essential to underline that 2D Newtonian sheets can either fall by gravity or retract towards the slot, depending on the ratio of the gravitational stress ($\rho g h_0$) to the capillary stress ($\sim \sigma/d_0$) at $t = 0$s. More specifically, as pointed out by Fig. 1(b) where each circle represents a numerical simulation, the sheet falls when $\rho g h_0$ exceeds $4\sigma/d_0$ (see the blue solid circles for which $\rho g h_0 > 4\sigma/d_0$). On the other hand, if $\rho g h_0 \le 4\sigma/d_0$ (open grey circles), the sheet retracts towards the slot due to surface tension effects, tending to assume a circular profile. A movie showing these two situations is available on youtube.com/watch?v=FPfXwdHsQN0 in which $\rho g h_0 \le 4\sigma/d_0$ for the left-hand side case (retraction) and $\rho g h_0 > 4\sigma/d_0$ for the right-hand side case (fall). Note that the fall criterion can be expressed as $\text{Bo} > 4h_0/d_0$ (blue region in Fig. 1b). Since we are interested in the stretching and breakup process, in the following section we just consider flow scenarios in which this flow criterion is fulfilled.
<figure><img src="image_3.png" /><figcaption>FIG. 2. Stretching and breakup of four Newtonian sheets for different combinations of 1/Ga and 1/Bo at $h_0/d_0 = 2.5$: (a) 1/Ga = 0.0003 and 1/Bo = 0.03; (b) 1/Ga = 1 and 1/Bo = 0.03; (c) 1/Ga = 2.55 and 1/Bo = 0.03; and (d) 1/Ga = 0.0003 and 1/Bo = 0.07. The sheets appear in black, while the air is in grey. The time interval between subsequent snapshots is denoted by $\Delta t$.</figcaption></figure>

Finally, because we consider a 2D Cartesian framework, we must emphasize that our theoretical and numerical study is devoted to Newtonian fluid sheets whose width $\gg$ height and width $\gg$ thickness [waterfalls, thin glass sheets, saliva films etc.; 11, 12, 32].

### III. RESULTS AND DISCUSSION

Figure 2 illustrates the stretching and breakup of four Newtonian sheets for different combinations of 1/Ga and 1/Bo at $h_0/d_0 = 2.5$: 1/Ga = 0.0003 and 1/Bo = 0.03 (Fig. 2a); 1/Ga = 1 and 1/Bo = 0.03 (Fig. 2b); 1/Ga = 2.55 and 1/Bo = 0.03 (Fig. 2c); and 1/Ga = 0.0003 and 1/Bo = 0.07 (Fig. 2d). The sheets appear in black, while the air is in grey. Each sub-figure contains a sequence of eight snapshots showing the time evolution of the thinning process, the sheet rupture, and the eventual formation of satellite objects. The time interval between subsequent snapshots is denoted by $\Delta t$.

As seen by comparing Figs. 2(a), 2(b) and 2(c), an increase in 1/Ga, which amplifies the viscous stresses relative to the gravitational stresses, results in a greater stretching before rupture. In other words, longer sheets are formed in high-viscosity fluids. Logically, higher viscosities delay the growth of perturbations within the liquid, favoring the formation of longer falling sheets [34, 55]. Interestingly, although the increase in 1/Bo marginally affects the breakup time, it produces shorter sheets before fragmentation, as observed by comparing Figs. 2(a) and 2(d). It is also worth noticing that satellite objects generated after the breakup vanish as viscous

and surface tension stabilizing effects are enlarged by the increase of 1/Ga and 1/Bo, respectively. This can be seen by comparing Figs. 2(a) and 2(b), and Figs. 2(a) and 2(d). The results suggest the existence of critical values of 1/Ga and 1/Bo from which secondary objects would be suppressed (we will come back to this point later in this section).

The influence of 1/Ga, 1/Bo and $h_0/d_0$ on the thinning dynamics is further illuminated in Fig. 3, where the instantaneous minimum thickness of the sheet made dimensionless by its initial diameter $d(t)/d_0$ is plotted against $t/t_b$. Clearly, as 1/Ga increases from 0.01 to 2.55 at fixed 1/Bo and $h_0/d_0$ (Fig. 3a), the thickness profile dramatically switches from a non-linear (at 1/Ga = 0.01) to a quasi-linear decay (at 1/Ga = 2.55), which indicates significant physical changes in the thinning process (follow the magenta arrow). In contrast, variations in 1/Bo and $h_0/d_0$ have marginal effects on $d(t)/d$, as highlighted by Figs. 3(b) and 3(c). The results displayed in Fig. 3 suggest then that, once the fall criterion is fulfilled (Bo > $4h_0/d_0$), the physical mechanisms dominating the thinning process are essentially affected by the Galileo number.

