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

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Abstract

Modeling interfacial dynamics with soluble surfactants in a multiphase system is a challenging task. Here, we consider the numerical approximation of a phase-field surfactant model with fluid flow. The nonlinearly coupled model consists of two Cahn–Hilliard-type equations and incompressible Navier–Stokes equation. With the introduction of two auxiliary variables, the governing system is transformed into an equivalent form, which allows the nonlinear potentials to be treated efficiently and semi-explicitly. By certain subtle explicit-implicit treatments to stress and convective terms, we construct first and second-order time marching schemes, which are extremely efficient and easy-to-implement, for the transformed governing system. At each time step, the schemes involve solving only a sequence of linear elliptic equations, and computations of phase-field variables, velocity and pressure are fully decoupled. We further establish a rigorous proof of unconditional energy stability for the first-order scheme. Numerical results in both two and three dimensions are obtained, which demonstrate that the proposed schemes are accurate, efficient and unconditionally energy stable. Using our schemes, we investigate the effect of surfactants on droplet deformation and collision under a shear flow, where the increase of surfactant concentration can enhance droplet deformation and inhibit droplet coalescence.

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Acknowledgements

Jun Yao and Guangpu Zhu acknowledge that this work is supported by the National Science and Technology Major Project (2016ZX05011-001), the NSF of China (51804325, 51504276, and 51674280). The work of Shuyu Sun and Jisheng Kou is supported by the KAUST research fund awarded to the Computational Transport Phenomena Laboratory at KAUST through the Grant BAS/1/1351-01-01.

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Authors and Affiliations

  1. Research Center of Multiphase Flow in Porous Media, School of Petroleum Engineering, China University of Petroleum (East China), Qingdao, 266580, China

    Guangpu Zhu, Jun Yao & Aifen Li

  2. School of Mathematics and Statistics, Hubei Engineering University, Xiaogan, 432000, Hubei, China

    Jisheng Kou

  3. Computational Transport Phenomena Laboratory, Division of Physical Science and Engineering, King Abdullah University of Science and Technology, Thuwal, 23955-6900, Kingdom of Saudi Arabia

    Shuyu Sun

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  1. Guangpu Zhu
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  2. Jisheng Kou
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Correspondence to Shuyu Sun.

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Zhu, G., Kou, J., Sun, S. et al. Numerical Approximation of a Phase-Field Surfactant Model with Fluid Flow. J Sci Comput 80, 223–247 (2019). https://doi.org/10.1007/s10915-019-00934-1

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