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Spatiotemporal evolution of evaporating liquid films sheared by a gas

Omair A. A. Mohamed*

Michael C. Dallaston†

Luca Biancofiore‡

  • Mechanical Engineering Department, Bilkent University, 06800 Ankara, Turkey

  • School of Mathematical Sciences, Queensland University of Technology, Brisbane, Queensland 4000, Australia

  • Mechanical Engineering Department, Bilkent University, 06800 Ankara, Turkey

  • *omair.mohamed@bilkent.edu.tr
  • †michael.dallaston@qut.edu.au
  • ‡Corresponding author: luca@bilkent.edu.tr
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Phys. Rev. Fluids 6, 114002 – Published 4 November, 2021

DOI: https://doi.org/10.1103/PhysRevFluids.6.114002

Abstract

We study the spatiotemporal evolution of an evaporating liquid film sheared by a gas considering both inertial and thermal instabilities, the latter arising from a combination of evaporation and Marangoni effects. The shearing gas is modeled by imposing a constant shear stress applied along the liquid's interface. Following in the footsteps of Joo et al. [S. W. Joo et al., J. Fluid Mech. 230, 117 (1991)], long-wave theory is used to derive a Benney-like equation governing the temporal evolution of the liquid interface under the effects of inertia, hydrostatic pressure, surface tension, thermocapillarity, evaporation, and gas shear. Linear stability theory is used to investigate the temporal and spatiotemporal characteristics of the flow, where it is found that the evaporation of the film promotes absolute instabilities and can cause convective-absolute transitions of the perturbations. It is also found that a strong enough counterflowing shearing gas can suppress the inertial instability, commonly known as the H mode, affirming similar conclusions found by previous studies for a strongly confined isothermal film. Additionally, our temporal stability analysis indicates that the thinning of the film reduces the phase speed of thermal perturbations, due to the increasing dominance of viscosity. However, our spatiotemporal analysis shows that the thinning of the film actually results in the growth of additional modes with higher group velocities resulting in faster contamination of the flow field. Moreover, the interface evolution equation is solved numerically to (i) simulate the film's interface evolution subject to finite perturbations and (ii) compare to the results of the linear stability analysis. We find qualitative agreement between the temporal dynamics of the linear and nonlinear instabilities. Our subsequent numerical nonlinear spatiotemporal stability analysis demonstrates that for weaker thermal instabilities, the wave-front dynamics are imposed by the nonlinearly saturated wave packet, while for stronger thermal instabilities, the wave-front dynamics are dictated by the linear dispersion relationship. We also study the effects of the dimensionless parameters on the rupture location and the time it takes for the fluid film to rupture. Finally, the shear stress's effect on the rupture mechanics of the film is studied using self-similarity analysis, where we identify the fate of the evolution equation's solutions.

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References (80)