To gain a deeper understanding of the physical mechanisms driving the thinning process, Fig. 4 presents the energy transfer during the sheet stretching at 1/Ga = 0.0003 (Fig. 4a) and 1/Ga = 1 (Fig. 4b) for 1/Bo = 0.03 and $h_0/d_0 = 2.5$. The power terms (per unit width) displayed against time are defined below:

$$ \dot{G} = \frac{\partial}{\partial t} \int_S \rho g z dS \quad (\text{gravitational energy variation}), \qquad (12) $$
<figure><img src="image_4.png" /><figcaption>FIG. 3. The instantaneous minimum thickness of the sheet made dimensionless by its initial diameter $d(t)/d_0$ is plotted against $t/t_b$. (a) 1/Ga increases from 0.01 to 2.55 at 1/Bo = 0.03 and $h_0/d_0 = 2.5$. (b) 1/Bo ranges from 0.03 to 0.06 at 1/Ga = 0.01 and $h_0/d_0 = 2.5$. (c) $h_0/d_0 = 2.5$ varies from 2 to 4 at 1/Ga = 0.01 and 1/Bo = 0.03.</figcaption></figure>

$$ \dot{S}E = \frac{\partial}{\partial t} \int_C \sigma dC \quad (\text{surface energy variation}), \quad (13) $$

$$ \dot{K}E = \frac{\partial}{\partial t} \int_S \frac{\rho |\mathbf{u}|^2}{2} dS \quad (\text{kinetic energy variation}), \quad (14) $$

$$ \dot{W} = \int_S \frac{\eta |\dot{\gamma}|^2}{2} dS \quad (\text{dissipation}), \quad (15) $$

where S and C represents the area and the perimeter of the sheet, respectively. Each power term is made dimensionless by $\rho g h_0 S \sqrt{g/h_0}$:

$$ \dot{G}^* = \frac{\dot{G}}{\rho g h_0 S \sqrt{g/h_0}}, \quad (16) $$

$$ \dot{S}E^* = \frac{\dot{S}E}{\rho g h_0 S \sqrt{g/h_0}}, \quad (17) $$

$$ \dot{K}E^* = \frac{\dot{K}E}{\rho g h_0 S \sqrt{g/h_0}}, \quad (18) $$

$$ \dot{W}^* = \frac{\dot{W}}{\rho g h_0 S \sqrt{g/h_0}}. \quad (19) $$

At low 1/Ga (Fig. 4a), the gravitational energy is primarily converted into kinetic energy during the sheet's fall, with a negligible energy dissipation. However, at 1/Ga = 1 (Fig. 4b), the dissipation becomes dominant in early times (note in the inset plots that $\dot{W}^*$ exhibits the highest positive values for $t/t_b < 0.5$), although it is over-passed by $\dot{K}E^*$ and $\dot{S}E^*$ at $t/t_b > 0.6$. In fact, in late

times, regardless of the value of 1/Ga, the sheet becomes balanced by a competition between inertial and capillary stresses, i.e. $\dot{K}E^*$ and $\dot{S}E^*$ are the relevant positive power terms while $\dot{W}^*$ tends to zero. The kinetic energy enhancement favors the growth of wavy instabilities caused by liquid-air interactions, whereas the capillary energy acts as a stabilizing entity. This can be demonstrated by considering the linear instability of the interface between a liquid of density $\rho$ and local downward vertical velocity $u_y$ and the surrounding air of density $\rho_{air}$ and local vertical velocity $u_{y,air}$. In the absence of viscous effects, one can show that the sheet is unstable for wavelengths larger than a critical value $\lambda_c$ [3, 22, 56, 57]

$$ \lambda_c \sim \left( \frac{\rho + \rho_{air}}{\rho \rho_{air}} \right) \frac{\sigma}{(u_y - u_{y,air})^2}. \quad (20) $$

