# Interfacial instability of liquid films coating the walls of a parallel-plate channel and sheared by a gas flow

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## Abstract

The stability and coupling of liquid films coating the walls of a parallel-plate channel and sheared by a pressure-driven gas flow along the channel center plane is studied. The films are susceptible to a long-wavelength instability. For sufficiently low Reynolds numbers and thick gas layers, the dynamic behavior is found to be described by two coupled nonlinear partial differential equations. A linear stability analysis is conducted under the condition that the material properties and the initial undisturbed liquid-film thicknesses are equal. The linear analysis is utilized to determine whether the interfaces are predominantly destabilized by the variations of the shear stress or by the pressure gradient acting upon them. The analysis of the weakly nonlinear equations performed for this case shows that instabilities corresponding to a vanishing Reynolds number are absent from the system. Moreover, for this configuration, the patterns emerging along the two interfaces are found to be identical in the long-time limit, implying that the films are fully synchronized. A different setup, where the liquid films have identical material properties but their undisturbed thicknesses differ, is studied numerically. The results show that, even for this configuration, the interfacial waves remain phase-synchronized and closely correlated for an extended period of time. These findings are particularly relevant for gaseous flow through narrow ducts with liquid-coated walls.

## Notes

### Acknowledgements

This study was supported by the Deutsche Forschungsgemeinschaft (DFG) under Grant number DI 1689/1-1, which is gratefully acknowledged.