  1. C. Marangoni, T. Peruzzo, I. G. B. Parisotto, V. W. Ricardo, C. A. Claumann, K. W. Milanez, M. B. H. Mantelli, M. B. Quadri, A. Bolzan, M. C. dos Santos, L. C. Medina, and R. A. F. Machado, Falling film distillation column with heat transfer by means of a vapor chamber. Part II: Operation with a temperature profile, Chem. Eng. Commun. 206, 1006 (2019).
  2. A. B. Erasmus and I. Nieuwoudt, Mass transfer in structured packing: A wetted-wall study, Ind. Eng. Chem. Res. 40, 2310 (2001).
  3. A. L. Kohl and R. Nielsen, Gas Purification (Gulf, Houston, 1997).
  4. A. H. Lefebvre and V. G. McDonell, Atomization and Sprays (Hemisphere, New York, 1989).
  5. G. A. Lawrence, F. K. Browand, and L. G. Redekopp, The stability of a sheared density interface, Phys. Fluids A 3, 2360 (1991).
  6. D. Lucas, C. P. Caulfield, and R. R. Kerswell, Layer formation in horizontally forced stratified turbulence: Connecting exact coherent structures to linear instabilities, J. Fluid Mech. 832, 409 (2017).
  7. P. L. Kapitza, Wave flow of thin layers of a viscous fluid: I. Free flow, J. Exp. Theor. Phys. 18, 3 (1948).
  8. P. L. Kapitza and S. P. Kapitza, Wave flow of thin layers of a viscous fluid: III. Experimental study of undulatory flow conditions, J. Exp. Theor. Phys. 19, 105 (1949).
  9. T. B. Benjamin, Wave formation in laminar flow down an inclined plane, J. Fluid Mech. 2, 554 (1957).
  10. C.-S. Yih, Stability of liquid flow down an inclined plane, Phys. Fluids 6, 321 (1963).
  11. D. J. Benney, Long waves on liquid films, J. Math. Phys. 45, 150 (1966).
  12. C. Ruyer-Quil and P. Manneville, Improved modeling of flows down inclined planes, Eur. Phys. J. B 15, 357 (2000).
  13. C. Ruyer-Quil and P. Manneville, Further accuracy and convergence results on the modeling of flows down inclined planes by weighted-residual approximations, Phys. Fluids 14, 170 (2002).
  14. M. K. Smith, The mechanism for the long-wave instability in thin liquid films, J. Fluid Mech. 217, 469 (1990).
  15. S. Kalliadasis, C. Ruyer-Quil, B. Scheid, and M. G. Velarde, Falling Liquid Films (Springer, London, 2012).
  16. G. Dietze, On the Kapitza instability and the generation of capillary waves, J. Fluid Mech. 789, 368 (2016).
  17. S. Sreenivasan and S. P. Lin, Surface tension driven instability of a liquid film flow down a heated incline, Int. J. Heat Mass Transfer 21, 1517 (1978).
  18. R. E. Kelly, S. H. Davis, and D. A. Goussis, in Proceedings of the Eighth International Heat Transfer Conference, San Francisco, 1986, edited by C. L. Tien, V. P. Carey, and J. K. Ferrell (Hemisphere, New York, 1986), p. 1937.
  19. D. A. Goussis and R. E. Kelly, Surface wave and thermocapillary instabilities in a liquid film flow, J. Fluid Mech. 223, 25 (1991).
  20. J. P. Burelbach, S. G. Bankoff, and S. H. Davis, Nonlinear stability of evaporating/condensing liquid films, J. Fluid Mech. 195, 463 (1988).
  21. S. W. Joo, S. H. Davis, and S. G. Bankoff, Long-wave instabilities of heated falling films: Two-dimensional theory of uniform layers, J. Fluid Mech. 230, 117 (1991).
  22. H. Mohamed and L. Biancofiore, Linear stability analysis of evaporating falling liquid films, Int. J. Multiphase Flow 130, 103354 (2020).
  23. V. S. Ajaev and G. M. Homsy, Steady vapor bubbles in rectangular microchannels, J. Colloid Interface Sci., 240, 259 (2001).
  24. V. S. Ajaev, Spreading of thin volatile liquid droplets on uniformly heated surfaces, J. Fluid Mech. 528, 279 (2005).
  25. V. S. Ajaev, Evolution of dry patches in evaporating liquid films, Phys. Rev. E 72, 031605 (2005).
  26. H. Ji and T. P. Witelski, Instability and dynamics of volatile thin films, Phys. Rev. Fluids 3, 024001 (2018).
  27. S. D. Howison, J. A. Moriarty, J. R. Ockendon, E. L. Terrill, and S. K. Wilson, A mathematical model for drying paint layers, J. Eng. Math. 32, 377 (1997).
  28. O. E. Shklyaev and E. Fried, Stability of an evaporating thin liquid film, J. Fluid Mech. 584, 157 (2007).
