Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-23T19:53:56.111Z Has data issue: false hasContentIssue false
  • English
  • Français

Role of odd viscosity in falling viscous fluid

Published online by Cambridge University Press:  09 March 2022

Arghya Samanta Open the ORCID record for Arghya Samanta [Opens in a new window]
Arghya Samanta*
Affiliation:
Department of Applied Mechanics, Indian Institute of Technology Delhi, Hauz Khas, New Delhi110016, India
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The aim of the present study is to investigate the linear and nonlinear wave dynamics of a falling incompressible viscous fluid when the fluid undergoes an effect of odd viscosity. In fact, such an effect arises in classical fluids when the time-reversal symmetry is broken. The motivation to study this dynamics was raised by recent studies (Ganeshan & Abanov, Phys. Rev. Fluids, vol. 2, 2017, p. 094101; Kirkinis & Andreev, J. Fluid Mech., vol. 878, 2019, pp. 169–189) where the odd viscosity coefficient suppresses thermocapillary instability. Here, we explore the linear surface wave and shear wave dynamics for the isothermal case by solving the Orr–Sommerfeld eigenvalue problem numerically with the aid of the Chebyshev spectral collocation method. It is found that surface and shear instabilities can be weakened by the odd viscosity coefficient. Furthermore, the growth rate of the wavepacket corresponding to the linear spatio-temporal response is reduced as long as the odd viscosity coefficient increases. In addition, a coupled system of a two-equation model is derived in terms of the fluid layer thickness $h(x,t)$ and the flow rate $q(x,t)$. The nonlinear travelling wave solution of the two-equation model reveals the attenuation of maximum amplitude and speed in the presence of an odd viscosity coefficient, which ensures the delay of transition from the primary parallel flow with a flat surface to secondary flow generated through the nonlinear wave interactions. This physical phenomenon is further corroborated by performing a nonlinear spatio-temporal simulation when a harmonic forcing is applied at the inlet.

