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Mean field control of droplet dynamics with high-order finite-element computations

Published online by Cambridge University Press:  20 November 2024

Guosheng Fu ,
Hangjie Ji Open the ORCID record for Hangjie Ji [Opens in a new window] ,
Will Pazner  and
Wuchen Li Open the ORCID record for Wuchen Li [Opens in a new window]
Guosheng Fu
Affiliation:
Department of Applied and Computational Mathematics and Statistics, University of Notre Dame, IN 46556, USA
Hangjie Ji
Affiliation:
Department of Mathematics, North Carolina State University, NC 27695, USA
Will Pazner
Affiliation:
Fariborz Maseeh Department of Mathematics and Statistics, Portland State University, Portland, OR 97207, USA
Wuchen Li*
Affiliation:
Department of Mathematics, University of South Carolina, Columbia, SC 29208, USA
*
†Email address for correspondence: [email protected]
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Abstract

Liquid droplet dynamics are widely used in biological and engineering applications, which contain complex interfacial instabilities and pattern formation such as droplet merging, splitting and transport. This paper studies a class of mean field control formulations for these droplet dynamics, which can be used to control and manipulate droplets in applications. We first formulate the droplet dynamics as gradient flows of free energies in modified optimal transport metrics with nonlinear mobilities. We then design an optimal control problem for these gradient flows. As an example, a lubrication equation for a thin volatile liquid film laden with an active suspension is developed, with control achieved through its activity field. Lastly, we apply the primal–dual hybrid gradient algorithm with high-order finite-element methods to simulate the proposed mean field control problems. Numerical examples, including droplet formation, bead-up/spreading, transport, and merging/splitting on a two-dimensional spatial domain, demonstrate the effectiveness of the proposed mean field control mechanism.

JFM classification

Flow Control: Control theory Low-Reynolds-number flows: Lubrication theory Interfacial Flows (free surface): Thin films
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 999 , 25 November 2024 , A76
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Copyright
© The Author(s), 2024. Published by Cambridge University Press

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Controlled droplet splitting dynamics
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