A numerical scheme for fluid-surfactant systems with data assimilation: Stability and energy behaviour analysis
Volume 3, Issue 1, Article Number: 261003 (2026)
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Prahlad Singh1,* | Neeraj Kumar1,* 
1Department of Mathematics, Govt Degree College Agra Cantt, Agra – 282001, U.P. (India)
1Department of Mathematics, Government Degree College, Goverdhan, Mathura – 281502, U.P. (India)
*Corresponding Author: neerajmath2010@gmail.com
Received: 31 March 2026 | Revised: 11 May 2026
Accepted: 14 May 2026 | Published Online: 26 May 2026
DOI: https://doi.org/10.5281/zenodo.20186104
© 2026 The Authors, under a Creative Commons license, Published by Scholarly Publication
Abstract
This study introduces a numerical implementation of fluid-surfactant systems that is based on a coupled phase-field model, combined with a nudging-based data assimilation mechanism. Governing equations are expressed on the basis of the Cahn–Hilliard formulation, along with a transport equation of a surfactant, and the velocity of the fluid is simplified through the use of a gradient-driven model to decrease the computational cost. A finite difference discretization with explicit time integration is used to discretize the system. Simulations are done to study the energy evolution, error dynamics, interface behavior and the surfactant distribution. The findings indicate that the total free energy grows and levels off, not due to energy dissipation but due to limited numerical behavior. The L2-error between the simulated and observed phase-field shows rapid early growth and then levels off, showing little efficiency of the data assimilation with the selected parameters. The analysis of the phase-field shows that explicit discretization causes checkerboard-type numerical oscillations that cause the loss of interface smoothness. Additionally, the surfactant distribution remains nearly uniform, indicating diffusion-dominated dynamics with weak coupling to the interface. Overall, the proposed framework provides a computationally efficient approach for modelling fluid–surfactant systems; however, the findings highlight the need for improved numerical stability and enhanced data assimilation strategies to achieve physically consistent and accurate simulations.
Keywords
Phase-field model, Fluid–surfactant systems, Data assimilation, Cahn–Hilliard equation, Numerical simulation, Surfactant transport, Computational modeling
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Cite This Article
P. Singh and N. Kumar, “A numerical scheme for fluid-surfactant systems with data assimilation: Stability and energy behaviour analysis,” Radius: Journal of Science and Technology 3(1) (2026) 261003. https://doi.org/10.5281/zenodo.20186104
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