# Inertia and slip effects on the instability of a liquid film coated on a
fibre

Chengxi Zhao  Ran Qiao  Kai Mu  Ting Si<EMAIL_ADDRESS>Xisheng Luo
Department of Modern Mechanics, University of Science and Technology of China,
Hefei 230026, China

###### Abstract

To investigate the influence of inertia and slip on the instability of a
liquid film on a fibre, a theoretical framework based on the axisymmetric
Navier-Stokes equations is proposed via linear instability analysis. The model
reveals that slip significantly enhances perturbation growth in viscous film
flows, whereas it exerts minimal influence on flows dominated by inertia.
Moreover, under no-slip boundary conditions, the dominant instability mode of
thin films remains unaltered by inertia, closely aligning with predictions
from a no-slip lubrication model. Conversely, when slip is introduced, the
dominant wavenumber experiences a noticeable reduction as inertia decreases.
This trend is captured by an introduced lubrication model with giant slip.
Direct numerical simulations of the Navier-Stokes equations are then performed
to further confirm the theoretical findings at the linear stage. For the
nonlinear dynamics, no-slip simulations show complex vortical structures
within films, driven by fluid inertia near surfaces. Additionally, in
scenarios with weak inertia, a reduction in the volume of satellite droplets
is observed due to slip, following a power-law relationship.

###### keywords:

liquid films, interfacial flows, slip boundary condition

##  1 Introduction

The investigation of the instability of liquid jets due to surface tension has
a long history, tracing back to the pioneering work of Plateau (1873) and
Rayleigh (1878, 1892). This instability phenomenon also holds significant
importance in understanding the dynamics of liquid films coated on solid
fibres, with additional complexities at the liquid-solid interface. This field
has attracted significant scientific attention (Quéré, 1999), owing to its
critical relevance across various technological domains, including additive
manufacturing (Deng et al., 2011; Oliveira et al., 2020), droplet transport
(Lee et al., 2022), chemical element extraction (Chen et al., 2023) and water
collection through fog harvesting (Chen et al., 2018; Zhang et al., 2022).

The instability of a film coated on a fibre has been extensively studied
across various stages. The early-time behaviours can be described by the
Rayleigh-Plateau instability (Quéré, 1999), showing that a film with an outer
radius $h_{0}$ becomes unstable to sufficiently long-wavelength disturbances,
specifically $\lambda>\lambda_{crit}=2\pi h_{0}$. Here, $\lambda_{crit}$
represents the critical wavelength beyond which the instability ceases to
grow. Additionally, the dominant (most unstable/fast growing) modes are
influenced by the ratio of $h_{0}$ to the fibre radius $a$, validated by
experiments (Goren, 1964). Subsequent nonlinear evolution is modelled using
the lubrication approximation (Hammond, 1983), resulting in a leading-order
lubrication equation that is applicable to films on both the inside and
outside of a cylinder. Due to its simplicity, this lubrication equation and
its higher-order versions (Craster & Matar, 2006; Ruyer-Quil et al., 2008)
have been used in studying various interface dynamics of annular films on
fibers. Examples include the transition from absolute unstable regimes to
convective ones (Kliakhandler et al., 2001; Duprat et al., 2007; Craster &
Matar, 2009) and capillary drainage involving complex interactions of ‘lobes’
and ‘collars’ formed on the interfaces (Lister et al., 2006). Recently, these
lubrication models have been extended to encompass more complex scenarios by
incorporating other physics, such as electric fields (Ding et al., 2014), heat
transfer (Zeng et al., 2017), thermal fluctuations (Zhang et al., 2021), and
Van der Waals forces (Tomo et al., 2022).

One of the important physical factors is liquid-solid slip, which has recently
attracted substantial research interest (Secchi et al., 2016; Zhang et al.,
2020; Kavokine et al., 2021, 2022) and been found to influence the dynamics of
various interfacial flows (Liao et al., 2014; Halpern et al., 2015; Martínez-
Calvo et al., 2020a; Zhao et al., 2022). For the case of cylindrical films,
Ding & Liu (2011) introduced a lubrication equation incorporating slip
conditions to investigate the instability of films descending along porous
vertical fibres. Their findings revealed that the instability is amplified by
the presence of a fluid-porous interface, which is modelled using a slip
boundary condition. Regarding annular films within slippery tubes, Liao et al.
(2013) numerically solved a lubrication equation with leading-order terms,
demonstrating that even a fractional amount of wall slip significantly
exaggerates the instability, leading to considerably faster drainage compared
to the no-slip scenario (Hammond, 1983). Haefner et al. (2015) conducted
experimental investigations into the influence of slip on the instability for
films coated on horizontal fibres. Similar to the observations of both Ding &
Liu (2011) and Liao et al. (2013), the wall slip was found to enhance the
instability, resulting in increased growth rates of perturbations. The
experimental results were also shown to match predictions of a slip-modified
lubrication equation. Halpern & Wei (2017) subsequently illustrated how wall
slip can amplify drop formation in a film descending a vertical fiber. This
observation provides a plausible explanation for the discrepancy between
experimentally predicted and theoretically derived critical Bond numbers for
drop formation. More recent investigations delved into the dynamics of films
on slippery fibres within non-isothermal conditions (Chao et al., 2018) and
under the influence of intermolecular forces (Ji et al., 2019), employing more
intricate lubrication models. Despite the extensive use of slip-modified
lubrication models, their constraints have been exposed by Zhao et al. (2023a)
through linear instability analysis applied to the axisymmetric Stokes
equations. The theoretical framework not only highlights an overestimation of
the slip-enhanced perturbation growth rate as compared to classical
lubrication models, but also reveals a slip-dependent dominant wavelength,
deviating from the constant value posited by prior works using the lubrication
method (Liao et al., 2013; Haefner et al., 2015; Chao et al., 2018; Halpern &
Wei, 2017).

Noticeably, in modern applications such as additive manufacturing in space
(Reitz et al., 2021; Van Ombergen et al., 2023) and 3D printing with liquid
metals (Assael et al., 2010; Kondic et al., 2020), the role of inertia in
governing the dynamics of liquid film interfaces has become notably apparent,
contrasting with the predominant neglect of inertia in most preceding
investigations. One exception is the work done by Goren (1962), who introduced
inertial effects by conducting instability analysis for the full NS equations.
The theoretical findings elucidated distinctions between two limiting cases,
i.e. the inviscid case and viscous case without inertia. However, the impact
of inertia within the intermediary regime between these two limits remained
uncharted. Ding et al. (2013), proposed two coupled equations governing the
film thickness and flow rate to study instability and dynamics of a film on a
fibre, considering both inertia and slip. Nonetheless, their focus was
primarily on scenarios characterized by small to moderate Reynolds numbers,
where the influence of inertia on interface dynamics seemed less pronounced.
While recent studies have extensively investigated the influence of inertia on
the instability of planar films (González et al., 2016; Moreno-Boza et al.,
2020a), its effect on the cylindrical films, especially on a fibre with slip,
remains unclear, thus motivating the present investigations.

In this work, linear instability analysis of the axisymmetric NS equations is
performed to investigate inertia and slip effects on the dynamics of a liquid
film on a fibre. Direct numerical simulations of the NS equations are also
employed to confirm the theoretical findings and provide more physical
insights. The article is laid out as follows. Non-dimensionalised governing
equations for a film on a fibre are introduced in § 2. Linear instability
analysis for the governing equations is performed in § 3, where the dispersion
relation is derived in § 3.1, followed by two limiting cases: jet flows in §
3.2 and film flows without inertia in § 3.3 respectively. Predictions arising
from the theoretical model are presented in § 3.4. Subsequently, direct
numerical simulations are performed in § 4. These simulations are compared
with the predictions of the theoretical model, specifically concerning the
influence on the dominant mode (§ 4.1) and the growth rate (§ 4.2) of
perturbations. Nonlinear dynamics extracted from the simulations is also
analysed in § 4.2.

##  2 Model formulation

We consider a Newtonian liquid film on a fibre of the radius $a$ with the
$z$-axis along the centre line (figure 1). The initial radius of the film
measured from the $z$-axis is $r=h_{0}$. Additionally, gravity is neglected,
and we assume uniform external pressure and surface tension.

Figure 1: Schematic of a liquid film on a slippery fibre

The incompressible NS equations are employed to predict the dynamics of the
flow inside the liquid film. To identify the governing dimensionless
parameters, we non-dimensionalise the NS equations with the rescaling
variables shown below:

$(r,z,h)=\frac{(\tilde{r},\tilde{z},\tilde{h})}{h_{0}},\quad(u,v,w)=\frac{(\tilde{u},\tilde{v},\tilde{w})}{U},\quad
t=\frac{U}{h_{0}}\tilde{t},\quad p=\frac{h_{0}}{\gamma}\tilde{p},$ (1)

where $\tilde{h}$, $\tilde{t}$, and $\tilde{p}$ represents the dimensional
interface height, time and pressure, respectively (note that the dimensional
material parameters are not given tildes). $(r,\phi,z)$ are the cylindrical
coordinates with the corresponding velocities ($u,v,w$). $U$ represents the
characteristic velocity inside the film and $\gamma$ is the surface tension of
the liquid–gas interface. After eliminating all the derivatives with with
respect to $\phi$ and setting $v=0$ in the cylindrical coordinates, the
axisymmetric incompressible equations can be written as

$\displaystyle\frac{\partial w}{\partial
z}+\frac{1}{r}\frac{\partial(ur)}{\partial r}=0\,,$ (2)
$\displaystyle\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial
r}+w\frac{\partial u}{\partial z}=-\frac{1}{We}\frac{\partial p}{\partial
r}+\frac{1}{Re}\left[\frac{\partial^{2}u}{\partial
z^{2}}+\frac{\partial}{\partial
r}\left(\frac{1}{r}\frac{\partial(ur)}{\partial r}\right)\right]\,,$ (3)
$\displaystyle\frac{\partial w}{\partial t}+u\frac{\partial w}{\partial
r}+w\frac{\partial w}{\partial z}=-\frac{1}{We}\frac{\partial p}{\partial
z}+\frac{1}{Re}\left[\frac{\partial^{2}w}{\partial
z^{2}}+\frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial w}{\partial
r}\right)\right]\,,$ (4)

where the non-dimensional quantity $We=\rho U^{2}h_{0}/\gamma$ is the Weber
number, which relates the inertial force to the capillary force. $Re=\rho
Uh_{0}/\mu$ is the Reynolds number, showing the ratio between inertial force
and the viscous force. Here, $\mu$ is the liquid dynamic viscosity, and $\rho$
is the liquid density.

Since the density of gas around the film is much smaller than that of liquid,
the gas flow outside can be assumed to be dynamically passive to simplify the
problem. The liquid–gas interface height $h(z,t)$ satisfies the kinematic
boundary condition

$\frac{\partial h}{\partial t}+w\frac{\partial h}{\partial z}-u=0\,.$ (5)

The normal stress balance at the interface $r=h$ gives

$p-\frac{We}{Re}\,\bm{n}\cdot\bm{\tau}\cdot\bm{n}=\nabla\cdot\bm{n}\,,$ (6)

where $\tau$ is the shear stress, which is proportional to the strain rate in
Newtonian fluids. $\bm{n}$ is the outward normal and $\nabla\cdot\bm{n}$
represents the dimensionless Laplace pressure. The tangential force balance is

$\bm{n}\cdot\bm{\tau}\cdot\bm{t}=0\,,$ (7)

where $\bm{t}$ is the tangential vector. With $\bm{n}$ and $\bm{t}$ expressed
in terms of the unit vectors in $z$-direction ($\hat{\bm{e}}_{z}$) and
$r$-direction ($\hat{\bm{e}}_{r}$),

$\bm{n}=-\frac{\partial_{z}h}{\sqrt{1+(\partial_{z}h)^{2}}}\hat{\bm{e}}_{z}+\frac{1}{\sqrt{1+(\partial_{z}h)^{2}}}\hat{\bm{e}}_{r}\textrm{\,\,\,and\,\,\,}\bm{t}=\frac{1}{\sqrt{1+(\partial_{z}h)^{2}}}\hat{\bm{e}}_{z}+\frac{\partial_{z}h}{\sqrt{1+(\partial_{z}h)^{2}}}\hat{\bm{e}}_{r},$

(6) and (7) explicitly give

$\displaystyle
p-\frac{We}{Re}\frac{2}{1+\left(\partial_{z}h\right)^{2}}\left[\frac{\partial
u}{\partial r}-\frac{\partial h}{\partial z}\left(\frac{\partial w}{\partial
r}+\frac{\partial u}{\partial z}\right)+\left(\frac{\partial h}{\partial
z}\right)^{2}\frac{\partial w}{\partial z}\right]$
$\displaystyle=\frac{1}{h\sqrt{1+\left(\partial_{z}h\right)^{2}}}-\frac{\partial_{z}^{2}h}{\left(1+\left(\partial_{z}h\right)^{2}\right)^{\frac{3}{2}}}$
(8)

for the normal forces, and

$2\frac{\partial h}{\partial z}\left(\frac{\partial u}{\partial
r}-\frac{\partial w}{\partial z}\right)+\left[1-\left(\frac{\partial
h}{\partial z}\right)^{2}\right]\left(\frac{\partial w}{\partial
r}+\frac{\partial u}{\partial z}\right)=0\,.$ (9)

for the tangential forces. Here $\partial_{z}$ and $\partial_{z}^{2}$ refers
to the first and second axial derivatives.