For $\rho \gg \rho_{air}$ and $u_{y,air} = \alpha u_y$, where $\alpha < 1$,

$$ \lambda_c \sim \frac{\sigma}{\rho_{air} (1 - \alpha)^2 u_y^2}. \quad (21) $$

Dividing the critical perturbation length by the sheet minimal thickness $d$,

$$ \frac{\lambda_c}{d} \sim \frac{1}{(1 - \alpha)^2} \left( \frac{\rho}{\rho_{air}} \right) \left( \frac{\sigma/d}{\rho u_y^2} \right), \quad (22) $$

where $(\sigma/d)/\rho u_y^2$ is the inverse local Weber number. This simple rationale underlines the strong connection between $\lambda_c/d$ and the liquid-air density ratio $\rho/\rho_{air}$, which is in agreement with recent instability analyses performed for Newtonian and viscoelastic liquids [34, 35], as well as with our numerical simulations (supplemental results stressing the impact of $\rho/\rho_{air}$ on the stretching and breakup processes are available in the Appendix). Furthermore, it clearly indicates that the range of wavelengths able to destabilize the sheets is an increasing
<figure><img src="image_5.png" /><figcaption>FIG. 4. (a-b) Dimensionless power terms as a function of $t/t_b$ at 1/Ga equal to 0.0003 (a) and 1 (b) for 1/Bo = 0.03 and $h_0/d_0 = 2.5$: gravitational (grey circles), surface (blue triangles), kinetic (green squares), and dissipative (red diamonds) term. | (c-d) The instantaneous minimum thickness of the sheet made dimensionless by its initial diameter $d(t)/d_0$ is plotted against $t/t_b$ at 1/Ga equal to 0.01 (c) and 2.55 (d) for 1/Bo = 0.03 and $h_0/d_0 = 2.5$. The numerical data (sky blue circles) is fitted by Eqs. 24 (in c) and 31 (in d) represented by the black curves.</figcaption></figure>

function of the local inertial stress (~$\rho u_y^2$), whereas the local capillary stress (~$\sigma/d$) plays an opposite role. As $\rho u_y^2$ becomes comparable to $\sigma/d$ eventually overpassing the latter, a wider range of instabilities grows quickly until the sheet breaks apart.

In short, low 1/Ga sheets are characterised by an initial gravito-inertial stretching stage, whereas high 1/Ga sheets exhibit an initial gravito-viscous stretching one. These stages are both followed by a final capillary-inertial stage, throughout which $\rho u_y^2$ increases, favouring the growth of perturbations induced by liquid-air interactions, ultimately leading to the liquid rupture. Lastly, it is essential to underline that marginal effects on the power curves are observed when varying both 1/Bo and $h_0/d_0$ at a fixed 1/Ga (not shown for brevity), a result in line with the findings pointed out by Figs. 3(b) and 3(c).

The power curves highlighted by Figs. 4(a) and 4(b) indicate that at least two stretching regimes can emerge in the early times of the sheet's fall: the gravito-inertial regime (Fig. 4a) and the gravito-viscous regime (Fig. 4b). In the first regime, viscosity and surface tension effects are negligible, and consequently the sheet falls freely. In the absence of viscous and capillary forces (free fall), the evolution of the position $h(t)$ of a particle at the end of the sheet can be described as

$$ \frac{d^2h(t)}{dt^2} = g. \qquad (23) $$

By integrating Eq. 23 with $dh(t)/dt = 0$ at $t = 0$, $h(t = 0) = h_0$, and $h(t) \sim h_0 d_0/d(t)$ by mass conservation (supposing a quasi-homogeneous sheet), we find

$$ \frac{d(t)}{d_0} \sim \frac{1}{1 + \frac{g}{2h_0} t^2} \quad (\text{gravito-inertial regime}), \quad (24) $$

a non-linear thickness decay in which $\sqrt{2h_0/g}$ represents a free-fall characteristic time denoted by $t_{g-i}$ [e.g., $t_{g-i} = \sqrt{2h_0/g}$].

In the second regime, the gravitational energy is mainly dissipated by viscous effects at the beginning of the thinning process, with negligible inertial and capillary effects. Consequently, we write that

$$ \nabla \cdot \boldsymbol{\tau} = -\rho g, \qquad (25) $$

with $\nabla \cdot \boldsymbol{u} = 0$ (incompressible flow). As a result, we find for a thin sheet under gravity-induced elongation

$$ 4\eta \frac{d^2u_y}{dy^2} = -\rho g, \qquad (26) $$

where $u_y$ is the downward vertical velocity. By solving Eq. 26 imposing that $u_y = 0$ at $y = 0$ and $du_y/dy = 0$ at $y = h$ (boundary conditions), we obtain

$$ u_y = -\frac{\rho g}{4\eta} \left( \frac{y^2}{2} - hy \right). \qquad (27) $$

Based on Eq. 27, the velocity of a particle at the end of the sheet ($y = h$) can be expressed as

$$ u_y = \frac{\rho gh^2}{8\eta}. \qquad (28) $$

Since, from the particle standpoint, $u_y = dh(t)/dt$, we find

$$ \frac{dh(t)}{dt} = \frac{\rho gh(t)^2}{8\eta}, \qquad (29) $$

which can be integrated

$$ \int_{h_0}^{h(t)} \frac{1}{h(t)^2} dh = \int_{0}^{t} \frac{\rho g}{8\eta} dt. \qquad (30) $$

