# The influence of an outer bath on the dewetting of an ultrathin liquid film

A. Martínez-Calvo  Princeton Center for Theoretical Science, Princeton
University, Princeton, NJ 08544, USA.  Department of Chemical and Biological
Engineering, Princeton University, Princeton, NJ 08544, USA.  D. Moreno-Boza
<EMAIL_ADDRESS>Departamento de Ingeniería Térmica y de Fluidos,
Universidad Carlos III de Madrid. Avda. de la Universidad 30, 28911, Leganés,
Madrid, Spain.  J. F. Guil-Pedrosa  Departamento de Ingeniería Térmica y de
Fluidos, Universidad Carlos III de Madrid. Avda. de la Universidad 30, 28911,
Leganés, Madrid, Spain.  A. Sevilla  Departamento de Ingeniería Térmica y de
Fluidos, Universidad Carlos III de Madrid. Avda. de la Universidad 30, 28911,
Leganés, Madrid, Spain.

(August 30, 2024)

###### Abstract

We report a theoretical and numerical investigation of the linear and
nonlinear dynamics of a thin liquid film of viscosity $\mu$ sandwiched between
a solid substrate and an unbounded liquid bath of viscosity $\lambda\mu$. In
the limit of negligible inertia, the flow depends on two non-dimensional
parameters, namely $\lambda$ and a dimensionless measure of the relative
strengths of the stabilizing surface tension force and the destabilizing van
der Waals force between the substrate and the film. We first analyze the
linear stability of the film, providing an analytical dispersion relation.
When the viscosity of the outer bath is much larger than that of the film,
$\lambda\gg 1$, the most amplified wavenumber decreases as
$k_{\textrm{m}}\sim\lambda^{-1/3}$, indicating that very slender dewetting
structures are expected when $\lambda$ becomes large. We then perform fully
nonlinear simulations of the complete Stokes equations to investigate the
spatial structure of the flow close to rupture revealing that the flow becomes
self-similar with the minimum film thickness scaling as
$h_{\textrm{min}}=K(\lambda)\tau^{1/3}$ when $\tau\to 0$, where $\tau$ is the
time remaining before the singularity. It is demonstrated that the presence of
an outer liquid bath affects the self-similar structure obtained by [16]
through the prefactor of the film thinning law, $K(\lambda)$, and the opening
angle of the self-similar film shape, which is shown to decrease with
$\lambda$.

††preprint: Preprint accepted in Phys. Fluids

##  I Introduction

Liquid films are encountered across scales and material properties, be it in
nature or in a large number of technological applications, ranging from
rheologically complex geological flows, tear films, or wine legs, to
nanometric metal coatings in plasmonic devices. For a panoramic view of the
field, the reader is referred to the reviews [20], [3], and [11]. A
configuration that is routinely used for the fabrication of such
nanostructures, is that of a viscous ultra-thin liquid film resting on a
substrate which is embedded in a viscous outer bath. The control of these
films is crucial to obtain the final product with a homogeneous coating, which
requires a fundamental understanding of the forces involved in their stability
and nonlinear dynamics.

This paper is devoted to study the linear and nonlinear regimes of this
particular configuration when the film is unstable to van der Waals (vdW)
forces between the solid substrate and the film, a destabilization process
that typically occurs when the thickness of the film drops below 100 nm
approximately. Below this thickness, the vdW force overcomes the stabilizing
surface tension force, which triggers a self-accelerated process leading to
the spinodal dewetting process. As the thickness of the film vanishes, a
finite-time singularity arises in the equations of motion, where the pressure,
stresses and velocity diverge as rupture is approached. These nonlinear near-
rupture dynamics have been extensively investigated when the film is embedded
in a passive ambient, by means of lubrication theory [26] and also considering
the full conservation equations [16, 15]. Close to the singularity, the
structure of the flow is self-similar and universal, in that it does not
depend on the initial or boundary conditions and lacks from a characteristic
length scale.

