# Coalescence of non-spherical drops with a liquid surface

Nagula Venkata Anirudha, Sachidananda<EMAIL_ADDRESS>and
Kirti Chandra<EMAIL_ADDRESS>aDepartment of Mechanical and
Aerospace Engineering, Indian Institute of Technology Hyderabad, Kandi - 502
284, Sangareddy, Telangana, India
bDepartment of Chemical Engineering, Indian Institute of Technology Hyderabad,
Kandi - 502 284, Sangareddy, Telangana, India

###### Abstract

We employ three-dimensional numerical simulations to explore the impact
dynamics of non-spherical drops in a deep liquid pool by varying the aspect
ratios $(A_{r})$ and Weber numbers $({\it We})$. We observe that when a non-
spherical drop is gently placed on a liquid pool, it exhibits a partial
coalescence phenomenon and the emergence of a daughter droplet for
$A_{r}>0.67$. In contrast to the prolate $(A_{r}<1)$ and spherical drops
$(A_{r}=1)$, an oblate $(A_{r}>1)$ drop with a high aspect ratio encapsulates
air in a ring-like bubble within the pool and emerges a liquid column that
undergoes Rayleigh-Plateau capillary instability, leading to the formation of
two daughter droplets with complex shapes. When the parent drop is impacted
with finite velocity, our observations indicate that increasing the Weber
number leads to elevated crater heights on the free surface for all aspect
ratios. A prolate drop produces a less pronounced wave swell and exhibits a
prolonged impact duration owing to its negligible impact area. Conversely, an
oblate drop generates a much wider wave swell than spherical and prolate
drops. We analyze the relationship between rim formation dynamics and the
kinetic and surface energies of the system. Finally, we establish an analogy
by comparing the dynamics of a freely falling non-spherical drop, undergoing
topological oscillations during its descent from a height, with the impact
dynamics of parent drops of various shapes striking the liquid surface with an
equivalent velocity. Our investigation involving non-spherical drops contrasts
the extensive studies conducted by various researchers on the impact of a
parent spherical drop just above the free surface of a liquid pool.

††journal: International Journal of Multiphase Flow

Keywords: Drop, coalescence, partial coalescence, interfacial flow, liquid-air
interface

##  1 Introduction

The dynamics of a drop impacting solid and liquid surfaces have been explored
for centuries and remain a topic of great interest even today. Leonardo Da
Vinci documented this phenomenon in the 1500s in Codex Leicester, as later
reported in Worthington [1877]. Over the past few decades, there have been
significant advances in understanding the interaction of two interfaces
[Josserand and Thoroddsen, 2016, Deka et al., 2019a, Sprittles, 2024]. This
progress has contributed to diverse fields, such as microfluidics, combustion,
coatings, inkjet printing, soil erosion, air trapping on the sea surface,
separation of the water phase from the oil phase in oil recovery [Stone et
al., 2004, Thoroddsen et al., 2008, Kavehpour, 2015], and natural phenomena,
such as raindrops falling on a liquid surface and ejection of marine
microplastics from lakes and oceans by raindrops [Low and List, 1982, Thomson
and Newall, 1886, Pumphrey et al., 1989, Veron, 2015, Liu, 2018], to name a
few. Apart from the wide range of practical applications, this subject
involves complex physics, such as the generation of capillary waves leading to
the formation of daughter droplets (partial coalescence) at low impact
velocities, as well as the formation of craters and splashing at high impact
velocities [Ray et al., 2015, Deka et al., 2019a, Kirar et al., 2022].

The impact velocity of the drop is characterised by the Weber number, defined
as ${\it We}=\rho_{l}v_{0}^{2}R_{eq}/\sigma$, where $\rho_{l}$ denotes the
density of the liquid, $v_{0}$ is the impact velocity, $R_{eq}$ represents the
radius of the parent/primary drop and $\sigma$ is the interfacial tension
between the drop and the surrounding medium. At a low Weber number, when a
primary droplet contacts the free surface of a liquid pool, it hovers until
the trapped surrounding fluid (air cushion between the drop and the free
surface) drains out. Subsequently, coalescence results in a rapidly expanding
neck at the contact point due to high capillary pressure. This triggers
upward-moving capillary waves, forming a liquid column and initiating necking.
Over time, the neck diameter decreases, causing a daughter droplet to pinch
off [Blanchette and Bigioni, 2006]. The competition between vertical and
horizontal collapse rates influences the pinch-off process. If capillary waves
sufficiently delay vertical collapse, horizontal collapse merges the neck,
producing a daughter/secondary droplet over the free surface. This process is
commonly known as the partial coalescence phenomenon [Charles and Mason,
1960]. In contrast to previous beliefs attributing partial coalescence to an
inviscid instability mechanism in the liquid column, later research indicates
that the dynamics of partial coalescence are predominantly influenced by
gravity, viscosity, and interfacial tension [Thoroddsen and Takehara, 2000,
Chen et al., 2006b]. Chen et al. [2006a] found gravity, inertial-capillary,
and viscosity-dominated regimes and described that gravity drives coalescence
at high Bond numbers $({\it Bo}=\rho_{l}g{R_{eq}}^{2}/\sigma)$, while
viscosity dominates at lower values of ${\it Bo}$. Here, $g$ represents the
acceleration due to gravity. At intermediate ${\it Bo}$ values, coalescence is
driven by inertia and capillary forces, with partial coalescence observed in
the inertial-capillary regime. Recently, Ivanova and Klyuev [2023]
demonstrated that the levitation of millimetre-sized droplets above a liquid
pool is caused by capillary convection within the liquid resulting from a
surface tension gradient created by the non-uniform distribution of vapour
molecules from the droplet at the pool surface. Paul et al. [2023] numerically
investigated the impact of two vertically aligned drops on a liquid surface,
exploring coalescence dynamics through variations in the ratio of diameters
between the lower and upper drops.

When a drop impacts a liquid pool at high velocity, it gives rise to several
distinct phenomena, including crater formation, air bubble entrainment inside
the pool, the release of a jet and splashing dynamics. In this situation,
immediately following impact, inertia triggers the creation of an ejecta
sheet, forming an edge. This edge, in turn, develops into a cylindrical-shaped
rim, evolving through a combination of Rayleigh–Taylor and Rayleigh–Plateau
instabilities. Nonlinearities then emerge, resulting in breakup through end
pinching and forming a cascade of droplets [Josserand and Thoroddsen, 2016,
Constante-Amores et al., 2023]. Such impacts trigger the expulsion of a sheet
above the pool surface, potentially resulting in a subsequent splash [Weiss
and Yarin, 1999, Thoroddsen, 2002, Agbaglah et al., 2015, Josserand and
Zaleski, 2003, Ray et al., 2015, Gielen et al., 2017]. The impact dynamics are
contingent on whether the interaction takes place with a thin liquid layer or
a deep pool. It was found that a droplet descending onto a thin liquid layer
manifests the generation of an upward Worthington jet [Gekle and Gordillo,
2010, Gordillo and Gekle, 2010], while in a deep liquid pool, it culminates in
the formation of the well-known milkdrop coronet [Josserand and Thoroddsen,
2016, Constante-Amores et al., 2023]. A few researchers have also examined the
oblique impact of a drop on a liquid surface [Gielen et al., 2017, Kirar et
al., 2022]. It has been observed that oblique impacts, unlike normal impacts,
suppress splashing and secondary droplet formation [Gielen et al., 2017, Ray
et al., 2012, Okawa et al., 2008]. Leneweit et al. [2005], Liu and Bothe
[2016] found that at large Weber numbers, capillary waves are absent, and
instead, a lamella is ejected at the droplet front, comprising partially of
the droplet fluid. The interplay between surface tension and inertial forces
in the contact-line region determines whether this region of high curvature is
influenced by surface tension, leading to the creation of capillary waves, or
if the stagnation pressure due to the oblique impact results in the extrusion
of a lamella-shaped surface. By performing axisymmetric simulations, Behera et
al. [2023] presented a regime map delineating the coalescence and splashing
regimes within the Weber and Ohnesorge numbers space. Deka et al. [2017]
employed axisymmetric simulations to elucidate the mechanism of large bubble
entrapment within the pool during the coalescence process and compared their
findings with the experimental observations of Wang et al. [2013]. Constante-
Amores et al. [2023] investigated the influence of surfactants during the
impact of a spherical droplet on thin liquid films by conducting three-
dimensional numerical simulations. Recently, Liu et al. [2021] experimentally
investigated the shape of the drop on the impact dynamics, albeit on a solid
substrate, and showed that altering the shape of a droplet can regulate its
spreading and splashing behaviour on a substrate.