In terms of the boundary conditions at the fibre surface $r=\alpha$ ($\alpha$
is the dimensionless fibre radius, i.e. $\alpha=a/h_{0}$), we introduce the
Navier slip boundary condition (Navier, 1823) in tangential direction and the
no-penetration boundary condition in the normal direction such that

$\displaystyle w$ $\displaystyle=l_{s}\frac{\partial w}{\partial r}\,,$ (10)
$\displaystyle u$ $\displaystyle=0\,.$ (11)

Here, $l_{s}$ represents the dimensionless slip length rescaled by $h_{0}$.

The governing equations above can be further simplified under the long-wave
approximation to give lubrication equations. Similar to the previous work on
planar films (Münch et al., 2005), different rescaling for the leading orders
give different forms of the lubrication model. When inertia is neglected
($Re\ll 1$), the one-equation model, which has been widely used in the
previous works (Craster & Matar, 2006; Haefner et al., 2015; Zhang et al.,
2020; Zhao et al., 2023a), can be obtained from the Stokes equations. The
dimensionless format of the no-slip lubrication model is

$\frac{\partial h}{\partial t}=\frac{1}{h}\frac{\partial}{\partial
z}\left[M(h)\frac{\partial}{\partial
z}\left(\frac{1}{h}-\frac{\partial^{2}h}{\partial z^{2}}\right)\right]\,,$
(12)

where $M(h)$ is the the mobility term

$M(h)=\frac{1}{16}\left[-3h^{4}-\alpha^{4}+4\alpha^{2}h^{2}+4h^{4}\mathrm{ln}\left(\frac{h}{\alpha}\right)\right]\,.$
(13)

When inertia is not negligible and $l_{s}\gg 1$, we propose another
lubrication model consisting of two equations (see Appendix A for the
derivation). The dimensionless format of this giant-slip lubrication model is

$\displaystyle\frac{\partial h^{2}}{\partial
t}+\frac{\partial\left(h^{2}w\right)}{\partial z}=$ $\displaystyle 0\,,$ (14a)
$\displaystyle\frac{\partial w}{\partial t}+w\frac{\partial w}{\partial z}=$
$\displaystyle-\frac{1}{We}\frac{\partial}{\partial
z}\left(\frac{1}{h}-\frac{\partial^{2}h}{\partial
z^{2}}\right)+\frac{1}{Re}\frac{3}{h^{2}-\alpha^{2}}\frac{\partial\left(h^{2}\partial_{z}w\right)}{\partial
z}$
$\displaystyle-\frac{1}{Re}\frac{2\alpha^{2}}{h^{2}-\alpha^{2}}\left(\frac{\partial^{2}w}{\partial
z^{2}}-\frac{w}{\alpha\,l_{s}}\right)\,.$ (14b)

When $\alpha=0$, the lubrication model for the jet flows (Eggers & Dupont,
1994) is recovered.

##  3 Instability analysis

In this section, linear instability analysis based on equations (2 \- 10) is
performed using the normal mode method, which has been widely used for the
instability in different fluid configurations (Rayleigh, 1878; Tomotika, 1935;
Craster & Matar, 2006; Li et al., 2008; Si et al., 2009; Liang et al., 2011;
González et al., 2016).

###  3.1 Derivation for the dispersion relation

To perform instability analysis, the dimensionless perturbed quantities are
set as

$\displaystyle u(r,z,t)=\hat{u}(r)e^{\omega
t+ikz},\,\,w(r,z,t)=\hat{w}(r)e^{\omega
t+ikz}\,\,\textrm{and}\,\,p(r,z,t)=1+\hat{p}(r)e^{\omega t+ikz}\,,$ (15)

where $\omega$ is the growth rate of perturbations and $k$ is the wavenumber.
Here, we assume that there is no base flow inside the film. The perturbed
quantities are linearly decomposed into the pressure term and the viscosity
term, $\hat{u}=\hat{u}_{p}+\hat{u}_{\nu}$ and
$\hat{w}=\hat{w}_{p}+\hat{w}_{\nu}$.

For the pressure term, the velocity potential $\phi$ (i.e.
$\partial_{r}\phi=\hat{u}_{p}\textrm{ and }\partial_{z}\phi=\hat{w}_{p}$) is
introduce to simplify the problem. The mass equation (2) becomes a zero-order
Bessel equation

$\frac{\mathrm{d}}{\mathrm{d}r}\left(r\frac{\mathrm{d}\phi}{\mathrm{d}r}\right)-k^{2}r\phi=0\,,$
(16)

whose solution can be expressed in terms of Bessel functions

$\phi=A_{1}\mathrm{I}_{0}(kr)+B_{1}\mathrm{K}_{0}(kr)\,.$ (17)

Here $\mathrm{I}_{0}$ and $\mathrm{K}_{0}$ are zero order modified Bessel
function of the first and second kinds. $A_{1}$ and $B_{1}$ are arbitrary
constants awaiting determination. Calculating the derivatives of $\phi$ gives
the solution of $\hat{u}_{p}$ and $\hat{w}_{p}$

$\displaystyle\hat{u}_{p}$
$\displaystyle=k\left[A_{1}\mathrm{I}_{1}(kr)-B_{1}\mathrm{K}_{1}(kr)\right]\,,$
(18) $\displaystyle\hat{w}_{p}$
$\displaystyle=ik\left[A_{1}\mathrm{I}_{0}(kr)+B_{1}\mathrm{K}_{0}(kr)\right]\,.$
(19)

Here $\mathrm{I}_{1}$ and $\mathrm{K}_{1}$ are first order modified Bessel
function of the first and second kinds. Substituting (17) into momentum
equation (4) yields the solution of $\hat{p}$, expressed as

$\hat{p}=-We\,\omega\left[A_{1}\mathrm{I}_{0}(kr)+B_{1}\mathrm{K}_{0}(kr)\right]\,.$

Considering the viscosity term of perturbed quantities, we simplify the
momentum equation (3) as a first-order Bessel equation,

$\frac{\mathrm{d}^{2}\hat{u}_{\nu}}{\mathrm{d}r^{2}}+\frac{1}{r}\frac{\mathrm{d}\hat{u}_{\nu}}{\mathrm{d}r}-\left(Re\omega+k^{2}+\frac{1}{r^{2}}\right)\hat{u}_{\nu}=0\,.$
(20)

So the solution of $\hat{u}_{\nu}$ is

$\hat{u}_{\nu}=A_{2}\mathrm{I}_{1}(lr)+B_{2}\mathrm{K}_{1}(lr)\,,$ (21)

where $l^{2}=k^{2}+Re\,\omega$. $A_{2}$ and $B_{2}$ are another two arbitrary
constants. According to equation (2),

$\hat{w}_{\nu}=\frac{il}{k}\left[A_{2}\mathrm{I}_{0}(lr)-B_{2}\mathrm{K}_{0}(lr)\right]\,.$
(22)

Combining the pressure parts and viscosity parts gives us the general solution
of perturbed variables, namely

$\displaystyle\hat{u}$
$\displaystyle=A_{1}k\mathrm{I}_{1}(kr)+A_{2}\mathrm{I}_{1}(lr)-B_{1}k\mathrm{K}_{1}(kr)+B_{2}\mathrm{K}_{1}(lr)\,,$
(23) $\displaystyle\hat{w}$
$\displaystyle=i\left[A_{1}k\mathrm{I}_{0}(kr)+A_{2}l\mathrm{I}_{0}(lr)/k+B_{1}k\mathrm{K}_{0}(kr)-B_{2}l\mathrm{K}_{0}(lr)/k\right]\,,$
(24) $\displaystyle\hat{p}$
$\displaystyle=-We\,\omega\left[A_{1}\mathrm{I}_{0}(kr)+B_{1}\mathrm{K}_{0}(kr)\right]\,.$
(25)

The dimensionless perturbed quantities for $\hat{u}$, $\hat{v}$, $\hat{p}$,
combined with $h(x,t)=1+\hat{h}e^{\omega t+ikx}$, then are substituted into
the boundary equations (5)-(10). For the boundary conditions at the interface
($r=1$), their linearisation gives

$\displaystyle\frac{\mathrm{d}\hat{w}}{\mathrm{d}r}+ik\hat{u}=0\,,$ (26)
$\displaystyle\hat{p}-2\frac{We}{Re}\frac{\mathrm{d}\hat{u}}{\mathrm{d}r}=\hat{h}\left(k^{2}-1\right)\,,$
(27) $\displaystyle\omega\hat{h}=\hat{u}\,.$ (28)

And for the boundary conditions on the fibre surface ($r=\alpha$), their
linearised forms are

$\displaystyle\hat{w}=l_{s}\frac{\mathrm{d}\hat{w}}{\mathrm{d}\hat{r}}\,,$
(29) $\displaystyle\hat{u}=0\,.$ (30)

According to equation (28), $\hat{h}$ in (27) can be eliminated to give the
final four equations of the boundary conditions, i.e. (26), (27), (29) and
(30). Substituting the Bessel functions (23 \- 25) into these perturbed
equations leads to a homogeneous system of linear equations for $A_{1}$,
$A_{2}$, $B_{1}$ and $B_{2}$, which has a non-trivial solution only if the
determinant of the coefficients vanishes. In this way, we have the final
equation

$\left|\begin{matrix}k\mathrm{I}_{1}(k\alpha)&\mathrm{I}_{1}(l\alpha)&-k\mathrm{K}_{1}(k\alpha)&\mathrm{K}_{1}(l\alpha)\\\
F_{21}&F_{22}&F_{23}&F_{24}\\\
2k^{3}\mathrm{I}_{1}(k)&(k^{2}+l^{2})\mathrm{I}_{1}(l)&-2k^{3}\mathrm{K}_{1}(k)&(k^{2}+l^{2})\mathrm{K}_{1}(l)\\\
F_{41}&F_{42}&F_{43}&F_{44}\\\ \end{matrix}\right|=0\,,$ (31)

where

$\displaystyle
F_{21}=k^{2}\mathrm{I}_{0}(k\alpha)-l_{s}k^{3}\mathrm{I}_{1}(k\alpha)\,,$
$\displaystyle
F_{22}=l\mathrm{I}_{0}(l\alpha)-l_{s}l^{2}\mathrm{I}_{1}(l\alpha)\,,$
$\displaystyle
F_{23}=k^{2}\mathrm{K}_{0}(k\alpha)+l_{s}k^{3}\mathrm{K}_{1}(k\alpha)\,,$
$\displaystyle
F_{24}=-l\mathrm{K}_{0}(l\alpha)-l_{s}l^{2}\mathrm{K}_{1}(l\alpha)\,,$
$\displaystyle
F_{41}=\mathrm{I}_{0}(k)\omega^{2}+2Re^{-1}k^{2}\mathrm{I}^{\prime}_{1}(k)\omega+\,We^{-1}(k^{2}-1)k\mathrm{I}_{1}(k)\,,$
$\displaystyle
F_{42}=2Re^{-1}l\mathrm{I}^{\prime}_{1}(l)\omega+We^{-1}(k^{2}-1)\mathrm{I}_{1}(l)\,,$
$\displaystyle
F_{43}=\mathrm{K}_{0}(k)\omega^{2}-2Re^{-1}k^{2}\mathrm{K}^{\prime}_{1}(k)\omega-
We^{-1}(k^{2}-1)k\mathrm{K}_{1}(k)\,,$ $\displaystyle
F_{44}=2Re^{-1}l\mathrm{K}^{\prime}_{1}(l)\omega+We^{-1}(k^{2}-1)\mathrm{K}_{1}(l)\,.$

As $\omega$ occurs in the argument of some Bessel functions, such as
$\mathrm{I}_{1}(l\alpha)$, equation (31) cannot be solved explicitly for
$\omega$, except in two limiting cases, which are presented in following
subsections.

