Enhanced numerical approaches for modeling insoluble surfactants in two-phase flows with the diffuse-interface method

TL;DR

This work addresses accurate interfacial transport of insoluble surfactants in two-phase flows using a diffuse-interface phase-field framework. It introduces two practical improvements: (i) favoring the $f$-type transport model over the $f_d$-type to reduce discretization error in gradient terms, and (ii) decoupling the delta-function width from the interface width by coupling a level-set based $\hat{\epsilon}$ to define $\hat{\phi}$ and $\hat{\delta_\Gamma}$, enabling independent control of $\hat{W}$ while preserving mass conservation via the ACDI formulation with $W=4\epsilon$. Numerical tests in 2D and 3D, including diffusion in uniform flow, vortical advection, drop deformation under shear, and a challenging highly deformed benchmark, show improved accuracy and stability, with near second-order accuracy in some cases, and demonstrate applicability to Marangoni forces. The study provides practical guidelines for delta-width selection, showcases the robustness of the proposed approaches, and offers a rigorous benchmark to evaluate future interfacial surfactant transport models.

Abstract

Surfactants reside at the interface of two-phase flows and significantly influence the flow dynamics. Numerical simulations are essential for a comprehensive understanding of such surfactant-laden flows and require a method that can accurately simulate surfactant transport along the interface. In this study, we focus on interfacial transport models for insoluble surfactants based on the diffuse-interface method and propose two approaches to improve their accuracy: (a) adopting a formulation that avoids the spatial derivatives of variables with sharp gradients and (b) allowing the width of the delta function to be specified independently of the interface width. These approaches are simple and practical in that they do not lead to significant increases in computational cost, implementation complexity, or degradation of interface-capturing accuracy. We conduct a series of numerical tests to demonstrate the effectiveness of the proposed approaches. Finally, we present a challenging test case that is difficult to solve accurately and has not been previously discussed. We expect this case to serve as a valuable benchmark for evaluating and comparing the performances of various methods proposed in the literature.

Figures (16)

• Figure 1: Schematic illustrating the representation of surfactant concentration using the diffuse-interface method.
• Figure 2: Schematic of the interfacial surfactant transport models compared in this study: the $f_d$-type model in Eq. (\ref{['eq:fd_type']}) and the $f$-type model in Eq. (\ref{['eq:f_type']}).
• Figure 3: Schematic of the proposed Approach 2 in Section \ref{['subsec:approach2']}. Conventionally, the delta function is computed directly from the phase-field variable $\phi$ using Eq. (\ref{['eq:delta_wide']}), which ties the width of the delta function $\delta_\Gamma$ to the interface width. In the proposed approach, the delta function is computed from the level-set function $\psi$, allowing the width of the delta function to be adjusted independently of the interface width by controlling $\hat{\epsilon}$.
• Figure 4: Schematic of surfactant diffusion on a circular interface advected by a uniform velocity field, as described in Section \ref{['subsec:diffusion_in_uniform_flow']}.
• Figure 5: Error in surfactant concentration at $t = 5$ for the case of surfactant diffusion in uniform flow, as described in Section \ref{['subsec:diffusion_in_uniform_flow']}. Two different diffusion coefficients are examined: (a) $D = 10^{-2}$ and (b) $D = 10^{-9}$. The test is performed using three different grid resolutions: $32^2$, $64^2$, and $128^2$, corresponding to $16$, $32$, and $64$ grid points per diameter $d_0$, respectively. $\hat{W}$ denotes the width of the delta function.