Research on the impact of spherical droplets into liquid pools relies on a
combination of experimental techniques involving high-speed imaging
[Thoroddsen et al., 2008, Blanchette and Bigioni, 2006, Bisighini et al.,
2010, Kirar et al., 2022] and axisymmetric numerical simulations [Ray et al.,
2010, 2012, 2015, Gielen et al., 2017, Josserand and Zaleski, 2003]. However,
many real-world scenarios, such as raindrops [Kostinski and Shaw, 2009],
charged drops [Ristenpart et al., 2009], and oscillating drops [Zhang et al.,
2019, Deka et al., 2019b, Agrawal et al., 2017, 2020, Balla et al., 2019]
exhibit a variety of non-spherical shapes. Apart from the axisymmetric study
by Deka et al. [2017], all other investigations focused on the impact of
spherical droplets into liquid pools. Moreover, the emphasis in the study of
Deka et al. [2017] was specifically on the large bubble entrapment observed in
prolate-shaped drops, attributed to their deeper penetration of the resultant
vortex ring into the liquid pool. Consequently, their study can not capture
the three-dimensional behaviour, as demonstrated experimentally by Liu et al.
[2021] in the context of impacts involving non-spherical drops on solid
surfaces. Thus, the present investigation aims to explore the impact dynamics
of a non-spherical drop in a deep liquid pool for different Weber numbers via
three-dimensional numerical simulations. Our findings reveal asymmetrical
dynamics and the formation of complex-shape daughter drops due to partial
coalescence. Additionally, an analogy is established by comparing the dynamics
of a freely falling non-spherical drop from height $L$, undergoing topological
oscillations, with parent drops of different shapes impacting the liquid
surface with an equivalent velocity ($\sqrt{2gL}$). This analysis enables us
to explore the influence of shape oscillations during the free fall of a drop,
resulting from the interplay between inertia and surface tension [Agrawal et
al., 2017], on the coalescence phenomenon. This is in contrast to the impact
of a parent drop just above the free surface, a phenomenon extensively studied
by various researchers [Thoroddsen et al., 2008, Blanchette and Bigioni, 2006,
Ray et al., 2015, Deka et al., 2017].

The rest of the manuscript is organised as follows. The problem is formulated
in §2, where we elaborate the numerical method for three-dimensional
simulations and extensively validate the flow solver. The results are
presented in §3. In this section, we first explore phenomena related to zero
impact velocity $({\it We}=0)$. Then, we examine the effect of the Weber
number on the dynamics of a drop impacting a deep liquid pool. Finally, we
present the equivalence between the dynamics of a freely falling drop from a
height and that of a drop impacting near the liquid pool. The conclusion
remarks are given in §4.

##  2 Formulation

We investigate the collision dynamics of a non-spherical primary drop
impacting a deep liquid pool by performing three-dimensional numerical
simulations of the Navier-Stokes and continuity equations. The drop and the
pool constitute the same liquid, and the surrounding medium is air. The
dynamic viscosities and the densities of liquid and air are denoted by
$(\mu_{l},\rho_{l})$ and $(\mu_{a},\rho_{a})$, respectively. The fluids are
assumed to be Newtonian and incompressible. The schematic diagram showing the
initial configuration of the drop is depicted in Figure 1$a$. To minimize the
boundary effect, we consider sufficiently large cubic computational domains
with dimensions of $16R_{eq}\times 16R_{eq}\times 16R_{eq}$ for ${\it We}=0$
and $30R_{eq}\times 30R_{eq}\times 30R_{eq}$ for the finite Weber numbers
examined in the current study. Here, $R_{eq}$ denotes the volume equivalent
spherical radius of the droplet, and $h$ represents the height of the liquid
pool, which is maintained at $5.5R_{eq}$ and $16R_{eq}$ for ${\it We}=0$ and
finite Weber numbers, respectively. Except for the case associated with the
freely falling droplet from a height $(L)$, the separation distance between
the bottom of the drop and the free surface of the liquid pool $(q)$ is
maintained at a constant value of $0.09R_{eq}$..

We consider three types of initial droplet shapes: oblate $(A_{r}>1)$,
spherical $(A_{r}=1)$, and prolate $(A_{r}<1)$, as depicted in Figure 1$b$. We
vary the aspect ratio of the primary non-spherical droplet $(A_{r}=a/b)$ while
keeping its volume constant. Here, $a$ and $b$ represent the initial diameters
of the primary droplet along its initial major and minor axes, respectively,
such that $R_{eq}=(a^{2}b/8)^{1/3}$. Our study employs a Cartesian coordinate
system $(x,y,z)$, with gravity $(g)$ acting in the negative $y$ direction.
Initially, the surrounding air is at rest.

($a$) ($b$)

Figure 1: ($a$) Schematic diagram of the cross-sectional $(xy)$ view at $z=0$
of the cubic computational domain of dimension $H^{3}$ showing the initial
configuration of a non-spherical drop impacting a liquid pool under gravity
$(g)$ acting in the negative $y$ direction. ($b$) Three shapes of drops
(oblate/spherical/prolate) have been considered in our study. The diameter of
the non-spherical drop in the horizontal $(x)$ and vertical $(y)$ directions
are $a$ and $b$, respectively. The diameter of the drop in the spanwise
direction $(z)$ is the same as that of the horizontal direction, such that the
equivalent spherical radius of the drop, $R_{eq}=(a^{2}b/8)^{1/3}$. The aspect
ratio of the drop, $A_{r}=a/b$, such that $A_{r}>1$, $A_{r}=1$ and $A_{r}<1$
represent oblate, spherical and prolate drops, respectively.

###  2.1 Governing equations

The three-dimensional Navier-Stokes and continuity equations and the
associated boundary conditions are non-dimensionalized using $R_{eq}$ as the
length scale, $t_{s}=\sqrt{\rho_{l}R_{eq}^{3}/\sigma}$ (the capillary time) as
the time scale and $V_{s}=R_{eq}/t_{s}$ as the velocity scale, such that

$\displaystyle(x,y,z)=R_{eq}(\widetilde{x},\widetilde{y},\widetilde{z}),~{}t=t_{s}\widetilde{t},~{}\bm{U}=V_{s}\bm{\widetilde{U}},~{}P=(\sigma/R_{eq})\widetilde{P},$
$\displaystyle\mu(\alpha)={\widetilde{\mu}(\alpha)}\mu_{l},~{}\rho(\alpha)={\widetilde{\rho}(\alpha)}\rho_{l},~{}{\rm
and}~{}~{}\delta_{s}={\widetilde{\delta}_{s}}R_{eq}.$ (1)

Here, the tildes designate dimensionless quantities; $\bm{U}=(u,v,w)$ is the
velocity field, wherein $u$, $v$ and $w$ denote the components of $\bm{U}$ in
the $x$, $y$ and $z$ directions, respectively; $P$ represents the pressure
field; $\sigma$ represents surface tension, and
$\delta_{s}(\bm{x}-\bm{x}_{f})$ is the delta distribution function which is
zero elsewhere except on the interface, where $\bm{x}=\bm{x}_{f}$.

After dropping tildes from all non-dimensional variables, the dimensionless
continuity and Navier–Stokes equations governing the coalescence dynamics of
the drop into the liquid pool are given by

$\displaystyle\nabla\cdot\bm{U}$ $\displaystyle=$ $\displaystyle 0,$ (2)
$\displaystyle\rho(\alpha)\left(\frac{\partial\bm{U}}{\partial
t}+\bm{U}.\nabla\bm{U}\right)$ $\displaystyle=$ $\displaystyle-\nabla
P+Oh\nabla\cdot\left[\mu(\alpha)(\nabla\bm{U}+\nabla\bm{U}^{T})\right]$ (3)
$\displaystyle+$
$\displaystyle\kappa\bm{n}\delta_{s}+Bo\rho(\alpha)\bm{g}\bm{j}.~{}~{}~{}~{}~{}~{}$

Here, $\bm{j}$ denotes the unit vector along the vertical (negative $y$)
direction, $\kappa~{}(\equiv-\nabla\cdot n)$ is the interfacial curvature,
wherein $n$ is the outward-pointing unit normal to the interface. Note that
the surface tension force is included as a body force term in Eq. (3) by
employing the continuum surface force (CSF) formulation proposed by Brackbill
et al. [1992]. The Ohnesorge number, $Oh=\mu_{l}/{\sqrt{\rho_{l}\sigma
R_{eq}}}$ represents the competition between the viscous and surface tension
forces, which can also be expressed as $Oh=\sqrt{We}/Re$, wherein
$Re=v_{0}\rho_{l}R_{eq}/\mu_{l}$ represents the Reynolds number.