JFM classification

Instability: Nonlinear instability Instability: Shear-flow instability Instability: Transition to turbulence
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 938 , 10 May 2022 , A9
Check for updates
Copyright
© The Author(s), 2022. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Alekseenko, S.V., Nakoryakov, V.E. & Pokusaev, B.G. 1985 Wave formation on a vertical falling liquid film. AIChE J. 31, 1446–1460.CrossRefGoogle Scholar
Avron, J.E. 1998 Odd viscosity. J. Stat. Phys. 92, 543–557.CrossRefGoogle Scholar
Avron, J.E., Seller, R. & Zograf, P. 1995 Viscosity of quantum Hall fluids. Phys. Rev. Lett. 75, 697–700.CrossRefGoogle ScholarPubMed
Balmforth, N.J. 1995 Solitary waves and homoclinic orbits. Annu. Rev. Fluid Mech. 27, 335–373.CrossRefGoogle Scholar
Banerjee, D., Souslov, A., Abanov, A.G. & Vitelli, V. 2017 Odd viscosity in chiral active fluids. Nat. Commun. 8, 1573–1584.CrossRefGoogle ScholarPubMed
Batchelor, G.K. 1993 An Introduction to Fluid Dynamics. Cambridge University Press.Google Scholar
Benjamin, T.B. 1957 Wave formation in laminar flow down an inclined plane. J. Fluid Mech. 2, 554–573.CrossRefGoogle Scholar
Bhat, F.A. & Samanta, A. 2019 Linear stability analysis of a surfactant-laden shear-imposed falling film. Phys. Fluids 31, 054103.CrossRefGoogle Scholar
Boyd, J.P. 2000 Chebyshev and Fourier Spectral Methods. Dover.Google Scholar
Brevdo, L., Laure, P., Dias, F. & Bridges, T.J. 1999 Linear pulse structure and signalling in a film flow on an inclined plane. J. Fluid Mech. 396, 37–71.CrossRefGoogle Scholar
Bruin, G.D. 1974 Stability of a layer of liquid flowing down an inclined plane. J. Engng Maths 8, 259–270.CrossRefGoogle Scholar
Chang, H.C. 1994 Wave evolution on a falling film. Annu. Rev. Fluid Mech. 42, 1553–1568.Google Scholar
Chang, H.C., Demekhin, E.A. & Kopelvitch, D.I. 1993 Nonlinear evolution of waves on a vertically falling film. J. Fluid Mech. 250, 433–480.CrossRefGoogle Scholar
Chang, H.C., Demekhin, E.A. & Kopelvitch, D.I. 1995 Interaction dynamics of a solitary waves on a falling film. J. Fluid Mech. 294, 123–154.CrossRefGoogle Scholar
Chattopadhyay, S. 2021 Influence of the odd viscosity on a falling film down a slippery inclined plane. Phys. Fluids 33, 062106.CrossRefGoogle Scholar
Chin, R., Abernath, F. & Bertschy, J. 1986 Gravity and shear wave stability of free surface flows. Part 1. Numerical calculations. J. Fluid Mech. 168, 501–513.CrossRefGoogle Scholar
Dietze, G.F., Leefken, A. & Kneer, R. 2008 Investigation of the backflow phenomenon in falling liquid films. J. Fluid Mech. 595, 435–459.CrossRefGoogle Scholar
Doedel, E.J., Champneys, A.R., Fairgrieve, T.F., Kuznetsov, Y.A., Sandstede, B. & Wang, X.-J. 2007 Auto07: continuation and bifurcation software for ordinary differential equations. Tech. Rep. Department of Computer Science, Concordia University, Montreal, Canada.Google Scholar
Finlayson, B.A. 1972 The Method of Weighted Residuals and Variational Principles, with Application in Fluid Mechanics. Academic Press.Google Scholar
Floryan, J.M., Davis, S.H. & Kelly, R.E. 1987 Instabilities of a liquid film flowing down a slightly inclined plane. Phys. Fluids 30, 983–989.CrossRefGoogle Scholar
Ganeshan, S. & Abanov, A.G. 2017 Odd viscosity in two-dimensional incompressible fluids. Phys. Rev. Fluids 2, 094101.CrossRefGoogle Scholar
Gao, D., Morley, N. & Dhir, V. 2003 Numerical simulation of wavy falling film flow using VOF method. J. Comput. Phys. 192, 624–642.CrossRefGoogle Scholar
Huerre, P. 2000 Open Shear Flow Instability. Cambridge University Press.Google Scholar
Huerre, P. & Roosi, M. 1998 Hydrodynamic Instability in Open Flows. Cambridge University Press.CrossRefGoogle Scholar
Indireshkumar, K. & Frenkel, A.L. 1997 Mutually penetrating motion of self-organized two-dimensional patterns of solitonlike structures. Phys. Rev. E 55, 1174–1177.CrossRefGoogle Scholar
Kalliadasis, S., Ruyer-Quil, C., Scheid, B. & Velarde, M. 2012 Falling Liquid Films, 1st edn. Springer.CrossRefGoogle Scholar
Kelly, R.E., Goussis, D.A., Lin, S.P. & Hsu, F.K. 1989 The mechanism for surface wave instability in film flow down an inclined plane. Phys. Fluids A 1, 819–828.CrossRefGoogle Scholar
Kirkinis, E. & Andreev, A.V. 2019 Odd-viscosity-induced stabilization of viscous thin liquid films. J. Fluid Mech. 878, 169–189.CrossRefGoogle Scholar
Landau, L.D. & Lifshitz, E.M. 1959 Fluid Mechanics. Pergamon.Google Scholar
Lapa, M.F. & Hughes, T.L. 2014 Swimming at low Reynolds number in fluids with odd, or Hall, viscosity. Phys. Rev. E 89, 043019.CrossRefGoogle ScholarPubMed
Lin, S.P. 1967 Instability of a liquid film flowing down an inclined plane. Phys. Fluids 10, 308–313.CrossRefGoogle Scholar
Liu, J. & Gollub, J.P. 1994 Solitary wave dynamics of film flows. Phys. Fluids 6, 1702–1712.CrossRefGoogle Scholar
Luchini, P. & Charru, F. 2010 Consistent section-averaged equations of quasi-one-dimensional laminar flow. J. Fluid Mech. 656, 337–341.CrossRefGoogle Scholar
