Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-20T03:27:37.716Z Has data issue: false hasContentIssue false
  • English
  • Français

Nonlinear dynamics of core-annular film flows in the presence of surfactant

Published online by Cambridge University Press:  10 May 2009

S. A. KAS-DANOUCHE ,
D. T. PAPAGEORGIOU  and
M. SIEGEL
S. A. KAS-DANOUCHE
Affiliation:
Departmento de Matemáticas, Universidad de Oriente, Cumaná, Sucre 6101, Venezuela
D. T. PAPAGEORGIOU*
Affiliation:
Department of Mathematical Sciences and Center for Applied Mathematics and Statistics, New Jersey Institute of Technology, Newark, NJ 07102, USA Department of Mathematics, Imperial College London, London SW7 2AZ, UK
M. SIEGEL
Affiliation:
Department of Mathematical Sciences and Center for Applied Mathematics and Statistics, New Jersey Institute of Technology, Newark, NJ 07102, USA
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The nonlinear stability of two-phase core-annular flow in a cylindrical pipe is studied. A constant pressure gradient drives the flow of two immiscible liquids of different viscosities and equal densities, and surface tension acts at the interface separating the phases. Insoluble surfactants are included, and we assess their effect on the flow stability and ensuing spatio-temporal dynamics. We achieve this by developing an asymptotic analysis in the limit of a thin annular layer – which is usually the relevant regime in applications – to derive a coupled system of nonlinear evolution equations that govern the dynamics of the interface and the local surfactant concentration on it. In the absence of surfactants the system reduces to the Kuramoto–Sivashinsky (KS) equation, and its modifications due to viscosity stratification (present when the phases have unequal viscosities) are derived elsewhere. We report on extensive numerical experiments to evaluate the effect of surfactants on KS dynamics (including chaotic states, for example), in both the absence and the presence of viscosity stratification. We find that chaos is suppressed in the absence of viscosity differences and that the new flow consists of successive windows (in parameter space) of steady-state travelling waves separated by time-periodic attractors. The intricate structure of the travelling pulses is also explained physically. When viscosity stratification is present we observe a transition from time-periodic dynamics, for instance, to steady-state travelling wave pulses of increasing amplitudes and speeds. Numerical evidence is presented that indicates that the transition occurs through a reverse Feigenbaum cascade in phase space.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 626 , 10 May 2009 , pp. 415 - 448
Copyright
Copyright © Cambridge University Press 2009