## Supplementary material

## References

- Anturkar NR, Papanastasiou TC, Wilkes JO (1990) Linear stability analysis of multilayer plane Poiseuille flow. Phys Fluids A Fluid (1989–1993) 2(4):530–541. https://doi.org/10.1063/1.857753 CrossRefzbMATHGoogle Scholar
- Brennen CE (2005) Fundamentals of multiphase flows. Cambridge University Press, CambridgeCrossRefGoogle Scholar
- Canic S, Plohr B (1995) Shock wave admissibility for quadratic conservation laws. J Differ Equ 118(2):293–335. https://doi.org/10.1006/jdeq.1995.1075 MathSciNetCrossRefzbMATHGoogle Scholar
- Chandrasekhar S (2013) Hydrodynamic and hydromagnetic stability. Courier Corporation, ChelmsfordzbMATHGoogle Scholar
- Chang H, Demekhin EA (2002) Complex wave dynamics on thin films. Elsevier, AmsterdamGoogle Scholar
- Charru F, Hinch EJ (2006) Ripple formation on a particle bed sheared by a viscous liquid. Part 1. Steady flow. J Fluid Mech 550:111–121. https://doi.org/10.1017/S002211200500786X MathSciNetCrossRefzbMATHGoogle Scholar
- Comsol (2014) COMSOL Multiphysics
^{®}. COMSOL Inc, GöttingenGoogle Scholar - Cross M, Greenside H (2009) Pattern formation and dynamics in nonequilibrium systems. Cambridge University Press, CambridgeCrossRefGoogle Scholar
- Cvitanovic P, Davidchack RL, Siminos E (2010) On the state space geometry of the Kuramoto–Sivashinsky flow in a periodic domain. SIAM J Appl Dyn Syst 9:1–33. https://doi.org/10.1137/070705623 MathSciNetCrossRefzbMATHGoogle Scholar
- Drazin PG, Reid WH (2004) Hydrodynamic stability. Cambridge University Press, CambridgeCrossRefGoogle Scholar
- Eifert A, Paulssen D, Varanakkottu SN, Baier T, Hardt S (2014) Simple fabrication of robust water-repellent surfaces with low contact-angle hysteresis based on impregnation. Adv Mater Interfaces 1(1300):138. https://doi.org/10.1002/admi.201300138 CrossRefGoogle Scholar
- Gondret P, Rabaud M (1997) Shear instability of two-fluid parallel flow in a hele-shaw cell. Phys Fluids 9(11):3267–3274CrossRefGoogle Scholar
- Grinthal AE, Aizenberg J (2013) Mobile interfaces: liquids as a perfect structural material for multifunctional, antifouling surfaces. Chem Mater 26(1):698–708. https://doi.org/10.1021/cm402364d CrossRefGoogle Scholar
- Halpern D, Fujioka H, Takayama S, Grotberg JB (2008) Liquid and surfactant delivery into pulmonary airways. Respir Physiol Neurobiol 163(1):222–231CrossRefGoogle Scholar
- Heil M, Hazel AL, Smith JA (2008) The mechanics of airway closure. Respir Physiol Neurobiol 163(1–3):214–221. https://doi.org/10.1016/j.resp.2008.05.013
**(Respiratory Biomechanics)**CrossRefGoogle Scholar - Hewitt G, Hall-Taylor N (1970) Annular two-phase flow. Pergamon, New York. https://doi.org/10.1016/B978-0-08-015797-9.50011-X CrossRefGoogle Scholar
- Hooper A, Boyd W (1983) Shear-flow instability at the interface between two viscous fluids. J Fluid Mech 128:507–528MathSciNetCrossRefGoogle Scholar
- Hooper AP, Grimshaw R (1985) Nonlinear instability at the interface between two viscous fluids. Phys Fluids (1958–1988) 28(1):37–45. https://doi.org/10.1063/1.865160 CrossRefzbMATHGoogle Scholar
- Houshmand F, Peles Y (2013) Convective heat transfer to shear-driven liquid-film flow in a microchannel. Int J Heat Mass Transf 64:42–52CrossRefGoogle Scholar
- Hu HH, Patankar N (1995) Non-axisymmetric instability of core-annular flow. J Fluid Mech 290:213–224. https://doi.org/10.1017/S0022112095002485 CrossRefzbMATHGoogle Scholar
- Hyman JM, Nicolaenko B (1986) The Kuramoto–Sivashinsky equation: a bridge between PDE’s and dynamical systems. Physica D 18(1–3):113–126. https://doi.org/10.1016/0167-2789(86)90166-1 MathSciNetCrossRefzbMATHGoogle Scholar
- Johnson M, Kamm RD, Ho LW, Shapiro AH, Pedley TJ (1991) The nonlinear growth of surface-tension-driven instabilities of a thin annular film. J Fluid Mech 233:141–156CrossRefGoogle Scholar
- Joseph D, Renardy M, Renardy Y (1984) Instability of the flow of two immiscible liquids with different viscosities in a pipe. J Fluid Mech 141:309–317CrossRefGoogle Scholar
- Joseph DD, Bai R, Chen KP, Renardy YY (1997) Core-annular flows. Annu Rev Fluid Mech 29(1):65–90MathSciNetCrossRefGoogle Scholar
- Kabov OA, Zaitsev DV, Cheverda VV, Bar-Cohen A (2011) Evaporation and flow dynamics of thin, shear-driven liquid films in microgap channels. Exp Therm Fluid Sci 35(5):825–831CrossRefGoogle Scholar
- Kandlikar SG (2012) History, advances, and challenges in liquid flow and flow boiling heat transfer in microchannels: a critical review. J Heat Transf 134(3):034001CrossRefGoogle Scholar
- Kevrekidis IG, Nicolaenko B, Scovel JC (1990) Back in the saddle again: a computer assisted study of the Kuramoto–Sivashinsky equation. SIAM J Appl Math 50(3):760–790MathSciNetCrossRefGoogle Scholar
