Abstract
In the framework of finite element analysis we propose fast and robust time integration scheme for viscoelastic fluid (the Oldroyd-B and Leonov models) flow by way of efficient decoupling of equations. Developed algorithms of the 1st and 2nd order are shown to disclose convergence characteristics equivalent to conventional methods of corresponding order when applied to 1D poiseuille and 2D creeping contraction flow problems. In comparison with fully coupled implicit technique, they notably enhance the computation speed. For the time dependent flow modeling with pressure difference imposed slightly below the steady limit, current as well as conventional approximation scheme has demonstrated fluctuating solution without approaching the steady state. From the result, we may conclude that the existence of upper limit for convergent steady solution implies flow transition to highly elastic time-fluctuating field without steady asymptotic. It is presumably associated with some real unstable elastic flow like re-entrant vortex oscillation and extrudate distortion outside the channel outlet.
Similar content being viewed by others
References
Baaijens, F. P. T., 1998, Mixed finite element methods for viscoelastic flow analysis: a review, J. Non-Newtonian Fluid Mech. 79, 361–385.
Bird, R. B., C. F. Curtiss, R. C. Armstrong, and O. Hassager, 1987, Dynamics of Polymeric Liquids, Wiley, New York.
Bogaerds, A. C. B., M. A. Hulsen, G. W. M. Peters, and F. P. T. Baaijens, 2004, Stability analysis of injection molding flows, J. Rheol. 48, 765–785.
Coronado, O. M., D. Arora, M. Behr, and M. Pasquali, 2007, A simple method for simulating general viscoelastic fluid flows with an alternate log-conformation formulation, J. Non-Newtonian Fluids Mech. 147, 189–199.
D’Avino, G. and M. A. Hulsen, 2010, Decoupled second-order transient schemes for the flow of viscoelastic fluids without a viscous solvent contribution, J. Non-Newtonian Fluid Mech. 165, 1602–1612.
Duarte, A. S. R., A. I. P. Miranda, and P. J. Oliveira, 2008, Numerical and analytical modeling of unsteady viscoelastic flows: The start-up and pulsating test case problems, J. Non-Newton. Fluid Mech., 154, 153–169.
Fattal, R. and R. Kupferman, 2004, Constitutive laws for the matrix-logarithm of the conformation tensor, J. Non-Newtonian Fluid Mech. 123, 281–285.
Fyrillas, M. M., G. C. Georgious, and D. Vlassopoulos, 1999, Time-dependent plane Poiseuille flow of a Johnson-Segalman fluid, J. Non-Newton. Fluid Mech. 82, 105–123.
Grillet, A. M., A. C. B. Bogaerds, G. W. M. Peters, and F. P. T. Baaijens, 2002, Numerical analysis of flow mark surface defects in injection molding flow, J. Rheol. 46, 651–669.
Hulsen, M. A., M. A., R. Fattal, and R. Kupferman, 2005, Flow of viscoelastic fluids past a cylinder at high Weissenberg number: Stabilized simulations using matrix logarithms, J. Non-Newton. Fluid Mech. 127, 27–39.
Joseph, D. D., 1990, Fluid dynamics of viscoelastic liquids, Springer-Verlag, New York.
Keshtiban, I. J., B. Puangkird, H. Tamaddon-Jahromi, and M. F. Webster, 2008, Generalised approach for transient computation of start-up pressure-driven viscoelastic flow, J. Non-Newton. Fluid Mech. 151, 2–20.
Kwon, Y. and A. I. Leonov, 1995, Stability constraints in the formulation of viscoelastic constitutive equations, J. Non-Newton. Fluid Mech. 58, 25–46.
Kwon, Y., 2004, Finite element analysis of planar 4:1 contraction flow with the tensor-logarithmic formulation of differential constitutive equations, Korea-Australia Rheol. J. 16, 183–191.
Kwon, Y., 2011, Numerical description of elastic flow instability and its dependence on liquid viscoelasticity in planar contraction, submitted to J. Rheol.
Lee, J., S. Yoon, Y. Kwon, and S. J. Kim, 2004, Practical comparison of differential viscoelastic constitutive equations in finite element analysis of planar 4:1 contraction flow, Rheol. Acta 44, 188–197.
Leonov, A. I., 1976, Nonequilibrium thermodynamics and rheology of viscoelastic polymer media, Rheol. Acta 15, 85–98.
McKinley, G. H., J. A. Byars, R. A. Brown, and R. C. Armstrong, 1991, J. Non-Newtonian Fluid Mech. 40, 201–229.
Owens, R. G. and T. N. Phillips, 2002, Computational Rheology, Imperial College Press, London.
Park, K. S. and Y. Kwon, 2009, Numerical description of startup viscoelastic plane poiseuille flow, Korea-Australia Rheol. J. 21, 47–58.
Poole, R. J., M. A. Alves, and P. J. Oliveira, 2007, Purely elastic flow asymmetries, Phys. Rev. Lett. 99, 164503.
Sato, T. and S. M. Richardson, 1994, Explicit numerical simulation of time-dependent viscoelastic flow problems by a finite element/finite volume method, J. Non-Newton. Fluid Mech. 51, 249–275.
Shaqfeh, E. S. G., 1996, Purely elastic instabilities in viscometric flows, Ann. Rev. Fluid Mech. 28, 129–185.
Soulages, J., M. S. N. Oliveira, P. C. Sousa, M. A. Alves, and G. H. McKinley, 2009, Investigating the stability of viscoelastic stagnation flows in T-shaped microchannels, J. Non-Newtonian Fluid Mech. 163, 9–24.
Truesdell, C. and W. Noll, 1992, The Non-Linear Field Theories of Mechanics, Springer-Verlag, Berlin.
Van Os, R. G. M. and T. N. Phillips, 2004, Spectral element methods for transient viscoelastic flow problems, J. Comp. Physics 201, 286–314.
Waters, N. D. and M. J. King, 1970, Unsteady flow of an elasticoviscous liquid, Rheol. Acta 9, 345–355.
Xue, S. C., R. I. Tanner, and N. Phan-Thien, 2004, Numerical modeling of transient viscoelastic flows, J. Non-Newton. Fluid Mech. 123, 33–58.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Kwon, Y., Park, K.S. Decoupled algorithm for transient viscoelastic flow modeling. Korea-Aust. Rheol. J. 24, 53–63 (2012). https://doi.org/10.1007/s13367-012-0006-1
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13367-012-0006-1