By solving the above equation with $h \sim h_0 d_0/d$ by mass conservation, we finally obtain

$$ \frac{d(t)}{d_0} \sim 1 - \frac{\rho g h_0}{8\eta} t \quad (\text{gravito-viscous regime}), \quad (31) $$
<figure><img src="image_6.png" /><figcaption>FIG. 5. (a) Diagram given the rescaled breakup time $t_b\sqrt{g/2h_0}$ as a function of $(8/\sqrt{2})(1/\text{Ga})$. The numerical data falls along a unified trend path that can be divided into three distinct zones: I (in blue) related to the gravito-inertial regime, for which $t_b$ scales with $\sqrt{2h_0/g}$ (blue line); II (red) which corresponds to the gravito-viscous regime, where $t_b$ scales with $8\eta/(\rho g h_0)$ (red dashed line); and III [white; $0.7 < (8/\sqrt{2})(1/\text{Ga}) < 3$] a transitional gravito-inertio-viscous regime, where gravity, inertia and viscosity play a significant role from an energy transfer standpoint. (b) Diagram showing that satellite objects are generated if $1/\text{Ga} < 0.2$ and $1/\text{Bo} < 0.04$ (green stars in the green zone). Nevertheless, these objects are suppressed by increasing either capillary effects through $1/\text{Bo}$ (black asterisks into the grey zone) or viscous effects via $1/\text{Ga}$ (black asterisks into the orange zone).</figcaption></figure>

a linear thickness decay in which $8\eta/(\rho g h_0)$ denotes a gravito-viscous characteristic time $t_{g-v}$ [e.g., $t_{g-v} = 8\eta/(\rho g h_0)$].

The ability of Eqs. 24 and 31 to predict the thickness profiles is stressed in Figs. 4(c) and 4(d), where $d(t)/d_0$ is plotted against $t/t_b$ at 1/Ga = 0.01 (gravito-inertial scenario) and 1/Ga = 2.55 (gravito-viscous scenario), respectively, for a fixed 1/Bo = 0.03 and $h_0/d_0 = 2.5$ (sky blue circles). The numerical data in Fig. 4(c) is fitted by Eq. 24, while the quasi-linear decay displayed by the numerical data in Fig. 4(d) is reproduced by Eq. 31 (black lines).

It is important to note that the thickness profiles given by the mentioned equations fit satisfactorily almost the totality of the $d(t)/d_0$ curves illustrated in Fig. 4. Such a good agreement suggests that $t_b$ can be expressed as a function of the gravito-inertial and the gravito-viscous characteristic times, which is confirmed by Fig. 5(a), where the rescaled breakup time $t_b/t_{g-i}$ is displayed as a function of $t_{g-v}/t_{g-i}$. In other words, we plot $t_b\sqrt{g/2h_0}$ against $(8/\sqrt{2})(1/\text{Ga})$. The data points (which include sheets of aspect ratio ranging from 2 to 5) fall along a unified trend path that can be divided into three distinct zones: I (in blue), characterized

by the gravito-inertial regime, for which $t_b$ scales with $\sqrt{2h_0/g}$ (blue line); II (red) which corresponds to the gravito-viscous regime, where $t_b$ scales with $8\eta/(\rho g h_0)$ (red dashed line); and III (white) a transitional gravito-inertio-viscous regime, where gravity, inertia and viscosity play a significant role from an energy transfer standpoint (not shown for brevity). Logically, this transition is expected to occur when $t_{g-i} \approx t_{g-v}$ or, said differently, when $(8/\sqrt{2})(1/\text{Ga}) \approx 1$. Indeed, a mixed gravito-inertio-viscous regime emerges for $0.7 < (8/\sqrt{2})(1/\text{Ga}) < 3$ (for which $\dot{K}E^* \approx \dot{W}^*$ at $t \le 0.5t_b$).

