Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-20T03:45:29.443Z Has data issue: false hasContentIssue false
  • English
  • Français

Interfacial instabilities in plane Poiseuille flow of two stratified viscoelastic fluids with heat transfer. Part 1. Evolution equation and stability analysis

Published online by Cambridge University Press:  26 April 2006

Sang W. Joo
Sang W. Joo
Affiliation:
Department of Mechanical Engineering, Wayne State University, Detroit, MI 48282, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

An evolution equation is derived that describes the nonlinear development of the interface between two viscoelastic fluids flowing, under the action of imposed pressure gradient and gravity, in a vertical channel. The channel walls are kept at different temperatures, resulting in heat transfer across the layers. The equation, based on the lubrication approximation, models the effects of stratifications in density, viscosity, elasticity, shear thinning, and thermal conductivity. It also describes the capillary and thermocapillary effects, as well as the sensitivity of viscosities to temperature. Linear-stability analysis is performed based on the evolution equation to understand the competing effects of viscous, elastic, and Marangoni instabilities. Particular attention is paid to the active control of the interfacial instabilities through the thermocapillarity.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 299 , 25 September 1995 , pp. 241 - 265
Copyright
© 1995 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

Benney, D. J. 1966 Long waves in liquid films. J. Math. Phys. 45, 150–155.Google Scholar
Bird, R. B., Armstrong, R. C. & Hassager, O. 1987 Dynamics of Polymeric Liquids, Vol. 1, Chapter 7. Wiley-Interscience.
Chang, H.-C. 1989 Onset of nonlinear waves on falling films. Phys. Fluids A 1, 1314–1327.Google Scholar
Charru, F. & Fabre J. 1994 Long waves at the interface between two viscous fluids. Phys. Fluids 6, 1223–1235.Google Scholar
Chen, K.-P. 1991 Interfacial instability due to elastic stratification in concentric coextrusion of two viscoelastic fluids. J. Non-Newtonian Fluid Mech. 40, 155–175.Google Scholar
Gjevik, B. 1970 Occurence of finite-amplitude surface waves on falling liquid films. Phys. Fluids 13, 1918–1925.Google Scholar
Goussis, D. A. & Kelly, R. W. 1990 On the thermocapillary instabilities in a liquid layer heated from below. Intl J. Heat Mass Transfer 33, 2237–2245.Google Scholar
Hooper, A. P. & Grimshaw, R. 1985 Nonlinear instability at the interface between two viscous fluids. Phys. Fluids 28, 37–45.Google Scholar
Joo, S. W. 1994 The stability and nonlinear flow developments of a viscoelastic draining film with shear thinning. J. Non-Newtonian Fluid Mech., 51, 125–140.Google Scholar
Joo, S. W. & Davis, S. H. 1992a Instabilities of three-dimensional viscous falling films. J. Fluid Mech. 242, 529–547.Google Scholar
Joo, S. W. & Davis, S. H. 1992b Irrigular waves on viscous falling films. Chem. Engng Commun. 118, 111–123.Google Scholar
Larson, R. G. 1992 Instabilities in viscoelastic fluids. Rheol. Acta 31, 213–239.Google Scholar
Li, C.-H. 1969 Stability of two superposed elasticoviscous liquids in plane Couette flow. Phys. Fluids 12, 531–538.Google Scholar
Lin, S.-P. & Wang, C.-Y. 1985 Encyclopedia of Fluid Mechanics, vol. 1, p. 931. Gulf.
Nordberg, M. E. & Winter, H. H. 1990 A simple model of nonisothermal coextrusion. Polymer Engng Sci. 30, 408–415.Google Scholar
Pearson, J. R. A. 1958 On convection cells induced by surface tension. J. Fluid Mech. 4, 489–500.Google Scholar
Renardy, M. & Renardy, Y. 1993 Derivation of amplitude equations and analysis of sideband instabilities in two-layer flows. Phys. Fluids A 5, 2738–2762.Google Scholar
Renardy, Y. 1988 Stability of the interface in two-layer Couette flow of upper convected Maxwell liquids. J. Non-Newtonian Fluid Mech. 28, 99–115.Google Scholar
Sornberger, G., Vergnes, B. & Agassant, J. F. 1986 Coextrusion flow of two molten polymers between parallel plates: non- isothermal computation and experimental study. Polymer Engng Sci. 26, 682–689.Google Scholar
Su, Y.-Y. & Khomami, B. 1992a Interfacial stability of multilayer viscoelastic fluids in slit and converging channel die geometries. J. Rheol. 36, 357.Google Scholar
Su, Y.-Y. & Khomami, B. 1992b Pure elastic interfacial instabilities in superposed flow of polymeric fluids. Rheol. Acta 31, 413–420.Google Scholar
Tilley, B. S., Davis, S. H. & Bankoff, S. G. 1994a Nonlinear long-wave stability of superposed fluids in an inclined channel. J. Fluid Mech. 277, 55–83.Google Scholar
Tilley, B. S., Davis, S. H. & Bankoff, S. G. 1994b Linear stability theory of two-layer flow in an inclined channel. Phys. Fluids 6, 3906–3922.Google Scholar
Yiantsios, S. G. & Higgins, B. G. 1988 Linear stability of plane Poiseuille flow of two superposed fluids. Phys. Fluids 31, 3225–3238.Google Scholar
Yih, C.-S. 1963 Stability of liquid flow down an inclined plane. Phys. Fluids 6, 321–334.Google Scholar
Yih, C.-S. 1967 Instability due to viscosity stratification. J. Fluid Mech. 27, 337–352.Google Scholar