  29. E. Fried, Sharp-interface nematic-isotropic phase transformations with flow, Arch. Ration. Mech. Anal. 190, 227 (2008).
  30. R. D. Deegan, O. Bakajin, T. F. Dupont, G. Huber, S. R. Nagel, and T. A. Witten, Capillary flow as the cause of ring stains from dried liquid drops, Nature (London) 389, 827 (1997).
  31. R. D. Deegan, O. Bakajin, T. F. Dupont, G. Huber, S. R. Nagel, and T. A. Witten, Contact line deposits in an evaporating drop, Phys. Rev. E 62, 756 (2000).
  32. E. Sultan, A. Boudaoud, and M. B. Amar, Diffusion-limited evaporation of thin polar liquid films, J. Eng. Math. 50, 209 (2004).
  33. E. Sultan, A. Boudaoud, and M. B. Amar, Evaporation of a thin film: Diffusion of the vapour and Marangoni instabilities, J. Fluid Mech. 543, 183 (2005).
  34. T. J. Hanratty and J. M. Engen, Interaction between a turbulent air stream and a moving water surface, AIChE J. 3, 299 (1957).
  35. A. D. D. Craik, Wind-generated waves in thin liquid films, J. Fluid Mech. 26, 369 (1966).
  36. L. O'Náraigh, P. D. M. Spelt, and T. A. Zaki, Turbulent flow over a liquid layer revisited: Multi-equation turbulence modelling, J. Fluid Mech. 683, 357 (2011).
  37. G. F. Dietze and C. Ruyer-Quil, Wavy liquid films in interaction with a confined laminar gas flow, J. Fluid Mech. 722, 348 (2013).
  38. G. Lavalle, Y. Li, S. Mergui, N. Grenier, and G. F. Dietze, Suppression of the Kapitza instability in confined falling liquid films, J. Fluid Mech. 860, 608 (2019).
  39. A. C. King, E. O. Tuck, and J. M. Vanden-Broeck, Air-blown waves on thin viscous sheets, Phys. Fluids A 5, 973 (1993).
  40. M. Sivapuratharasu, S. Hibberd, M. E. Hubbard, and H. Power, Inertial effects on thin-film wave structures with imposed surface shear on an inclined plane, Physica D 325, 86 (2016).
  41. C. J. Shearer and J. F. Davidson, The investigation of a standing wave due to gas blowing upwards over a liquid film; its relation to flooding in wetted-wall columns, J. Fluid Mech. 22, 321 (1965).
  42. A. G. Cetinbudaklar and G. J. Jameson, The mechanism of flooding in vertical countercurrent two-phase flow, Chem. Eng. Sci. 24, 1669 (1969).
  43. R. Vellingiri, D. Tseluiko, and S. Kalliadasis, Absolute and convective instabilities in counter-current gas–liquid film flows, J. Fluid Mech. 763, 166 (2015).
  44. R. J. Briggs, Electron-Stream Interaction with Plasmas (MIT Press, Cambridge, 1964).
  45. P. Huerre and P. A. Monkewitz, Local and global instabilities in spatially developing flows, Annu. Rev. Fluid Mech. 22, 473 (1990).
  46. L. Brevdo, P. Laure, F. Dias, and T. J. Bridges, Linear pulse structure and signalling in a film flow on an inclined plane, J. Fluid Mech. 396, 37 (1999).
  47. S. W. Joo and S. H. Davis, Instabilities of three-dimensional viscous falling films, J. Fluid Mech. 242, 529 (1992).
  48. L. O'Náraigh, P. D. M. Spelt, and S. J. Shaw, Absolute linear instability in laminar and turbulent gas-liquid two-layer channel flow, J. Fluid Mech. 714, 58 (2013).
  49. D. Tseluiko and S. Kalliadasis, Nonlinear waves in counter-current gas-liquid film flow, J. Fluid Mech. 673, 19 (2011).
  50. M. B. Williams and S. H. Davis, Nonlinear theory of film rupture, J. Colloid Interface Sci. 90, 220 (1982).
  51. S. Krishnamoorthy, B. Ramaswamy, and S. W. Joo, Spontaneous rupture of thin liquid films due to thermocapillarity: A full-scale direct numerical simulation, Phys. Fluids 7, 2291 (1995).
  52. W. Boos and A. Thess, Cascade of structures in long-wavelength Marangoni instability, Phys. Fluids 11, 1484 (1999).
  53. S. Shklyaev, A. V. Straube, and A. Pikovsky, Superexponential droplet fractalization as a hierarchical formation of dissipative compactons, Phys. Rev. E 82, 020601(R) (2010).
  54. T. Wei and F. Duan, Interfacial stability and self-similar rupture of evaporating liquid layers under vapor recoil, Phys. Fluids 28, 124106 (2016).
  55. M. C. Dallaston, M. A. Fontelos, D. Tseluiko, and S. Kalliadasis, Discrete Self-Similarity in Interfacial Hydrodynamics and the Formation of Iterated Structures, Phys. Rev. Lett. 120, 034505 (2018).
