Volume 56, Number 2, March-April 2022
|Page(s)||651 - 678|
|Published online||15 March 2022|
Fully-discrete, decoupled, second-order time-accurate and energy stable finite element numerical scheme of the Cahn-Hilliard binary surfactant model confined in the Hele-Shaw cell
Department of Mathematics, University of South Carolina, Columbia, SC 29208, USA
* Corresponding author: email@example.com
Accepted: 4 January 2022
We consider the numerical approximation of the binary fluid surfactant phase-field model confined in a Hele-Shaw cell, where the system includes two coupled Cahn-Hilliard equations and Darcy equations. We develop a fully-discrete finite element scheme with some desired characteristics, including linearity, second-order time accuracy, decoupling structure, and unconditional energy stability. The scheme is constructed by combining the projection method for the Darcy equation, the quadratization approach for the nonlinear energy potential, and a decoupling method of using a trivial ODE built upon the “zero-energy-contribution” feature. The advantage of this scheme is that not only can all variables be calculated in a decoupled manner, but each equation has only constant coefficients at each time step. We strictly prove that the scheme satisfies the unconditional energy stability and give a detailed implementation process. Various numerical examples are further carried out to prove the effectiveness of the scheme, in which the benchmark Saffman-Taylor fingering instability problems in various flow regimes are simulated to verify the weakening effects of surfactant on surface tension.
Mathematics Subject Classification: 65N12 / 65M12 / 65M70
Key words: Finite element / fully-discrete / second-order / fluid-surfactant / Cahn-Hilliard / Hele-Shaw cell
© The authors. Published by EDP Sciences, SMAI 2022
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