Hostname: page-component-7dd5485656-c8zdz Total loading time: 0 Render date: 2025-10-31T13:58:44.325Z Has data issue: false hasContentIssue false
  • English
  • Français

Linear stability of two-layer Couette flows

Published online by Cambridge University Press:  04 August 2017

Alireza Mohammadi Open the ORCID record for Alireza Mohammadi [Opens in a new window]  and
Alexander J. Smits
Alireza Mohammadi*
Affiliation:
Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA
Alexander J. Smits
Affiliation:
Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA Mechanical and Aerospace Engineering, Monash University, Monash, VIC 3800, Australia
*
†Email address for correspondence: alirezam@princeton.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

The stability of two-layer Couette flow is investigated under variations in viscosity ratio, thickness ratio, interfacial tension and density ratio. The effects of the base flow on eigenvalue spectra are explained. A new type of interfacial mode is discovered at low viscosity ratio with properties that are different from Yih’s original interfacial mode (Yih, J. Fluid Mech., vol. 27, 1967, pp. 337–352). No unstable Tollmien–Schlichting waves were found over the range of parameters considered in this work. The results for thin films with different thicknesses can be collapsed onto a single curve if the Reynolds number and wavenumber are suitably defined based on the parameters of the thin layer. Interfacial tension always has a stabilizing effect, but the effects of density ratio cannot be so easily generalized. Neutral stability curves for water–alkane and water–air systems are presented as an initial step towards better understanding the effects of flow stability on the longevity and performance of liquid-infused surfaces and superhydrophobic surfaces.

JFM classification

Waves/Free-surface Flows: Channel flow Instability: Instability Multiphase and Particle-laden Flows: Multiphase flow

Information

Type
Papers
Information
Journal of Fluid Mechanics , Volume 826 , 10 September 2017 , pp. 128 - 157
Check for updates
Copyright
© 2017 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.)