In our previous work [16], we showed that the self-similar thinning of the
liquid film predicted by lubrication theory is only observed during a
transient that depends on the initial height of the film, but it is not
established as rupture is approached and the slenderness of the flow breaks
down. The complete Stokes equations predict a non-slender universal structure
of the flow associated with a film shape that exhibits an opening angle of
37${}^{\textrm{o}}$ with respect to the substrate. In this work we extend the
results of [16] and study the dynamics of an ultra-thin liquid film embedded
in a viscous outer bath.

Flow configurations related with the one reported herein have been studied in
the context of the dewetting of two-layer thin films [21, 7, 14, 19], although
the finite thickness of the outer layer has an essential effect on the
dynamics. A similar flow also takes place during the late stages prior to drop
coalescence, which are locally driven by vdW interactions close to contact [1,
2]. . However, to the best of our knowledge, the dynamics of a thin film
sandwiched between a solid substrate and an unbounded liquid bath has not been
reported before, and therefore constitutes the main objective of the present
contribution.

The paper is structured as follows. In section §II we detail the governing
equations of motion, which are then used in section §III to study the linear
stability of the film, and in section §IV to perform direct numerical
simulations. In particular, in the latter section we investigate the structure
of the flow near rupture and the existence of self-similarity. Conclusions are
finally drawn in §V.

Figure 1: Sketch of the flow configuration and controlling parameters.

##  II Equations of motion

We consider the Newtonian flow of a thin liquid film of viscosity $\mu$ and
initial height $h^{*}_{\textrm{o}}$, resting on a solid substrate and immersed
in a bath of viscosity $\lambda\mu$ (see sketch in figure 1), with $\sigma$
denoting the surface tension coefficient. To describe the flow we make use of
the incompressible Stokes equations, which are non-dimensionalized with the
following characteristic length, velocity, time, pressure, and intermolecular
potential scales, respectively,

$\ell_{\textrm{c}}=a=\sqrt{\frac{A}{6\pi\sigma}},\quad
v_{\textrm{c}}=\frac{\sigma}{\mu},\quad t_{\textrm{c}}=\frac{\mu
a}{\sigma},\quad p_{\textrm{c}}=\phi_{\textrm{c}}=\frac{A}{6\pi a^{3}},$ (1)

where $a$ is the molecular length scale [4, 26, 16] and $A$ the Hamaker
constant [8], to yield

$\bm{\nabla}\bm{\cdot}\bm{v}_{i}=0,\quad\text{and}\quad\bm{0}=\bm{\nabla}\bm{\cdot}\bm{T}_{i}\quad\bm{x}\in\mathcal{V}_{i},$
(2)

where $i=\\{\textrm{in},\textrm{out}\\}$ denotes inner and outer bath fluid,
respectively,
$\bm{T}_{\textrm{in}}=-p_{\textrm{in}}\bm{I}+\bm{\nabla}\bm{v}_{\textrm{in}}+\bm{\nabla}\bm{v}_{\textrm{in}}^{\textrm{T}}$
and
$\bm{T}_{\textrm{out}}=-p_{\textrm{out}}\bm{I}+\lambda\left(\bm{\nabla}\bm{v}_{\textrm{out}}+\bm{\nabla}\bm{v}_{\textrm{out}}^{\textrm{T}}\right)$
are the stress tensors, $\bm{I}$ is the identity tensor, and $p$ is the
pressure field. Here $\mathcal{V}_{i}$ denotes the liquid domains,
$\bm{x}=(x,y)$ is the positional vector field, and $\bm{v}=(u,v)$ is the
velocity field, assumed to be two-dimensional. At the free surface
$\partial\mathcal{V}$ we impose the continuity of velocities, the kinematic
and stress balance boundary conditions, which read