###  3.2 Limiting case of jet flows

When the viscosity of the liquid is neglected ($Re\rightarrow\infty$), the
viscosity terms governing the instability become zero, i.e.
$u_{\nu}=w_{\nu}=0$. As a result, the dispersion relation is simplified to:

$\left|\begin{matrix}k\mathrm{I}_{1}(k\alpha)&-k\mathrm{K}_{1}(k\alpha)\\\
\mathrm{I}_{0}(k)\omega^{2}+\,We^{-1}(k^{2}-1)k\mathrm{I}_{1}(k)&\mathrm{K}_{0}(k)\omega^{2}-We^{-1}(k^{2}-1)k\mathrm{K}_{1}(k)\\\
\end{matrix}\right|=0\,.$ (32)

Here $\omega$ can be expressed explicitly as

$\omega=\sqrt{\frac{(1-k^{2})k}{We}\frac{\left[\mathrm{K}_{1}(k\alpha)\mathrm{I}_{1}(k)-\mathrm{I}_{1}(k\alpha)\mathrm{K}_{1}(k)\right]}{\mathrm{I}_{1}(k\alpha)\mathrm{K}_{0}(k)+\mathrm{K}_{1}(k\alpha)\mathrm{I}_{0}(k)}}\,.$
(33)

As the fibre radius approaches infinitesimally small values, the flows inside
the film are expected to resemble jet flows. Consequently, substituting
$\alpha=0$ into (33) yields the dispersion relation for the instability of
inviscid jets, originally proposed by Rayleigh (1878), which is expressed as

$\omega=\sqrt{\frac{(1-k^{2})k}{We}\frac{\mathrm{I}_{1}(k)}{\mathrm{I}_{0}(k)}}\,.$
(34)

When viscosity is taken into account, relying solely on the condition
$\alpha=0$ is no longer adequate to simplify (31) into the dispersion relation
of jet flows. Hence, it becomes imperative to introduce ultra-slip boundary
conditions ($l_{s}\rightarrow\infty$), resulting in

$\left|\begin{matrix}2k^{3}\mathrm{I}_{1}(k)&(k^{2}+l^{2})\mathrm{I}_{1}(l)\\\
F_{41}&F_{42}\\\ \end{matrix}\right|=0\,.$ (35)

This relation can be rearranged as

$\omega^{2}+\frac{2}{Re}\frac{k^{2}}{\mathrm{I}_{0}(k)}\left[\mathrm{I}_{1}^{\prime}(k)-\frac{2kl\mathrm{I}_{1}(k)\mathrm{I}_{1}^{\prime}(l)}{(l^{2}+k^{2})\mathrm{I}_{1}(l)}\right]\omega=\frac{1}{We}(1-k^{2})k\frac{\mathrm{I}_{1}(k)}{\mathrm{I}_{0}(k)}\frac{l^{2}-k^{2}}{l^{2}+k^{2}}\,,$
(36)

which is presented by Goldin et al. (1969). Using
$(l^{2}+k^{2})/(l^{2}-k^{2})=1+2k^{2}/(Re\omega)$ and
$\mathrm{I}^{\prime}_{1}(k)=\mathrm{I}_{0}(k)-\mathrm{I}_{1}(k)/k$, we obtain
the equivalent representation of (36)

$\omega^{2}+\frac{2k^{2}}{Re}\left[2-\frac{\mathrm{I}_{1}(k)}{k\mathrm{I}_{0}(k)}+\frac{2k^{2}}{l^{2}-k^{2}}\left(1-\frac{l\,\mathrm{I}_{1}(k)\mathrm{I}_{0}(l)}{k\,\mathrm{I}_{0}(k)\mathrm{I}_{1}(l)}\right)\right]\omega=\frac{1}{We}(1-k^{2})k\frac{\mathrm{I}_{1}(k)}{\mathrm{I}_{0}(k)}\,,$
(37)

which is widely-used form of the dispersion relation for the temporal
instability of a viscous Newtonian jet, first proposed by Weber (1931). When
$Re\rightarrow\infty$, (34) is also recovered.

Figure 2: The dispersion relation between the growth rate $\omega$ and the
wavenumber $k$ for the limiting cases of inviscid and viscous fluids. (a) The
inviscid liquid film ($Oh=10^{-3}$) on fibres with different radii
$\alpha=0.8$ (green dash-dotted line), $0.5$ (blue dashed line), $0.2$ (red
dotted line), $0.01$ (black solid line); (b) The viscous liquid film
($Oh=0.5$) on extremely thin fibres ($\alpha=0.01$) with different slip
lengths $l_{s}=0.0$ (green dash-dotted line), $0.1$ (blue dashed line), $1.0$
(red dotted line), $10$ (black solid line). The circles are the predictions of
Rayleigh (1878) and Goldin et al. (1969). The lines are predictions from the
NS dispersion relation (31).

These findings are further confirmed through numerical solutions of equation
(31) using the FindRoot function of MATHEMATICA. In the analysis, the
capillary velocity is adopted as the characteristic velocity for non-
dimensionalisation, namely $U=\gamma/\mu$. Consequently, we arrive at
$Re=We=Oh^{-2}$, where the non-dimensional quantity
$Oh=\mu\big{/}\sqrt{\rho\gamma h_{0}}$ represents the Ohnesorge number,
serving as a linkage between viscous forces, inertial forces, and surface-
tension forces. For inviscid flows, we set $Oh=10^{-3}$. As depicted in figure
2 (a), the results from the NS dispersion relation (31) gradually converge
towards the predictions of Rayleigh’s model as the fibre radius diminishes.
This outcome is consistent with the theoretical analysis, thus further
validating the numerical solutions of (31). Although the asymptotic behaviours
of inviscid cases are realised on the no-slip boundary condition, for the
viscous cases, they only manifest when an ultra-slippery fibre is considered,
as elucidated in figure 2 (b). Here, the solutions of (31) converge towards
Goldin’s model (36) as $l_{s}$ increases. This divergence can be attributed to
the differential impacts of shear stresses from the fibre surfaces, influenced
by the slip length.

###  3.3 Limiting case of film flows without inertia

In the regime where inertia of the liquid film is disregarded, i.e. $Re\ll 1$
(or $Oh\gg 1$), $l$ approximates $k$. This leads to the first column in (31)
coinciding with the second column, and the third with the fourth, resulting in
an indeterminate form. To address this issue, we employ the method proposed by
Tomotika (1935), which involves expanding the Bessel functions in Taylor
series with respect to $l$. For instance,
$I_{1}(l)=I_{1}(k)+I_{1}^{\prime}(k)(l-k)+\mathrm{O}\left[(l-k)^{2}\right]$.
By eliminating the zero-order terms and neglecting higher-order terms (greater
than the second order), we arrive at a determinant form. , expressed as

$\left|\begin{matrix}k\mathrm{I}_{1}(k\alpha)&k\alpha\mathrm{I}_{1}^{\prime}(k\alpha)&-k\mathrm{K}_{1}(k\alpha)&k\alpha\mathrm{K}_{1}^{\prime}(k\alpha)\\\
G_{21}&G_{22}&G_{23}&G_{24}\\\
2k^{3}\mathrm{I}_{1}(k)&2k^{2}\left[\mathrm{I}_{1}(k)+k\mathrm{I}_{1}^{\prime}(k)\right]&-2k^{3}\mathrm{K}_{1}(k)&2k^{2}\left[\mathrm{K}_{1}(k)+k\mathrm{K}_{1}^{\prime}(k)\right]\\\
G_{41}&G_{42}&G_{43}&G_{44}\\\ \end{matrix}\right|=0.$ (38)

The definitions of the functions $G_{ij}$ can be found in Appendix C. Note
that $\omega$ appears only linearly in the fourth line of the determinant, the
dispersion relation between $\omega$ and $k$ can be expressed explicitly.
After replacing the differentiation of the Bessel functions by
$\mathrm{I}^{\prime}_{1}(k)=\mathrm{I}_{0}(k)-\mathrm{I}_{1}(k)/k$ and
$\mathrm{K}^{\prime}_{1}(k)=-\mathrm{K}_{0}(k)-\mathrm{K}_{1}(k)/k$, we have

$\omega=\frac{k^{2}-1}{2}\frac{-\mathrm{I}_{1}(k)\Delta_{1}+\mathrm{I}_{0}(k)\Delta_{2}-\mathrm{K}_{1}(k)\Delta_{3}+\mathrm{K}_{0}(k)\Delta_{4}}{\left[k\mathrm{I}_{0}(k)-\mathrm{I}_{1}(k)\right]\Delta_{1}-k\mathrm{I}_{1}(k)\Delta_{2}-\left[k\mathrm{K}_{0}(k)+\mathrm{K}_{1}(k)\right]\Delta_{3}+k\mathrm{K}_{1}(k)\Delta_{4}}\,,$
(39)

where details of $\Delta_{ij}$ are shown in Appendix C.

Figure 3: The dispersion relation between the growth rate $\omega$ and the
wavenumber $k$ for the limiting cases of thin-film flows ($\alpha=0.8$). (a)
No-slip cases with different inertial effects, $Oh=2\times 10^{-3}$ (green
dash-dotted line), $8\times 10^{-3}$ (blue dashed line), $3.2\times 10^{-2}$
(red dotted line), $0.128$ (black solid line). (b) Slip cases ($l_{s}=10$)
with different inertial effects, $Oh=0.01$ (green dash-dotted line), $0.1$
(blue dashed line), $1$ (red dotted line), $10$ (black solid line). The
circles are the predictions from the slip-modified Stokes model (Zhao et al.,
2023a) and the lines are predictions from the NS dispersion relation (31).

This dispersion relation (39) is identical to the slip-modified Stokes model
proposed by Zhao et al. (2023a), which was derived directly from the Stokes
equations (neglecting inertia in the NS equations). Numerical investigations
are also performed to further support the theoretical analysis, illustrated in
figure 3, where the predictions generated by (31) tend to converge towards the
Stokes model as $Oh$ increases (inertia declines). Remarkably, the rate of
convergence for the no-slip cases surpasses that of the slip cases
significantly. Specifically, when $Oh=3.2\times 10^{-2}$ (as indicated by the
red dotted lines in figure 3 a), predictions from the NS dispersion relation
closely align with the results of the no-slip Stokes model. However, for the
slip cases, $Oh\geq 1$ is required for a similar convergence. One plausible
explanation for this observation, considering the omission of the base flow,
is that the no-slip boundary conditions constrain the fluid motion within the
liquid film more effectively than the slip boundary conditions, thereby
mitigating the influence of inertia.

###  3.4 Predictions of the dispersion relation

Based on the insights gained from our analysis of limiting cases, we turn to
the examination of inertia and slip effects in more general scenarios in this
subsection.

In figure 4, we present dispersion relations from (31) for various slip
lengths, while holding specific values of $Oh$ (columns) and $\alpha$ (rows).
The inertial effects are shown along a given row (fixed $\alpha$), with the
first column being inertia-dominated flows and third column corresponding to
viscosity-dominated cases. The deviations observed across different $l_{s}$
values indicate that slip predominantly governs the dynamics in viscous flows
within the film but has a comparatively minor impact on inertia-dominated
cases. This observation is consistent with the outcomes obtained from the
limiting case analysis of jet flows (figure 2 a). We also investigate the
influence of fibre radii (film thickness) on the instability within each
column. As film thickness ($1-\alpha$) increases, the deviations diminish,
suggesting that slip effects become less pronounced in thicker films.

