Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-21T00:14:52.988Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

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

Apelian, M. R., Armstrong, R. C. & Brown, R. A. 1988 Impact of the constitutive equation and singularity on the calculation of stick-slip flow: the modified upper-convected Maxwell model (MUCM). J. Non-Newtonian Fluid Mech. 27, 299–321.Google Scholar
Babchin, A. J., Frenkel, A. L., Levich, B. G. & Sivashinsky, G. I. 1983 Nonlinear saturation of Rayleigh-Taylor instability in thin films. Phys. Fluids 26, 3159–3161.Google Scholar
Bird, R. B., Armstrong, R. C. & Hassager, O. 1987 Dynamics of Polymeric Liquids, Vol. I, Fluid Mechanics. John Wiley.
Bousfield, D. W., Keunings, R., Marucci, G. & Denn, M. M. 1986 Non-linear analysis of surface tension driven breakup of viscoelastic filaments. J. Non-Newtonian Fluid Mech. 21, 79–97.Google Scholar
Brooks, A. N. & Hughes, T. J. R. 1982 Streamline upwind/Petrov-Galerkin formulation for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations. Comput. Meth. Appl. Mech. Engng 32, 199–259.Google Scholar
Brown, R. A., Armstrong, R. C., Beris, A. N. & Yeh, P.-W. 1986 Galerkin finite element analysis of complex viscoelastic flows. Comput. Meth. Appl. Mech. Engng 58, 201–226.Google Scholar
Rurdette, S. R., Coates, P. J., Armstrong, R. C. & Brown, R. A. 1989 Calculations of viscoelastic flow through an axisymmetric corrugated tube using the explicitly elliptic momentum equation formulation (EEME). J. Non-Newtonian Fluid Mech. 33, 1–23.Google Scholar
Carey, G. F. & Oden, J. T. 1986 Finite Elements: Fluid Mechanics Vol. VI. Prentice-Hall.
Christodoulou, K. N. & Scriven, L. E. 1992 Discretization of free surface flows and other moving boundary problems. J. Comput. Phys. (in press).Google Scholar
Coates, P. J., Armstrong, R. C. & Brown, R. A. 1992 Calculation of steady viscoelastic flow through axisymmetric contractions using the EEME formulation. J. Non-Newtonian Fluid Mech. (in press).Google Scholar
Crochet, M. J., Davies, A. R. & Walters, K. 1984 Numerical Simulation of Non-Newtonian Flow. Elsevier.
Crochet, M. J., Delvaux, V. & Marchal, M. J. 1990 On the convergence of the streamline-upwind mixed finite element. J. Non-Newtonian Fluid Mech. 34, 261–268.Google Scholar
Deiber, J. A. & Cerro, R. L. 1976 Viscous flow with a free surface inside a horizontal rotating drum. I. hydrodynamics. Indust. Engng Chem. Fundam. 15, 102–110.Google Scholar
Dheur, J. & Crochet, M. J. 1989 Stratified flows of newtonian and viscoelastic fluids. J. Non-Newtonian Fluid Mech. 32, 1–18.Google Scholar
Giesekus, H. 1982 A simple constitutive equation for polymer fluids based on the concept of deformation dependent tensorial mobility. J. Non-Newtonian Fluid Mech. 11, 69–109.Google Scholar
Green, A. E. & Rivlin, R. S. 1957 The mechanics of non-linear materials with memory, Part I. Arch. Rat. Mech. Anal. 1, 1–21.Google Scholar
Hood, P. 1976 Frontal solution program for unsymmetric matrices. Intl J. Numer. Meth. Engng 10, 379–399.Google Scholar
Iooss, G. & Joseph, D. D. 1981 Elementary Stability and Bifurcation Theory. Springer.
Johnson, C., Navert, U. & Pitkaranta, J. 1984 Finite element methods for linear hyperbolic problems. Comput. Meth. Appl. Mech. Engng 45, 285–312.Google Scholar
Johnson, R. E. 1988 Steady-state coating flows inside a rotating horizontal cylinder. J. Fluid Mech. 190, 321–342.Google Scholar
Joseph, D. D., Renardy, M. & Saut, J.-C. 1985 Hyperbolicity and change of type in the flow of viscoelastic fluids. Arch. Rat. Mech. Anal. 87, 213–251.Google Scholar
Karweit, M. J. & Corrsin, S. 1975 Observations of cellular patterns in a partly filled, horizontal rotating drum. Phys. Fluids 18, 111–112.Google Scholar
Keunings, R. 1986 An algorithm for the simulation of transient viscoelastic flows with free surfaces. J. Comput. Phys. 62, 199–220.Google Scholar
Keunings, R. 1989 Simulation of viscoelastic fluid flow. In Fundamental of Computer Modeling for Polymer Processing (ed. C. L. Tucker), 403–469. Carl Hanser.