$\displaystyle\bm{v}_{\textrm{in}}=\bm{v}_{\textrm{out}},\quad\bm{x}\in\partial\mathcal{V}$
(3a)
$\displaystyle\bm{n}\bm{\cdot}\left(\partial_{t}\bm{x}_{\textrm{s}}-\bm{v}\right)=0,\quad\bm{x}\in\partial\mathcal{V}$
(3b)
$\displaystyle\left(\bm{T}_{\textrm{in}}-\bm{T}_{\textrm{out}}+h^{-3}\bm{I}\right)\bm{\cdot}\bm{n}=-\bm{n}(\bm{\nabla}\bm{\cdot}\bm{n}),\quad\bm{x}\in\partial\mathcal{V}$
(3c)

respectively, where $\bm{x}_{\textrm{s}}$ is the parametrization of the
interface, located at $y=h(x,t)$, and $\bm{n}$ is the unit normal vector to
the interface. At the solid substrate, $y=0$, the no-slip boundary condition
is enforced, $\bm{v}_{\textrm{in}}=\bm{0}$, and at $y\to\infty$ we impose
$\bm{T}_{\textrm{out}}\bm{\cdot}\bm{e}_{y}=\bm{0}$ . As for the initial
conditions, in the numerical simulations we consider half a wavelength of a
spatially periodic liquid film, and thus we impose the symmetry condition
$u_{\mathrm{in}}=0$ at $x=0$ and $x=\pi/k$, where
$k<k_{\textrm{c}}=\sqrt{3}/h_{o}$ is the dimensionless wavenumber of the
initially perturbed interface
$\bm{x}_{\textrm{s}}=\left[x,h_{\textrm{o}}\,(1-\epsilon\cos{kx})\right]$,
imposed at $t=0$. Here, $k_{\textrm{c}}$ is the dimensionless cut-off
wavenumber predicted by linear instability theory [25],
$h_{\textrm{o}}=h^{*}_{\textrm{o}}/a$ is the initial film thickness normalised
with the molecular length scale, and $\epsilon$ is a small constant that
triggers the instability and induces the rupture of the liquid film at $x=0$
and $t=t_{\textrm{R}}$.

Equations (2)–(3) can be rewritten in terms of the alternative scalings

$\ell_{\textrm{c}}=h^{*}_{\textrm{o}},\quad
v_{\textrm{c}}=\frac{A}{6\pi\mu{h^{*}_{\textrm{o}}}^{2}},\quad
t_{\textrm{c}}=\frac{6\pi\mu{h^{*}_{\textrm{o}}}^{3}}{A},\quad
p_{\textrm{c}}=\phi_{\textrm{c}}=\frac{A}{6\pi{h^{*}_{\textrm{o}}}^{3}},$ (4)

based on the initial film thickness $h_{\textrm{o}}^{*}$, which are suitable
for the analysis of the linear stage of the film evolution. The capillary
number
$\mbox{{Ca}}=A/(6\pi\sigma{h_{\textrm{o}}^{*}}^{2})=h_{\textrm{o}}^{-2}$
comparing vdW-effects to surface tension is seen as the controlling parameter
using the scales given in (4).

##  III Linear stability analysis

We first investigate the linear stability of the film by deducing the growth
rate of disturbances from the complete Stokes equations (2)–(3c). To this end,
by making use of the scales given in (4), we expand all the flow variables
normal modes of the form where $\omega$ is the growth rate of perturbations, ,
and $k$ the associated axial wavenumber, and solve the corresponding
eigenvalue problem which yields the following dispersion relation

$\omega=\frac{(3-\mbox{{Ca}}^{-1}k^{2})[\lambda(1-\cosh(2k))-\sinh(2k)+2k(1+k\lambda)]}{2k[(1+2k^{2})(\lambda^{2}-1)-(1+\lambda^{2})\cosh(2k)-2\lambda\sinh(2k)]}.$
(5)

In the limit $\lambda\to 0$ we recover the well-known result of a thin film
surrounded by a solid substrate and a passive gaseous ambient [9, 16].

Figure 2: Amplification curves ($a$) $\omega(k)$ , and ($b$)
$\omega_{\textrm{o}}(k)=\lambda\omega(k)$ for $h_{\textrm{o}}=3.16$ ($Ca=0.1$)
and different values of the viscosity ratio $\lambda$ indicated in the legend.
The inset in ($b$) shows the function $k\omega_{\textrm{o}}(k)$. ($c$) Maximum
growth rate $\omega_{\textrm{m}}$ and ($d$) associated wavenumber
$k_{\textrm{m}}$, as functions of $\lambda$ for different values of
$h_{\textrm{o}}$ indicated in the legend.