The interface between the air and liquid phases is monitored by solving an
advection equation for the volume fraction of the liquid phase, denoted as
$\alpha$ (where $\alpha=0$ and $1$ correspond to the air and liquid phases,
respectively). This is expressed as:

$\displaystyle{\partial\alpha\over\partial t}+\bm{U}\cdot\nabla\alpha=0.$ (4)

The density field, $\rho(\alpha)$, and the viscosity field, $\mu(\alpha)$, are
given by

$\displaystyle\rho(\alpha)=\rho_{r}(1-\alpha)+\alpha,$ (5)
$\displaystyle\mu(\alpha)=\mu_{r}(1-\alpha)+\alpha,$ (6)

where $\rho_{r}(=\rho_{a}/\rho_{l})$ and $\mu_{r}(=\mu_{a}/\mu_{l})$ represent
the density and viscosity ratios, respectively. In the present study, as we
consider a water-air system, $\rho_{r}=10^{-3}$ and $\mu_{r}=0.018$, unless
otherwise specified. In subsequent sections, we normalize the time $(t)$, such
that the rescaled time, $\tau=0$, signifies the initiation of coalescence
between the drop and the free surface of the liquid pool.

###  2.2 Numerical method

We adopt a two-phase flow solver within the OpenFOAM framework to simulate the
impact dynamics of a drop into a liquid pool governed by eqs. (2) and (3) in a
collocated grid arrangement, where the velocity vector, pressure and volume
fraction are defined at the cell centers. A non-uniform mesh that assigns a
large number of grids near the interfacial regions and slightly coarser mesh
away from the interface is employed. The numerical schemes used are second-
order accurate in both space and time. The time integration is performed using
the Crank-Nicolson scheme. In the momentum equation (3), the diffusion and
gradient terms are discretized using Gauss linear, and the convective terms
are discretized employing a van Leer scheme. The pressure-velocity coupling is
handled using the PIMPLE algorithm. This method is a combination of PISO
(Pressure Implicit with Splitting of Operator) and SIMPLE (Semi-Implicit
Method for Pressure-Linked Equations) algorithms.

The simulation employs a volume-of-fluid (VoF) method incorporating a
numerical interface compression method based on the multidimensional universal
limiter with explicit solution (MULES) to track the interface separating the
liquid and air phases. Ideally, the interface between the two fluids must
represent a sharp discontinuity. However, a standard VOF method diffuses the
interface, resulting in the volume fraction ($\alpha$) varying between 0 and 1
in a thin region across the interface. To address this issue, we use the MULES
technique that enables the numerical compression of the interface by
introducing the extra compression term into the standard VOF equation. This is
given by

$\frac{\partial\alpha}{\partial
t}+\nabla\cdot(\bm{U}\alpha)\underbrace{-\nabla\cdot\left\\{U_{r,f}(1-\alpha)\alpha\right\\}}_{\text{compression
term}}=0.$ (7)

It is to be noted that the compression term acts exclusively at the interface
and is computed in the normal direction of the interface. The term $U_{r,f}$
signifies the relative velocity between the two fluids, which is modeled as

$U_{r,f}=\min\left(C_{\rm
compr.}\frac{|\phi_{f}|}{|S_{f}|},\max\left[\frac{|\phi_{f}|}{|S_{f}|}\right]\right)n_{f}.$
(8)

Here, $C_{\rm compr.}$ indicates a user-defined interface compression factor,
and its value is set to 1 in the present study as suggested by Deshpande et
al. [2012] to avoid errors in the calculation of interfacial curvatures and
interface smearing. The term $\phi_{f}$ represents the volumetric flux through
the cell face, $S_{f}$ is the outward pointing face area vector, and $n_{f}$
is the face-centered interface normal vector. For a more comprehensive
discussion on the numerical implementation of the MULES method, refer
Deshpande et al. [2012].

We employ the following boundary conditions in order to solve the governing
equations. The free-slip boundary condition is imposed on all side boundaries
of the computational domain. The no-slip and no-penetration boundary
conditions are used at the bottom wall, and the Neumann boundary condition is
used at the top of the computational domain.

###  2.3 Validation

In order to validate the present solver, we compare the results obtained from
our numerical simulations with the experimental observations of Chen et al.
[2006b] and Wang and Chen [2000], Cossali et al. [1997]. In the experiment by
Chen et al. [2006b], a water droplet of diameter 1.1 mm was gently deposited
onto the free surface of a liquid pool containing 20% polybutene in decane and
water, and the dynamics were recorded using a high-speed camera. The
associated dimensionless numbers in our simulation are $\rho_{r}=0.763$,
$\mu_{r}=2$, ${\it Oh}=0.008$, ${\it Bo}=0.10$ and ${\it We}=0$. It can be
seen in Figure 2 that the coalescence initiates at the re-scaled dimensionless
time, $\tau=0$, with the drop draining into the pool for $\tau>0$. This
process gives rise to a cylindrical column due to the upward-moving capillary
wave. The neck of this column constricts due to surface tension, leading to
the pinching off of a daughter drop as discussed in §1. Figure 2 also reveals
that the dynamics obtained from our numerical simulation at various instants
agree with the experimental findings of Chen et al. [2006b].

Figure 2: Comparison of the results obtained from our numerical simulation
(depicted by the red line) with the experimental data from Chen et al. [2006b]
(shown in the background). Panel ($a$) corresponds to the instant $(\tau=0)$
when the drop comes into contact with the free surface of the liquid pool.
Subsequent panels follow at intervals of 0.542 ms. The values of the non-
dimensional numbers considered in our simulations are $\rho_{r}=0.763$,
$\mu_{r}=2$, ${\it Oh}=0.008$, ${\it Bo}=0.10$ and ${\it We}=0$. Figure 3:
Comparison of the results obtained from our numerical simulation (panel $b$)
with the experimental results of Wang and Chen [2000] (panel $a$). The values
of the non-dimensional numbers considered in our simulations are
$\rho_{r}=10^{-3}$, $\mu_{r}=8.3\times 10^{-4}$, ${\it Oh}=0.054$, ${\it
Bo}=0.79$ and ${\it We}=1001.35$.

Figures 3 and 4 illustrate the comparison of our numerical results with the
experimental observations of Wang and Chen [2000], Cossali et al. [1997] in
the context of finite impact velocities of a drop colliding with a deep liquid
pool. In the investigation conducted by Wang and Chen [2000], the drop and the
pool liquid is a mixture of 70% glycerol in water by weight. Consequently, the
values of the various dimensionless numbers in our formulation become
$\rho_{r}=10^{-3}$, $\mu_{r}=8.3\times 10^{-4}$, ${\it Oh}=0.054$, ${\it
Bo}=0.79$ and ${\it We}=1001.35$. It can be seen that, upon impact, the drop
exhibits splashing behavior, and the free surface manifests crown structures
and a milkdrop coronet. Our results qualitatively match with the experimental
observations quite well. For a quantitative validation of our solver, in
Figure 4$b$, we compare the temporal evolution of the crown diameter, $D_{c}$
(as illustrated in Figure 4$a$), with the experimental findings of Cossali et
al. [1997]. The close agreement observed further strengthens the validation of
our solver. We further validate our solver by comparing the shape oscillations
of a freely falling oblate drop with an initial aspect ratio ($A_{r}=1.95$)
with the results from Agrawal et al. [2017]. The comparison is presented in
the appendix (Figure 20).

$(a)$ $(b)$

Figure 4: ($a$) A typical crown formation during the impact of a drop on a
liquid pool. ($b$) Comparison of the temporal evolution of the dimensionless
crown diameter ($D_{c}$) obtained from our numerical simulation with the
experimental results of Cossali et al. [1997]. The values of the non-
dimensional numbers considered in our simulations are $\rho_{r}=0.0012$,
$\mu_{r}=0.014$, ${\it Oh}=0.003$, ${\it Bo}=0.87$ and ${\it We}=160.15$.
Figure 5: The topology of a non-spherical primary droplet with $A_{r}=0.5$
coalescing with the liquid pool at $\tau=1.22$ obtained using three different
grids with the smallest (dimensionless) cell sizes of $\Delta=0.041R_{eq}$,
$0.033R_{eq}$, and $0.028R_{eq}$. The rest of the dimensionless numbers are
$\rho_{r}=10^{-3}$, $\mu_{r}=0.018$, ${\it Oh}=7.82\times 10^{-3}$, ${\it
Bo}=0.04$ and ${\it We}=0$. Panel $(a)$ shows the overlap profiles, and panels
$(b,c)$ depict the cross-sectional views and the iso-surfaces of the drop
obtained using different grids, respectively. Panel $(d)$ shows the temporal
evolutions of the maximum height of the column ($h_{cl}$) for $A_{r}=0.5$
obtained using different grid sizes.