  56. H. Ji and T. P. Witelski, Finite-time thin film rupture driven by modified evaporative loss, Physica D 342, 1 (2017).
  57. J. M. Delhaye, Jump conditions and entropy sources in two-phase systems. Local instant formulation, Int. J. Multiphase Flow 1, 395 (1974).
  58. H. J. Palmer, The hydrodynamic stability of rapidly evaporating liquids at reduced pressure, J. Fluid Mech. 75, 487 (1976).
  59. T. F. Lin, C. J. Chang, and W. M. Yan, Analysis of combined buoyancy effects of thermal and mass diffusion on laminar forced convection heat transfer in a vertical tube, J. Heat Transfer 110, 337 (1988).
  60. W. M. Yan and C. Y. Soong, Convective heat and mass transfer along an inclined heated plate with film evaporation, Int. J. Heat Mass Transfer 38, 1261 (1995).
  61. H. Ding and P. D. M. Spelt, Onset of motion of a three-dimensional droplet on a wall in shear flow at moderate Reynolds numbers, J. Fluid Mech. 599, 341 (2008).
  62. H. Ding, M. N. H. Gilani, and P. D. M. Spelt, Sliding, pinch-off and detachment of a droplet on a wall in shear flow, J. Fluid Mech. 644, 217 (2010).
  63. C. Paterson, S. K. Wilson, and B. R. Duffy, Rivulet flow round a horizontal cylinder subject to a uniform surface shear stress, Q. J. Mech. Appl. Math. 67, 567 (2014).
  64. J. M. Sullivan, C. Paterson, S. K. Wilson, and B. R. Duffy, A thin rivulet or ridge subject to a uniform transverse shear stress at its free surface due to an external airflow, Phys. Fluids 24, 082109 (2012).
  65. J. M. Sullivan, S. K. Wilson, and B. R. Duffy, A thin rivulet of perfectly wetting fluid subject to a longitudinal surface shear stress, Q. J. Mech. Appl. Math. 61, 25 (2008).
  66. S. K. Wilson, J. M. Sullivan, and B. R. Duffy, The energetics of the breakup of a sheet and of a rivulet on a vertical substrate in the presence of a uniform surface shear stress, J. Fluid Mech. 674, 281 (2011).
  67. S. Davis, in Waves on Fluid Interfaces, edited by R. E. Meyer (Academic, Cambridge, 1983), p. 291.
  68. S. H. Davis, The stability of time-periodic flows, Annu. Rev. Fluid Mech. 8, 57 (1976).
  69. L. Biancofiore and F. Gallaire, Counterpropagating Rossby waves in confined plane wakes, Phys. Fluids 24, 074102 (2012).
  70. L. Biancofiore, F. Gallaire, and E. Heifetz, Interaction between counterpropagating Rossby waves and capillary waves in planar shear flows, Phys. Fluids 27, 044104 (2015).
  71. G. K. Batchelor, H. K. Moffatt, and M. G. Worster, Perspectives in Fluid Dynamics: A Collective Introduction to Current Research (Cambridge University Press, Cambridge, 2002).
  72. MATLAB, Version 9.9.0.1592791 (R2020b), Update 5 (Mathworks, Inc., Natick, 2020).
  73. P. Huerre and P. A. Monkewitz, Absolute and convective instabilities in free shear layers, J. Fluid Mech. 159, 151 (1985).
  74. I. Delbende and J.-M. Chomaz, Nonlinear convective/absolute instabilities in parallel two-dimensional wakes, Phys. Fluids 10, 2724 (1998).
  75. G. Dee and J. S. Langer, Propagating Pattern Selection, Phys. Rev. Lett. 50, 383 (1983).
  76. T. P. Witelski and A. J. Bernoff, Stability of self-similar solutions for van der Waals driven thin film rupture, Phys. Fluids 11, 2443 (1999).
  77. W. W. Zhang and J. R. Lister, Similarity solutions for van der Waals rupture of a thin film on a solid substrate, Phys. Fluids 11, 2454 (1999).
  78. G. I. Sivashinsky and D. M. Michelson, On irregular wavy flow of a liquid film down a vertical plane, Prog. Theor. Phys. 63, 2112 (1980).
  79. A. Pumir, P. Manneville, and Y. Pomeau, On solitary waves running down an inclined plane, J. Fluid Mech. 135, 27 (1983).
  80. M. Irfan and M. Muradoglu, A front tracking method for direct numerical simulation of evaporation process in a multiphase system, J. Comput. Phys. 337, 132 (2017).

Outline

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Phys. Rev. Fluids 6, 114002– Published 4 November, 2021
  • Received 5 May 2021
  • Accepted 11 October 2021
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