Figure 4: The dispersion relation between the growth rate $\omega$ and the
wavenumber $k$ on different boundary conditions of various fibre radii:
(a,b,c) $\alpha=0.8$, (d,e,f) $\alpha=0.5$, (g,h,i) $\alpha=0.2$. For the
inertial effects: (a,d,g) $Oh=10^{-3}$, (b,e,h) $Oh=0.1$, (c,f,i) $Oh=10$.
Line types represent different values of the slip length: $l_{s}=0$ (green
dash-dotted line), $0.1$ (blue dashed line), $1$ (red dotted line), $10$
(black solid line).

These two observations can be explained qualitatively by considering
variations in velocity profiles within the liquid films, influenced by both
slip and inertia. As the slip length increases, the flow field near the solid
wall undergoes a transition from parabolic flow with a non-uniform velocity
profile to plug flow with a uniform velocity profile (Münch et al., 2005). The
parabolic flow field decreases and constitutes only a small fraction of the
film thickness with an increase in inertia (Schlichting & Kestin, 1961),
leading to more uniform velocity profiles. This explains why slip does not
significantly impact the instability with $Oh=10^{-3}$ (first column in figure
4). However, when $Oh=10$, most of the flow fields within the films are
expected to resemble parabolic profiles, making them more susceptible to the
effects of slip. Moreover, as the film thickness increases, the proportion of
the parabolic flow field in the films diminishes, and the velocity profiles
become more uniform, resulting in weaker influences of slip on the
instability.

The critical wavenumbers shown in figure 4 align with the findings of Plateau
(1873), i.e. $k_{crit}=2\pi h_{0}/\lambda_{crit}=1$, indicating that slip
conditions do not impact these values. These values are determined by the
interplay between two curvature terms governing the Laplace pressure on the
right-hand side of (2). The circumferential curvature term,
$1\big{/}(h\sqrt{1+(\partial_{x}h)^{2}})$, acts as the driving force, while
the tangential curvature term,
$\partial_{x}^{2}h\big{/}(1+\left(\partial_{x}h\right)^{2})^{3/2}$, acts as
the resisting force. The balance of forces outlined in (2) yields the term
$k^{2}-1$ in $F_{4j}$ of the final dispersion relation (31), ultimately
determining the critical wavenumber as $k_{crit}=1$.

Figure 5: Influence of inertia (different values of $Oh$) on the dominant
wavenumber $k_{max}$ on fibres of two radii: (a) $\alpha=0.8$, (b)
$\alpha=0.2$. The solid lines are the predictions of the NS dispersion
relation (31) for different slip lengths: $l_{s}=0$ (green), $0.1$ (purple),
$1.0$ (blue), $10$ (black), $100$ (red). The dotted lines and dashed lines
represent the predictions from the no-slip lubrication model (12) and the
giant-slip one (14)

.

Figure 5 further elucidates the relationship between the dominant wavenumber
$k_{max}$ and $Oh$. Remarkably, inertia appears to exert minimal influence on
$k_{max}$ in no-slip cases. Specifically, for a thin film with $\alpha=0.8$
(figure 5 a), $k_{max}$ for small $l_{s}$ remains unchanged as $Oh$ increases.
This value is close to the analytical expression $k_{max}=\sqrt{1/2}$ derived
from (71) for no-slip cases. This finding offers a plausible explanation for
why the no-slip lubrication model (12), which neglects inertia, has
demonstrated remarkable capabilities in predicting the wavelengths observed in
numerous experimental studies (Quéré, 1990; Duprat et al., 2007; Craster &
Matar, 2009, 2006; Ji et al., 2019). Conversely, in slip cases, $k_{max}$
exhibits a decline with increasing $Oh$. This trend holds across different
values of slip length ($l_{s}$), with more pronounced decreases in $k_{max}$
for larger slip lengths. Encouragingly, the predictions of the giant-slip
lubrication model (70) closely align with the results of cases with
substantial slip ($l_{s}>10$). As $Oh$ becomes sufficiently large for all
$l_{s}$, $k_{max}$ predicted by the NS dispersion relation converges to a
constant, whereas in the giant-slip lubrication model, $k_{max}$ consistently
decreases rapidly. For a thick film with $\alpha=0.2$ (figure 5 b), though
$k_{max}$ for the no-slip case decline from $0.7$ to $0.61$, which cannot be
predicted by the no-slip lubrication model (71), the variation trend of
$k_{max}$ with $Oh$ is similar to that observed in thin films ($\alpha=0.8$).
Furthermore, slip is found not to significantly impact $k_{max}$ in film flows
dominated by inertia ($Oh<10^{-2}$), consistent with the findings in figure 4.
However, in viscous cases ($Oh>1$), $k_{max}$ decreases significantly as
$l_{s}$ increases, corroborating the conclusions drawn in the work of Zhao et
al. (2023a).

##  4 Direct numerical simulations

In this section, direct numerical simulations are performed to corroborate the
theoretical findings in § 3 and gain deeper physical understanding of how
inertia and slip impact the instability of films on fibres.

The numerical solution of the NS equations is achieved through the Finite
Element method within a computational framework facilitated by COMSOL
Multiphysics 6.1. The simulations for the film flows are conducted using the
Arbitrary Lagrangian–Eulerian (ALE) approach. In this method, free surface
nodes are moved in a Lagrangian manner, deforming the computational domain,
while nodes inside the film follow a predefined evolution. This approach
offers a distinct advantage over other techniques, such as level set or phase
field methods. Unlike these alternatives, the ALE approach ensures an exact
capture of the free surface, akin to the Lagrangian approach, while retaining
the primary advantage of Eulerian methods that mesh elements are less prone to
distortion. Consequently, it has gained widespread use in predicting the
dynamics of free-surface flows across various phenomena such as droplet
dynamics (Chubynsky et al., 2020; Chakraborty et al., 2022), dynamics of a
ligament (Wei et al., 2021), jet breakup (Martínez-Calvo et al., 2020b) and
the instability of planar films (González et al., 2016; Moreno-Boza et al.,
2020b). However, it is worth noting that this method is not suitable for
scenarios where the topology of the domain might change, such as in cases
involving fluids inside the film after rupture. This limitation arises from
the necessity to maintain consistent mesh connectivity throughout the
simulation, and the only workaround is to manually change the mesh topology.

Figure 6: Numerical settings: (a) quadrilateral computational domains with
different boundary conditions (BCs); (b) non-uniform triangular mesh.

The computational domain is a quadrilateral (the section of a hollow fibre in
cylindrical coordinates) with a size $[\alpha,h_{0}+\hat{h}]\times[0,L]$,
illustrated in figure 6 (a). Here $L$ is the length of the film/fibre. we
assign a value of $\alpha=0.5$ to the radius of the fibre, and the initial
radius of the film is $h_{0}=1$. Small perturbations ($\hat{h}$) are
introduced at the liquid-gas interface. The left and right boundary conditions
of the computational domain are considered periodic. The top represents the
free surface, following (5)-(7). The bottom is treated as the slip-wall
boundary modelled by (10) and (11), where the slip length $l_{s}$ serves as an
input parameter for this boundary condition. The axial velocity on the
boundary $w_{b}=l_{s}\partial_{r}w_{b}$. When $l_{s}=0$, the no-slip boundary
condition ($w_{b}=0$) is recovered. The computational mesh for the liquid
domain utilises non-uniform triangular Lagrange elements, shown in figure 6
(b). Special attention is given to placing finer grid elements near the solid
boundary to accurately capture the fluid behaviours in the non-uniform
velocity profiles. The minimum grid size employed is $10^{-3}$. Additionally,
the variable-order backward differentiation formula is utilised for the
temporal integration of the NS equations. All simulations are conducted using
dimensionless units, which are established through rescaling variables as
described in equation (1).

We explore two different configurations in this study, each varying in film
length and initial perturbations, allowing us to investigate the combined
effects of inertia and slip on the dominant wavelength (§ 4.1) and the
evolution of perturbation growth (§ 4.2)

###  4.1 Dominant wavelengths of perturbations

To investigate the influence of inertia and slip effects on the dominant
wavelengths of perturbations, we perform simulations involving long films with
a length of $L=200$ on various slippery fibres. We explore twenty different
cases by considering five values of the Ohnesorge number, specifically
$Oh=10^{-3}$, $10^{-2}$, $0.1$, $1$, and $10$, on fibres characterized by four
slip lengths, namely $l_{\mathrm{s}}=0$, $1$, $10$, and $100$. In these
simulations, the system is initiated with random initial perturbations
described as $h(z,0)=1+\varepsilon N(z)$, where $\varepsilon=10^{-3}$, and
$N(z)$ is a random variable following a normal distribution with a mean of
zero and a unit variance. These initial perturbations are designed to
replicate the arbitrary disturbances commonly encountered in reality.

Figure 7:  Interface profiles at four time instants, illustrated in different
colours, on the fibres of the radius $\alpha=0.5$. The inertial effects are
presented by different values of $Oh$: (a) $Oh=10^{-3}$, (b) $Oh=0.1$, (c)
$Oh=10$. The upper panels in each figure are the predictions of no-slip cases
($l_{s}=0$) and lower ones are the results of slip cases ($l_{s}=10$).

Driven by surface tension, small random perturbations gradually evolve over
time, giving rise to significant capillary waves, as illustrated in figure 7,
which shows the evolution of interface profiles $h(z,t)$ for six different
cases. To assess the impact of inertial effects, identical initial conditions
are assigned for all six cases. For the inertia-dominated cases
($Oh=10^{-3}$), the evolution of capillary waves in slip cases is nearly
indistinguishable from that in no-slip cases, except for the slightly faster
growth of perturbations in the slip case compared to the no-slip case (figure
7 a). Conversely, when viscosity becomes significant (figure 7 b,c), slip
noticeably affects $h(z,t)$, resulting in faster perturbation growth and
longer capillary waves. Furthermore, the wavelengths of the no-slip cases do
not appear to be significantly influenced by inertia (see the six waves in the
upper panels of figure 7 a,b,c), despite the presence of deviations in local
interface profiles. All these findings align qualitatively with the
theoretical predictions outlined in § 3.4. Additionally, we conduct a
comparison between the interface profiles obtained through simulations for the
NS equations and those calculated numerically from the lubrication models. The
details are presented in Appendix D.

To quantitatively compare the numerical observations with the theoretical
predictions derived from (31), we conduct multiple independent simulations (10
for each case) with different initial conditions to collect statistical data
of the dominant modes. This statistical approach was proposed by Zhao et al.
(2019) and has been employed in evaluating dominant modes of instability in
various films (Zhao et al., 2021, 2023b, 2023a). For each simulation, a
discrete Fourier transform is applied to the interface position $h(z,t)$ to
obtain the power spectral density (PSD) of the perturbations. The square root
of the ensemble-averaged PSD ($H_{rms}$) at each time step is depicted in
figure 8, with a Gaussian function used to fit the modal distribution
(spectrum). The peak of this fitted spectrum corresponds to the dominant
wavenumber $k_{max}$, as indicated by the black dash-dotted lines. Extracting
$k_{max}$ from the fitted spectrum at each time instant yields the insets in
figure 8. Promisingly, $k_{max}$ converges to a constant rapidly in all the
cases. Consistent with the findings in figure 7, the two spectra in figure 8
(a,c) are nearly identical, suggesting that slip does not significantly impact
the instability in the inertia-dominated regime. However, in figure 8 (d), the
spectrum for the slip case exhibits a smaller dominant wavenumber compared to
the no-slip case in figure 8 (c). This discrepancy indicates that, in the
viscous regime, slip leads to an increase in the wavelength.

Figure 8: The root mean square (rms) of nondimensional perturbation amplitude
versus nondimensional wave number on fibres with two slip lengths at four time
instants: (a) $10583$ (green), $15875$ (blue), $19050$ (red) and $21166$
(black); (b) $264.6$ (green), $476.2$ (blue), $582.1$ (red) and $635.0$
(black); (c) $10583$ (green), $15875$ (blue), $18521$ (red) and $20632$
(black); (d) $26.5$ (green), $52.9$ (blue), $66.1$ (red) and $74.1$ (black).
The circles are extracted from numerical simulations fitted by the Gaussian
function. The inset shows the time history of the dominant wavenumbers
extracted from the spectra.