Furthermore, in Figures 5 and 6, we conduct grid convergence and domain
dependence tests for a typical set of parameters considered in our study
($\rho_{r}=10^{-3}$, $\mu_{r}=0.018$, ${\it Oh}=7.82\times 10^{-3}$, ${\it
Bo}=0.04$, and ${\it We}=0$). An octree mesh, generated using SnappyHexMesh,
an OpenFOAM tool specifically designed for octree mesh creation, is employed
to discretize the computational domain. A detailed discussion on octree-based
hexahedral mesh generation can be found in Schneiders [2000]. The octree mesh
facilitates a flexible and efficient representation of complex geometries,
adapting the mesh resolution to specific regions based on the problem
requirement, and is particularly advantageous in simulating two-phase flows
where intricate physics dominates near the interface. Consequently, a finer
mesh zone is established in the proximity of both the droplet and the free
surface of the liquid pool by refining the grid close to the interface, while
larger-sized grids are employed in the outer regions. This strategic approach
effectively minimizes computational costs. In Figure 5(a-c), the topology of a
non-spherical primary droplet with $A_{r}=0.5$ coalescing with the liquid pool
at $\tau=1.22$ is depicted using three different grids. These grids are
characterized by the smallest (dimensionless) cell sizes of
$\Delta=0.041R_{eq}$, $0.033R_{eq}$, and $0.028R_{eq}$, respectively. The
corresponding far-field maximum grid sizes of $\Delta_{max}=0.64R_{eq}$,
$0.53R_{eq}$, and $0.46R_{eq}$, respectively. Notably, the results obtained
with $\Delta=0.033R_{eq}$ and $\Delta=0.028R_{eq}$ are practically
indistinguishable. Additionally, an inspection of the temporal evolution of
the maximum column height ($h_{cl}$), as illustrated in Figure 5$(d)$ for
$A_{r}=0.5$ across three different grid sizes, reveals minimal disparities.
Similarly, Figure 6 demonstrates that results obtained with different sizes of
the computational domain considered for this test are also indistinguishable.
Given these observations, we adopt the grid with $\Delta=0.033R_{eq}$, and a
computational domain size of $16R_{eq}\times 16R_{eq}\times 16R_{eq}$ for
generating the rest of the results presented in the following section.

Figure 6: The topology of a non-spherical primary droplet with $A_{r}=0.5$
coalescing into the liquid pool at $\tau=1.22$ obtained using three different
computational domains. The rest of the dimensionless numbers are
$\rho_{r}=10^{-3}$, $\mu_{r}=0.018$, ${\it Oh}=7.82\times 10^{-3}$, ${\it
Bo}=0.04$ and ${\it We}=0$. Panel $(a)$ shows the overlap profiles, and panels
$(b,c)$ depict the cross-sectional views and the isosurfaces of the drop
obtained using different computational domains, respectively.

##  3 Results and discussion

###  3.1 Coalescence of non-spherical drops gently placed on a pool

The shape of the parent droplet is expected to play a crucial role in the
coalescence process and subsequent dynamics. This aligns with the observations
made by Liu et al. [2021] in the context of the collision of non-spherical
drops on a solid substrate. Thus, we begin the presentation of our results by
examining the effect of the aspect ratio $(A_{r})$ of the non-spherical parent
droplet gently placed on a liquid pool $(We=0)$. Figure 7 illustrates the
temporal evolution of the coalescence phenomenon for the parent non-spherical
drops with $A_{r}=0.2$, 0.5, 1, 2, and 5. The temporal evaluations of the
pressure and the flow field (cross-sectional views in the $x-y$ plane crossing
the center of the drop) for different aspect ratios are presented in Figure 8.
It is important to note that $A_{r}<1$, $A_{r}=1$, and $A_{r}>1$ represent
parent drops with prolate, spherical, and oblate shapes, respectively. To
exclusively examine the impact of the aspect ratio of the parent droplet on
the coalescence phenomenon, we maintain the other dimensionless numbers
constant at $\rho_{r}=10^{-3}$, $\mu_{r}=0.018$, ${\it Oh}=7.82\times 10^{-3}$
and ${\it Bo}=0.04$, while varying the aspect ratio of the parent drop in
Figures 7 and 8.

Figure 7: Temporal evolution of the coalescence of the parent drop of
different aspect ratios $(A_{r})$ gently placed on a deep liquid pool
$(We=0)$. The values of the rest of the dimensionless parameters are ${\it
Oh}=7.82\times 10^{-3}$ and ${\it Bo}=0.04$. The coalescence dynamics for
$A_{r}=0.2$, 0.5, 1.0, 2.0 and 5.0 are included as supplementary video 1-5,
respectively.

For a spherical drop ($A_{r}=1$), as depicted in Figure 7, the drop initially
contacts the liquid pool at $\tau=0$, following the evacuation of the air
cushion between the drop and the free surface. As time progresses ($\tau>0$),
the drop undergoes drainage into the pool due to the pressure difference
created by the negative curvature of the interface (Figure 7 at $\tau=0.56$).
The drop exhibits high capillary pressure near the contact region, generating
two capillary waves, one upwards and the other into the pool. Once the wave
within the droplet regime converges at the top, it induces the formation of a
dent-like structure, observed at $\tau=0.89$, followed by a whipping action of
the tip at the free surface, initiating the formation of a cylindrical column.
This whipping action leads to a sudden increase in the column height
($h_{cl}$) just after $\tau$ = 0.89, then increases slowly till it reaches a
maximum height of $h_{cl}=2.45R_{eq}$ at $\tau=1.35$, which is also seen in
Figure 10$(c)$. The illustration defining the column height ($h_{cl}$) as well
as the neck radius ($R_{n}$) is presented in Figure 10$(a)$. The capillary
wave, after rebounding, causes the droplet to neck, which, if dominated by the
horizontal surface tension force, prevents secondary droplet formation. In the
case of $A_{r}=1.0$, the momentum induced by the capillary wave rebound
dominates the horizontal surface tension force, resulting in the pinching off
of a daughter drop (partial coalescence) with an inverted-tear drop shape at
$\tau=1.88$. This phenomenon is further highlighted in Figure 10$(b)$, where
the neck radius diminishes at $\tau=1.88$, signifying the occurrence of
partial coalescence. Subsequently, the daughter drop deforms back towards a
spherical shape, coalesces again into the pool, and undergoes a coalescence
cascade until fully submerged. As discussed in §1, this phenomenon has been
extensively studied experimentally and numerically by previous researchers
[Thoroddsen and Takehara, 2000, Chen et al., 2006b, Blanchette and Bigioni,
2006, Ray et al., 2010].

Figure 8: Evolution of pressure field and representation of flow field in the
$x-y$ plane crossing the center of the drop for different values of the aspect
ratio $(A_{r})$ of the parent drop. (a) $A_{r}=0.2$, (b) $A_{r}=0.5$, (c)
$A_{r}=1.0$ and $A_{r}=5.0$. The color bar represents the pressure counters.
The values of the rest of the dimensionless parameters are ${\it
Oh}=7.82\times 10^{-3}$, ${\it Bo}=0.04$ and ${\it We}=0$.

In contrast, a significantly prolate primary drop ($A_{r}=0.2$), touching the
liquid pool at almost a single point at $\tau=0$, reverts to a spherical shape
due to exceptionally high-pressure gradients within the droplet, a
characteristic unique to $A_{r}=0.2$. Upon contact with the liquid pool,
competition arises between the capillary pressure due to negative curvature
near the contact point and the Laplace pressure inside the parent drop. The
prevailing Laplace pressure induces a low-pressure zone at the center, with
increased pressure at the top and bottom ends (Figure 8$a$). Consequently, a
velocity field is initiated inside the droplet, directed towards the center,
where the flow remains nearly stationary. This central stationary flow
obstructs the flow from the polar ends, leading to bulges forming away from
the center ($\tau=1.12$ in Figure 8$a$). The resulting crater exhibits low
pressure attributed to negative curvature. The droplet morphology exhibits two
high-pressure regions on each bulge, prompting fluid flow towards the center,
causing the crater to overshoot ($\tau=1.58$). This sequence results in the
droplet adopting a dumbbell shape at $\tau=2.0$ and eventually transforming
into a shape resembling a bicone at $\tau=2.63$ and later, reverting to a
shape similar to the initial form of the drop, albeit less predominantly
prolate (observe $\tau=3.19$), before undergoing oscillations and returning to
a spherical shape or making contact with the free surface.

