Coalescence of surfactant-laden drops by Phase Field Method

Abstract

In this work, we propose and test the validity of a modified Phase Field Method (PFM), which was specifically developed for large scale simulations of turbulent flows with large and deformable surfactant-laden droplets. The time evolution of the phase field, φ, and of the surfactant concentration field, ψ, are obtained from two Cahn-Hilliard-like equations together with a two-order-parameter Time-Dependent Ginzburg-Landau (TDGL) free energy functional. The modifications introduced circumvent existing limitations of current approaches based on PFM and improve the well-posedness of the model. The effect of surfactant on surface tension is modeled via an Equation Of State (EOS), further improving the flexibility of the approach. This method can efficiently handle topological changes, i.e. breakup and coalescence, and describe adsorption/desorption of surfactant. The capabilities of the proposed approach are tested in this paper against previous experimental results on the effects of surfactant on the deformation of a single droplet and on the interactions between two droplets. Finally, to appreciate the performances of the model on a large scale complex simulation, a qualitative analysis of the behavior of surfactant-laden droplets in a turbulent channel flow is presented and discussed.

Deformation of clean and surfactant-laden droplets in shear flow

“…The dynamics of a multiphase flow with a surfactant is modeled coupling pseudo-spectral-based solution of the Navier-Stokes (NS) equation with a Phase Field Method (PFM), used in a two-order-parameter formulation, to compute the interface dynamics and the surfactant concentration [15,26,39]. In the following, the governing equations of the two order parameters, phase field / and surfactant concentration w, will be derived and then coupled with continuity and Navier-Stokes (NS) equations to describe the hydrodynamics of the system.…”

“…In the phase field chemical potential, l / , the terms depending on the surfactant concentration have been removed. These terms induce an unphysical behavior of the interface [26,39] and, thus, to restore the correct interfacial behavior they have been neglected. Further details on this point can be found in Yun et al [39] and Soligo et al [26].…”

“…These terms induce an unphysical behavior of the interface [26,39] and, thus, to restore the correct interfacial behavior they have been neglected. Further details on this point can be found in Yun et al [39] and Soligo et al [26]. From the expressions of the chemical potentials, the equilibrium profiles of the two order parameters can be obtained.…”

“…Later, the accuracy of Taylor's formula was improved by Shapira and Haber [24], who introduced a correction for confinement effects. These sets of prediction tools are now a well assessed and easy to use benchmark for numerical methods and codes for multiphase flows [12,26,31,35,38,40].…”

“…In this work we use Direct Numerical Simulation (DNS) of the Navier-Stokes equations coupled with a two-order-parameter phase field method (PFM) to describe the interface topology and the surfactant concentration [15,26,39]. We start by comparing the deformation of circular (2D) and spherical (3D) droplets and we extend previous works analyses [32,41] to investigate the limits of Taylor's formula.…”