This statistical analysis is applied for all the cases to generate the symbols
($\lambda_{max}=2\pi/k_{max}$) in figure 9, which exhibit a good agreement
with theoretical predictions. Consequently, we can draw the conclusion that
the dominant modes of the thin-film instability remain largely unaffected by
inertia in no-slip cases. However, they become significantly influenced by
inertia on slippery surfaces, leading to the formation of longer perturbation
waves.

Figure 9: Inertial effects on the dominant wavelengths: a comparison between
the theoretical predictions of (31) and numerical solutions with different
slip lengths.

###  4.2 Evolutions of perturbation growth

To investigate the impact of inertia and slip on perturbation growth, we
conduct simulations of a relatively short film with $L=10$ on slippery fibres.
The film is initially perturbed with
$h(z,t)=1+\varepsilon\mathrm{cos}\left[2\pi(z/L-1/2)\right]$, where
$\varepsilon=0.01$.

Figure 10 presents the time evolution of the film interface on a no-slip fibre
with a radius of $\alpha=0.5$, with the corresponding fluid structure. The
contour in the lower panel of figure 10 depicts the radial velocity $u$. The
upper half displays the axial velocity $w$, showing that opposite fluxes
occur, directed towards the left and right boundaries, as the perturbation at
the free surface grows due to instability. Notably, while the magnitudes of
both $u$ and $w$ increase during the process, the fluid structure remains
similar at the linear stage (the amplitude of disturbances is typically less
than 20% of initial radius, i.e. $\hat{h}<0.2$), as evident from the contour
distribution in figure 10.

Figure 10: Film thinning of one perturbation wave in an axially symmetric
domain. For this case, $L=10$, $\alpha=0.5$, $Oh=0.1$ and $l_{s}=0$. Contours
of the axial velocity $w(r,z,t)$ (upper half) and radial velocity $u(r,z,t)$
(lower half) inside the film are shown at three time instants: (a)
$t_{1}=158$; (b) $t_{2}=264$; (c) $t_{3}=370$. $h_{\mathrm{min}}$ represents
the minimum radius of the film.

In figure 11, we present more velocity fields for four distinct cases,
considering two different values of $Oh$: $10^{-3}$ for the inertia-dominated
cases and $0.1$ for the viscous cases, on both no-slip (left panel) and slip
(right panel) fibres. According to the variations of the contours in the upper
half of figure 11 (b), slip not only alters the velocity distribution near the
surface of the fibres but also accelerates instability in the viscous cases,
as evidenced by the larger axial velocity component $w$. However, in the case
of inertia-dominated flow, slip has a negligible impact on instability,
corroborating our previous findings. The lower half of figure 11 illustrates
the velocity vectors near $z=\pm 0.9$ for different cases. It is apparent that
slip reduces the velocity gradient $\partial_{r}w$ near the surface.
Furthermore, the parabolic flow field in the no-slip case with $Oh=10^{-3}$ is
observed to be considerably smaller than that in the case with $Oh=0.1$. This
finding lends additional support to the explanation provided following figure
4. In essence, the more uniform velocity profile in the inertia-dominated case
serves to limit the impact of slip, resulting in nearly identical dynamics of
the instability.

Figure 11: Inertia and slip effects on axial velocity fields: (a) $Oh=10^{-3}$
at $t=2027$, (b) $Oh=0.1$ at $t=106$. Here, the fibre radius $\alpha=0.5$. The
left panels in each figure illustrate the contours of the entire configuration
(upper half) and velocity vectors of the local field near $\left|z\right|=0.9$
(lower half) of the no-slip cases. The right panels show the results of the
slip case ($l_{s}=10$).

Figure 12 illustrates the growth rates of perturbations in the twenty
simulated cases. Based on the instability analysis (§ 3.1) that employs the
expression $h(z,t)=1+\hat{h}e^{ikz+\omega t}$, the initial perturbation,
modelled by a cosine function with a fixed wavenumber $k=2\pi/L$, experiences
exponential growth. So $\mathrm{ln}(1-h_{min})$ is utilised as the
$y$-coordinate in figure 12 to present the linear growth of perturbations. The
numerical results closely align with the theoretical predictions,
demonstrating the effectiveness of the NS dispersion relation (31) in
describing the inertial effects on the instability of films on slippery
fibres. Furthermore, it becomes evident that deviations due to slip become
more pronounced as inertial effects diminish, thus providing quantitative
confirmation of previous findings.

Figure 12: Linear time evolution of the minimum radii of films $h_{min}(t)$
with four $Oh$ values: (a) $Oh=10^{-3}$, (b) $Oh=10^{-2}$, (c) $Oh=0.1$, (d)
$Oh=10$. The numerical solutions (dotted lines with symbols) are compared to
the predictions of the NS dispersion relation (solid lines) for three four
lengths: $l_{\mathrm{s}}=0$ (green), $0.1$ (blue), $1$ (red) and $10$ (black).
The theoretical values of $\omega$ are: (a) $2.81\times 10^{-4}$ (green),
$2.91\times 10^{-4}$ (blue), $2.94\times 10^{-4}$ (red), $2.95\times 10^{-4}$
(black); (b) $2.41\times 10^{-3}$ (green), $2.61\times 10^{-3}$ (blue),
$2.84\times 10^{-3}$ (red), $2.89\times 10^{-3}$ (black); (c) $8.12\times
10^{-3}$ (green), $1.10\times 10^{-2}$ (blue), $1.99\times 10^{-2}$ (red),
$2.34\times 10^{-2}$ (black); (d) $8.90\times 10^{-3}$ (green), $1.35\times
10^{-2}$ (blue), $3.62\times 10^{-2}$ (red), $6.32\times 10^{-2}$ (black).

In addition to examining the evolution of perturbation growth at the linear
stage, we also delve into the nonlinear dynamics, illustrated in figure 13.
Due to the dramatic changes in the interface profile at the nonlinear stage,
the initial dense mesh experiences significant deformation, resulting in poor
grid quality and potential numerical errors. To address this challenge, we
implement an approach of remeshing, regenerating the mesh (reducing nodes) in
the vicinity near the point of $h_{min}$ when $h_{min}<0.55$. The simulations
encompass both inertia-dominated and viscosity-dominated cases on the no-slip
(left panel) and slip (right panel) boundary conditions. Notably, the
nonlinear evolution reveals substantial distinctions from the dynamics at the
linear stage. The interface shapes are found to deviate from their initial
cosine, forming plateau structures at their lowest points. Ultimately,
satellite droplets emerge between the two main drops. Additionally, fluid
structure within the film no longer exhibits ‘similarity’ at different time
instants owing to the drastic changes in interface profiles. In scenarios
dominated by inertia, although slip has a minimal effect on the interface
shape, it substantially alters the flow structure. Figures 13 (c,d) illustrate
the evolution of vortical structures within the liquid film on a no-slip wall,
resulting in significant oscillations of the interface before rupture.
Conversely, no vortices appear within the film on the slippery fibre due to
the weak shear forces acting on the fluid near the surface. In viscosity-
dominated scenarios, slip not only accelerates perturbation growth
significantly, as observed at the linear stage, but also affects the interface
profiles near rupture. This alteration results in the formation of filaments,
rather than satellite droplets, between the two main drops. One plausible
explanation is that the dominant perturbation wavelength of the instability in
the slip case ($\lambda_{max}=15.32$) is notably larger than that in the no-
slip case ($\lambda_{max}=9.38$), resulting in the flatter profile in figure
13 (h). The observation also suggests that the volume of satellite droplets is
influenced by both inertia and slip.

Figure 13: Evolutions of perturbation growth at the nonlinear stage with two
$Oh$ values: (a-d) $Oh=10^{-3}$, (e-h) $Oh=10$. The contours represent the
velocity magnitude $\left|U\right|=\sqrt{u^{2}+w^{2}}$ . The left panels in
each figure illustrate the contours and streamlines of no-slip cases.The right
panels show the results of the slip cases ($l_{s}=10$).

Figure 14 presents variations of satellite droplets, extracted from more than
seventy simulations with varying values of $Oh$ and $l_{s}$. In figure 14 (a),
we depict the interface profiles of satellite droplets. These profiles clearly
demonstrate that in viscosity-dominated scenarios, slip significantly reduces
the volumes of satellite droplets. However, in the case of inertia-dominated
scenarios where $l_{s}>1$, slip has no discernible effect on the droplet
volumes. Note that these profiles were obtained when $h_{min}\leq
0.01\,h_{0}$. Furthermore, the volume of the satellite droplets is quantified
by calculating the area between the two lowest points of the profiles, as
shown in figure 14 (b). It is evident that in viscous cases, the volume of
satellite droplets decreases as $l_{s}$ increases, and higher viscosity
(larger $Oh$) leads to a more rapid rate of decrease. When viscosity dominates
the fluids ($Oh>=10$), the relationship between volume ($V_{sat}$) and slip
length ($l_{s}$) approximately follows $V_{sat}\sim l_{s}^{-5}$.

Figure 14: (a) The interface profiles of satellite droplets on different
boundary conditions with: $l_{s}=0$ (black dash-dotted lines), $1$ (green
solid lines), $10$ (blue dashed lines) and $100$ (red dotted lines). The upper
and lower panel shows that the results with $Oh=10^{-3}$ and $Oh=10$,
respectively. (b) Variations of the satellite droplets with slip lengths. The
inset provides a schematic of the volume of satellite droplets.

##  5 Conclusions

In this study, a theoretical model is developed based on the linear
instability analysis of the axisymmetric NS equations to investigate the
influence of inertia and slip on dynamics of liquid films on fibres. The model
is verified theoretically by the limiting cases for jet flows and film flows
without inertia. The resulting dispersion relation (31) reveals some
intriguing insights. Firstly, it shows that slip has a relatively minor impact
on the instability of flows dominated by inertia, whereas it significantly
accelerates the growth of perturbations in viscous film flows. Moreover, the
influence of slip appears to be contingent on the thickness of the liquid
film, with thinner films exhibiting more pronounced slip effects. We also
extract the dominant perturbation modes, denoted as $k_{max}$, from the
dispersion relation. In cases with no-slip boundary conditions, $k_{max}$
remains largely unaffected by inertia. Remarkably, for thin films
characterized by $\alpha=0.8$, $k_{max}$ maintains a nearly constant value,
closely aligning with the predictions of the no-slip lubrication model (12),
even as $Oh$ varies. Conversely, when slip is introduced, $k_{max}$ exhibits a
noticeable decrease as inertia decreases. In the limiting case where
$l_{s}\rightarrow\infty$, the giant-slip lubrication model (14) offers an
approximate prediction for $k_{max}$.

To substantiate our theoretical findings, direct numerical simulations of the
NS equations are conducted via using two distinct fluid configurations: (i) a
long film with random initial perturbations to investigate the dominant modes
of perturbations and (ii) a short film with a fixed wavelength to examine the
evolution of perturbation growth. The velocity fields extracted from these
simulations yield valuable insights. Notably, the parabolic field of the
velocity profiles in inertia-dominated cases are observed to be significantly
smaller than those in cases governed by viscosity. Given that slip primarily
influences non-uniform velocity profiles, this observation provides an
explanation for why the impact of slip is dampened by inertia. Furthermore, we
delve into the realm of nonlinear dynamics in perturbation growth. We found
that nonlinear dynamics, in contrast to the linear stage, leads to the
generation of intricate vortical structures near no-slip surfaces in inertia-
dominated cases. Interestingly, slip was found to mitigate the impact of these
shear stresses, resulting in smoother flows without the presence of vortices.
In the context of viscous cases, slip was identified to reduce the volume of
satellite droplets ($V_{sat}$) at the final stages before rupture. When
viscosity dominates the fluid dynamics, this reduction in volume followed an
approximate power law relationship, specifically $V_{sat}\sim l_{s}^{-5}$.

Given that experimental evidence has already confirmed the existence of wall
slip in films on fibres, as demonstrated in previous studies (Haefner et al.,
2015; Ji et al., 2019), and that slip length can be directly quantified
through methods such as those described in (Huang et al., 2006; Maali &
Bhushan, 2012; Maali et al., 2016), it is our hope that the wall slip can be
controlled experimentally to validate our predictions, especially regarding
the impact of inertia on wavelengths (drop size) for various liquids. For
instance, experiments involving water can be conducted to examine inertia-
dominated scenarios, while experiments with silicone oil can be carried out to
investigate viscosity-dominated cases. There are numerous potential extensions
to this framework. One avenue is to incorporate the influence of other
physical factors, such as electric fields (Ding et al., 2014) and
intermolecular forces (Ji et al., 2019; Tomo et al., 2022), both of which are
known to affect critical wavenumbers ($k_{crit}$) and dominant modes
($k_{max}$) in the instability. Additionally, exploring related flow
configurations, such as a liquid film flowing down a fibre driven by a body
force, as discussed in (Liu & Ding, 2021), presents intriguing opportunities.
An open problem in this context is to comprehend the impact of inertial
effects on the dynamics of traveling waves in various flow regimes.

