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Instability of an oscillatory fluid layer with insoluble surfactants

Published online by Cambridge University Press:  08 January 2008

PENG GAO  and
XI-YUN LU
PENG GAO
Affiliation:
Department of Modern Mechanics, University of Science and Technology of China, Hefei, Anhui 230026, China
XI-YUN LU*
Affiliation:
Department of Modern Mechanics, University of Science and Technology of China, Hefei, Anhui 230026, China
*
†Author to whom correspondence should be addressed: [email protected]
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Abstract

The linear stability of an infinite fluid layer with a deformable free surface covered by an insoluble surfactant and bounded below by a horizontal rigid plate oscillating in its own plane is studied based on the Floquet theory. The differential system governing the stability problem for perturbations of arbitrary wavenumbers is solved numerically by a Chebyshev collocation method. Stability boundaries are obtained in a wide range of amplitude and frequency of the modulation as well as surfactant elasticity. Results show that the presence of the surfactant may significantly stabilize (destabilize) the flow by raising (lowering) the critical Reynolds number associated with the onset of instability. The effect of the surfactant plays a stabilizing role for small surfactant elasticity and a destabilizing one for relatively large surfactant elasticity. The destabilizing effect of the surfactant on the stability of flows with a zero-shear surface is found for the first time. The disturbance modes in the form of travelling waves may be induced by the surfactant and dominate the instability of the flow.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 595 , 25 January 2008 , pp. 461 - 490
Copyright
Copyright © Cambridge University Press 2008