For $A_{r}=0.5$, the capillary pressure near the contact point and the Laplace
pressure inside the parent drop become comparable, unlike a detachment
condition observed for $A_{r}=0.2$. In the initial stages of coalescence, the
parent drop experiences high pressure at its uppermost region, leading to
liquid migration towards the center and subsequent horizontal bulging.
Simultaneously, liquid continuously drains into the pool due to the negative
curvature caused by the neck ($\tau=0.10$ in Figure 8$b$). A capillary wave
manifests above and below the neck of the droplet, displaying dynamics similar
to the $A_{r}=1.0$ case, albeit with delayed whipping action and column
formation observed at $\tau=1.15$ (Figure 8$b$) and in Figure 10$(c)$. Unlike
other $A_{r}$ values, the increase in column height persists within
$\tau=1.15-1.5$, indicating a lesser induced upward momentum by the capillary
wave—i.e., the change in $h_{cl}$ is less for an extended period, resulting in
significantly slower column height increase. Consequently, the droplet does
not undergo pinch-off but instead undergoes a cycle of expansion and
contraction, as depicted in Figure 10$(b)$, where $R_{n}$ oscillates until all
the liquid is drained into the pool. This phenomenon is known as total
coalescence. Parent drops with $A_{r}>0.67$ exhibit partial coalescence
behavior, displaying dynamics similar to that of a spherical drop.

Figure 9: $(a)$ Temporal evolution of the splitting of the liquid column due
to Rayleigh-Plateau capillary instability above the free surface of the liquid
pool. $(b)$ Evolution of the entrapped air bubble inside the liquid pool for
$A_{r}=5.0$. The rest of the dimensionless parameters are $We=0$, ${\it
Oh}=7.82\times 10^{-3}$ and ${\it Bo}=0.04$.

An intriguing phenomenon unfolds with $A_{r}=5.0$, where the emergence of two
daughter droplets is observed. In this scenario, a significant volume of air
between the parent oblate drop and the pool is displaced outward upon contact
with the free surface. Consequently, the drop establishes contact across a
large area, introducing inherent asymmetry into the coalescence process and
trapping multiple air bubbles inside the liquid pool. The parent drop
undergoes a horizontal Laplace pressure gradient, inducing horizontal collapse
and liquid migration from the ends toward the center, reminiscent of the
dynamics observed in $A_{r}=0.2$. However, this process leads to faster
bulging near the horizontal periphery, forming an annular contact region with
the pool and trapping air in a ring-like bubble, as depicted in Figure 9$(b)$.
The instability of the ring-shaped bubble causes fragmentation into smaller
air bubbles, which move towards the free surface asymmetrically. The free
surface develops a three-dimensional morphology, and surface-wave structures
become chaotic at $\tau=1.88$, eventually settling over time. Furthermore, a
significant capillary wave above the free surface contributes to the formation
of a pronounced depth in the crater ($\tau=0.43$ in Figure 8$d$) just before
the capillary waves converge. This leads to an intense whipping action at the
topmost tip of the droplet regime, evident in Figure 10$(c)$ during the time
period of $\tau=0.43-0.89$. The formation of the column is remarkably rapid
compared to any other case discussed, resulting in a very high upward momentum
within the column, i.e., $h_{cl}$ varies from 1.1$R_{eq}$ to 2.7$R_{eq}$. This
heightened upward momentum is the primary factor behind the formation of the
tall column structure at $\tau=0.89$. The liquid column undergoes a Rayleigh-
Plateau capillary instability, eventually pinching off at $\tau=1.61$ to form
one daughter droplet. The remaining liquid in the column also forms another
non-spherical blob of liquid due to surface tension, whose evolution is
depicted in Figure 9$(a)$. The phenomenon of multiple daughter droplets due to
Rayleigh-Plateau instability has been observed by several researchers [Ray et
al., 2010, Fedorchenko and Wang, 2004, Charles and Mason, 1960]. As the aspect
ratio of the parent drop increases, the width of the column structure
decreases while the maximum height of migration for the daughter drop
increases. The detachment time $(\tau_{d})$ of the daughter drop and its
volume $(V_{d})$ appear to be strongly influenced by the aspect ratio of the
parent drop.

($a$)

($b$) ($c$)

Figure 10: $(a)$ A schematic representation of the column configuration,
depicting the neck radius $(R_{n})$ and column height $(h_{cl})$ with respect
to the undisturbed free surface of the liquid pool. $(b,c)$ Temporal
evolutions of the neck radius $(R_{n})$ and height of the column $(h_{cl})$
for different values of $A_{r}$.

The temporal evolution of the neck radius ($R_{n}$) and the column height
($h_{cl}$) for different $A_{r}$ values is presented in Figure 10$(b,c)$. In
the case of $A_{r}=0.5$, as shown in Figure 10$(b)$, the oscillation of the
neck radius suggests no formation of a secondary droplet, whereas $(R_{n})$
approaches zero for $A_{r}$ = 1.0 and 2.0, indicating the formation of a
secondary droplet in these instances. Figure 10$(c)$ illustrates the variation
of the column height $(h_{cl}$) with time for different $A_{r}$. It is evident
that the rate of increase in $h_{cl}$ is higher with an increase in $A_{r}$
for an oblate droplet. This heightened rate is attributed to the higher
capillary pressure gradient for larger $A_{r}$, which generates a horizontal
velocity field directed towards the center. This induces a rush of fluid in
the vertical direction, causing an increase in $h_{cl}$. Conversely, for
$A_{r}=0.5$, the maximum capillary pressure is concentrated at the vertical
poles of the droplet, resulting in a vertical flow towards the center and a
consistent decrease in $h_{cl}$. For a spherical droplet ($A_{r}=1.0$), the
change in $h_{cl}$ appears considerably less compared to a non-spherical
droplet. This is because there is no internal capillary pressure gradient
independently caused by the droplet. A sudden but slight change in $h_{cl}$
observed around $\tau=0.5\sim 1.0$ indicates the capillary wave meeting at the
topmost end of the droplet, causing a sudden catapulting action. The rate of
this sudden increase in $h_{cl}$ determines the occurrence of either the
phenomenon of total coalescence or partial coalescence. Further insights into
these findings are provided in §3.1, where the dynamics of coalescence with
zero impact velocity of the droplet are described in detail.

Figure 11($a$) and ($b$) depict the variations of the detachment time
($\tau_{d}$) and the normalised volume of the daughter drop $(V_{d}/V_{p})$
emerging due to the partial coalescence phenomenon with the aspect ratio of
the parent drop $(A_{r})$. As discussed above, the parent drops with
$A_{r}<0.67$ do not experience the partial coalescence phenomenon; thus, no
daughter droplet is generated. For $A_{r}>0.67$, the presence of upward-moving
capillary waves is significant enough to delay the vertical collapse of the
liquid column with the dominating horizontal collapse that promotes the neck
formation near the free surface and the detachment of liquid mass to form a
daughter/secondary drop. Inspection of Figure 8 reveals that as the aspect
ratio ($A_{r}$) increases, the lateral velocity also increases. This lateral
velocity then extends across the longitudinal axis of the liquid column,
resulting in a higher upward momentum at the top. This, in turn, leads to an
earlier pinch-off of the droplet, as depicted in Figure 11($a$). In this
situation, the parent droplet will not have sufficient time to release the
fluid into the pool, forming a larger daughter droplet. For $A_{r}<1$, the
initial curvature difference in the droplet results in a notable pressure
gradient. In other words, the droplet exhibits more pressure at the
longitudinal tips and less pressure at the lateral ends. This creates a net
fluid flow towards the lateral direction, causing the parent droplet to bulge
sideways. Despite having a longer detachment time, this bulge prevents the
fluid from draining into the pool due to the bulk momentum being in the
lateral direction. Consequently, the daughter droplet attains a minimum size
near $A_{r}=1$, as depicted in Figure 11$(b)$. It is essential to highlight
that as the capillary wave moves across the surface of the droplet in an
upward direction, the variation of the detachment time with aspect ratio
follows a similar trend as that of the circumference of the droplet. The
detachment requires the capillary wave to traverse the entire droplet surface
twice, covering a total travel length of $\pi(a+b)$, with $a$ and $b$
representing the major and minor diameters of the parent drop.