To investigate the limit of a passive ultra-thin gaseous film confined by a
solid substrate below and by a viscous bath above, corresponding to the limit
$\lambda\gg 1$, it is convenient to first re-scale the growth rate using the
viscosity of the outer bath, $\omega_{\textrm{o}}=\omega\lambda$. Taking the
limit $\lambda\to\infty$ in the dispersion relation (5), we find

$\omega_{\textrm{o}}=\frac{3-\mbox{{Ca}}^{-1}k^{2}}{2k}$ (6)

for $k\ll 1$, displaying a singular growth rate for $k\to 0$. A common feature
of inertialess unstable interfacial flows that lack from shear associated with
a solid boundary is that the most unstable wavenumber is $k=0$, as happens,
for instance, in cylindrical liquid threads [23] and free liquid films [5,
24]. Here, in addition, the absence of a transverse length scale within the
bath makes the growth rate diverge as $k\to 0$.

Figure 2 summarizes the main results obtained from the linear stability
analysis. The temporal growth rate scaled with the film viscosity,
$\omega(k)$, is seen to decrease monotonically as $\lambda$ increases, as does
the most dangerous wavenumber. In particular, as deduced from Fig. 2(d), the
wavenumber of maximum growth rate $k_{\textrm{m}}\sim\lambda^{-1/3}$ for
$\lambda\to\infty$, irrespective of the value of $h_{\textrm{o}}$. In view of
the results of the linear stability analysis it is expected that the
longitudinal length scale of the dewetting structures increases with
$\lambda$, an aspect that will be examined next by means of numerical
simulations and similarity theory.

##  IV Nonlinear thinning

In this section we study the nonlinear and near-rupture dynamics of the thin
liquid film. To that end, it proves convenient to take back the original
scales (1), that will be assumed hereafter. First we perform numerical
simulations of the full conservation equations (2)–(3) in §IV.1, and then we
investigate the self-similar structure of the flow close to the singularity in
§IV.2.

###  IV.1 Description of the flow evolution

Figure 3: Top panels: minimum film thickness, $h_{\textrm{min}}$, as a
function of the time to rupture, $\tau$, for $h_{\textrm{o}}=10$ (left) and
$h_{\textrm{o}}=31.6$ (right), and different values of the viscosity ratio,
$\lambda$, indicated in the legend. The insets show the instantaneous power-
law exponents, $n={\rm d}\log(h_{\textrm{min}})/{\rm d}\log(\tau)$, as
functions of $h_{\textrm{min}}$. Bottom panels: corresponding free-surface
shapes for $h_{\textrm{min}}\approx 1$.

The temporal evolution of the unstable film was obtained by integrating the
set of equations (2)-(3c) using the same finite-element technique explained in
detail by Moreno-Boza et al. [16]. To that end, a small-amplitude harmonic
disturbance with optimal wavenumber is imposed as initial condition, and
evolved in time up to times close to the instant of the rupture singularity,
$t_{\textrm{R}}$. Figure 3 shows the results obtained from simulations
performed for two different values of the initial film height,
$h_{\textrm{o}}=10$ (left) and $h_{\textrm{o}}=31.6$ (right), and
$\lambda=(1,100,1000)$ as indicated in the legend. The upper panels show the
minimum film thickness, $h_{\textrm{min}}$, as a function of the time to
rupture, $\tau=t_{\textrm{R}}-t$. The simulations show that
$h_{\textrm{min}}\propto\tau^{1/3}$ as $\tau\to 0$ in all cases, revealing
that the same power-law exponent found by Moreno-Boza et al. [16] for the case
without outer bath, $\lambda\to 0$, is actually valid for any value of
$\lambda$. The viscosity ratio $\lambda$ is seen to only affect the prefactor
of the thinning law, such that $h_{\textrm{min}}=K(\lambda)\tau^{1/3}$. The
function $K(\lambda)$, represented in Fig. 4, has the limit $K\to 0.66$ as
$\lambda\to 0$, corresponding to a negligible influence of the outer bath
[16], and decreases monotonically with $\lambda$, reaching a power-law
behavior $K\propto\lambda^{-2/9}$ for $\lambda\gg 1$. The instantaneous power-
law exponents $n={\rm d}\log(h_{\textrm{min}})/{\rm d}\log(\tau)$, represented
as function of $h_{\textrm{min}}$ in the insets, are seen to monotonically
increase towards the $1/3$ power law, which is fully established for values of
$h_{\textrm{min}}\approx 1$, which correspond to scales close to the molecular
one. In the case with larger initial thickness, $h_{\textrm{o}}=31.6$, the
$1/5$ power law obtained by Zhang and Lister [26] using lubrication theory for
the case $\lambda\to 0$, is established during a brief stage in the cases with
$\lambda=1$ and $\lambda=100$.