[Acknowledgements] Useful discussions with Dr. Yixin Zhang, Dr. Zhaodong Ding
and Mr. Jingkun Xiong are gratefully acknowledged.

[Funding]This work was supported by the National Natural Science Foundation of
China (grant no. 12202437, 12027801, 12388101, 12272372), the Youth Innovation
Promotion Association CAS (grant no. 2018491, 2023477), the Fundamental
Research Funds for the Central Universities, the China Postdoctoral Science
Foundation (grant no. 2022M723044) and the China International Postdoctoral
Exchange Program Fellowship (grant no. YJ20210177) [Declaration of interests]
The authors report no conflict of interest.

[Author ORCIDs]
Chengxi Zhao https://orcid.org/0000-0002-3041-0882
Ran Qiao https://orcid.org/0000-0002-4445-1255
Kai Mu https://orcid.org/0000-0002-4743-2332
Ting Si https://orcid.org/0000-0001-9071-8646
Xisheng Luo https://orcid.org/0000-0002-4303-8290

##  Appendix A Derivation for the giant-slip lubrication model

In this appendix, we follow the approach employed by Münch et al. (2005) to
derive the giant-slip lubrication model. To get this lubrication equation from
the axisymmetric Navier-Stokes equations, we need to establish the leading
order terms by their asymptotic expansion in $\varepsilon$, for which we use
the rescaling shown below:

$\displaystyle\tilde{x}=\tilde{\lambda}x,\,\,\tilde{r}=\varepsilon\tilde{\lambda}r,\,\,\tilde{w}=\tilde{U}w,\,\,\tilde{u}=\varepsilon\tilde{U}u,\,\,\tilde{t}=\frac{\tilde{\lambda}}{\tilde{U}}t,\,\,\tilde{p}=\frac{\mu
U}{\lambda}p\,.$ (40)

Here, $\tilde{\lambda}=h_{0}/\varepsilon$. Substituting all these scalings
into the dimensional NS equations yields,

$\frac{\partial w}{\partial z}+\frac{1}{r}\frac{\partial(ur)}{\partial
r}=0\,,\\\ $ (41) $\varepsilon^{2}Re\left(\frac{\partial u}{\partial
t}+w\frac{\partial u}{\partial z}+u\frac{\partial u}{\partial
r}\right)=-\frac{\partial p}{\partial
r}+\varepsilon^{2}\frac{\partial^{2}u}{\partial
z^{2}}+\frac{\partial}{\partial
r}\left[\frac{1}{r}\frac{\partial(ur)}{\partial r}\right]\,.$ (42)
$\varepsilon^{2}Re\left(\frac{\partial w}{\partial t}+w\frac{\partial
w}{\partial z}+u\frac{\partial w}{\partial
r}\right)=-\varepsilon^{2}\frac{\partial p}{\partial
z}+\varepsilon^{2}\frac{\partial^{2}w}{\partial
z^{2}}+\frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial w}{\partial
r}\right)\,,$ (43)

The equations at the liquid-gas interface ($r=h$) are scaled as

$\frac{\partial h}{\partial t}+w\frac{\partial h}{\partial z}-u=0\,,$ (44)
$\begin{split}p-\frac{2}{1+\varepsilon^{2}\left(\partial_{z}h\right)^{2}}\left[\frac{\partial
u}{\partial r}-\frac{\partial h}{\partial z}\left(\frac{\partial w}{\partial
r}+\varepsilon^{2}\frac{\partial u}{\partial
z}\right)+\varepsilon^{2}\left(\frac{\partial h}{\partial
z}\right)^{2}\frac{\partial w}{\partial z}\right]\\\
=\frac{Re}{We}\left[\frac{1}{\varepsilon^{2}h\sqrt{1+\varepsilon^{2}\left(\partial_{z}h\right)^{2}}}-\frac{\partial_{z}^{2}h}{\left(1+\varepsilon^{2}\left(\partial_{z}h\right)^{2}\right)^{\frac{3}{2}}}\right]\,,\end{split}$
(45) $2\varepsilon^{2}\frac{\partial h}{\partial z}\left(\frac{\partial
u}{\partial r}-\frac{\partial w}{\partial
z}\right)+\left[1-\varepsilon^{2}\left(\frac{\partial h}{\partial
z}\right)^{2}\right]\left(\frac{\partial w}{\partial
r}+\varepsilon^{2}\frac{\partial u}{\partial z}\right)=0\,.$ (46)

For the boundary conditions at the liquid-solid interface ($r=\alpha$) the
scaled form are

$\displaystyle w=\frac{l_{s}}{\varepsilon^{2}}\frac{\partial w}{\partial
r}\,,$ (47) $\displaystyle u=0\,.$ (48)

The rescaled equations can be approximately solved by the perturbation
expansion, expressed as:

$\left(w,u,p,h\right)=\left(w_{0},u_{0},p_{0},h_{0}\right)+\varepsilon^{2}\left(w_{1},u_{1},p_{1},h_{1}\right)+...$
(49)

After eliminating all the high order terms of $\varepsilon$ and only keeping
the terms of the leading order, we obtain

$\partial_{z}w_{0}+\partial_{r}(u_{0}r)/r=0\,,$ (50)
$\partial_{r}p_{0}=\partial_{r}\left[\partial_{r}\left(u_{0}r\right)/r\right]\,,$
(51) $\partial_{r}\left(r\partial_{r}w_{0}\right)=0\,,$ (52)
$\partial_{t}h_{0}+w_{0}\,\partial_{z}h_{0}-u_{0}=0\,,$ (53)
$p_{0}-2\left(\partial_{r}u_{0}-\partial_{z}h_{0}\,\partial_{r}w_{0}\right)=\frac{Re}{We}\left(1/h_{0}-\partial_{z}^{2}h_{0}\right),\quad(r=h)\,,$
(54) $\partial_{r}w_{0}=0,\quad(r=h)\,,$ (55)
$\partial_{r}w_{0}=0,\quad(r=\alpha)\,,$ (56) $u_{0}=0,\quad(r=\alpha)\,.$
(57)

According to the equations (52), (56) and (57), $w$ is independent of the $r$,
i.e. $w_{0}=w_{0}(z,t)$. So equation (50) is rearranged as

$u_{0}=-r\partial_{z}w_{0}/2\,.$ (58)

Substituting (58) into (53) and (54) yields

$\displaystyle\partial_{t}h_{0}+w_{0}\,\partial_{z}h_{0}+h_{0}\partial_{z}w_{0}/2=0\,,$
(59) $\displaystyle
p_{0}+\partial_{z}w_{0}=\frac{Re}{We}\left(1/h_{0}-\partial_{z}^{2}h_{0}\right)\,,$
(60)

For the terms of the next order, equation (43) becomes

$Re\left(\frac{\partial w_{0}}{\partial t}+w_{0}\frac{\partial w_{0}}{\partial
z}\right)=-\frac{\partial p_{0}}{\partial z}+\frac{\partial^{2}w_{0}}{\partial
z^{2}}+\frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial
w_{1}}{\partial r}\right)\,,$ (61)

and equations (46) and (47) are applied for the boundary conditions

$\displaystyle-3\frac{\partial h_{0}}{\partial z}\frac{\partial
w_{0}}{\partial z}+\frac{\partial w_{1}}{\partial
r}-\frac{r}{2}\frac{\partial^{2}w_{0}}{\partial z^{2}}$
$\displaystyle=0\,,\quad(r=h)\,,$ (62) $\displaystyle
w_{0}-l_{s}\frac{\partial w_{1}}{\partial r}$
$\displaystyle=0\,,\quad(r=\alpha)\,,$ (63)

Integrating (61) from $\alpha$ to $h$ and using boundary conditions (62) and
(63) leads to

$Re\left(\frac{\partial w_{0}}{\partial t}+w_{0}\frac{\partial w_{0}}{\partial
z}\right)\int_{\alpha}^{h}rdr=\left(-\frac{\partial p_{0}}{\partial
z}+\frac{\partial^{2}w_{0}}{\partial
z^{2}}\right)\int_{\alpha}^{h}rdr+r\frac{\partial w_{1}}{\partial
r}\bigg{|}_{r=\alpha}^{r=h}\,,$ (64)

Rearranging (64) yields

$Re\left(\frac{\partial w_{0}}{\partial t}+w_{0}\frac{\partial w_{0}}{\partial
z}\right)=-\frac{\partial p_{0}}{\partial z}+\frac{\partial^{2}w_{0}}{\partial
z^{2}}+\frac{h_{0}}{h_{0}^{2}-\alpha^{2}}\left(6\frac{\partial h_{0}}{\partial
z}\frac{\partial w_{0}}{\partial z}+h_{0}\frac{\partial^{2}w_{0}}{\partial
z^{2}}\right)-\frac{2\alpha}{h_{0}^{2}-\alpha^{2}}\frac{w_{0}}{l_{s}}\,.$ (65)

Combining (59), (60) and (65) gives the final lubrication model for the film
on a fibre with a giant slip length, written as

$\displaystyle\frac{\partial h^{2}}{\partial
t}+\frac{\partial\left(h^{2}w\right)}{\partial z}=$ $\displaystyle 0\,,$ (66a)
$\displaystyle\frac{\partial w}{\partial t}+w\frac{\partial w}{\partial z}=$
$\displaystyle-\frac{1}{We}\frac{\partial}{\partial
z}\left(\frac{1}{h}-\frac{\partial^{2}h}{\partial
z^{2}}\right)+\frac{1}{Re}\frac{3}{h^{2}-\alpha^{2}}\frac{\partial\left(h^{2}\partial_{z}w\right)}{\partial
z}$
$\displaystyle-\frac{1}{Re}\frac{2\alpha^{2}}{h^{2}-\alpha^{2}}\left(\frac{\partial^{2}w}{\partial
z^{2}}-\frac{w}{\alpha\,l_{s}}\right)\,.$ (66b)

If $U=\gamma/\mu$, $Re=We=Oh^{-2}$, the momentum equation (66b) becomes

$\frac{\partial w}{\partial t}+w\frac{\partial w}{\partial
z}=-Oh^{2}\frac{\partial}{\partial
z}\left(\frac{1}{h}-\frac{\partial^{2}h}{\partial
z^{2}}\right)+Oh^{2}\frac{3}{h^{2}-\alpha^{2}}\frac{\partial\left(h^{2}\partial_{z}w\right)}{\partial
z}-Oh^{2}\frac{2\alpha^{2}}{h^{2}-\alpha^{2}}\left(\frac{\partial^{2}w}{\partial
z^{2}}-\frac{w}{\alpha\,l_{\mathrm{s}}}\right)\,.$ (67)

It is worth noting that $h^{2}-\alpha^{2}$ introduces singularity as
$h\rightarrow\alpha$. A similar singularity is also observed in other
lubrication equations describing free-surface flows with significant inertial
effects, such as liquid jets (Eggers & Dupont, 1994), liquid sheets (Erneux &
Davis, 1993), and planar films on ultra-slip walls (Münch et al., 2005).
However, this singularity disappears in lubrication models that describe
inertialess flows of bounded thin films (Kang et al., 2017; Oron et al.,
1997). The absence is primarily attributed to the distinct scaling used in the
lubrication approximation.

##  Appendix B Linear instability analysis for lubrication equations

In this appendix, the instability analysis is performed for the lubrication
equations (14) and (12) using the normal mode method.

