Interfacial dynamics with soluble surfactants: A phase-field two-phase flow model with variable densities

Abstract

In this work, we present a hydrodynamics coupled phase-field surfactant model with variable densities. Two scalar auxiliary variables are introduced to transform the original free energy functional into an equivalent form, and then a new thermodynamically consistent model can be obtained. In this model, evolutions of two phase-field variables are described by two Cahn-Hilliard-type equations, and the fluid flow is dominated by incompressible Navier-Stokes equation. The finite difference method on staggered grid is used to solve the above model. Then a classical droplet rising case and a droplet merging case are used to validate our model. Finally, we study the effect of surfactants on droplet deformation and merging. A more prolate profile of droplet is observed under a higher surfactant bulk concentration, which verifies the effect of surfactant in reducing the interfacial tension. Increases in surface Péclet number and initial surfactant bulk concentration can enhance the non-uniformity of surfactant distribution around the interface, which will arise the Marangoni force. The Marangoni force acts as an additional repulsive force to delay the droplet merging.

Thermodynamically consistent modeling of two-phase incompressible flows in heterogeneous and fractured media

“…where m is the sample size and p c;i is the experimental value of capillary pressure measured at S w;i . Figure 1 shows the capability and performance of the capillary pressure function equation (22) in approximating the experimental data of three samples in [58]. Here, the wetting-phase residual saturations are taken as S rw ¼ 0:098; 0:039; 0:03 for three samples respectively, while S rn ¼ 0 is taken for all samples.…”

“…Taking into account equations (21), (22) and 28, we deduce the formulations of Darcy's velocities as,…”

“…As shown in equation 22, the capillary pressure can be obtained from the difference between chemical potentials of two phases. To validate the model, we employ the capillary pressure formulation equation (22) to fit the experimental data of water-oil systems provided in [58]. Using the constraint S w þ S n ¼ 1, we can express the capillary pressure function given in equation (22) as a function of S w , i.e., p c ðS w Þ.…”

“…To validate the model, we employ the capillary pressure formulation equation (22) to fit the experimental data of water-oil systems provided in [58]. Using the constraint S w þ S n ¼ 1, we can express the capillary pressure function given in equation (22) as a function of S w , i.e., p c ðS w Þ. The energy parameters can be calculated by the least squares method: solve the minimizer of,…”

“…However, the energy reconstructed from the classical model involves the ideal contribution of the wetting-phase saturation only, but the ideal fluid energy of both phases and attraction effect between two phases have not been taken into account, so the classical two-phase flow model is not completely thermodynamically consistent. Phase-field models for twophase flows have been extensively studied in the literature, for instance [18][19][20][21][22]. Thermodynamically consistent phasefield-based diffuse interface models and simulation of twophase flows in porous media have been reported in the literature, for instance [23][24][25][26][27][28][29][30], which introduce free energy potentials to characterize capillarity effect caused by the surface tension.…”