Besides its influence on the temporal scaling law for the thinning process,
the viscosity ratio $\lambda$ is also seen to profoundly affect the geometry
of the interface near rupture. Indeed, the lower panels of Fig. 3 display the
free-surface shapes corresponding to the same cases of the upper panels,
evaluated at $h_{\textrm{min}}\approx 1$. The typical slope of the interface
is seen to decrease with $\lambda$, revealing that the flow becomes more
slender as the viscosity ratio is increased as anticipated by the linear
stability theory of the previous section. In the next section, the actual flow
structure close to the rupture singularity will be deduced by means of a
similarity analysis.

###  IV.2 The self-similar flow close to rupture

Following our previous work [16], dimensional analysis suggests the similarity
Ansatz for the near-rupture flow:

$\displaystyle x=\tau^{1/3}\xi,\quad y=\tau^{1/3}\eta,\quad
h=\tau^{1/3}f(\xi),\quad u_{i}=\tau^{-2/3}U^{i}(\xi,\eta),$ (7a)
$\displaystyle v_{i}=\tau^{-2/3}V^{i}(\xi,\eta),\quad
p_{i}=\tau^{-1}P^{i}(\xi,\eta),$ (7b)

which when introduced into Eqs. (2) yields, at leading order in $\tau$, the
set of equations

$\displaystyle\begin{rcases}U^{\textrm{in}}_{\xi}+V^{\textrm{in}}_{\eta}=0,\\\
U^{\textrm{in}}_{\xi\xi}+U^{\textrm{in}}_{\eta\eta}=P^{\textrm{in}}_{\xi}-3f^{-4}f_{\xi},\\\
V^{\textrm{in}}_{\xi\xi}+V^{\textrm{in}}_{\eta\eta}=P^{\textrm{in}}_{\eta},\end{rcases}\quad\text{in}\quad
0\leq\eta\leq f(\xi),\quad 0\leq\xi<\infty,$ (8a)
$\displaystyle\begin{rcases}U^{\textrm{out}}_{\xi}+V^{\textrm{out}}_{\eta}=0,\\\
\lambda(U^{\textrm{out}}_{\xi\xi}+U^{\textrm{out}}_{\eta\eta})=P^{\textrm{out}}_{\xi},\\\
\lambda(V^{\textrm{out}}_{\xi\xi}+V^{\textrm{out}}_{\eta\eta})=P^{\textrm{out}}_{\eta},\end{rcases}\quad\text{in}\quad
f(\xi)\leq\eta<\infty,\quad 0\leq\xi<\infty,$ (8b)

to be solved with the no-slip boundary condition
$U^{\textrm{in}}=V^{\textrm{in}}=0$ at the wall, $\eta=0$, the symmetry
condition
$U^{\textrm{in}}=V^{\textrm{in}}_{\xi}=U^{\textrm{out}}=V^{\textrm{out}}_{\xi}=0$
at $\xi=0$, and the following normal and tangential surface-stress balance,
kinematic condition, and continuity of velocities at the as-yet-unknown free
surface, $\eta=f(\xi)$, respectively,