For the giant-slip model (14), substituting $h(z,t)=1+\hat{h}e^{\omega t+ikz}$
and $w(z,t)=\hat{w}e^{\omega t+ikz}$ into the linearised lubrication equation
gives

$\displaystyle\omega\hat{h}=-ik\hat{w}/2\,,$ (68) $\displaystyle
Oh^{-2}\,\omega\hat{w}=ik(1-k^{2})\hat{h}-\frac{3k^{2}\hat{w}}{1-\alpha^{2}}+\frac{2\alpha^{2}}{1-\alpha^{2}}\left(k^{2}+\frac{1}{\alpha\,l_{\mathrm{s}}}\right)\hat{w}\,.$
(69)

Eliminating the $\hat{h}$ in (69) yields the dispersion relation

$\omega^{2}+Oh^{2}\left[\frac{3-2\alpha^{2}}{1-\alpha^{2}}k^{2}-\frac{2\alpha}{(1-\alpha^{2})l_{\mathrm{s}}}\right]\omega+Oh^{2}\frac{k^{2}(k^{2}-1)}{2}=0$
(70)

With a similar approach, we can have the dispersion relation from the no-slip
lubrication model (12), namely

$\omega=(k^{2}-1)k^{2}(3+\alpha^{4}-4\alpha^{2}+4\,\mathrm{ln}\,\alpha)/16.$
(71)

##  Appendix C Auxiliary functions for the dispersion relations

In this appendix, we present the definitions of the auxiliary functions for
the dispersion relations (38) and (39).

The $G_{ij}$ in (38) are

$\displaystyle G_{21}$
$\displaystyle=k^{2}\mathrm{I}_{0}(k\alpha)-l_{s}k^{3}\mathrm{I}_{1}(k\alpha),$
$\displaystyle G_{22}$
$\displaystyle=k\mathrm{I}_{0}(k\alpha)+k^{2}\alpha\mathrm{I}_{0}^{\prime}(k\alpha)-2l_{s}k^{2}\mathrm{I}_{1}(k\alpha)-l_{s}k^{3}\alpha\mathrm{I}_{1}^{\prime}(k\alpha),$
$\displaystyle G_{23}$
$\displaystyle=k^{2}\mathrm{K}_{0}(k\alpha)+l_{s}k^{3}\mathrm{K}_{1}(k\alpha),$
$\displaystyle G_{24}$
$\displaystyle=-k\mathrm{K}_{0}(k\alpha)-k^{2}\alpha\mathrm{K}_{0}^{\prime}(k\alpha)-2l_{s}k^{2}\mathrm{K}_{1}(k\alpha)-l_{s}k^{3}\alpha\mathrm{K}_{1}^{\prime}(k\alpha),$
$\displaystyle G_{41}$
$\displaystyle=2k^{2}\mathrm{I}^{\prime}_{1}(k)\omega+(k^{2}-1)k\mathrm{I}_{1}(k),$
$\displaystyle G_{42}$
$\displaystyle=2\left[k^{2}\mathrm{I}_{0}^{\prime}(k)-k\mathrm{I}_{1}^{\prime}(k)\right]\omega+(k^{2}-1)k\mathrm{I}_{1}^{\prime}(k)\,,$
$\displaystyle G_{43}$
$\displaystyle=-2k^{2}\mathrm{K}^{\prime}_{1}(k)\omega-(k^{2}-1)k\mathrm{K}_{1}(k)\,,$
$\displaystyle G_{44}$
$\displaystyle=-2\left[k^{2}\mathrm{K}_{0}^{\prime}(k)+k\mathrm{K}_{1}^{\prime}(k)\right]\omega+(k^{2}-1)k\mathrm{K}_{1}^{\prime}(k)\,.$

For (39), we have

$\displaystyle\Delta_{1}$
$\displaystyle=\begin{array}[]{|ccc|}\alpha\mathrm{I}_{0}(k\alpha)&\mathrm{K}_{1}(k\alpha)&\alpha\mathrm{K}_{0}(k\alpha)\\\
H_{22}&H_{23}&H_{24}\\\
k\mathrm{I}_{0}(k)+\mathrm{I}_{1}(k)&k\mathrm{K}_{1}(k)&k\mathrm{K}_{0}(k)-\mathrm{K}_{1}(k)\\\
\end{array}\,,$ (75) $\displaystyle\Delta_{2}$
$\displaystyle=\begin{array}[]{|ccc|}\mathrm{I}_{1}(k\alpha)&\mathrm{K}_{1}(k\alpha)&\alpha\mathrm{K}_{0}(k\alpha)\\\
H_{21}&H_{23}&H_{24}\\\
k\mathrm{I}_{1}(k)&k\mathrm{K}_{1}(k)&k\mathrm{K}_{0}(k)-\mathrm{K}_{1}(k)\\\
\end{array}\,,$ (79) $\displaystyle\Delta_{3}$
$\displaystyle=\begin{array}[]{|ccc|}\mathrm{I}_{1}(k\alpha)&\alpha\mathrm{I}_{0}(k\alpha)&\alpha\mathrm{K}_{0}(k\alpha)\\\
H_{21}&H_{22}&H_{24}\\\
k\mathrm{I}_{1}(k)&k\mathrm{I}_{0}(k)+\mathrm{I}_{1}(k)&k\mathrm{K}_{0}(k)-\mathrm{K}_{1}(k)\\\
\end{array}\,,$ (83) $\displaystyle\Delta_{4}$
$\displaystyle=\begin{array}[]{|ccc|}\mathrm{I}_{1}(k\alpha)&\alpha\mathrm{I}_{0}(k\alpha)&\mathrm{K}_{1}(k\alpha)\\\
H_{21}&H_{22}&H_{23}\\\
k\mathrm{I}_{1}(k)&k\mathrm{I}_{0}(k)+\mathrm{I}_{1}(k)&k\mathrm{K}_{1}(k)\\\
\end{array}\,,$ (87)

where

$\displaystyle H_{21}$
$\displaystyle=k\mathrm{I}_{0}(k\alpha)-l_{s}k^{2}\mathrm{I}_{1}(k\alpha)\,,$
$\displaystyle H_{22}$
$\displaystyle=(2-l_{s}k^{2}\alpha)\mathrm{I}_{0}(k\alpha)+(\alpha-2l_{s})k\mathrm{I}_{1}(k\alpha)\,,$
$\displaystyle H_{23}$
$\displaystyle=-k\mathrm{K}_{0}(k\alpha)-l_{s}k^{2}\mathrm{K}_{1}(k\alpha)\,,$
$\displaystyle H_{24}$
$\displaystyle=(2-l_{s}k^{2}\alpha)\mathrm{K}_{0}(k\alpha)+(2l_{s}-\alpha)k\mathrm{K}_{1}(k\alpha)\,.$

##  Appendix D Comparisons of the interface profiles

In this appendix, we compare the interface profiles obtained through
simulations for the NS equations with those calculated numerically from the
lubrication equations (LEs). Here, we focus on inertialess flows ($Oh\gg 1$)
on no-slip fibres. So we perform numerical investigations for the no-slip LE
(12) introduced in § 2. Note that Lister et al. (2006) used a different LE
(88) to explore nonlinear dynamics of film interfaces, particularly those
exhibiting characteristic collar-and-lobe structures, written as

$\frac{\partial h}{\partial t}=-\frac{1}{3}\frac{\partial}{\partial
z}\left[(h-\alpha)^{3}\left(\frac{\partial h}{\partial
z}+\frac{\partial^{3}h}{\partial z^{3}}\right)\right]\,.$ (88)

Here, the assumption $h-\alpha\ll\alpha$ simplifies the form of (88) compared
to (12). Both of these LEs, with periodic boundary conditions, are solved
using a simple second-order finite-difference scheme in both time and space.
In the direct numerical simulations of the NS equations, we set $Oh=10$ and
$l_{s}=0$ to enable a straightforward comparison with the numerical solutions
of the LEs. Figure 15 (a) illustrates excellent agreement between the
numerical predictions of the NS and LEs, starting from the same initial
interface profile, for a thin film ($\alpha=0.8$). However, deviations become
more pronounced for a thicker film ($\alpha=0.5$), consistent with previous
theoretical results from instability analysis (Zhao et al., 2023a).

Figure 15:  Interface profiles of inertialess flows on two fibres of different
radii at three time instants: (a) $\alpha=0.8$, $t_{1}=0$, $t_{2}=5700$,
$t_{3}=8400$; (b) $\alpha=0.5$, $t_{1}=0$, $t_{2}=400$, $t_{3}=600$. The blue
dotted lines and red dashed lines represent the results predicted by (88) and
(12), respectively. The solid lines are obtained from direct numerical
simulations for the NS equations.

