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Electrostatic control of the Navier-Stokes equations for thin films

Alexander W. Wray*

Radu Cimpeanu† and Susana N. Gomes‡

  • Department of Mathematics and Statistics, University of Strathclyde, Livingstone Tower, 26 Richmond Street, Glasgow G1 1XH, United Kingdom

  • Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom

  • *alexander.wray@strath.ac.uk
  • †radu.cimpeanu@warwick.ac.uk
  • ‡susana.gomes@warwick.ac.uk
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Phys. Rev. Fluids 7, L122001 – Published 26 December, 2022

DOI: https://doi.org/10.1103/PhysRevFluids.7.L122001

Abstract

A robust control scheme is derived and tested for the Navier-Stokes equations for two-dimensional multiphase flow of a thin film underneath an inclined solid surface. Control is exerted via the use of an electrode parallel to the substrate, which induces an electric field in the gas phase, and a resultant Maxwell stress at the liquid-gas interface. The imposed potential at the second electrode is derived using a model predictive control loop, together with optimal control of a high-fidelity reduced-dimensional model. In this implementation the interfacial shape of the fluid is successfully controlled; however, the algorithm is sufficiently general to control any other quantity of interest.

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References (42)

  1. A. Oron, S. H. Davis, and S. G. Bankoff, Long-scale evolution of thin liquid films, Rev. Mod. Phys. 69, 931 (1997).
  2. R. V. Craster and O. K. Matar, Dynamics and stability of thin liquid films, Rev. Mod. Phys. 81, 1131 (2009).
  3. S. Kalliadasis, C. Ruyer-Quil, B. Scheid, and M. G. Velarde, Falling Liquid Films, Applied Mathematical Sciences (Springer Science & Business Media, London, 2011), Vol. 176.
  4. C. Ruyer-Quil and P. Manneville, Improved modeling of flows down inclined planes, Eur. Phys. J. B 15, 357 (2000).
  5. B. Scheid, C. Ruyer-Quil, and P. Manneville, Wave patterns in film flows: Modelling and three-dimensional waves, J. Fluid Mech. 562, 183 (2006).
  6. S. Chakraborty, P.-K. Nguyen, C. Ruyer-Quil, and V. Bontozoglou, Extreme solitary waves on falling liquid films, J. Fluid Mech. 745, 564 (2014).
  7. A. B. Thompson, D. Tseluiko, and D. T. Papageorgiou, Falling liquid films with blowing and suction, J. Fluid Mech. 787, 292 (2016).
  8. A. B. Thompson, S. N. Gomes, F. Denner, M. C. Dallaston, and S. Kalliadasis, Robust low-dimensional modelling of falling liquid films subject to variable wall heating, J. Fluid Mech. 877, 844 (2019).
  9. A. W. Wray, O. K. Matar, and D. T. Papageorgiou, Accurate low-order modeling of electrified falling films at moderate Reynolds number, Phys. Rev. Fluids 2, 063701 (2017).
  10. D. Saville, Electrohydrodynamics: The Taylor-Melcher leaky dielectric model, Annu. Rev. Fluid Mech. 29, 27 (1997).
  11. D. T. Papageorgiou, Film flows in the presence of electric fields, Annu. Rev. Fluid Mech. 51, 155 (2019).
  12. F. Tröltzsch, Optimal Control of Partial Differential Equations: Theory, Methods, and Applications, Graduate Studies in Mathematics (American Mathematical Society, Providence, RI, 2010), Vol. 112.
  13. E. Cerpa, P. Guzmán, and A. Mercado, On the control of the linear Kuramoto-Sivashinsky equation, ESAIM 23, 165 (2017).
  14. S. N. Gomes, D. T. Papageorgiou, and G. A. Pavliotis, Stabilizing non-trivial solutions of the generalized Kuramoto–Sivashinsky equation using feedback and optimal control: Lighthill–Thwaites prize, IMA J. Appl. Math. 82, 158 (2017).
  15. A. Armaou and P. D. Christofides, Feedback control of the Kuramoto–Sivashinsky equation, Physica D 137, 49 (2000).
  16. A. Armaou and P. D. Christofides, Wave suppression by nonlinear finite-dimensional control, Chem. Eng. Sci. 55, 2627 (2000).
  17. R. Al Jamal and K. Morris, Linearized stability of partial differential equations with application to stabilization of the Kuramoto–Sivashinsky equation, SIAM J. Control Optim. 56, 120 (2018).
  18. R. J. Tomlin and S. N. Gomes, Point-actuated feedback control of multidimensional interfaces, IMA J. Appl. Math. 84, 1112 (2019).