$\displaystyle(1+f_{\xi}^{2})(P^{\textrm{in}}-P^{\textrm{out}})-2(V^{\textrm{in}}_{\eta}-\lambda
V^{\textrm{out}}_{\eta})+$ $\displaystyle
2[V^{\textrm{in}}_{\xi}-f_{\xi}U^{\textrm{in}}_{\xi}+U^{\textrm{in}}_{\eta}-\lambda(V^{\textrm{out}}_{\xi}-f_{\xi}U^{\textrm{out}}_{\xi}+U^{\textrm{out}}_{\eta})]f_{\xi}=0,$
(9a)
$\displaystyle(1-f_{\xi}^{2})(V^{\textrm{in}}_{\xi}+U^{\textrm{in}}_{\eta})+2(V^{\textrm{in}}_{\eta}-U^{\textrm{in}}_{\xi})f_{\xi}=\lambda[(1-f_{\xi}^{2})(V^{\textrm{out}}_{\xi}+U^{\textrm{out}}_{\eta})+2(V^{\textrm{out}}_{\eta}-U^{\textrm{out}}_{\xi})f_{\xi}],$
(9b) $\displaystyle
f/3-\left(\xi/3+U^{\textrm{in}}\right)f_{\xi}+V^{\textrm{in}}=0,$ (9c)
$\displaystyle
U^{\textrm{in}}-U^{\textrm{out}}=V^{\textrm{in}}-V^{\textrm{out}}=0.$ (9d)

The solution to the elliptic problem (8)–(9) would determine the asymptotic
film shape $f=f(\xi)$ as $\tau\to 0$. Figure 4 shows the results of the
numerical integration of (8)–(9) compared with the time marched film evolution
represented in self similar scales, showing a very good agreement.

For $\lambda\ll 1$, the solution deduced by Moreno-Boza et al. [16] is seen to
be effectively recovered, with an opening angle $\theta_{\textrm{o}}\simeq
37^{\circ}$ . Indeed, as seen in [16], the kinematic condition (9c) shows that
a wedge-shaped far field of the form
$f(\xi)=(\xi-\xi_{\textrm{o}})\tan\theta_{\textrm{o}}$ is compatible with
vanishing velocities, valid for $r^{2}=\xi^{2}+\eta^{2}\gg 1$, with
$(r,\theta)$ polar coordinates such that $\xi-\xi_{\textrm{o}}=r\cos{\theta}$
and $\eta=r\sin{\theta}$. The opening angle $\theta_{\textrm{o}}$ and the
constant $\xi_{\textrm{o}}$ are to be determined numerically. However, insight
can be gained by examining the far field on assuming radially decaying
velocities with associated stream functions $\psi^{i}\simeq r^{-\ell}$, with
$\ell>0$, such that $V_{r}^{i}\sim V_{\theta}^{i}\sim r^{-(1+\ell)}$ along
rays $\theta=\text{constant}$, where $(V_{r}^{i},V_{\theta}^{i})$ are the
associated radial and polar components of the velocity field. On assuming
$\psi^{i}=F_{i}(\theta)r^{-\ell}$, the following fourth-order homogeneous
problem for $F$ arises:

Figure 4: ($a$) Self-similar flow structure for $\lambda=0.1$ (upper panels),
$1$ (middle panels) and $100$ (lower panels), for $h_{\textrm{o}}=10$,
illustrated by means of temporal integrations of (2) for $\tau\to 0$ (left
panels) and the solution of the self-similar elliptic problem (8a)–(9d) (right
panels). ($b$) Angle and ($c$) prefactor $K$ of the 1/3 power-law as functions
of the viscosity ratio, $\lambda$.

$F_{i}^{\text{(iv)}}+\left[4+2\ell(2+\ell)\right]F_{i}^{\prime\prime}+\ell^{2}(2+\ell)^{2}F_{i}=0,$
(10)

