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Through this so-called Marangoni effect, surfactants can play an important role in several physical phenomena such as vortex pair interaction (e.g., [1]), fingering (e.g., [2]), drop break-up and coalescence (e.g., [3, 4, 5]) and tip-streaming (e.g., [6, 4]).

From a numerical point of view, solving the problem of soluble surfactants is highly challenging.

A non-linear equation of state is used to relate the surface tension to the interfacial surfactant concentration.

The method is based on the use of a diffuse interface, which allows a simple implementation using standard finite difference or finite element techniques.

The presence of surfactant typically alters the interface dynamics by a reduction in the surface tension of the interface.

An inhomogeneous distribution of surfactants produces gradients in surface tension, which again gives rise to tangential forces along the interface.

Multiphase flows, interfacial dynamics, surfactant, soluble surfactant, surface phase, bulk phase, adsorption, desorption, complex geometry, diffuse interface, phase field, multigrid, adaptive grid, finite difference The presence of surface active agents (surfactants) at fluid interfaces can have a considerable effect on flow dynamics.

This method was extended to arbitrary viscosity ratios in [8], to 3D in [9, 4] and to soluble surfactants in [10]. Another tracking method is the front-tracking method (see the review [12]), where a fixed grid is used to compute the flow, while a set of connected marker particles is used to track the interface and any interfacial quantities.

Further, the surface concentration may only be soluble in either the exterior or interior of the domain (e.g., amphiphilic nature of surfactants).

The available numerical methods for solving these problems can roughly be divided into two categories: interface tracking and interface capturing methods.

Interface tracking methods use either a separate grid for the interface, or a set of interconnected points to mark the interface.

For example, boundary integral methods use a surface mesh to track the interface.