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Nonlinear long-wave stability of superposed fluids in an inclined channel

Published online by Cambridge University Press:  26 April 2006

B. S. Tilley ,
S. H. Davis  and
S. G. Bankoff
B. S. Tilley
Affiliation:
Department of Engineering Sciences and Applied Mathematics, Northwestern University, Evanston, IL 60208, USA
S. H. Davis
Affiliation:
Department of Engineering Sciences and Applied Mathematics, Northwestern University, Evanston, IL 60208, USA
S. G. Bankoff
Affiliation:
Department of Chemical Engineering, Northwestern University, Evanston, IL 60208, USA
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Abstract

We consider the two-layer flow of immiscible, viscous, incompressible fluids in an inclined channel. We use long-wave theory to obtain a strongly nonlinear evolution equation which describes the motion of the interface. This equation includes the physical effects of viscosity stratification, density stratification, and shear. A weakly nonlinear analysis of this equation yields a Kuramoto–Sivashinsky equation, which possesses a quadratic nonlinearity. However, certain physical situations exist in two-layer flow for which modifications of the Kuramoto–Sivashinsky equation are physically pertinent. In particular, the presence of the second layer can mediate the wave-steepening instability found in single-phase falling films, requiring the inclusion of a cubic nonlinearity in the weakly nonlinear analysis. The introduction of the cubic nonlinearity destroys the symmetry-breaking bifurcations of the Kuramoto–Sivashinsky equation, and new isolated solution branches emerge as the strength of the cubic nonlinearity increases. Bistability between these new solutions and those associated with the Kuramoto–Sivashinsky equation is found, as well as the formation of a hysteresis loop from smaller-amplitude travelling waves to larger-amplitude travelling waves. The physical implications of these dynamics to the phenomenon of laminar flooding in a channel are discussed.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 277 , 25 October 1994 , pp. 55 - 83
Copyright
© 1994 Cambridge University Press

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References

Aston, P. J. 1991 Physica D 52, 415.
Aston, P. J., Spence, A. & Wu, W. 1992a SIAM J. Appl. Maths 52, 792.
Aston, P. J., Spence, A. & Wu, W. 1992b Intl Ser. Numer. Maths 104, 35.
Bankoff, S. G. & Lee, S. C. 1986 In Multiphase Science and Technology (ed. G. F. Hewitt, J. M. Delhaye & N. Zuber) p. 95. Hemisphere.
Benjamin, T. B. 1957 J. Fluid Mech. 2, 554.
Benney, D. J. 1966 J. Maths Phys. 45, 150.
Chang, H. C. 1986 Chem. Engng Sci. 41, 2463.
Chang, H. C., Demekhin, E. A. & Kopelevich, D. I. 1993 J. Fluid Mech. 250, 433.
Charru, F. & Fabre, J. 1994 Phys. Fluids 10, 1223.
Chen, L. H. & Chang, H. C. 1986 Chem. Engng Sci. 41, 2477.
Demekhin, Y. A., Todarev, G. Yu. & Shkadov, V. Ya. 1991 Physica D 52, 338.
Doedel, E. J. 1981 Congressus Numerantium 30, 265.
Drazin, P. G. 1992 Nonlinear Systems. Cambridge University Press.
Fowler, A. C. & Lisseter, P. E. 1992 SIAM J. Appl. Maths 52, 15.
Frisch, U., She, Z. S. & Thual, O. 1986 J. Fluid Mech. 168, 221.
Golubitsky, M. & Schaeffer, D. G. 1985 Singularities and Groups in Bifurcation Theory. Springer.
Hooper, A. P. & Grimshaw, R. 1985 Phys. Fluids 28, 37.
Hyman, J. M. & Nicolaenko, B. 1986 Physica D 18, 113.
Joo, S. W., Davis, S. H. & Bankoff, S. G. 1991 J. Fluid Mech. 230, 117.
Kevrekidis, I. G., Nicolaenko, B. & Scovel, J. C. 1990 SIAM J. Appl. Maths 50, 760.
Matkowsky, B. J. & Reiss, E. L. 1977 SIAM J. Appl. Maths 33, 230.
Papageorgiou, D. T., Maldarelli, C. & Rumschitzki, D. S. 1990 Phys. Fluids A 2, 340.
Papageorgiou, D. T. & Smyrlis, Y. S. 1991 Theoret. Comput. Fluid Dyn. 3, 15.
Schlang, T. 1984 Nonlinear stability analysis of problems in thin film fluid theory. PhD thesis, Department of Applied Mathematics, Tel Aviv University.
Scovel, J. C., Kevrekidis, I. G. & Nicolaenko, B. 1988 Phys. Lett. A 130, 73.
Smith, M. K. 1990 J. Fluid Mech. 217, 469.
Tilley, B. S. 1994 Stability of two-layer flow in an inclined channel. PhD thesis, Department of Engineering Sciences and Applied Mathematics, Northwestern University.
Tilley, B. S., Davis, S. H. & Bankoff, S. G. 1994 Linear stability theory of two-layer fluid flow in an inclined channel. Phys. Fluids (to appear).Google Scholar
Wallis, G. B. 1969 One-Dimensional Two-Phase Flow. McGraw-Hill.
Yiantsios, S. G. & Higgins, B. G. 1988 Phys. Fluids 31, 3225.
Yiantsios, S. G. & Higgins, B. G. 1989 Phys. Fluids A 1, 1484.
Yih, C. S. 1955 Proc. 2nd US Congr. Appl. Mech., Am. Soc. Mech. Engrs. p. 623.
Yih, C. S. 1967 J. Fluid Mech. 27, 337.