Hostname: page-component-8448b6f56d-c47g7 Total loading time: 0 Render date: 2024-04-16T21:43:21.161Z Has data issue: false hasContentIssue false
  • English
  • Français

Nonlinear interfacial dynamics in stratified multilayer channel flows

Published online by Cambridge University Press:  08 October 2013

E. S. Papaefthymiou ,
D. T. Papageorgiou  and
G. A. Pavliotis
E. S. Papaefthymiou
Affiliation:
Department of Mathematics, Imperial College London, London SW7 2AZ, UK
D. T. Papageorgiou*
Affiliation:
Department of Mathematics, Imperial College London, London SW7 2AZ, UK
G. A. Pavliotis
Affiliation:
Department of Mathematics, Imperial College London, London SW7 2AZ, UK
*
†Email address for correspondence: d.papageorgiou@imperial.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

The dynamics of viscous immiscible pressure-driven multilayer flows in channels are investigated using a combination of modelling, analysis and numerical computations. More specifically, the particular system of three stratified layers with two internal fluid–fluid interfaces is considered in detail in order to identify the nonlinear mechanisms involved due to multiple fluid surface interactions. The approach adopted is analytical/asymptotic and is valid for interfacial waves that are long compared with the channel height or individual undisturbed liquid layer thicknesses. This leads to a coupled system of fully nonlinear partial differential equations of Benney type that contain a small slenderness parameter that cannot be scaled out of the problem. This system is in turn used to develop a consistent coupled system of weakly nonlinear evolution equations, and it is shown that this is possible only if the underlying base-flow and fluid parameters satisfy certain conditions that enable a synchronous Galilean transformation to be performed at leading order. Two distinct canonical cases (all terms in the equations are of the same order) are identified in the absence and presence of inertia, respectively. The resulting systems incorporate all of the active physical mechanisms at Reynolds numbers that are not large, namely, nonlinearities, inertia-induced instabilities (at non-zero Reynolds number) and surface tension stabilization of sufficiently short waves. The coupled system supports several instabilities that are not found in single long-wave equations including, transitional instabilities due to a change of type of the flux nonlinearity from hyperbolic to elliptic, kinematic instabilities due to the presence of complex eigenvalues in the linearized advection matrix leading to a resonance between the interfaces, and the possibility of long-wave instabilities induced by an interaction between the flux function of the system and the surface tension terms. All of these instabilities are followed into the nonlinear regime by carrying out extensive numerical simulations using spectral methods on periodic domains. It is established that instabilities leading to coherent structures in the form of nonlinear travelling waves are possible even at zero Reynolds number, in contrast to single interface (two-layer) systems; in addition, even in parameter regimes where the flow is linearly stable, the coupling of the flux functions and their hyperbolic–elliptic transitions lead to coherent structures for initial disturbances above a threshold value. When inertia is present an additional short-wave instability enters and the systems become general coupled Kuramoto–Sivashinsky-type equations. Extensive numerical experiments indicate a rich landscape of dynamical behaviour including nonlinear travelling waves, time-periodic travelling states and chaotic dynamics. It is also established that it is possible to regularize the chaotic dynamics into travelling wave pulses by enhancing the inertialess instabilities through the advective terms. Such phenomena may be of importance in mixing, mass and heat-transfer applications.

JFM classification

Interfacial Flows (free surface): Interfacial Flows (free surface)
Type
Papers
Information
Journal of Fluid Mechanics , Volume 734 , 10 November 2013 , pp. 114 - 143
Copyright
©2013 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

