\section*{Context}
The coalescence of liquid droplets has significant implications for various industrial applications, such as clean energy production through liquid-liquid extraction processes. 
Understanding the dynamics of coalescence is challenging due to the problem's multi-scale nature. It encompasses colloidal forces at the molecular level and hydrodynamic forces at the flow scale, which is often turbulent or buoyancy-driven.
The dimensionless parameter space governing these dynamics is vast, yet a phenomenological understanding of the coalescence process can be given \citep{chesters1991}. In the context of IFPEN's processes of interest, the droplets are inertial, with sizes exceeding the dissipative scale, and are immersed in highly inertial and unsteady flows. 
The viscosity ratio between the two phases is typically close to unity, suggesting partially mobile interfaces in the absence of surfactants \citep{davis1989}. 
Given the moderate droplet volume fractions, it seems reasonable to limit the analysis to pairwise interactions. 
Additionally, the droplets are generally spherical (much smaller than the Kolmogorov-Hinze scale) and exhibit only slight deformability.

\section*{Bibliography}
Within this framework, droplet collisions leading to coalescence typically involve two stages \citep{yiantsios1990}. 
The first stage, characterised by negligible deformation, involves a significant reduction in the relative normal velocity of the droplets due to lubrication forces \citep{davis1989}. 
The second stage, known as film drainage, begins when deformations become significant, specifically, when the pressure within the thin film reaches the capillary pressure-and the relative velocity between the droplets is sufficiently low \citep{jones1978,nemer2013}.
From a purely hydrodynamic perspective, these two stages have been extensively analysed in quasi-static or Stokes flow regimes for normal collisions but are less understood in inertial flows \citep{sambath2019} or under conditions representative of "real-world" collisions, particularly with tangential relative velocities \citep{rother1997,loewenberg1997}. 
In addition, there is experimental evidence that flow-driven coalescence frequently occurs after the drops have already rotated to a configuration where they are being pulled apart by the external flow (see figure on the right).


\section*{Working plan}

The novelty of this study lies in using the adaptive mesh refinement technique offered by the open-source software \href{http://basilisk.fr/}{Basilisk} to address the collision process and inital stage of the film drainage process. 
This solver has been shown to accurately predict the dynamics of non-axisymmetric collisions between two bubbles and partially capture the thin film drainage dynamics \footnote{The primary limitation in the latter being due to the capillary time step.} \citep{zhang2021}.

We will focus on two problems. 
The first involves buoyancy-driven coalescence of two unequal sizes droplets, as investigated by \citet{rother1997} in viscous-dominated flows. 
The second focuses on the shear flow configuration studied by \citet{loewenberg1997}. 
For both configurations, we will examine the influence of inertia, droplet deformation (characterized by the Weber number), and the viscosity ratio.
The findings from these configurations, particularly the hydrodynamic forces acting on the droplets, will serve as outer boundary conditions for an ongoing Ph.D. project focusing specifically on solving thin-film drainage equations.

The proposed postdoctoral research is part of a CEFIPRA project in collaboration with Hiranya Deka (IIT Dharwad) and Susmita Dash (IISc Bangalore). It specifically addresses inertial coalescence of droplets.