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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: d.papageorgiou@imperial.ac.uk
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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

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References

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