It is worth mentioning that coefficients of determination ($R^2$) ranging from 0.92 and 0.99 are obtained when fitting $d(t)/d_0$ with Eqs. 24 and 31 for the cases landing respectively within zones I and II in Fig. 5(a). As a result, the breakup length $h_b$ (e.g., maximum length exhibited by the sheet before its breakup) can be estimated by combining $t_b$ given by Fig. 5(a) with Eqs. 23 and 30 for $t = t_b$.

Lastly, the appearance of satellite objects is summarized by the $1/\text{Ga}-1/\text{Bo}$ diagram illustrated in Fig. 5(b), where each point represents a 2D numerical simulation. Once again, the data include aspect ratio values ranging from 2 to 5. The green stars denote flow cases for which
satellite objects emerge after the sheet rupture, while the asterisks indicate the absence of such objects. The latter typically appear when 1/Ga < 0.2 and 1/Bo < 0.04 (A region in green), i.e., when capillary and viscous stresses are not high enough to mitigate the supplemental growth of instabilities after the sheet's first rupture (similar to Fig. 2a). In contrast, satellite objects are suppressed by further increasing either $\sigma/h_0$ (B region, in grey; similar to Fig. 2d) or $\eta\sqrt{g/h_0}$ (C region in orange; similar to Figs. 2b, 2c and 2d). Nevertheless, it is important to emphasize that, as stressed by Eq. 22, the decrease of $\rho/\rho_{air}$ is also expected to affect the formation of satellite objects and consequently modify the critical values highlighted by Fig. 5(b). Although $\rho/\rho_{air}$ effects are not deeply investigated here, we briefly discuss how the density ratio changes the sheet stretching and fragmentation in the Appendix.

## IV. CONCLUDING REMARKS

We have presented a theoretical and numerical study devoted to the physical mechanisms driving the stretching and fragmentation of two-dimensional a falling Newtonian sheet. The latter falls from a slot in ambient air under gravity, being elongated and subsequently breaking apart. Our 2D numerical simulations were based on a variational multi-scale approach devoted to multiphase flows. The results obtained were analyzed by considering the sheet thinning dynamics, energy transfers and scaling laws.

Three flow regimes associated with the early times of the stretching process have been highlighted: the gravito-inertial regime, in which the sheet tends to fall freely while its gravitational potential energy is converted into kinetic energy; the gravito-viscous regime, resulting from a balance between gravitational and viscous stresses during the sheet fall; and the mixed gravito-inertio-viscous regime, for which gravity, inertia and viscosity are all important. In late times, the sheet becomes balanced by a competition between inertial and capillary stresses. These results were synthesized in the form of a master curve giving the sheet dimensionless breakup time (e.g., the breakup time made dimensionless by the characteristic stretching time $\sqrt{h_0/g}$) as a function of the inverse Galileo number $1/Ga = \eta\sqrt{g/h_0}/(\rho gh_0)$. Furthermore, the development of satellite objects after the sheet breakup can be suppressed either due to surface tension effects when $1/Bo > 0.04$, where $1/Bo = (\sigma/h_0)/(\rho gh_0)$, or to viscosity effects when $1/Ga > 0.2$.

In short, our study expands previous works [27–31, 34–37] by providing supplemental features on the sheet’s thinning process for different flow regimes, as well as on the generation of satellite objects following the sheet’s rupture. The results have direct implications in several industrial contexts, including the manufacturing of thin glass sheets, paper fabrication, the control of container filling processes from the liquid injection to its rupture,

analyses of destabilization process in saliva films during speech, and the stabilization of a thin and uniform fluid film (or curtain) during its deposition on a moving solid substrate [11–15, 32].

Finally, since non-Newtonian fluids are associated with several industrial and everyday life situations, it would also be interesting to incorporate other sources of complexity in future works, including viscoplasticity, thixotropy, and viscoelasticity. Further (and deeper) analyses on the atomization process would be equally more than welcome.

**Acknowledgements:** The authors would like to acknowledge the support from the UCAJEDI program (IDEX of Université Côte d'Azur), the PSL Research University under the program ‘Investissements d'Avenir’ launched by the French Government and implemented by the French National Research Agency (ANR) with the reference ANR-10-IDEX-0001-02 PSL, and the ANR for supporting the INNpact project (ANR-21-CE06-0036) under the ‘Jeunes chercheuses et jeunes chercheurs’ program.