## References

  * Assael et al. (2010) Assael, M. J., Kalyva, A. E., Antoniadis, K. D., Michael Banish, R., Egry, I., Wu, J., Kaschnitz, E. & Wakeham, W. A. 2010 Reference data for the density and viscosity of liquid copper and liquid tin. J. Phys. Chem. Ref. Data 39 (3), 033105.
  * Chakraborty et al. (2022) Chakraborty, I., Chubynsky, M. V. & Sprittles, J. E. 2022 Computational modelling of leidenfrost drops. J. Fluid Mech. 936, A12.
  * Chao et al. (2018) Chao, Y., Ding, Z. & Liu, R. 2018 Dynamics of thin liquid films flowing down the uniformly heated/cooled cylinder with wall slippage. Chem. Eng. Sci. 175, 354–364.
  * Chen et al. (2018) Chen, H., Ran, T., Gan, Y., Zhou, J., Zhang, Y., Zhang, L., Zhang, D. & Jiang, L. 2018 Ultrafast water harvesting and transport in hierarchical microchannels. Nat. Mater. 17 (10), 935–942.
  * Chen et al. (2023) Chen, X., Yang, M., Zheng, S., Temprano-Coleto, F., Dong, Q., Cheng, G., Yao, N., Stone, H. A., Hu, L. & Ren, Zh. J. 2023 Spatially separated crystallization for selective lithium extraction from saline water. Nat. Water pp. 1–10.
  * Chubynsky et al. (2020) Chubynsky, M. V., Belousov, K. I., Lockerby, D. A. & Sprittles, J. E. 2020 Bouncing off the walls: the influence of gas-kinetic and van der waals effects in drop impact. Phys. Rev. Lett. 124 (8), 084501.
  * Craster & Matar (2006) Craster, R. V. & Matar, O. K. 2006 On viscous beads flowing down a vertical fibre. J. Fluid Mech. 553, 85–105.
  * Craster & Matar (2009) Craster, R. V. & Matar, O. K. 2009 Dynamics and stability of thin liquid films. Rev. Mod. Phys 81 (3), 1131\.
  * Deng et al. (2011) Deng, D., Nave, J., Liang, X., Johnson, S. G. & Fink, Y. 2011 Exploration of in-fiber nanostructures from capillary instability. Opt. Express 19 (17), 16273–16290.
  * Ding & Liu (2011) Ding, Z. & Liu, Q. 2011 Stability of liquid films on a porous vertical cylinder. Phys. Rev. E 84 (4), 046307.
  * Ding et al. (2013) Ding, Z., Wong, T. N., Liu, R. & Liu, Q. 2013 Viscous liquid films on a porous vertical cylinder: Dynamics and stability. Phys. Fluids 25 (6), 064101.
  * Ding et al. (2014) Ding, Z., Xie, J., Wong, T. N. & Liu, R. 2014 Dynamics of liquid films on vertical fibres in a radial electric field. J. Fluid Mech. 752, 66–89.
  * Duprat et al. (2007) Duprat, C., Ruyer-Quil, C., Kalliadasis, S. & Giorgiutti-Dauphiné, F. 2007 Absolute and convective instabilities of a viscous film flowing down a vertical fiber. Phys. Rev. Lett. 98 (24), 244502.
  * Eggers & Dupont (1994) Eggers, J. & Dupont, T. F. 1994 Drop formation in a one–dimensional approximation of the navier–stokes equation. J. Fluid Mech. 262, 205–221.
  * Erneux & Davis (1993) Erneux, T. & Davis, S. H. 1993 Nonlinear rupture of free films. Physics of Fluids A: Fluid Dynamics 5 (5), 1117–1122.
  * Goldin et al. (1969) Goldin, M., Yerushalmi, J., Pfeffer, R. & Shinnar, R. 1969 Breakup of a laminar capillary jet of a viscoelastic fluid. J. Fluid Mech. 38 (4), 689–711.
  * González et al. (2016) González, A. G., Diez, J. A. & Sellier, M. 2016 Inertial and dimensional effects on the instability of a thin film. J. Fluid Mech. 787, 449–473.
  * Goren (1962) Goren, S. L. 1962 The instability of an annular thread of fluid. J. Fluid Mech. 12 (2), 309–319.
  * Goren (1964) Goren, S. L. 1964 The shape of a thread of liquid undergoing break–up. J. Colloid Sci. 19 (1), 81–86.
  * Haefner et al. (2015) Haefner, S., Benzaquen, M., Bäumchen, O., Salez, T., Peters, R., McGraw, J. D, Jacobs, K., Raphaël, E. & Dalnoki-Veress, K. 2015 Influence of slip on the Plateau–Rayleigh instability on a fibre. Nat. Commun. 6 (1), 1–6.
  * Halpern et al. (2015) Halpern, D., Li, Y. & Wei, H. 2015 Slip–induced suppression of marangoni film thickening in surfactant–retarded Landau–Levich–Bretherton flows. J. Fluid Mech. 781, 578–594.
  * Halpern & Wei (2017) Halpern, D. & Wei, H. 2017 Slip–enhanced drop formation in a liquid falling down a vertical fibre. J. Fluid Mech. 820, 42–60.
  * Hammond (1983) Hammond, P. S. 1983 Nonlinear adjustment of a thin annular film of viscous fluid surrounding a thread of another within a circular cylindrical pipe. J. Fluid Mech. 137, 363–384.
  * Huang et al. (2006) Huang, P., Guasto, J. S. & Breuer, K. S. 2006 Direct measurement of slip velocities using three–dimensional total internal reflection velocimetry. J. Fluid Mech. 566, 447–464.
  * Ji et al. (2019) Ji, H., Falcon, C., Sadeghpour, A., Zeng, Z., Ju, Y. S. & Bertozzi, A. L. 2019 Dynamics of thin liquid films on vertical cylindrical fibres. J. Fluid Mech. 865, 303–327.
  * Kang et al. (2017) Kang, D., Nadim, A. & Chugunova, M. 2017 Marangoni effects on a thin liquid film coating a sphere with axial or radial thermal gradients. Phys. Fluids 29 (7), 072106.
  * Kavokine et al. (2022) Kavokine, N., Bocquet, M. & Bocquet, L. 2022 Fluctuation–induced quantum friction in nanoscale water flows. Nature 602 (7895), 84–90.
  * Kavokine et al. (2021) Kavokine, N., Netz, R. R. & Bocquet, L. 2021 Fluids at the nanoscale: From continuum to subcontinuum transport. Annu. Rev. Fluid Mech. 53, 377–410.
  * Kliakhandler et al. (2001) Kliakhandler, I. L., Davis, S. H. & Bankoff, S. G. 2001 Viscous beads on vertical fibre. J. Fluid Mech. 429, 381–390.
  * Kondic et al. (2020) Kondic, L., González, A. G., Diez, J. A., Fowlkes, J. D. & Rack, P. 2020 Liquid-state dewetting of pulsed-laser-heated nanoscale metal films and other geometries. Annu. Rev. Fluid Mech 52, 235–262.
  * Lee et al. (2022) Lee, C. L., Chan, T. S., Carlson, A. & Dalnoki-Veress, K. 2022 Multiple droplets on a conical fiber: formation, motion, and droplet mergers. Soft Matter 18, 134495.
  * Li et al. (2008) Li, F., Yin, X. & Yin, X. 2008 Instability of a viscous coflowing jet in a radial electric field. J. Fluid Mech. 596, 285–311.
  * Liang et al. (2011) Liang, X., Deng, D., Nave, J. & Johnson, S. G. 2011 Linear stability analysis of capillary instabilities for concentric cylindrical shells. J. Fluid Mech. 683, 235–262.
  * Liao et al. (2014) Liao, Y., Li, Y., Chang, Y., Huang, C. & Wei, H. 2014 Speeding up thermocapillary migration of a confined bubble by wall slip. J. Fluid Mech. 746, 31–52.
  * Liao et al. (2013) Liao, Y., Li, Y. & Wei, H. 2013 Drastic changes in interfacial hydrodynamics due to wall slippage: slip-intensified film thinning, drop spreading, and capillary instability. Phys. Rev. Lett. 111 (18), 1364–1370.
  * Lister et al. (2006) Lister, J. R., Rallison, J. M., King, A. A., Cummings, L. J. & Jensen, O. E. 2006 Capillary drainage of an annular film: the dynamics of collars and lobes. J. Fluid Mech. 552, 311–343.
  * Liu & Ding (2021) Liu, R. & Ding, Z. 2021 Coating flows down a vertical fibre: towards the full Navier–Stokes problem. J. Fluid Mech. 914, A30.
  * Maali & Bhushan (2012) Maali, A. & Bhushan, B. 2012 Measurement of slip length on superhydrophobic surfaces. Philos. Trans. Royal Soc. A 370 (1967), 2304–2320.
  * Maali et al. (2016) Maali, A., Colin, S. & Bhushan, B. 2016 Slip length measurement of gas flow. Nanotechnology 27 (37), 374004\.
  * Martínez-Calvo et al. (2020a) Martínez-Calvo, A., Moreno-Boza, D. & Sevilla, A. 2020a The effect of wall slip on the dewetting of ultrathin films on solid substrates: Linear instability and second–order lubrication theory. Phys. Fluids 32 (10), 102107.
  * Martínez-Calvo et al. (2020b) Martínez-Calvo, A., Rivero-Rodríguez, J., Scheid, B. & Sevilla, A. 2020b Natural break-up and satellite formation regimes of surfactant-laden liquid threads. J. Fluid Mech. 883, A35.
  * Moreno-Boza et al. (2020a) Moreno-Boza, D., Martínez-Calvo, A. & Sevilla, A. 2020a The role of inertia in the rupture of ultrathin liquid films. Phys. Fluids 32 (11), 112114.
  * Moreno-Boza et al. (2020b) Moreno-Boza, D., Martínez-Calvo, A. & Sevilla, A. 2020b Stokes theory of thin-film rupture. Phys. Rev. Fluids 5 (1), 014002.
  * Münch et al. (2005) Münch, A., Wagner, B. A. & Witelski, T. P. 2005 Lubrication models with small to large slip lengths. J. Eng. Math. 53, 359–383.
  * Navier (1823) Navier, C. 1823 Mémoire sur les lois du mouvement des fluides, mémoire de l’académie royale des siences de l’institut de france. Sci. de L’Institut de France, VI pp. 389–440.
  * Oliveira et al. (2020) Oliveira, J. P., Santos, T. G. & Miranda, R. M. 2020 Revisiting fundamental welding concepts to improve additive manufacturing: From theory to practice. Prog. Mater. Sci. 107, 100590\.
  * Oron et al. (1997) Oron, A., Davis, S. H. & Bankoff, S. G. 1997 Long-scale evolution of thin liquid films. Rev. Mod. Phys 69 (3), 931–980.
  * Plateau (1873) Plateau, J. A. F. 1873 Statique expérimentale et théorique des liquides soumis aux seules forces moléculaires, , vol. 2. Gauthier-Villars.
  * Quéré (1990) Quéré, D. 1990 Thin films flowing on vertical fibers. Europhys. Lett. 13 (8), 721.
  * Quéré (1999) Quéré, D. 1999 Fluid coating on a fiber. Annu. Rev. Fluid Mech. 31, 347–384.
  * Rayleigh (1878) Rayleigh, L. 1878 On the instability of jets. Proc. London Math. Soc. 1, 4–13.
  * Rayleigh (1892) Rayleigh, L. 1892 Xvi. on the instability of a cylinder of viscous liquid under capillary force. London, Edinburgh, Dublin Philos. Mag. J. Sci. 34, 145–154.
  * Reitz et al. (2021) Reitz, B., Lotz, C., Gerdes, N., Linke, S., Olsen, E., Pflieger, K., Sohrt, S., Ernst, M., Taschner, P., Neumann, J. & others 2021 Additive manufacturing under lunar gravity and microgravity. Microgravity Sci. Technol. 33, 1–12.
  * Ruyer-Quil et al. (2008) Ruyer-Quil, C., Treveleyan, P., Giorgiutti-Dauphiné, F., Duprat, C. & Kalliadasis, S. 2008 Modelling film flows down a fibre. J. Fluid Mech. 603, 431–462.
  * Schlichting & Kestin (1961) Schlichting, H. & Kestin, J. 1961 Boundary layer theory, , vol. 121. Springer.
  * Secchi et al. (2016) Secchi, E., Marbach, S., Niguès, A., Stein, D., Siria, A. & Bocquet, L. 2016 Massive radius–dependent flow slippage in carbon nanotubes. Nature 537 (7619), 210–213.
  * Si et al. (2009) Si, T., Li, F., Yin, X. & Yin, X. 2009 Modes in flow focusing and instability of coaxial liquid–gas jets. J. Fluid Mech. 629, 1–23.
  * Tomo et al. (2022) Tomo, Y., Nag, S. & Takamatsu, H. 2022 Observation of interfacial instability of an ultrathin water film. Phys. Rev. Lett. 128 (14), 144502.
  * Tomotika (1935) Tomotika, S. 1935 On the instability of a cylindrical thread of a viscous liquid surrounded by another viscous fluid. Proc. R. Soc. Lond. A Math. Phys. Sci. 150 (870), 322–337.
  * Van Ombergen et al. (2023) Van Ombergen, A., Chalupa-Gantner, F., Chansoria, P., Colosimo, B. M., Costantini, M., Domingos, M., Dufour, A., De Maria, C., Groll, J., Jungst, T. & others 2023 3d bioprinting in microgravity: Opportunities, challenges, and possible applications in space. Adv. Healthc. Mater. p. 2300443.
  * Weber (1931) Weber, C. 1931 Zum zerfall eines flüssigkeitsstrahles. ZAMM-J. Appl. Math. Mech. 11, 136–154.
  * Wei et al. (2021) Wei, X., Rivero-Rodríguez, J., Zou, J. & Scheid, B. 2021 Statics and dynamics of a viscous ligament drawn out of a pure-liquid bath. J. Fluid Mech. 922, A14.
  * Zeng et al. (2017) Zeng, Z., Sadeghpour, A., Warrier, G. & Ju, Y. S. 2017 Experimental study of heat transfer between thin liquid films flowing down a vertical string in the Rayleigh–Plateau instability regime and a counterflowing gas stream. Int. J. Heat Mass Transf. 108, 830–840.
  * Zhang et al. (2022) Zhang, M., Zheng, Z., Zhu, Y., Zhu, Z., Si, T. & Xu, R. X. 2022 Combinational biomimetic microfibers for high-efficiency water collection. Chem. Eng. J. 433, 134495\.
  * Zhang et al. (2020) Zhang, Y., Sprittles, J. E. & Lockerby, D. A. 2020 Nanoscale thin-film flows with thermal fluctuations and slip. Phys. Rev. E 102 (5), 053105.
  * Zhang et al. (2021) Zhang, Y., Sprittles, J. E. & Lockerby, D. A. 2021 Thermal capillary wave growth and surface roughening of nanoscale liquid films. J. Fluid Mech. 915, A135.
  * Zhao et al. (2022) Zhao, C., Liu, J., Lockerby, D. A. & Sprittles, J. E. 2022 Fluctuation–driven dynamics in nanoscale thin–film flows: physical insights from numerical investigations. Phys. Rev. Fluids 7 (2), 024203.
  * Zhao et al. (2019) Zhao, C., Sprittles, J. E. & Lockerby, D. A. 2019 Revisiting the rayleigh–plateau instability for the nanoscale. J. Fluid Mech. 861, R3.
  * Zhao et al. (2023a) Zhao, C., Zhang, Y. & Si, T. 2023a Slip-enhanced rayleigh–plateau instability of a liquid film on a fibre. J. Fluid Mech. 954, A46.
  * Zhao et al. (2023b) Zhao, C., Zhang, Z. & Si, T. 2023b Fluctuation-driven instability of nanoscale liquid films on chemically heterogeneous substrates. Phys. Fluids 35 (7), 072016\.
  * Zhao et al. (2021) Zhao, C., Zhao, J., Si, T. & Chen, S. 2021 Influence of thermal fluctuations on nanoscale free-surface flows: A many-body dissipative particle dynamics study. Phys. Fluids 33 (11), 112004.