($a$) ($b$)

Figure 11: The variation of the ($a$) detachment time ($\tau_{d}$) and ($b$)
normalized volume of the daughter droplet $(V_{d}/V_{p})$ with the aspect
ratio of the parent drop $(A_{r})$. The rest of the dimensionless parameters
are ${\it Oh}=7.82\times 10^{-3}$, ${\it Bo}=0.04$ and ${\it We}=0$. Figure
12: Temporal evolution of the shapes of the secondary drop after detachment
for different aspect ratios $(A_{r})$. The rest of the dimensionless
parameters are ${\it Oh}=7.82\times 10^{-3}$, ${\it Bo}=0.04$ and ${\it
We}=0$.

Figure 12 shows the temporal evolution of the morphology of the daughter
droplet (cross-sectional view) associated with the spherical ($A_{r}=1$) and
oblate parent drops with $A_{r}=2$ and 5. The observation spans from just
after the detachment instant ($\tau-\tau_{d}=0$) until the daughter droplet
reconnects with the pool. As depicted in Figure 7, drops with $A_{r}=1$ and 2
exhibit nearly identical behaviour, oscillating significantly from inverted-
teardrop to teardrop shapes. However, for $A_{r}=2$, the daughter droplet
ascends to a higher height than $A_{r}=1$. Consequently, the daughter droplet
for $A_{r}=2$ requires significantly more time to return to the liquid pool
while falling under the action of gravity compared to the spherical drop
($A_{r}=1$). In contrast, $A_{r}=5$ exhibits two daughter drops. The liquid
column, generated by the upward-moving capillary wave, experiences a
detachment process near the free surface, primarily driven by the prevailing
horizontal collapse over the vertical collapse. This process is similar to
that observed for lower aspect ratios. Concurrently, the top part of the
liquid column undergoes fragmentation due to Rayleigh-Plateau instability,
forming a slightly oblate-shaped daughter drop that subsequently undergoes
oblate-prolate shape oscillations. The remaining liquid in the column
amalgamates into a blob of complex shape (as also depicted in Figure 9$a$.
This drop also exhibits shape oscillations, albeit asymmetrical, that result
from the interplay between inertia and surface tension. These observations
also highlight the importance of the three-dimensional numerical simulations
conducted in the present study.

###  3.2 Splashing

Next, we investigate the collision of non-spherical parent drops with varying
aspect ratios into the liquid pool, taking into account different impact
velocities characterized by the Weber number (${\it
We}=\rho_{l}v_{0}^{2}R_{eq}/\sigma$). Here, we consider the impact velocities,
$v_{0}=3.5$ m/s and 5 m/s, while maintaining the remaining parameters
consistent with those considered in §3.1. They are associated with $We=93.4$
and 190.6, respectively. In order to minimize the boundary effects, we employ
a larger computational domain of size $30R_{eq}\times 30R_{eq}\times 30R_{eq}$
in our simulations.

Figure 13: Temporal evolution of the interface morphology resulting from the
impact of a parent drop with different aspect ratios $(A_{r})$ and different
impact velocities. The rest of the dimensionless parameters are $Oh=7.82\times
10^{-3}$ and $Bo=0.04$. Figure 14: A schematic representation illustrating the
crater configuration, depicting the crater depth $(h_{cr})$, diameter
$(D_{cr})$, and wave swell size $(h_{w})$ with respect to the undisturbed free
surface of the liquid pool.

In Figure 13, two effects are investigated: (i) the effect of the impact
velocity $(We)$ of a water droplet with $A_{r}=0.2$, 1, and 5.0, and (ii) the
influence of $A_{r}$ for $We=93.4$ and 190.6 on the temporal evolution of the
morphology of the air-water interface due to the impact of the drops. The
remaining dimensionless parameters are ${\it Oh}=7.82\times 10^{-3}$ and ${\it
Bo}=0.04$. We first discuss the effect of the Weber number for $A_{r}=1$.
Figure 13 (for $A_{r}=1.0$) depicts the manifestation of three distinct
phenomena in the collision process, namely, (i) the expansion of the crater
and the rim of the wave (wave swell), (ii) the retraction of the wave swell
leading to the retraction of the crater sides, and (iii) the retraction of the
crater base. It can be seen that at $\tau=0$, the drop comes in contact with
the liquid pool and forms a hemispherical crater surrounded by a rim-like
structure. As expected, the rim appears sharper at higher Weber numbers (see,
at $\tau=0.33$). Over time, the rim expands, transforming into a wave swell.
In Figure 14, a comparison between craters at ${\it We}=93.4$ and $190.6$
illustrates that increasing the impact velocity increases crater depth
$(h_{cr})$, diameter $(D_{cr})$ and wave swell size $(h_{w})$. Figure
15$(a,b)$ and $(c,d)$ depict the temporal variation of non-dimensional crater
depth and wave swell height for various aspect ratios and Weber numbers. For
$A_{r}=1.0$ i.e., the spherical droplet, they clearly show that the crater
depth is greater for ${\it We}=190.6$ in comparison to ${\it We}=93.4$.
Consequently, the wave swell retracts earlier for ${\it We}=93.4$ compared to
${\it We}=190.6$. Thus, in the case of ${\it We}=93.4$, the rapid retraction
of the crater prevents it from rebounding fully, resulting in an air pocket
before collapse. The wave swell height attains its peak much sooner for ${\it
We}=93.4$ in comparison to ${\it We}=190.6$, indicating that the wave
retraction occurs earlier for ${\it We}=93.4$ than for ${\it We}=190.6$.
Conversely, at ${\it We}=190.6$, the larger crater amplitude delays the
retraction, allowing the crater to rebound sufficiently, preventing air
entrapment. This leads to a high pressure gradient at the crater base in both
cases, with ${\it We}=190.6$ exhibiting a stronger gradient. The pressure
gradient prompts jet formation at the crater base. The strength of the jet
plays an important role in the development of the Rayleigh-Plateau
instability, which triggers the breakup of the jet, giving rise to smaller
daughter droplets. In the case of ${\it We}=93.4$, the jet is comparatively
weaker due to air bubble entrapment, resulting in a less pronounced pressure
gradient than ${\it We}=190.6$, where no air bubble entrapment occurs. The
strength of the jet is also influenced by the maximum height reached by the
wave swell. Despite achieving an earlier maximum in wave swell height, the
actual height is not as substantial for ${\it We}=93.4$ compared to ${\it
We}=190.6$. This height plays a crucial role in determining the intensity of
the crater collapse, consequently leading to a stronger catapulting effect of
the crater base. The dissimilarity in jet strength becomes apparent at
$\tau=8.71$ for ${\it We}=190.6$ in Figure 13, where the Rayleigh-Plateau
instability induces jet breakup and droplet formation. In contrast, at
$\tau=8.22$ for ${\it We}=93.4$ in Figure 13, no droplet formation takes
place. Subsequently, the jet thickens and transforms into a flat interface due
to Laplace pressures and viscous dissipation. These phenomena also resemble
those observed by Ray et al. [2015] for the impact of a spherical drop
($A_{r}=1.0$) on a liquid pool.

The impact area of a prolate drop i.e., $A_{r}=0.2$ is significantly smaller
than that of a spherical drop due to the lower radius of curvature in the
prolate shape. Consequently, in the case of the impact of a prolate drop
($A_{r}=0.2$), Figure 13 reveals a less pronounced wave swell during impact.
The wave height resulting from the droplet impact reaches its maximum earlier
for ${\it We}=190.6$ compared to ${\it We}=93.4$ (Figure 15). Moreover, due to
its bullet-like shape, the impact duration is prolonged, resulting in the
persistence of a shearing zone within the liquid pool and the generation of
more intense vortices. The impact zone forms a crater that delves into the
pool until the crater collapses inward, driven by the wave swell retraction
phenomenon. This process creates an air pocket that gets smaller with time as
the crater walls retract within the pool. It can be seen that the size of this
air pocket is proportional to the impact velocity (Weber number) at time
$\tau=0.6$ and $\tau=0.9$ for $We=190.6$ and 93.4 respectively (Figure 17$b$).
The collapse of the wave height prevents the crater from expelling all the air
from the crater base, forming larger air bubbles within the liquid. A similar
phenomenon was also observed in the axisymmetric simulations of Deka et al.
[2017]. As the crater collapses entirely, the air pocket transforms into a
bubble, fully submerged by the collapsing crater. For ${\it We}=190.6$, the
maximum crater height is achieved sooner compared to ${\it We}=93.4$, leading
to the entrapment of larger-volume air pockets in the pool. As the crater
collapses, a jet is formed, shooting upwards (see, $\tau=0.98$ for ${\it
We}=93.4$ and $\tau=0.82$ for ${\it We}=190.6$ in Figure 13). This jet
entrains the surrounding air, causing it to recirculate (see, $\tau=1.32$ for
${\it We}=93.4$ and $\tau=0.98$ for ${\it We}=190.6$).