Article purchase

Temporarily unavailable

References

Barthelet, P., Charru, F. & Fabre, J. 1995 Experimental study of interfacial long waves in a two-layer shear flow. J. Fluid Mech. 303, 23–53.CrossRefGoogle Scholar
Boomkamp, P. A. M. & Miesen, R. H. M. 1996 Classification of instabilities in parallel two-phase flow. Intl J. Multiphase Flow 22, 67–88.Google Scholar
Busse, A., Sandham, N. D., McHale, G. & Newton, M. I. 2013 Change in drag, apparent slip and optimum air layer thickness for laminar flow over an idealised superhydrophobic surface. J. Fluid Mech. 727, 488–508.CrossRefGoogle Scholar
Canuto, C., Hussaini, M. Y., Quarteroni, A. & Zang, T. A. 2006 Spectral Methods: Fundamentals in Single Domains. Springer.CrossRefGoogle Scholar
Charru, F. & Hinch, E. J. 2000 Phase diagram of interfacial instabilities in a two-layer Couette flow and mechanism of the long-wave instability. J. Fluid Mech. 414, 195–223.Google Scholar
Craster, R. V. & Matar, O. K. 2009 Dynamics and stability of thin liquid films. Rev. Mod. Phys. 81 (3), 1131–1198.CrossRefGoogle Scholar
Floryan, J. M. 2002 Centrifugal instability of Couette flow over a wavy wall. Phys. Fluids 14 (1), 312–322.Google Scholar
Govindarajan, R. & Sahu, K. C. 2014 Instabilities in viscosity-stratified flow. Annu. Rev. Fluid Mech. 46 (1), 331–353.Google Scholar
Hinch, E. J. 1984 A note on the mechanism of the instability at the interface between two shearing fluids. J. Fluid Mech. 144, 463–465.CrossRefGoogle Scholar
Hooper, A. P. 1985 Long-wave instability at the interface between two viscous fluids: thin layer effects. Phys. Fluids 28 (6), 1613–1618.Google Scholar
Hooper, A. P. & Boyd, W. G. C. 1983 Shear-flow instability at the interface between two viscous fluids. J. Fluid Mech. 128, 507–528.CrossRefGoogle Scholar
Hooper, A. P. & Boyd, W. G. C. 1987 Shear-flow instability due to a wall and a viscosity discontinuity at the interface. J. Fluid Mech. 179, 201–225.Google Scholar
Jacobi, I., Wexler, J. S. & Stone, H. A. 2015 Overflow cascades in liquid-infused substrates. Phys. Fluids 27 (8), 082101.Google Scholar
Joseph, D. D., Bai, R., Chen, K. P. & Renardy, Y. Y. 1997 Core–annular flows. Annu. Rev. Fluid Mech. 29 (1), 65–90.CrossRefGoogle Scholar
Joseph, D. D. & Renardy, Y. Y. 1993 Fundamentals of Two-Fluid Dynamics. Springer.Google Scholar
Kaffel, A. & Riaz, A. 2015 Eigenspectra and mode coalescence of temporal instability in two-phase channel flow. Phys. Fluids 27 (4), 042101.Google Scholar
Malik, S. V. & Hooper, A. P. 2007 Three-dimensional disturbances in channel flows. Phys. Fluids 19 (5), 052102.CrossRefGoogle Scholar
Mason, J. C. & Handscomb, D. C. 2002 Chebyshev Polynomials. CRC Press.Google Scholar
Mohammadi, A.2013 Flows in grooved channels. PhD thesis, University of Western Ontario, London, Ontario, Canada.Google Scholar
Mohammadi, A. & Smits, A. J. 2016 Stability of two-immiscible-fluid systems: a review of canonical plane parallel flows. Trans. ASME J. Fluids Engng 138 (10), 100803.CrossRefGoogle Scholar
Özgen, S., Degrez, G. & Sarma, G. S. R. 1998 Two-fluid boundary layer stability. Phys. Fluids 10 (11), 2746–2757.CrossRefGoogle Scholar
Redapangu, P. R., Sahu, K. C. & Vanka, S. P. 2013 A lattice Boltzmann simulation of three-dimensional displacement flow of two immiscible liquids in a square duct. Trans. ASME J. Fluids Engng 135 (12), 121202.Google Scholar
Rosenberg, B. J., Van Buren, T., Fu, M. K. & Smits, A. J. 2016 Turbulent drag reduction over air- and liquid-impregnated surfaces. Phys. Fluids 28 (1), 015103.Google Scholar
Rothstein, J. P. 2010 Slip on superhydrophobic surfaces. Annu. Rev. Fluid Mech. 42 (1), 89–109.Google Scholar
Saffman, P. G. & Taylor, G. 1958 The penetration of a fluid into a porous medium or a Hele-Shaw cell containing a more viscous liquid. Proc. R. Soc. Lond. A 245 (1242), 312–329.Google Scholar
Schmid, P. J. & Henningson, D. S. 2001 Stability and Transition in Shear Flows. Springer.Google Scholar
Schrenk, W. J. & Alfrey, T. Jr. 1978 Coextruded multilayer polymer films and sheets. In Polymer Blends (ed. Paul, D. R. & Newman, S.), vol. 2, chap. 15, p. 129. Academic.Google Scholar
Solomon, B. R., Khalil, K. S. & Varanasi, K. K. 2014 Drag reduction using lubricant-impregnated surfaces in viscous laminar flow. Langmuir 30 (36), 10970–10976.CrossRefGoogle ScholarPubMed
Squire, H. B. 1933 On the stability for three-dimensional disturbances of viscous fluid flow between parallel walls. Proc. R. Soc. Lond. A 142 (847), 621–628.Google Scholar
Timoshin, S. N. 1997 Instabilities in a high Reynolds-number boundary-layer on a film coated surface. J. Fluid Mech. 353, 163–195.Google Scholar
Timoshin, S. N. & Hooper, A. P. 2000 Mode coalescence in a two-fluid boundary-layer stability problem. Phys. Fluids 12 (8), 1969–1978.Google Scholar
Weinstein, S. J. & Ruschak, K. J. 2004 Coating flows. Annu. Rev. Fluid Mech. 36 (1), 29–53.Google Scholar
Wexler, J. S., Grosskopf, A., Chow, M., Fan, Y., Jacobi, I. & Stone, H. A. 2015a Robust liquid-infused surfaces through patterned wettability. Soft Matt. 11 (25), 5023–5029.Google Scholar
Wexler, J. S., Jacobi, I. & Stone, H. A. 2015b Shear-driven failure of liquid-infused surfaces. Phys. Rev. Lett. 114 (16), 168301.CrossRefGoogle ScholarPubMed
Wong, T.-S., Kang, S. H., Tang, S. K. Y., Smythe, E. J., Hatton, B. D., Grinthal, A. & Aizenberg, J. 2011 Bioinspired self-repairing slippery surfaces with pressure-stable omniphobicity. Nature 477 (7365), 443–447.CrossRefGoogle ScholarPubMed
Yiantsios, S. G. & Higgins, B. G. 1988 Linear stability of plane Poiseuille flow of two superposed fluids. Phys. Fluids 31 (11), 3225–3238.Google Scholar
Yih, C.-S. 1963 Stability of liquid flow down an inclined plane. Phys. Fluids 6 (3), 321–334.Google Scholar
Yih, C.-S. 1967 Instability due to viscosity stratification. J. Fluid Mech. 27 (02), 337–352.Google Scholar
Yih, C.-S. 1990 Wave formation on a liquid layer for de-icing airplane wings. J. Fluid Mech. 212, 41–53.CrossRefGoogle Scholar