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References

REFERENCES

Blennerhassett, P. J. & Bassom, A. P. 2002 The linear stability of flat Stokes layers. J. Fluid Mech. 464, 393–410.CrossRefGoogle Scholar
Blennerhassett, P. J. & Bassom, A. P. 2006 The linear stability of high-frequency oscillatory flow in a channel. J. Fluid Mech. 556, 1–25.CrossRefGoogle Scholar
Blyth, M. G. & Pozrikidis, C. 2004 a Effect of surfactant on the stability of film flow down an inclined plane. J. Fluid Mech. 521, 241–250.CrossRefGoogle Scholar
Blyth, M. G. & Pozrikidis, C. 2004 b Effect of surfactants on the stability of two-layer channel flow. J. Fluid Mech. 505, 59–86.CrossRefGoogle Scholar
Canuto, C., Hussaini, M. Y., Quarteroni, A. & Zang, T. A. 1988 Spectral Methods in Fluid Dynamics. 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.CrossRefGoogle Scholar
Coward, A. V. & Papageorgiou, D. T. 1994 Stability of oscillatory two-phase Couette flow. IMA J. Appl. Maths 53, 75–93.CrossRefGoogle Scholar
Coward, A. V. & Renardy, Y. Y. 1997 Small-amplitude oscillatory forcing on two-layer plane channel flow. J. Fluid Mech. 334, 87–109.CrossRefGoogle Scholar
Dolph, C. L. & Lewis, D. C. 1958 On the application of infinite systems of ordinary differential equations to perturbations of plane Poiseuille flow. Q. Appl. Maths 16, 97–110.CrossRefGoogle Scholar
Fornberg, B. 1996 A Practical Guide to Pseudospectral Methods. Cambridge University Press.CrossRefGoogle Scholar
Frenkel, A. L. & Halpern, D. 2002 Stokes-flow instability due to interfacial surfactant. Phys. Fluids 14, L45–L48.CrossRefGoogle Scholar
Frenkel, A. L. & Halpern, D. 2005 Effect of inertia on the insoluble-surfactant instability of a shear flow. Phys. Rev. E 71, 016302.Google ScholarPubMed
Gao, P. & Lu, X.-Y. 2006 a Effect of surfactants on the long-wave stability of oscillatory film flow. J. Fluid Mech. 562, 345–354.CrossRefGoogle Scholar
Gao, P. & Lu, X.-Y. 2006 b Effects of wall suction/injection on the linear stability of flat Stokes layers. J. Fluid Mech. 551, 303–308.CrossRefGoogle Scholar
Gao, P. & Lu, X.-Y. 2006 c Instability of channel flow with oscillatory wall suction/blowing. Phys. Fluids 18, 034102.CrossRefGoogle Scholar
Gao, P. & Lu, X.-Y. 2007 Effect of surfactants on the inertialess instability of a two-layer film flow. J. Fluid Mech. 591, 495–507.CrossRefGoogle Scholar
Halpern, D. & Frenkel, A. L. 2001 Saturated Rayleigh–Taylor instability of an oscillating Couette film flow. J. Fluid Mech. 446, 67–93.CrossRefGoogle Scholar
Halpern, D. & Frenkel, A. L. 2003 Destabilization of a creeping flow by interfacial surfactant: linear theory extended to all wavenumbers. J. Fluid Mech. 485, 191–220.CrossRefGoogle Scholar
Hinch, E. J. 1984 A note on the mechanism of the instability at the interface between two shearing flows. J. Fluid Mech. 144, 463–465.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
von Kerczek, C. & Davis, S. H. 1974 Linear stability theory of oscillatory Stokes layers. J. Fluid Mech. 62, 753–773.CrossRefGoogle Scholar
King, M. R., Leighton, D. T. & McCready, M. J. 1999 Stability of oscillatory two-phase Couette flow: theory and experiment. Phys. Fluids 11, 833–844.CrossRefGoogle Scholar
Li, X. F. & Pozrikidis, C. 1997 The effect of surfactants on drop deformation and on the rheology of dilute emulsions in Stokes flow. J. Fluid Mech. 341, 165–194.CrossRefGoogle Scholar
Lin, S. P. 1970 Stabilizing effects of surface-active agents on a film flow. AIChE J. 16, 375–379.CrossRefGoogle Scholar
Or, A. C. 1997 Finite-wavelength instability in a horizontal liquid layer on an oscillating plane. J. Fluid Mech. 335, 213–232.CrossRefGoogle Scholar
Or, A. C. & Kelly, R. E. 1998 Thermocapillary and oscillatory-shear instabilities in a layer of liquid with a deformable surface. J. Fluid Mech. 360, 21–39.CrossRefGoogle Scholar
Ostrach, S. 1982 Low-gravity fluid flows. Annu. Rev. Fluid. Mech. 14, 313–345.CrossRefGoogle Scholar
Phillips, W. R. C. 2001 On an instability to Langmuir circulations and the role of Prandtl and Richardson numbers. J. Fluid Mech. 442, 335–358.CrossRefGoogle Scholar
Pozrikidis, C. 2003 Effect of surfactants on film flow down a periodic wall. J. Fluid Mech. 496, 105–127.CrossRefGoogle Scholar
Sarpkaya, T. 1996 Vorticity, free surface, and surfactants. Annu. Rev. Fluid. Mech. 28, 83–128.CrossRefGoogle Scholar
Shen, L., Yue, D. K. P. & Triantafyllou, G. S. 2004 Effect of surfactants on free-surface turbulent flows. J. Fluid Mech. 506, 79–115.CrossRefGoogle Scholar
Skarda, J. R. L. 2001 Instability of a gravity-modulated fluid layer with surface tension variation. J. Fluid Mech. 434, 243–271.CrossRefGoogle Scholar
Stone, H. A. 1990 A simple derivation of the time-dependent convection–diffusion equation for surfactant transport along a deforming interface. Phys. Fluids A 2, 111–112.CrossRefGoogle Scholar
Thiele, U., Vega, M. & Knobloch, E. 2006 Long-wave Marangoni instability with vibration. J. Fluid Mech. 546, 61–87.CrossRefGoogle Scholar
Wei, H.-H. 2005 a Effect of surfactant on the long-wave instability of a shear-imposed liquid flow down an inclined plane. Phys. Fluids 17, 012103.CrossRefGoogle Scholar
Wei, H.-H. 2005 b On the flow-induced Marangoni instability due to the presence of surfactant. J. Fluid Mech. 544, 173–200.CrossRefGoogle Scholar
Wei, H.-H., Halpern, D. & Grotberg, J. B. 2005 Linear stability of a surfactant-laden annular film in a time-periodic pressure-driven flow through a capillary. J. Colloid Interface Sci. 285, 769–780.CrossRefGoogle Scholar
Weideman, J. A. C. & Reddy, S. C. 2000 A matlab differentiation matrix suite. ACM Trans. Math. Software 26, 465–519.CrossRefGoogle Scholar
Whitaker, S. & Jones, L. O. 1966 Stability of falling liquid films. Effect of interface and interfacial mass transport. AIChE J. 12, 421–431.CrossRefGoogle Scholar
Wong, H., Rumschitzki, D. & Maldarelli, C. 1996 On the surfactant mass balance at a deforming fluid interface. Phys. Fluids 8, 3203–3204.CrossRefGoogle Scholar
Yih, C. S. 1963 Stability of liquid flow down an inclined plane. Phys. Fluids 6, 321–334.CrossRefGoogle Scholar
Yih, C. S. 1968 Instability of unsteady flows or configurations. Part 1. Instability of a horizontal liquid layer on an oscillating plane. J. Fluid Mech. 31, 737–751.CrossRefGoogle Scholar