Abstract
This study, employing a numerical approximation, computationally describes 2D melt fracture as elastic instability in the flow along and outside a straight channel. In the preceding research (Kwon, 2018, Numerical modeling of two-dimensional melt fracture instability in viscoelastic flow, J. Fluid Mech. 855, 595–615) several types of unique instability and corresponding bifurcations such as subcritical and chaotic transitions have been illustrated with possible mechanism presumed. However, the 1st bifurcation from stable steady to unstable periodic state could not be accurately characterized even though its existence was proven evident. The analysis herein aims at verification of this 1st transition to temporally (and also spatially) periodic instability, utilizing the same numerical technique with attentive control of flow condition. As a result of scrutinizing the solutions, the steady elastic flow described by the Leonov rheological model passes through supercritical Hopf bifurcation at the Deborah number of 10.42 and then transforms to the state of the 1st weak periodic instability. It has also been confirmed that near this bifurcation point it takes extremely long to completely develop into either steady state (in the stable case) or periodic instability, which obstructed immediate characterization of the transition in the previous work.
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References
Adewale, K.P. and A.I. Leonov, 1997, Modeling spurt and stress oscillations in flows of molten polymers, Rheol. Acta 36, 110–127.
Boger, D.V. and K. Walters, 1993, Rheological Phenomena in Focus, Elsevier, Amsterdam.
Denn, M.M., 2001, Extrusion instabilities and wall slip, Ann. Rev. Fluid Mech. 33, 265–287.
Fattal, R. and R. Kupferman, 2004, Constitutive laws for the matrix-logarithm of the conformation tensor, J. Non-Newtonian Fluid Mech. 123, 281–285.
Graham, M.D., 1999, The sharkskin instability of polymer melt flows, Chaos 9, 154–163.
Kiss, N.E. and J.M. Piau, 1994, Adhesion of linear low density polyethylene for flow regimes with sharkskin, J. Rheol. 38, 1447–1463.
Koopmans, R., J.D. Doelder, and J. Molenaar, 2010, Polymer Melt Fracture, CRC Press.
Kwon, Y. and A.I. Leonov, 1995, Stability constraints in the formulation of viscoelastic constitutive equations, J. Non-Newtonian Fluid Mech. 58, 25–46.
Kwon, Y., 2014, Numerical aspects in modeling high Deborah number flow and elastic instability, J. Comput. Phys. 265, 128–144.
Kwon, Y., 2015, Melt fracture modeled as 2D elastic flow instability, Rheol. Acta 54, 445–453.
Kwon, Y., 2018, Numerical modelling of two-dimensional melt fracture instability in viscoelastic flow, J. Fluid Mech. 855, 595–615.
Larson, R.G., 1992, Instabilities in viscoelastic flows, Rheol. Acta 31, 213–263.
Leonov, A.I., 1976, Nonequilibrium thermodynamics and rheology of viscoelastic polymer media, Rheol. Acta 15, 85–98.
Leonov, A.I. and A. N. Prokunin, 1994, Nonlinear Phenomena in Flows of Viscoelastic Polymer Fluids, Chapman and Hall.
Piau, J.M., N.E. Kiss, and B. Tremblay, 1990, Influence of upstream instabilities and wall slip on melt fracture and sharkskin phenomena during silicones extrusion through orifice dies, J. Non-Newtonian Fluid Mech. 34, 145–180.
Piau, J.M., S. Nigen, and N.E. Kissi, 2000, Effect of die entrance filtering on mitigation of upstream instability during extrusion of polymer melts, J. Non-Newtonian Fluid Mech. 91, 37–57.
Shaqfeh, E.S.G., 1996, Purely elastic instabilities in viscometric flows, Annu. Rev. Fluid Mech. 28, 129–185.
Simhambhatla, M. and A.I. Leonov, 1995, On the rheological modeling of viscoelastic polymer liquids by stable constitutive equations, Rheol. Acta 34, 259–273.
Tordella, J.P., 1956, Fracture in the extrusion of amorphous polymer through capillaries, J. Appl. Phys. 27, 454–458.
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Kwon, Y. Supercritical bifurcation to periodic melt fracture as the 1st transition to 2D elastic flow instability. Korea-Aust. Rheol. J. 32, 309–317 (2020). https://doi.org/10.1007/s13367-020-0029-y
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DOI: https://doi.org/10.1007/s13367-020-0029-y