## APPENDIX

In this Appendix, we use 2D numerical simulations to underline density ratio effects on the stretching and fragmentation of falling Newtonian sheets.

According to Eq. 22, as the gas density increases (and consequently $\rho/\rho_{air}$ decreases), a lower local kinetic energy ($\sim \rho v_y^2$) is required to reach a fixed critical perturbation length $\lambda_c$. This theoretical prediction is corroborated by the results displayed in Fig. 6 where the stretching/fragmentation process is illustrated by fourteen subsequent snapshots related to four different $\rho/\rho_{air}$ values at $1/Ga = 0.0003$, $1/Bo = 0.03$, $h_0/d_0 = 2.5$: 1000 (Fig. 6a); 100 (Fig. 6b); 20 (Fig. 6c); 2 (Fig. 6d). The interval between consecutive snapshots is $\Delta t = 10$ms.

As observed, the maximum length reached by the sheet before its rupture is an increasing function of $\rho/\rho_{air}$, an effect that becomes non-negligible for $\rho/\rho_{air} \lesssim 100$. Consequently, the sheet exhibits lower kinetic energy at the breakup instant when its density becomes comparable to $\rho_{air}$. This is confirmed by Figs. 7(a) and 7(b), where the instantaneous minimum thickness of the sheet made dimensionless by its initial diameter $d(t)/d_0$ is plotted against the area-averaged kinetic energy of the sheet $\rho|\bar{u}|^2$ made dimensionless by $\rho gh_0$ at $1/Ga = 0.0003$, $1/Bo = 0.03$ and $h_0/d_0 = 2.5$ (Fig. 7a); and $1/Ga = 0.0003$, $1/Bo = 0.07$ and $h_0/d_0 = 2.5$ (Fig. 7b). Each symbol is related to a density ratio value. Clearly, for $\rho/\rho_{air} \lesssim 100$, the kinetic energy profile becomes highly affected by $\rho/\rho_{air}$, behaving as a decreasing function of the latter.

Interestingly, Figs. 7(c) and 7(d) show that the density ratio has marginal effects on the $d(t)/d_0$ variation with $t/t_b$. The results suggest that the $d(t)/d_0$ profiles illustrated in Fig. 3 are density-ratio independent. On
<figure><img src="image_7.png" /><figcaption>FIG. 6. Sheet contours illustrating the stretching/fragmentation process for four $ρ/ρ_{air}$ values at 1/Ga = 0.0003, 1/Bo = 0.03, $h_0/d_0$ = 2.5: (a) 1000; (b) 100; (c) 20; and (d) 2. The interval between consecutive snapshots is $\Delta t$ = 10ms.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>FIG. 7. (a-b) Instantaneous minimum thickness of the sheet made dimensionless by its initial diameter $d(t)/d_0$ plotted against the area-averaged kinetic energy of the sheet $ρ|\bar{u}|^2$ made dimensionless by $ρgh_0$ at 1/Ga = 0.0003, 1/Bo = 0.03 and $h_0/d_0$ = 2.5 (a), and 1/Ga = 0.0003, 1/Bo = 0.07 and $h_0/d_0$ = 2.5 (b). Each symbol is related to a density ratio value. | (c-d) Instantaneous minimum thickness of the sheet made dimensionless by its initial diameter $d(t)/d_0$ plotted against $t/t_b$. Each symbol is related to a density ratio value.</figcaption></figure>
<figure><img src="image_9.png" /><figcaption>FIG. 8. Sensitivity study: computations linking the rescaled breakup time $t_b\sqrt{g/2h_0}$ with the number of mesh elements per unit height ($\times 10^{-5}$) at $(8/\sqrt{2})(1/\text{Ga})$ equal to 0.01 (gravito-inertial regime) and 10 (gravito-viscous regime).</figcaption></figure>

the other hand, the formation of satellite drops is highly affected by $\rho/\rho_{air}$, as observed by comparing Figs. 6(a) and 6(d).

Lastly, Fig. 8 summarises a sensitivity study linking the rescaled breakup time $t_b\sqrt{g/2h_0}$ with the number of mesh elements per unit height ($\times 10^{-5} \text{m}^{-1}$) at $(8/\sqrt{2})(1/\text{Ga})$ equal to 0.01 (gravito-inertial regime) and 10 (gravito-viscous regime). Clearly, the rescaled breakup time becomes mesh-independent when the quantity of mesh elements per unit height becomes higher then $250000 \text{m}^{-1}$ (yellow region containing the mesh used in for the simulations presented here).

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