Keunings, R. & Bousfield, D. W. 1987 Analysis of surface tension driven leveling in viscoelastic flows. J. Non-Newtonian Fluid Mech. 22, 219–233.Google Scholar
King, R. C., Apelian, M. R., Armstrong, R. C. & Brown, R. A. 1988 Numerically stable finite element techniques for viscoelastic calculations in smooth and singular geometries. J. Non-Newtonian Fluid Mech. 29, 147–216.Google Scholar
Kistler, S. F. & Scriven, L. E. 198 Coating flows. In Computational Analysis of Polymer Processing (eds J. R. A. Pearson & S. M. Richardson), p. 243 Essex: Applied Science Publishers.
Marchal, J. M. & Crochet, M. J. 1987 A new mixed finite element method for calculating viscoelastic flow. J. Non-Newtonian Fluid Mech. 26, 26–77.Google Scholar
Musarra, S. & Keunings, R. 1989 Co-current exisymmetric flow in complex geometries: Numerical simulation. J. Non-Newtonian Fluid Mech. 32, 253–268.Google Scholar
Northey, P. J., Armstrong, R. C. & Brown, R. A. 1990 Finite-element calculation of time-dependent two-dimensional viscoelastic with the EEME formulation. J. Non-Newtonian Fluid Mech. 36, 109–133.Google Scholar
Orr, F. M. & Scriven, L. E. 1978 Rimming flow: numerical simulation of steady, viscous, free-surface flow with surface tension. J. Fluid Mech. 84, 145–165.Google Scholar
Pearson, J. R. A. 1985 Mechanics of Polymer Processing. Elsevier.
Pilitsis, S. & Beris, A. N. 1989 Calcaulations of steady-state viscoelastic flow in an undulating tube. J. Non-Newtonian Fluid Mech. 31, 231–287.Google Scholar
Quinzani, L. M., McKinley, G. H., Brown, R. A. & Armstrong, R. C. 1990 Modeling the rheology of polyisobutylene solutions. J. Rheol. 34, 705–748.Google Scholar
Rajagopalan, D., Armstrong, R. C. & Brown, R. A. 1990a Calculation of steady viscoelastic flow using a multimode maxwell model: application of the explicitly elliptic momentum equation (EEME) formulation. J. Non-Newtonian Fluid Mech. 36, 135–158.Google Scholar
Rajagopalan, D., Armstrong, R. C. & Brown, R. A. 1990b Finite element methods for calculation of steady, viscoelastic flow using constitutive equations with a Newtonian viscosity. J. Non-Newtonian Fluid Mech. 36, 159–193.Google Scholar
Rao, M. A. & Thorne, J. L. 1972 Principles of rotational moulding. Polymer. Engng Sci. 12, 237–264.Google Scholar
Ruschak, K. J. & Scriven, L. E. 1976 Rimming flow of liquid in a rotating horizontal cylinder. J. Fluid Mech. 76, 113–125.Google Scholar
Ryskin, G. & Leal, L. G. 1983 Orthogonal mapping. J. Comput. Phys. 50, 71–100.Google Scholar
Sackinger, P. A., Brown, R. A. & Derby, J. J. 1989 A finite element method for analysis of fluid flow, heat transfer and free interfaces in Czochralski crystal growth. Intl J. Numer. Meth. Fluids 9, 453–492.Google Scholar
Sanders, J., Joseph, D. D. & Beavers, G. S. 1981 Rimming flow of a viscoelastic liquid inside a rotating horizontal cylinder. J. Non-Newtonian Fluid Mech. 9, 269–300.Google Scholar
Strang, G. & Fix, G. F. 1973 An Analysis of the Finite Element Method. Prentice-Hall.
Tsiveriotis, K. G. & Brown, R. A. 1991 Boundary conforming mapping applied to computations of highly deformed solidification interfaces. Intl J. Numer. Meth. Fluids (in press).Google Scholar
Weatherburn, C. E. 1927 Differntial Geometry of Three Dimensions. Cambridge University Press.
Yamaguchi, Y., Chang, C. J. & Brown, R. A. 1984 Multiple buoyancy driven flow in a vertical cylinder heated from below. Phil. Trans. R. Soc. Lond. A 312, 519–552.Google Scholar