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The influence of viscoelasticity on the existence of steady solutions in two-dimensional rimming flow

Published online by Cambridge University Press:  26 April 2006

Dilip Rajagopalan ,
Ronald J. Phillips ,
Robert C. Armstrong ,
Robert A. Brown  and
Arijit Bose
Dilip Rajagopalan
Affiliation:
Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Ronald J. Phillips
Affiliation:
Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Robert C. Armstrong
Affiliation:
Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Robert A. Brown
Affiliation:
Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Arijit Bose
Affiliation:
Department of Chemical Engineering, University of Rhode Island, Kingston, RI 02881, USA
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Abstract

The steady, two-dimensional flows and interface shapes of the rimming flow of Newtonian and viscoelastic liquid films are studied by finite-element analysis. The viscoelastic flow calculations are based on the elastic-viscous split stress (EVSS) formulation for differential constitutive models. The EVSS formulation is derived by taking into account the mathematical type of the momentum, continuity and constitutive equations and is extended in this paper to calculation of free-surface flows. Calculations for a viscous Newtonian fluid demonstrate the balance between viscous forces and gravity which sets the shape of the interface of the liquid film coating the inside of the rotating cylinder. The liquid shape evolves from a concentric and circular film at high rotation rates to become thicker on the rising surface as the rotation rate is lowered. No steady flows with continuous films are found to exist below a minimum rotation rate, Ω = Ωc, where the family of flows evolves back toward higher values of Ω. Multiple solutions are predicted for a range of rotation rates, Ω > Ωc, and unstable flows develop a pronounced bulge on the rising side of the film. Asymptotic analysis for a thin film predicts this limiting rotation rate. Adding viscoelasticity to the liquid, as modelled by the Giesekus constitutive equation, leads to the existence of steady solutions at lower rotation rates and causes the bulge to appear on stable films. The minimum rotation rate for steady, viscoelastic flow shifts to lower values as the time constant of the fluid is increased.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 235 , February 1992 , pp. 611 - 642
Copyright
© 1992 Cambridge University Press

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