Numerical Approximation of a Phase-Field Surfactant Model with Fluid Flow

Zhu, Guangpu; Kou, Jisheng; Sun, Shuyu; Ye, Jun; Li, Aifen

doi:10.1007/s10915-019-00934-1

Cited by 38 publications

(13 citation statements)

References 66 publications

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“…Recently, we coupled this model with the hydrodynamic equations to simulate the interfacial flows with surfactants. Several linear, decoupled and energy stable schemes were also constructed to solve this complex system effectively (Zhu et al 2019b). Our three-dimensional (3D) results successfully demonstrate the effect of surfactants on the interfacial dynamics.…”

Section: Introductionmentioning

confidence: 87%

Thermodynamically consistent modelling of two-phase flows with moving contact line and soluble surfactants

et al. 2019

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Droplet dynamics on a solid substrate is significantly influenced by surfactants. It remains a challenging task to model and simulate the moving contact line dynamics with soluble surfactants. In this work, we present a derivation of the phase-field moving contact line model with soluble surfactants through the first law of thermodynamics, associated thermodynamic relations and the Onsager variational principle. The derived thermodynamically consistent model consists of two Cahn–Hilliard type of equations governing the evolution of interface and surfactant concentration, the incompressible Navier–Stokes equations and the generalized Navier boundary condition for the moving contact line. With chemical potentials derived from the free energy functional, we analytically obtain certain equilibrium properties of surfactant adsorption, including equilibrium profiles for phase-field variables, the Langmuir isotherm and the equilibrium equation of state. A classical droplet spread case is used to numerically validate the moving contact line model and equilibrium properties of surfactant adsorption. The influence of surfactants on the contact line dynamics observed in our simulations is consistent with the results obtained using sharp interface models. Using the proposed model, we investigate the droplet dynamics with soluble surfactants on a chemically patterned surface. It is observed that droplets will form three typical flow states as a result of different surfactant bulk concentrations and defect strengths, specifically the coalescence mode, the non-coalescence mode and the detachment mode. In addition, a phase diagram for the three flow states is presented. Finally, we study the unbalanced Young stress acting on triple-phase contact points. The unbalanced Young stress could be a driving or resistance force, which is determined by the critical defect strength.

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Section: Introductionmentioning

confidence: 87%

Thermodynamically consistent modelling of two-phase flows with moving contact line and soluble surfactants

et al. 2019

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“…An efficient energy-stable time-marching scheme can be easily constructed for the above governing equation, and details can refer to Zhu et al 2019b; Zhu et al (2020). Note that the current work is the extension of matched density case in (Zhu et al, 2019b). The SAV approach used in this study is more accurate and efficient than the IEQ approach used in (Zhu et al, 2019b).…”

Section: Governing Equationmentioning

confidence: 99%

“…Note that the current work is the extension of matched density case in (Zhu et al, 2019b). The SAV approach used in this study is more accurate and efficient than the IEQ approach used in (Zhu et al, 2019b). Also, the variable density case considered in this study presents new challenges to the development of energy stable time-marching scheme, and we can only construct a nonlinearly coupled scheme for the above governing equation.…”

Section: Governing Equationmentioning

confidence: 99%

“…This model allows us to investigate the interfacial dynamics with soluble surfactants. However, nonlinear terms in chemical potentials, arising from the double well potential and Flory-Huggins potential, will bring great difficulties to the construction of numerical schemes and solution of the whole governing system (Zhu et al, 2019b). A strategy is needed to transform the nonlinear potentials in the free energy functional, then we can get a new model, which allows us to design efficient schemes to solve the whole governing system.…”

Section: Introductionmentioning

confidence: 99%

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

Zhu

2020

Adv. Geo-Energ. Res.

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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.

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“…The main advantage of this approach is to easily design a linear scheme satisfying the energy stability. The other two famous Lagrange multiplier-type methods are the invariant energy quadratization (IEQ) (Yang 2016) and scalar auxiliary variable (SAV) approaches (Gao et al 2020), they have been widely used for various gradient flow problems (Zhu et al 2019;Liu and Yin 2019;Li et al 2018). However, classical IEQ and SAV methods require the nonlinear parts or its integral to be bounded from below.…”

Section: Introductionmentioning

confidence: 99%

Numerical simulation and analysis of the Swift–Hohenberg equation by the stabilized Lagrange multiplier approach

Yang

Kim

2021

Comp. Appl. Math.

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In this work, we develop linear, first-and second-order accurate, and energy-stable numerical scheme for the Swift-Hohenberg (SH) equation. An auxiliary variable (Lagrange multiplier) is used to control the nonlinear term so that the linear temporal scheme can be easily constructed. To further achieve the accuracy with large time steps, a proper stabilized term is adopted. For the first-order time-accurate scheme, the backward Euler approximation is adopted. The Crank-Nicolson (CN) and explicit Adams-Bashforth (AB) approximations are applied to achieve temporally second-order accuracy. We analytically perform the estimations of the semi-discrete solvability, the energy stability with respect to the original and pseudo-energy functionals, and the convergence error. To numerically solve the resulting discrete system of equations, we use an efficient linear multigrid method. We present various two-(2D) and three-dimensional (3D) computational examples to demonstrate the accuracy and energy stability of the proposed scheme.

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Numerical Approximation of a Phase-Field Surfactant Model with Fluid Flow

Cited by 38 publications

References 66 publications

Thermodynamically consistent modelling of two-phase flows with moving contact line and soluble surfactants

Thermodynamically consistent modelling of two-phase flows with moving contact line and soluble surfactants

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

Numerical simulation and analysis of the Swift–Hohenberg equation by the stabilized Lagrange multiplier approach

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