Akrivis, G, Papageorgiou, D. T. & Smyrlis, Y.-S. 2009 Linearly implicit methods for a semilinear parabolic system arising in two-phase flows. IMA J. Numer. Anal. 31, 299–321.CrossRefGoogle Scholar
Akrivis, G., Papageorgiou, D. T. & Smyrlis, Y.-S. 2012 Computational study of the dispersively modified Kuramoto–Sivashinsky equation. SIAM J. Sci. Comput. 34, A792–A813.CrossRefGoogle Scholar
Beebe, D. J., Mensing, G. A. & Walker, G. M. 2002 Physics and applications of microfluidics in biology. Annu. Rev. Biomed. Engng 4, 261–286.CrossRefGoogle Scholar
Bell, J. B., Trangenstein, J. A. & Shubin, G. R. 1986 Conservation laws of mixed type describing three-phase flow in porous media. SIAM J. Appl. Maths 46, 1000–1017.CrossRefGoogle Scholar
Benjamin, T. B. 1957 Wave formation in laminar flow down an inclined plane. J. Fluid Mech. 2, 554–574.CrossRefGoogle Scholar
Benney, D. J. 1966 Long waves in liquid films. J. Math. Phys. 45, 150–155.CrossRefGoogle Scholar
Canic, S. & Plohr, B. J. 1995 Shock wave admissibility for quadratic conservation laws. J. Differ. Equ. 118, 293–335.CrossRefGoogle Scholar
Chen, K. P. 1993 Wave formation in the gravity driven low Reynolds number flow of two liquid films down an inclined plane. Phys. Fluids A 5, 3038–3048.CrossRefGoogle Scholar
Chen, K. P. 1995 Interfacial instabilities in stratified shear flows involving multiple viscous and viscoelastic fluids. Appl. Mech. Rev. 48, 763–776.CrossRefGoogle Scholar
Chumakova, L., Menzaque, F. E., Milewski, P. A., Rosales, R. R., Tabak, E. G. & Turner, C. V. 2009 Shear instability for stratified hydrostatic flows. Commun. Pure Appl. Maths 62, 183–197.CrossRefGoogle Scholar
Cole, J. D. & Cook, L. P. 1996 Transonic Aerodynamics. Elsevier Science.Google Scholar
Evans, L. C. 2010 Partial Differential Equations. American Mathematical Society.Google Scholar
Figeys, D. l & Pinto, D. 2000 Lab-on-a-chip: a revolution in biological and medical sciences. Analyt. Chem. 72, 330A–335A.CrossRefGoogle ScholarPubMed
Gao, P. & Lu, X.-Y. 2008 Mechanism of the long-wave inertialess instability of a two-layer film flow. J. Fluid Mech. 608, 379–391.CrossRefGoogle Scholar
Hibara, A., Tokeshi, M., Uchiyama, K., Hisamoto, H. & Kitamori, T. 2001 Integrated multilayer flow system on a microchip. Anal. Sci. 17, 89–93.CrossRefGoogle ScholarPubMed
Hooper, A. P. & Grimshaw, R. 1985 Nonlinear instability at the interface between two viscous fluids. Phys. Fluids 28, 37–45.CrossRefGoogle Scholar
Jiang, W. Y., Helenbrook, B. T., Lin, S. P. & Weinstein, S. J. 2005 Low-Reynolds-number instabilities in three-layer flow down an inclined wall. J. Fluid Mech. 539, 387–416.CrossRefGoogle Scholar
Joseph, D. D. & Renardy, Y. Y. 1991 Fundamentals of Two-Fluid Dynamics, Part I: Mathematical Theory and Applications. Springer.Google Scholar
Kalogirou, A., Papageorgiou, D. T. & Smyrlis, Y.-S. 2012 Surfactant destabilization and nonlinear phenomena in two-fluid shear flows at small Reynolds numbers. IMA J. Appl. Maths 77, 351–360.CrossRefGoogle Scholar
Kao, T. W. 1968 Inertialess instability of a two-layer liquid film flow. J. Fluid Mech. 33, 561.CrossRefGoogle Scholar
Kliakhandler, I. 1999 Long interfacial waves in multilayer thin films and coupled Kuramoto–Sivashinsky equations. J. Fluid Mech. 391, 45–65.CrossRefGoogle Scholar
Kliakhandler, I. & Sivashinsky, G. 1995 Kinetic alpha effect in viiscosity stratified creeping flows. Phys. Fluids 7, 1866–1871.CrossRefGoogle Scholar
Kliakhandler, I. & Sivashinsky, G. 1996 Inertial effects and long waves in multilayer plane Poiseuille flows. Phys. Scr. T67, 90–95.CrossRefGoogle Scholar
Kogan, M. N. 1961 On magnetohydrodynamic flows of mixed type. PMM 25, 132–137.Google Scholar
Li, C.-H. 1969 Instability of three-layer viscous stratified flow. Phys. Fluids 12, 2473–2481.CrossRefGoogle Scholar
Loewenherz, D. S. & Lawrence, C. J. 1989 The effect of viscosity stratification on the instability of a free surface flow at low Reynolds number. Phys. Fluids A 1, 1689.CrossRefGoogle Scholar