Malamataris, N.A., Vlachogiannis, M. & Bontozoglou, V. 2002 Solitary waves on inclined films: flow structure and binary interactions. Phys. Fluids 14, 1082–1094.CrossRefGoogle Scholar
Mudunuri, R.R. & Balakotaiah, V. 2006 Solitary waves on thin falling films in the very low forcing frequency limit. AIChE J. 52, 3995–4003.CrossRefGoogle Scholar
Mukhopadhyay, S. & Mukhopadhyay, A. 2021 Hydrodynamic instability and wave formation of a viscous film flowing down a slippery inclined substrate: effect of odd-viscosity. Eur. J. Mech. (B/Fluids) 89, 161–170.CrossRefGoogle Scholar
Nakaya, C. 1989 Waves on a viscous fluid film down a vertical wall. Phys. Fluids 1, 1143–1154.CrossRefGoogle Scholar
Nguyen, L.T. & Balakotaiah, V. 2000 Modeling and experimental studies of wave evolution on free falling viscous films. Phys. Fluids 12, 2236–2256.CrossRefGoogle Scholar
Onsager, L. 1931 Reciprocal relations in irreversible processes. Phys. Rev. 37, 405–426.CrossRefGoogle Scholar
Ooshida, T. 1999 Surface equation of falling film flows with moderate Reynolds number and large but finite Weber number. Phys. Fluids 11, 3247–3269.Google Scholar
Oron, A., Davis, S.H. & Bankoff, S.G. 1997 Long scale evolution of thin films. Rev. Mod. Phys. 69, 931–980.CrossRefGoogle Scholar
Oron, A. & Gottlieb, O. 2004 Subcritical and supercritical bifurcations of the first- and second- order benney equations. J. Engng Maths 50, 121–140.CrossRefGoogle Scholar
Orszag, S.A. 1971 Accurate solution of the Orr–Sommerfeld stability equation. J. Fluid Mech. 50, 689–703.CrossRefGoogle Scholar
Pradas, M., Tseluiko, D. & Kalliadasis, S. 2011 Rigorous coherent structure theory for falling liquid films: viscous dispersion effects on bound-state formation and self-organization. Phys. Fluids 23, 044104.CrossRefGoogle Scholar
Pumir, A., Manneville, P. & Pomeau, Y. 1983 On solitary waves running down an inclined plane. J. Fluid Mech. 135, 27–50.CrossRefGoogle Scholar
Ramaswamy, B., Chippada, S. & Joo, S.W. 1996 A full-scale numerical study of interfacial instabilities in thin-film flows. J. Fluid Mech. 325, 163–194.CrossRefGoogle Scholar
Richtmeyer, R.D. & Morton, K.W. 1967 Difference Methods for Initial Value Problems. Interscience.Google Scholar
Ruyer-Quil, C. & Manneville, P. 2000 Improved modeling of flows down inclined planes. Eur. Phys. J. B 15, 357–369.CrossRefGoogle Scholar
Ruyer-Quil, C. & Manneville, P. 2005 On the speed of solitary waves running down a vertical wall. J. Fluid Mech. 531, 181–190.CrossRefGoogle Scholar
Ruyer-Quil, C., Trevelyan, P.M.J., Giorgiutti-Dauphiné, F., Duprat, C. & Kalliadasis, S. 2008 Modelling film flows down a fiber. J. Fluid Mech. 603, 431–462.CrossRefGoogle Scholar
Salamon, T.R., Armstrong, R.C. & Brown, R.A. 1994 Traveling waves on vertical films: numerical analysis using the finite element method. Phys. Fluids 6, 2202–2220.CrossRefGoogle Scholar
Samanta, A. 2014 Shear-imposed falling film. J. Fluid Mech. 753, 131–149.CrossRefGoogle Scholar
Samanta, A. 2016 Spatiotemporal instability of an electrified falling film. Phys. Rev. E 93, 013125.CrossRefGoogle ScholarPubMed
Samanta, A. 2020 Linear stability of a plane Couette–Poiseuille flow overlying a porous layer. Intl J. Multiphase Flow 123, 103160.CrossRefGoogle Scholar
Samanta, A., Goyeau, B. & Ruyer-Quil, C. 2013 A falling film on a porous medium. J. Fluid Mech. 716, 414–444.CrossRefGoogle Scholar
Samanta, A., Ruyer-Quil, C. & Goyeau, B. 2011 Falling film down a slippery inclined plane. J. Fluid Mech. 684, 353–383.CrossRefGoogle Scholar
Schmid, P. & Henningson, D. 2001 Stability and Transition in Shear Flows. Springer.CrossRefGoogle Scholar
Shkadov, V.Y. 1967 Wave flow regimes of a thin layer of viscous fluid subject to gravity. Izv. Akad. Nauk SSSR Mekh. Zhidk. Gaza 1, 43–51.Google Scholar
Smith, M.K. 1990 The mechanism for the long-wave instability in thin liquid films. J. Fluid Mech. 217, 469–485.CrossRefGoogle Scholar
Soni, V., Bililign, E.S., Magkiriadou, S., Sacanna, S., Bartolo, D., Shelley, M.J. & Irvine, W.T.M. 2019 The odd free surface flows of a colloidal chiral fluid. Nat. Phys. 15, 1188–1194.CrossRefGoogle Scholar
Tan, M.J., Bankoff, S.G. & Davis, S.H. 1990 Steady thermocapillary flows of thin liquid layers. Phys. Fluids 2, 313–321.CrossRefGoogle Scholar
Vlachogiannis, M. & Bontozoglou, V. 2001 Observations of solitary wave dynamics of lm flows. J. Fluid Mech. 435, 191–215.CrossRefGoogle Scholar
Yih, C.S. 1963 Stability of liquid flow down an inclined plane. Phys. Fluids 6, 321–334.CrossRefGoogle Scholar
Zhao, J. & Jian, Y. 2021 a Effect of odd viscosity on the stability of a falling thin film in presence of electromagnetic field. Fluid Dyn. Res. 53, 015510.CrossRefGoogle Scholar
Zhao, J. & Jian, Y. 2021 b Effect of odd viscosity on the stability of thin viscoelastic liquid film flowing along an inclined plate. Phys. Scr. 96, 055214.CrossRefGoogle Scholar