  19. V. Belus, J. Rabault, J. Viquerat, Z. Che, E. Hachem, and U. Reglade, Exploiting locality and translational invariance to design effective deep reinforcement learning control of the 1-dimensional unstable falling liquid film, AIP Adv. 9, 125014 (2019).
  20. M. Sellier, Inverse problems in free surface flows: A review, Acta Mech. 227, 913 (2016).
  21. E. Boujo and M. Sellier, Pancake making and surface coating: Optimal control of a gravity-driven liquid film, Phys. Rev. Fluids 4, 064802 (2019).
  22. D. Sipp and P. J. Schmid, Linear closed-loop control of fluid instabilities and noise-induced perturbations: A review of approaches and tools, Appl. Mech. Rev. 68, 020801 (2016).
  23. R. Cimpeanu, S. N. Gomes, and D. T. Papageorgiou, Active control of liquid film flows: Beyond reduced-order models, Nonlin. Dyn. 104, 267 (2021).
  24. F. Denner, A. Charogiannis, M. Pradas, C. N. Markides, B. G. Van Wachem, and S. Kalliadasis, Solitary waves on falling liquid films in the inertia-dominated regime, J. Fluid Mech. 837, 491 (2018).
  25. A. W. Wray and R. Cimpeanu, Reduced-order modelling of thick inertial flows around rotating cylinders, J. Fluid Mech. 898, A1 (2020).
  26. W. Rohlfs, P. Pischke, and B. Scheid, Hydrodynamic waves in films flowing under an inclined plane, Phys. Rev. Fluids 2, 044003 (2017).
  27. N. Kofman, W. Rohlfs, F. Gallaire, B. Scheid, and C. Ruyer-Quil, Prediction of two-dimensional dripping onset of a liquid film under an inclined plane, Int. J. Multiphase Flow 104, 286 (2018).
  28. T. G. Anderson, R. Cimpeanu, D. T. Papageorgiou, and P. G. Petropoulos, Electric field stabilization of viscous liquid layers coating the underside of a surface, Phys. Rev. Fluids 2, 054001 (2017).
  29. R. Cimpeanu, D. T. Papageorgiou, and P. G. Petropoulos, On the control and suppression of the Rayleigh-Taylor instability using electric fields, Phys. Fluids 26, 022105 (2014).
  30. A. B. Thompson, S. N. Gomes, G. A. Pavliotis, and D. T. Papageorgiou, Stabilising falling liquid film flows using feedback control, Phys. Fluids 28, 012107 (2016).
  31. L. Grüne and J. Pannek, Nonlinear model predictive control, in Nonlinear Model Predictive Control, Communications and Control Engineering (Springer, Cham, 2017), pp. 45–69.
  32. D. S. Pillai and R. Narayanan, Nonlinear dynamics of electrostatic Faraday instability in thin films, J. Fluid Mech. 855, R4 (2018).
  33. R. Craster and O. Matar, Electrically induced pattern formation in thin leaky dielectric films, Phys. Fluids 17, 032104 (2005).
  34. W. Rohlfs, L. M. Cammiade, M. Rietz, and B. Scheid, On the effect of electrostatic surface forces on dielectric falling films, J. Fluid Mech. 906, A18 (2021).
  35. E. Polak and G. Ribiere, Note sur la convergence de méthodes de directions conjuguées, Rev. Fran. Inf. Rech. Oper. Ser. Rouge 3, 35 (1969).
  36. J. Kiefer, Sequential minimax search for a maximum, Proc. Am. Math. Soc. 4, 502 (1953).
  37. N. Fabbiane, O. Semeraro, S. Bagheri, and D. S. Henningson, Adaptive and model-based control theory applied to convectively unstable flows, Appl. Mech. Rev. 66, 060801 (2014).
  38. M. Morari, C. E. Garcia, and D. M. Prett, Model predictive control: Theory and practice, IFAC Proc. 21, 1 (1988).
  39. S. Popinet, A quadtree-adaptive multigrid solver for the Serre–Green–Naghdi equations, J. Comput. Phys. 302, 336 (2015).
  40. J.M. López-Herrera, S. Popinet, and M. Herrada, A charge-conservative approach for simulating electrohydrodynamic two-phase flows using volume-of-fluid, J. Comput. Phys. 230, 1939 (2011).
  41. S. Popinet, An accurate adaptive solver for surface-tension-driven interfacial flows, J. Comput. Phys. 228, 5838 (2009).
  42. A. W. Wray, D. T. Papageorgiou, and O. K. Matar, Reduced models for thick liquid layers with inertia on highly curved substrates, SIAM J. Appl. Math. 77, 881 (2017).

Outline

Information

Phys. Rev. Fluids 7, L122001– Published 26 December, 2022
  • Received 26 July 2022
  • Accepted 16 December 2022
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