In contrast, an oblate drop with $A_{r}=5.0$, despite having an earlier
maximum wave height than the spherical drop (Figure 15) has a significantly
higher impact area, which in turn produces a much wider wave swell (see, at
$\tau=0.66$ for ${\it We}=93.4$ and ${\it We}=190.6$ in Figure 13). Despite
this, the droplet lacks sufficient polar length to sustain the wave swell for
an extended period, causing it to retract quickly. Meanwhile, the crater depth
continues to increase, with ${\it We}=190.6$ exhibiting a greater depth
compared to ${\it We}=93.4$.

This results in the entrapment of a substantial air pocket when the crater
walls collapse, a phenomenon observed in ${\it We}=190.6$, unlike in ${\it
We}=93.4$, where the crater has not descended as far. This distinction is
evident at $\tau=3.95$ for ${\it We}=93.4$ and $\tau=5.92$ for ${\it
We}=190.6$. A jet forms upward once the crater fully collapses, as indicated
at $\tau=4.6$ for ${\it We}=93.4$ and $\tau=6.58$ for ${\it We}=190.6$. The
subsequent stages are similar to that observed for $A_{r}=1.0$, except in this
case, the jet lacks the strength to generate a secondary droplet due to the
Rayleigh-Plateau instability.

($a$) ($b$)

($c$) ($d$)

Figure 15: Temporal variation of non-dimensional crater depth $(h_{cr})$ and
wave swell height $(h_{w})$ for different aspect ratios and Weber numbers.
Panels $(a,b)$ and $(c,d)$ are associated with $h_{cr}$ and $h_{w}$,
respectively. Panels $(a,c)$ and $(b,d)$ correspond to $We=93.4$ ($v_{0}=3.5$
m/s) and $We=190.6$ ($v_{0}=5$ m/s), respectively. The rest of the parameters
are $Oh=7.82\times 10^{-3}$ and $Bo=0.04$.

($a$) ($b$) ($c$)

Figure 16: Temporal variation of non-dimensional crater diameter $(D_{cr})$
for different values of the aspect ratio and Weber number. $(a)$ $A_{r}=0.2$,
$(b)$ $A_{r}=1.0$ and $(c)$ $A_{r}=5.0$. The rest of the parameters are ${\it
Oh}=7.82\times 10^{-3}$ and ${\it Bo}=0.04$.

The temporal evolutions of the crater diameter (spreading of rim) are
illustrated in Figure 16$(a-c)$ for $A_{r}=0.2$, 1.0 and 5.0, respectively.
Upon contact with the free surface of the liquid pool, the droplet imparts
velocity to the pool, forming a rim around the droplet. This rim steadily
expands until its energy is entirely dissipated. For $A_{r}=0.2$, the rim
exhibits a constant spread rate due to its small impact area, as observed for
both ${\it We}=93.4$ and $190.6$. In contrast, for $A_{r}=1.0$ and 5.0, the
initial spreading rate is higher for higher ${\it We}$, but it eventually
decays due to the elevated vertical component of the velocity. This trend is
evident in Figure 16$(b,c)$, where the solid line decays more rapidly than the
dashed line. In Figure 16$b$, it can be observed that the diameter of the rim
for ${\it We}=93.4$ and 190.6 in the case of the spherical drop clearly
undergoes a crossover around $\tau\approx 3$. Below, to understand the
behaviour, we examine the kinetic $(KE)$ and surface energy $(SE)$ of the
system due to the impact of the drop on the free surface of the pool, which
are defined as

$KE=\frac{1}{2}\sum\alpha\rho_{l}(u^{2}+v^{2}+w^{2})dV_{cell},~{}{\rm
and}~{}SE=\sum\sigma A_{s},$ (9)

where $A_{s}$ denotes the surface area of the interface and $dV_{cell}$
represents the volume of the computational cell.

($a$) ($b$)

($c$) ($d$)

($e$) ($f$)

Figure 17: Temporal variations of $KE/KE_{0}$ and $SE/SE_{0}$ for different
values of the aspect ratio and Weber number. The panels $(a,b)$, $(c,d)$ and
$(e,f)$ are for $A_{r}=0.2$, $A_{r}=1.0$ and $A_{r}=5.0$, respectively. The
solid and dashed lines represent the results for $We=93.4$ and $We=190.6$,
respectively. The rest of the parameters are ${\it Oh}=7.82\times 10^{-3}$ and
${\it Bo}=0.04$. The interface shapes at different instants (mentioned inside
the circle) are inserted in panels $b$, $d$ and $f$.

($a$) ($b$)

($c$) ($d$)

($e$) ($f$)

Figure 18: The temporal variations of the normalized kinetic energy $(KE)$,
surface energy $(SE)$, and dissipation energy $(DE)$ with the total energy
$(TE)$. Here, $SE^{*}=SE/TE$, $KE^{*}=KE/TE$, $DE^{*}=DE/TE$. Panels $(a,b)$,
$(c,d)$ and $(e,f)$ are for $A_{r}=0.2$, $A_{r}=1.0$ and $A_{r}=5.0$,
respectively. The solid and dashed lines represent the results for $We=93.4$
and $We=190.6$, respectively. The rest of the parameters are ${\it
Oh}=7.82\times 10^{-3}$ and ${\it Bo}=0.04$.

Figure 17$(a-f)$ depicts the temporal variation of normalized kinetic energy
$(KE/KE_{0})$ and surface energy $(SE/SE_{0})$ for different values of $A_{r}$
and ${\it We}$. Here, $KE_{0}$ and $SE_{0}$ represent the kinetic and surface
energies of the system at $\tau=0$. Initially, for $A_{r}=0.2$, the smaller
contact area of the droplet on the free surface leads to the creation of a
narrower crater (Figure 13). Consequently, there is minimal change in surface
energy, while kinetic energy gradually dissipates over time for ${\it
We}=93.4$ and 190.6. In contrast, for $A_{r}=1.0$ and $5.0$, the impact of the
droplet on the water pool results in the formation of a rim-like structure,
leading to an increase in surface area. This, in turn, increases surface
energy and decreases kinetic energy with time. This trend persists until the
wave height recedes, causing the crater to collapse (around $\tau\approx 3$
for $A_{r}=1.0$). Notably, the value of $SE$ and the time of its peak are
lower for $A_{r}=5.0$ due to the oblate drop creating a wider yet shallower
crater than $A_{r}=1.0$ (Figure 13). An intriguing observation is the
crossover in rim diameter for ${\it We}=93.4$ and 190.6 around $\tau\approx 3$
in Figure 16$(b)$ for $A_{r}=1$. It is interesting to observe that the
crossover in rim diameter for ${\it We}=93.4$ and 190.6, as depicted in Figure
16$(b)$ for $A_{r}=1$, coincides with the instant when $SE$ reaches its
maximum. This suggests a close relationship between the dynamics of the rim
formation and the variation in surface energy. The absence of the crossover in
rim diameter for $A_{r}=5.0$ is likely attributed to its significantly lower
surface energy than $A_{r}=1$. However, the apparent features in Figure
16$(c)$ hint at a similar underlying dynamic, even if the crossover itself is
not observed. The subsequent collapse of the crater initiates a reduction in
the surface area of the air-water interface, leading to a decrease in surface
energy, while the kinetic energy increases slightly. Once the crater is
collapsed, the catapulting effect induces a whipping motion in the fluid
interface, resulting in the formation of a jet that increases the surface area
slightly. Subsequently, surface tension attempts to pull the jet downward,
causing a slight decrease in $KE$, which stabilizes over time.