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

Aul, R. W. & Olbricht, W. L. 1990 Stability of thin annular film in pressure-driven, low-Reynolds-number flow through a capillary. J. Fluid Mech. 215, 585–599.CrossRefGoogle Scholar
Blyth, M. G., Hall, P. & Papageorgiou, D. T. 2003 Chaotic flows in pulsating cylindrical tubes: a class of exact solutions of the Navier–Stokes equations. J. Fluid Mech. 481, 187–213.CrossRefGoogle Scholar
Blyth, M. G., Luo, H. & Pozrikidis, C. 2006 Stability of axisymmetric core-annular flow in the presence of an insoluble surfactant. J. Fluid Mech. 548, 207–235.CrossRefGoogle Scholar
Blyth, M. G. & Pozrikidis, C. 2004 Effect of surfactants on the stability of two-layer channel flow. J. Fluid Mech. 505, 59–86.CrossRefGoogle Scholar
Chang, H.-C. & Franses, E. I. 1995 Adsorption dynamics of surfactants at the air/water interface: a critical review of mathematical models, data, and mechanisms. Colloids and Surfaces A 100, 1–45.CrossRefGoogle Scholar
Charru, F. & Hinch, E. J. 2000 ‘Phase diagram’ of interfacial instabilities in two-layer Couette flow and mechanism of the long-wave instability. J. Fluid Mech. 414, 195–223.CrossRefGoogle Scholar
Collet, P. & Eckmann, J.-P. 1980 Iterated Maps on the Interval as Dynamical Systems. Birkhauser.Google Scholar
Craster, R. V., Matar, O. K. & Papageorgiou, D. T. 2002 Pinchoff and satellite formation in surfactant covered viscous threads. Phys. Fluids 14, 1364–1376.CrossRefGoogle Scholar
Feigenbaum, M. J. 1978 Quantitive universality for a class of nonlinear transformations. J. Stat. Phys. 19, 25–52.CrossRefGoogle Scholar
Feigenbaum, M. J. 1983 Universal behavior in non-linear systems. Physica D 7, 16–39.CrossRefGoogle Scholar
Frenkel, A. L., Babchin, A. J., Levich, B. G., Shlang, T. & Sivashinsky, G. I. 1987 Annular flows can keep unstable films from breakup: nonlinear saturation of capillary instability. J. Colloid Interface Sci. 115, 225.CrossRefGoogle Scholar
Frenkel, A. L. & Halpern, D. 2002 Stokes-flow instability due to interfacial surfactant. Phys. Fluids 14, L45–L48.CrossRefGoogle Scholar
Georgiou, E., Maldarelli, C., Papageorgiou, D. T. & Rumschitzki, D. S. 1992 An asymptotic theory for the linear stability of core-annular flow in the thin annular limit. J. Fluid Mech. 243, 653–677.CrossRefGoogle Scholar
Goren, S. L. 1962 The instability of annular thread of fluid. J. Fluid Mech. 12, 309–319.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
Halpern, D. & Grotberg, J. B. 1993 Surfactant effects on fluid–elastic instabilities of liquid-lined flexible tubes: a model of airway closure. J. Biomech. Engng 115, 271–277.CrossRefGoogle Scholar
Hammond, P. S. 1983 Nonlinear adjustment of a thin annular film of viscous fluid surrounding a thread of another within a circular cylinder pipe. J. Fluid Mech. 137, 363–384.CrossRefGoogle Scholar
Hansen, S., Peters, G. W. M. & Meijer, H. E. H. 1999 The effect of surfactant on the stability of a fluid filament embedded in a viscous fluid. J. Fluid Mech. 382, 331–349.CrossRefGoogle Scholar
Hickox, C. E. 1971 Stability due to viscosity and density stratification in axisymmetric pipe flow. Phys. Fluids 14, 251.CrossRefGoogle Scholar
Joseph, D. D. & Renardy, Y. Y. 1993 Fundamentals of Two-Fluid Dynamics. Part 2. Lubricated Transport, Drops and Miscible Liquids. Springer.Google Scholar
Kas-Danouche, S. 2002 Nonlinear Interfacial Stability in Core-Annular Films Flows in the Presence of Surfactants. New Jersey Institute of Technology.Google Scholar
Kerchman, V. 1995 Strongly nonlinear interfacial dynamics in core-annular flows. J. Fluid Mech. 290, 131–166.CrossRefGoogle Scholar
Kwak, S., Fyrillas, M. M. & Pozrikidis, C. 2001 Effect of surfactants on the instability of a liquid thread. Part 2. Extensional flow. Intl J. Multiphase Flow 1–22.CrossRefGoogle Scholar
Kwak, S. & Pozrikidis, C. 2001 Effect of surfactants on the instability of a liquid thread or annular layer. Part 1. Quiescent fluids. Intl J. Multiphase Flow 27, 1–37.CrossRefGoogle Scholar
Newhouse, L. A. & Pozrikidis, C. 1992 The capillary instability of annular layers and liquid threads. J. Fluid Mech. 242, 193–209.CrossRefGoogle Scholar
Otis, D. R. Jr., Johnson, M., Pedley, T. J. & Kamm, R. D. 1990 The effect of surfactant on liquid film stability in the peripheral airways. Adv. Bioengng 17, 55–57.Google Scholar
Otis, D. R. Jr., Johnson, M., Pedley, T. J. & Kamm, R. D. 1993 Role of pulmonary surfactant airway closure: a computational study. J. Appl. Physiol. 75, 1323–1333.CrossRefGoogle ScholarPubMed
Papageorgiou, D. T., Maldarelli, C. & Rumschitzki, D. S. 1990 Nonlinear interfacial stability of core-annular film flows. Phys. Fluids A 2 (3), 340–352.CrossRefGoogle Scholar
Papageorgiou, D. T. & Smyrlis, Y. S. 1991 The route to chaos for the Kuramoto–Sivashinsky equation. Theoret. Comput. Fluid Dyn. 3, 15–42.CrossRefGoogle Scholar
Park, C. W. & Homsy, G. M. 1984 Two-phase displacement in hele-shaw cells: theory. J. Fluid Mech. 141, 257–287.CrossRefGoogle Scholar
Preziosi, L., Chen, K. & Joseph, D. D. 1989 Lubricated pipelining: stability of core-annular flow. J. Fluid Mech. 201, 323–356.CrossRefGoogle Scholar
Slattery, J. C. 1974 Interfacial effects in the entrapment and displacement of residual oil. AIChE J. 20, 1145–1154.CrossRefGoogle Scholar
Smyrlis, Y. S. & Papageorgiou, D. T. 1991 Predicting chaos for infinite dimensional dynamical systems: the Kuramoto–Sivashinsky equation, a case of study. Proc. Natl Acad. Sci. USA 88, 11129–11132.CrossRefGoogle Scholar
Smyrlis, Y. S. & Papageorgiou, D. T. 1996 Computational study of chaotic and ordered solutions of the Kuramoto–Sivashinsky equation. Tech Rep. 96-12. Institute for Computer Applications in Science and Engineering, NASA Langley Research Center.Google Scholar
Smyrlis, Y. S. & Papageorgiou, D. T. 1998 The effects of generalized dispersion on dissipative dynamical systems. Appl. Math. Lett. 11 (6), 93–99.CrossRefGoogle Scholar
Timmermans, M. L. E. & Lister, J. R. 2002 The effect of surfactant on the stability of a liquid thread. J. Fluid Mech. 459, 289–306.CrossRefGoogle Scholar
Tomotika, S. 1935 On the instability of a cylindrical thread of a viscous liquid surrounded by another viscous liquid. Proc. R. Soc. Lond. A 150, 322–337.Google Scholar
Wei, H. H. & Rumschitzki, D. S. 2005 The effects of insoluble surfactants on the linear stability of core-annular flow. J. Fluid Mech. 541, 115–142.CrossRefGoogle Scholar
Wong, H., Rumschitzki, D. & Maldarelli, C. 1996 On the surfactant mass balance at a deforming fluid interface. Phys. Fluids 8 (11), 3203–3204.CrossRefGoogle Scholar
Yih, C. S. 1967 Instability due to viscosity stratification. J. Fluid Mech. 27, 337–352.CrossRefGoogle Scholar