Lovick, J. & Angeli, P. 2004 Experimental studies on the dual continuous flow pattern in oil–water flows. Intl J. Multiphase Flow 30, 139–157.CrossRefGoogle Scholar
Majda, A. & Pego, R. L. 1985 Stable viscosity matrices for systems of conservation laws. J. Differ. Equ. 56, 229–262.CrossRefGoogle Scholar
Milewski, P., Tabak, E., Turner, C., Rosales, C. & Menzaque, F. 2004 Nonlinear stability of two-layer flows. Commun. Math. Sci. 2, 427–442.Google Scholar
Nordberg, M. E. & Winter, H. H. 1988 Fully developed multilayer polymer flows in slits and annuli. Polym. Engng Sci. 28, 444–452.CrossRefGoogle Scholar
Papageorgiou, D. T. & Orellana, O. 1998 Study of cylindrical jet breakup using one-dimensional approximations of the Euler equations. SIAM J. Appl. Maths 59, 286–317.CrossRefGoogle Scholar
Papageorgiou, D. T., Maldarelli, C. & Rumschitzki, D. S. 1990 Nonlinear interfacial stability of core-annular film flows. Phys. Fluids A 2, 340–352.CrossRefGoogle Scholar
Peters, G. R. & Canic, S. 2000 On the oscillatory solutions in hyperbolic conservation laws. Nonlinear Anal.-Real. 1, 287–314.CrossRefGoogle Scholar
Polsin, K. A. & Choueiri, E. Y. 2002 A passive propellant feeding mechanism for micropropulsion using capillarity. In 38th AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit.CrossRefGoogle Scholar
Pozrikidis, C. 2004 Instability of multi-layer channel and film flows. Adv. Appl. Mech. 40, 179–239.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
Rosenau, R. J., Oron, A. & Hyman, J. M. 1992 Bounded and unbounded patterns of the Benney equation. Phys. Fluids A 4, 1102.CrossRefGoogle Scholar
Schaeffer, D.-G. & Shearer, M. 1987 The classification of $2\times 2$ systems of non-strictly hyperbolic conservation laws, with application to oil recovery. Comm. Pure Appl. Maths 40, 141–178.CrossRefGoogle Scholar
Shlang, T. & Sivashinsky, G. I. 1985 Irregular wavy flow due to viscous stratification. J. Phys. 46, 863–866.CrossRefGoogle Scholar
Skurtys, O. & Aguilera, J. 2008 Applications of microfluidic devices in food engineering. Food Biophys. 3, 1–15.CrossRefGoogle Scholar
Slemrod, M. 1983 Admissibility criteria for propagating phase boundaries in a van der Waals fluid. Arch. Rat. Mech. 81, 301–315.CrossRefGoogle Scholar
Smyrlis, Y.-S. & Papageorgiou, D. T. 1990 Computational study of chaotic and ordered solutions for the Kuramoto–Sivashinsky equation. ICASE Rep. 96–12, 1–32.Google Scholar
Smyrlis, Y.-S. & Papageorgiou, D. T. 1991 Computational study of chaotic and ordered solutions for the Kuramoto–Sivashinsky equation. Proc. Natl Acad. Sci. USA 88, 11129–11132.CrossRefGoogle Scholar
SooHoo, J. R. & Walker, G. M. 2009 Microfluidic aqueous two phase system for leukocyte concentration from whole blood. Biomed. Microdevices 11, 323–329.CrossRefGoogle ScholarPubMed
Surmeian, M., Slyadnev, M. N., Hisamoto, H., Hibara, A., Uchiyama, K. & Kitamori, T. 2002 Three-layer flow membrane system on a microchip for investigation of molecular transport. Analyt. Chem. 74, 2014–2020.CrossRefGoogle ScholarPubMed
Tilley, B. S., Davis, S. H. & Bankoff, S. G. 1994a Linear stability theory of two-layer fluid flow in an inclined channel. Phys. Fluids 6, 3906–3922.CrossRefGoogle Scholar
Tilley, B. S., Davis, S. H. & Bankoff, S. G. 1994b Nonlinear long-wave stability of superposed fluids in an inclined channel. J. Fluid Mech. 277, 55–83.CrossRefGoogle Scholar
Weinstein, S. J. & Ruschak, K. J. 2004 Coating flows. Annu. Rev. Fluid Mech. 36, 29–53.CrossRefGoogle Scholar
Yih, C.-S. 1963 Stability of liquid flow down an inclined plane. Phys. Fluids 6, 321–335.CrossRefGoogle Scholar
Yih, C.-S. 1967 Instability due to viscosity stratification. J. Fluid Mech. 27, 337–352.CrossRefGoogle Scholar
Zauderer, E. 2006 Partial Differential Equations of Applied Mathematics. John Wiley & Sons.CrossRefGoogle Scholar