to be integrated with the no slip conditions
$F_{\textrm{in}}(0)=F^{\prime}_{\textrm{in}}(0)=0$, the symmetry conditions
$F_{i}(\pi/2)=F_{i}^{\prime\prime}(\pi/2)=0$, and the normal stress balance
$\lambda
F_{\textrm{out}}^{\prime\prime\prime}-F_{\textrm{in}}^{\prime\prime\prime}+\left[4+3\ell\left(2+\ell\right)\right]\lambda
F_{\textrm{out}}^{\prime}-\left[4+3\ell\left(2+\ell\right)\right]F^{\prime}_{\textrm{in}}=0$
at $\theta=\theta_{\textrm{o}}$. Since the problem is homogeneous, the system
furnished by the prescription of the above boundary conditions for the
corresponding constants of integration is also homogeneous. Therefore, the
determinant of the so-formed system being zero can be recast into the
compatibility condition $\varphi(\theta_{\textrm{o}},\ell;\lambda)=0$, where
the algebraic function $\varphi$ is omitted here for conciseness. The function
$\varphi$ can be solved numerically to give the far-field angle
$\theta_{\textrm{o}}$ as a function of the radial decay of the stream
functions $\ell$ for given values of $\lambda$, similarly to what was done in
[16] for $\lambda=0$. After retrieving the different values of $\ell$ from the
computations in figure 4 one obtains, for instance, $\theta_{\textrm{o}}\simeq
37^{\circ}$ for $\lambda=0$ and $\ell\simeq 0.19$, $\theta_{\textrm{o}}\simeq
44.0^{\circ}$ for $\lambda=1$ and $\ell\simeq 0.5$, and
$\theta_{\textrm{o}}\simeq 19.1^{\circ}$ for $\lambda=100$ and $\ell\simeq
0.1$.

The far-field angle of the interface, $\theta_{\textrm{o}}$, and the prefactor
of the $1/3$ power-law, $K$, are plotted as functions of $\lambda$ in Figs.
4(b) and 4(c), respectively. Interestingly, the function
$\theta_{\textrm{o}}(\lambda)$ presents a global maximum of $44.4^{\circ}$ at
an intermediate value of $\lambda=1.9$. It is also deduced that
$\theta_{\textrm{o}}\to 37^{\circ}$ as $\lambda\to 0$, consistent with the
result of Moreno-Boza et al. [16], and that $\theta_{\textrm{o}}$ decreases
rapidly for $\lambda\gtrsim 10$, eventually achieving a power-law
$\theta_{\textrm{o}}\sim\lambda^{-1/3}$, for $\lambda\gg 1$, demonstrating
that the shapes close to rupture become extremely slender for large values of
the viscosity ratio.

##  V Concluding remarks

We have unraveled the influence of an outer immiscible bath on the two-
dimensional dewetting dynamics of an ultrathin non-wetting liquid film resting
on a solid substrate. A linear stability analysis of the governing Stokes
equations has revealed that the most unstable wavelength increases as the
bath-to-film viscosity ratio $\lambda$ increases, pointing to the existence of
more slender dewetting structures compared to the case without an outer bath,
which is recovered as a particular case of our analysis in the limit
$\lambda\to 0$. Indeed, nonlinear numerical simulations have confirmed that
the local shape of the interface close to rupture becomes increasingly slender
as $\lambda$ is increased, and have revealed that the film thinning process is
ultimately governed by the same self-similar dynamics as that discovered by
Moreno-Boza et al. [16] for the case without an outer bath, $\lambda=0$. The
self-similar dynamics depends on a balance between the van der Waals and
viscous forces, with asymptotically negligible surface tension forces, and
provides a minimum film thickness $h_{\textrm{min}}$ obeying the power law
$h_{\textrm{min}}=K(\lambda)\tau^{1/3}$, where $\tau$ is the time remaining
until rupture and the prefactor $K$ is a monotonically decreasing function of
$\lambda$. We have also revealed that the structure of the near-rupture flow
is self-similar, and that the interface is wedge-shaped with an opening angle
$\theta_{\textrm{o}}(\lambda)$ which decreases with $\lambda$, in agreement
both with the linear stability analysis and with the numerical evidence.

As prospects for future works, we point out the need to perform experiments
and three-dimensional numerical simulations of the flow configuration studied
herein. In addition, a number of effects of practical relevance, like the slip
at the solid substrate [10, 17, 6, 12] or the viscoelasticity of the liquid
[22, 18, 13], should also be taken into account, since both are known to play
an important role in the dewetting of polymeric liquids.

## Acknowledgments

This research was funded by the Spanish MCIU-Agencia Estatal de Investigación
through project PID2020-115655GB-C22, partly financed through FEDER European
funds.

## Data availability

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

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