- Kliakhandler I, Sivashinsky G (1995) Kinetic alpha effect in viscosity stratified creeping flows. Phys Fluids 7(8):1866–1871. https://doi.org/10.1063/1.868501 MathSciNetCrossRefzbMATHGoogle Scholar
- Kudryashov N (1990) Exact solutions of the generalized Kuramoto–Sivashinsky equation. Phys Lett A 147(5–6):287–291. https://doi.org/10.1016/0375-9601(90)90449-X MathSciNetCrossRefGoogle Scholar
- Kuramoto Y, Tsuzuki T (1976) Persistent propagation of concentration waves in dissipative media far from thermal equilibrium. Prog Theor Phys 55(2):356–369. https://doi.org/10.1143/PTP.55.356 CrossRefGoogle Scholar
- Li CH (1969) Instability of three-layer viscous stratified flow. Phys Fluids (1958–1988) 12(12):2473–2481. https://doi.org/10.1063/1.1692383 CrossRefzbMATHGoogle Scholar
- Lide DR, Haynes WM (eds) (2010) CRC handbook of chemistry and physics, 90th edn. CRC Press, Boca RatonGoogle Scholar
- Majda A, Pego RL (1985) Stable viscosity matrices for systems of conservation laws. J Differ Equ 56(2):229–262. https://doi.org/10.1016/0022-0396(85)90107-X MathSciNetCrossRefzbMATHGoogle Scholar
- Nicolis G, Prigogine I (1977) Self-organization in nonequilibrium systems. Wiley, New YorkzbMATHGoogle Scholar
- Oron A, Davis SH, Bankoff SG (1997) Long-scale evolution of thin liquid films. Rev Mod Phys 69:931–980. https://doi.org/10.1103/RevModPhys.69.931 CrossRefGoogle Scholar
- Papaefthymiou ES, Papageorgiou DT (2017) Nonlinear stability in three-layer channel flows. J Fluid Mech 829:R2MathSciNetCrossRefGoogle Scholar
- Papaefthymiou ES, Papageorgiou DT, Pavliotis GA (2013) Nonlinear interfacial dynamics in stratified multilayer channel flows. J Fluid Mech 734:114–143. https://doi.org/10.1017/jfm.2013.443 MathSciNetCrossRefzbMATHGoogle Scholar
- Reisfeld B, Bankoff SG, Davis SH (1991) The dynamics and stability of thin liquid films during spin coating. I. Films with constant rates of evaporation or absorption. J Appl Phys 70(10):5258–5266. https://doi.org/10.1063/1.350235 CrossRefGoogle Scholar
- Renardy Y (1987) Viscosity and density stratification in vertical Poiseuille flow. Phys Fluids (1958–1988) 30(6):1638–1648. https://doi.org/10.1063/1.866228 CrossRefzbMATHGoogle Scholar
- Saisorn S, Wongwises S (2008) A review of two-phase gas–liquid adiabatic flow characteristics in micro-channels. Renew Sustain Energy Rev 12(3):824–838. https://doi.org/10.1016/j.rser.2006.10.012 CrossRefGoogle Scholar
- Shlang T, Sivashinsky G, Babchin A, Frenkel A (1985) Irregular wavy flow due to viscous stratification. J Phys Paris 46(6):863–866. https://doi.org/10.1051/jphys:01985004606086300 CrossRefGoogle Scholar
- Sivashinsky GI, Michelson DM (1980) On irregular wavy flow of a liquid film down a vertical plane. Prog Theor Phys 63:2112–2114. https://doi.org/10.1143/PTP.63.2112 CrossRefGoogle Scholar
- Taitel Y, Dukler AE (1976) A model for predicting flow regime transitions in horizontal and near horizontal gas-liquid flow. AIChE J 22(1):47–55. https://doi.org/10.1002/aic.690220105 CrossRefGoogle Scholar
- Talimi V, Muzychka Y, Kocabiyik S (2012) A review on numerical studies of slug flow hydrodynamics and heat transfer in microtubes and microchannels. Int J Multiph Flow 39:88–104CrossRefGoogle Scholar
- Triplett K, Ghiaasiaan S, Abdel-Khalik S, Sadowski D (1999) Gas–liquid two-phase flow in microchannels part I: two-phase flow patterns. Int J Multiph Flow 25(3):377–394. https://doi.org/10.1016/S0301-9322(98)00054-8 CrossRefzbMATHGoogle Scholar
- VanHook SJ, Schatz MF, Swift JB, McCormick WD, Swinney HL (1997) Long-wavelength surface-tension-driven Bénard convection: experiment and theory. J Fluid Mech 345:45–78. https://doi.org/10.1017/S0022112097006101 MathSciNetCrossRefzbMATHGoogle Scholar
- Vécsei M, Dietzel M, Hardt S (2014) Coupled self-organization: thermal interaction between two liquid films undergoing long-wavelength instabilities. Phys Rev E 89(053):018. https://doi.org/10.1103/PhysRevE.89.053018 CrossRefGoogle Scholar
- Weisman J (1983) Two-phase flow patterns. In: Cheremisinoff NP, Gupta R (eds) Handbook of fluids in motion, chap 15. Ann Arbor Science Publishers, Ann Arbor, pp 409–425 Google Scholar
- Wong TS, Kang SH, Tang SK, Smythe EJ, Hatton BD, Grinthal A, Aizenberg J (2011) Bioinspired self-repairing slippery surfaces with pressure-stable omniphobicity. Nature 477:443–447. https://doi.org/10.1017/S0022112097006101 CrossRefGoogle Scholar
- Yiantsios SG, Higgins BG (1988) Linear stability of plane poiseuille flow of two superposed fluids. Phys Fluids 31(11):3225–3238MathSciNetCrossRefGoogle Scholar
- Yih CS (1967) Instability due to viscosity stratification. J Fluid Mech 27:337–352. https://doi.org/10.1017/S0022112067000357 CrossRefzbMATHGoogle Scholar