Figure 18 illustrates the temporal variations of the normalized kinetic energy
$(KE)$, surface energy $(SE)$, and dissipation energy $(DE)$ with the total
energy $(TE)$, such that $SE^{*}=SE/TE$, $KE^{*}=KE/TE$, $DE^{*}=DE/TE$, and
$TE=SE_{0}+KE_{0}+DE_{0}$, where $SE_{0}$, $KE_{0}$, and $DE_{0}$ represent
the surface energy, kinetic energy, and dissipated energy at $\tau=0$. The
dissipation energy is defined as:

$\begin{split}&DE=\int_{0}^{\tau}\sum\mu_{l}\Bigg{[}2\mu_{r}\left(\left(\frac{\partial
u}{\partial x}\right)^{2}+\left(\frac{\partial v}{\partial
y}\right)^{2}+\left(\frac{\partial w}{\partial z}\right)^{2}\right)\\\
&\quad+\mu_{r}\left(\left(\frac{\partial u}{\partial y}+\frac{\partial
v}{\partial x}\right)^{2}+\left(\frac{\partial v}{\partial z}+\frac{\partial
w}{\partial y}\right)^{2}+\left(\frac{\partial u}{\partial z}+\frac{\partial
w}{\partial x}\right)^{2}\right)\Bigg{]}dV_{\text{cell}}d\tau.\end{split}$
(10)

In Figure 18, it is evident that the energy dissipation rate (represented by
the slope of the dissipation curve), while nearly constant with time, is
significantly higher for $A_{r}$ = 0.2 compared to $A_{r}$ = 1.0 and 5.0. This
difference can be attributed to the shear forces induced by the elongated
shape of the droplet, as emphasized by Deka et al. [2017] in their
observations of prolate droplets. Conversely, in the case of oblate and
spherical droplets, the broader impact area leads to a greater disturbance in
the fluid volume, resulting in a more evenly distributed velocity profile and,
subsequently, lower dissipation rates. The disparity in dissipation rate
between the prolate shape drop and the non-prolate shape drops (spherical and
oblate drops) is further amplified as the dissipation rate is dependent on the
square of velocity gradients, which are inherently higher in prolate shapes.
Additionally, higher Weber numbers correspond to elevated dissipation rates,
owing to the heightened velocity gradients associated with greater impact
velocities. It is worth noting that potential energy, being significantly less
compared to the other energies, is disregarded in the energy budget
calculations.

###  3.3 Comparison with the freely falling non-spherical drop impacting a
liquid pool

Figure 19: The comparison of the impact dynamics of a freely falling droplet
from height $L=92.7R_{eq}$ from the free surface of the liquid pool with the
impact dynamics of parent drops with $A_{r}=0.5$ (prolate shape), $A_{r}=1$
(spherical shape) and $A_{r}=2$ (oblate shape) impacting with an equivalent
velocity of $\sqrt{2gL}$. This corresponds to ${\it We}=7.62$. The rest of the
dimensionless parameters are ${\it Oh}=7.82\times 10^{-3}$ and ${\it
Bo}=0.04$.

Finally, in Figure 19, we compare the impact dynamics of an oblate drop with
an initial aspect ratio of $A_{r}=2.0$, descending from a height of
$L=92.7R_{eq}$ above the free surface of the liquid pool, with the impact
dynamics of parent drops of various shapes subjected to impact just above the
free surface, with an equivalent velocity of $\sqrt{2gL}$. For instance, a
drop falling from a height of $L=5.1$ cm impacts the free surface with a
velocity of 1 m/s, which is considered in Figure 19. For the set of parameters
considered (for ${\it We}=7.62$ defined based on the velocity of the falling
drop just above the free surface), it can be seen in Figure 19$a$ that the
oscillations in the free-falling oblate drop mostly dampened out, resulting in
its impact with an almost spherical shape. Consequently, the subsequent
coalescence dynamics closely resembles that observed in the case of a
spherical drop impacting the liquid pool with ${\it We}=7.62$ (Figure 19$c$).
By conducting an inviscid analysis in the microgravity condition, Tsamopoulos
and Brown [1983], Lamb [1924] showed that a drop exhibits periodic shape
oscillations. Recently, by conducting three-dimensional numerical simulations
and experiments, Agrawal et al. [2017, 2020] demonstrated that a non-spherical
drop falling from a height experiences oblate-spherical-prolate shape
oscillations due to the interplay between inertia and surface tension forces.
However, the viscosity acts to dampen these oscillations. Thus, depending on
fluid properties, a freely falling drop can impact the pool with various
shapes, as observed in the diverse shapes of raindrops falling on the ocean
surface [Szakall et al., 2009]. Figure 19$(b)$ and $(d)$ depict the
coalescence behaviour of a prolate, and an oblate shape parent drop with the
same impact velocity (${\it We}=7.62$). A closer examination of these results
reveals distinct dynamics for the prolate and oblate drops compared to the
spherical drop (Figure 19$c$) and the freely falling oblate drop (Figure
19$a$) despite their same impact velocity and the same volume (i.e., the same
Weber number). Specifically, due to its elongated bullet-like shape, the
prolate drop results in a cylindrical crater (Figure 19$b$). On the other
hand, the crater formed by the oblate drop is predominantly hemispherical
(Figure 19$d$). The diameter of the rim formed after impact increases as we
transition from prolate to spherical to oblate-shaped drops, as shown in
Figure 19. This analysis offers insights into the distinct coalescence
dynamics influenced by drop shape and impact conditions.

##  4 Concluding remarks

We conduct three-dimensional numerical simulations to investigate the impact
dynamics of non-spherical drops in a deep liquid pool, varying aspect ratios
and Weber numbers. The numerical solver is validated by comparing our results
with previous numerical and experimental investigations. Our findings
emphasize the significant influence of the shape of the parent droplet
(keeping the volume the same) on coalescence dynamics, resulting in distinct
morphologies of the liquid-air interface compared to spherical drops studied
in earlier literature. First, we investigate the coalescence dynamics of non-
spherical parent droplets gently placed on a liquid pool $({\it We}=0)$. We
observe that a prolate drop makes contact with the liquid pool at almost a
single point as it tries to return to a spherical shape under the influence of
surface tension. This initiates a competition between the capillary pressure
resulting from the negative curvature near the contact point and the Laplace
pressure within the parent drop, setting off a velocity field inside the
droplet. Due to the substantially higher Laplace pressure than the capillary
pressure arising from negative curvature, the drop immediately detaches from
the free surface upon contact, showcasing intricate three-dimensional shape
oscillations for the high aspect ratio $A_{r}=0.2$. Subsequently, under the
influence of gravity, the drop touches the free surface again, initiating the
coalescence process. We found that the parent drop exhibits a partial
coalescence phenomenon and the emergence of a daughter droplet for
$A_{r}>0.67$. In contrast to the prolate $(A_{r}<1)$ and spherical drop
$(A_{r}=1)$, an oblate parent drop with $A_{r}=5$ encapsulates air in a ring-
like bubble within the pool and emerges a liquid column that undergoes
Rayleigh-Plateau capillary instability, leading to the formation of two
daughter droplets. Further, our observations indicate that the variation of
detachment time with aspect ratio exhibits a similar trend to that of the
magnitude of the circumference of the droplet (i.e., $\approx\pi(a+b)$).

Then, we explore the collision dynamics of non-spherical parent drops with
various aspect ratios, impacting the liquid pool at different velocities
(Weber numbers). It is well known that the impact of a spherical drop with a
high velocity is characterised by the crater formation, air bubble entrainment
inside the pool, the release of a jet and splashing dynamics. Our observations
indicate that increasing the Weber number leads to elevated crater heights on
the free surface for all shapes of parent drops. As the impact area of a
prolate drop is significantly smaller than that of spherical and oblate drops,
a prolate drop produces a less pronounced wave swell and a prolonged impact
duration. Conversely, an oblate drop generates a much wider wave swell than
spherical and prolate drops. For spherical drop ($A_{r}=1$), we observe an
interesting behaviour of the crossover in rim diameter for ${\it We}=93.4$ and
190.6 at around $\tau\approx 3$, which coincides with the instant when surface
energy reaches its maximum. This suggests a close relationship between rim
formation dynamics and variations in surface energy. Although $A_{r}=5.0$ does
not show any crossover in rim diameter due to its significantly lower surface
energy than $A_{r}=1$, our results hint at a similar underlying behaviour.

Finally, we establish an analogy by comparing the dynamics of a freely falling
non-spherical drop, experiencing topological oscillations as it descends from
a height, with the impact dynamics of parent drops of various shapes striking
the liquid surface with an equivalent velocity. This comparative analysis
allows us to delve into the impact of shape oscillations, arising from the
interplay between inertia and surface tension, on the coalescence phenomenon
during the free fall of a drop. This approach contrasts with the extensive
studies conducted by various researchers on the impact of a spherical parent
drop just above the free surface.

Acknowledgement: K.C.S. thanks the Science & Engineering Research Board,
India, and IIT Hyderabad for their financial support provided through grants
CRG/2020/000507 and IITH/CHE/F011/SOCH1. S.B. thanks for the support and
resources provided by PARAM Seva under the National Supercomputing Mission,
Government of India.

## Appendix

Figure 20: Comparison of the temporal evolution of $(a-b)$ of an oblate
droplet of initial $A_{r}=1.95$ during its free fall obtained from our
numerical simulation with that of Agrawal et al. [2017]. The values of the
non-dimensional numbers considered in our simulations are $Oh=5.5\times
10^{-3}$, $We=0$, $Bo=1$, $\rho_{r}=10^{-3}$ and $\mu_{r}=0.018$.

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