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A timestepper approach for the systematic bifurcation and stability analysis of polymer extrusion dynamics. (English) Zbl 1388.76015

Summary: We discuss how matrix-free/timestepper algorithms can efficiently be used with dynamic non-Newtonian fluid mechanics simulators in performing systematic stability/bifurcation analysis. The timestepper approach to bifurcation analysis of large-scale systems is applied to the plane Poiseuille flow of an Oldroyd-B fluid with non-monotonic slip at the wall, in order to further investigate a mechanism of extrusion instability based on the combination of viscoelasticity and non-monotonic slip. Due to the non-monotonicity of the slip equation the resulting steady-state flow curve is non-monotonic and unstable steady states appear in the negative-slope regime. It has been known that self-sustained oscillations of the pressure gradient are obtained when an unstable steady state is perturbed [M. Fyrillas et al., “A mechanism for extrusion instabilities in polymer melts”, Polym. Eng. Sci. 39, No. 12, 2498–2504 (1999; doi:10.1002/pen.11637)].
Treating the simulator of a distributed parameter model describing the dynamics of the above flow as an input-output “black-box” timestepper of the state variables, stable and unstable branches of both equilibrium and periodic oscillating solutions are computed and their stability is examined. It is shown for the first time how equilibrium solutions lose stability to oscillating ones through a subcritical Hopf bifurcation point which generates a branch of unstable limit cycles and how the stable periodic solutions lose their stability through a critical point which marks the onset of the unstable limit cycles. This implicates the coexistence of stable equilibria with stable and unstable periodic solutions in a narrow range of volumetric flow rates.

MSC:

76A05 Non-Newtonian fluids
37N10 Dynamical systems in fluid mechanics, oceanography and meteorology
82D60 Statistical mechanics of polymers

Software:

KELLEY; MATCONT
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Hatztkiriakos, S. G.; Migler, K. B.: Polymer processing instabilities: control and understanding, (2004)
[2] Hyun, J. C.; Kim, H.; Lee, S.; Song, H. -S.; Jung, H. W.: Transient solutions of the dynamics in film blowing processes, J. non-Newtonian fluid mech. 121, 157-162 (2004) · Zbl 1115.76329 · doi:10.1016/j.jnnfm.2004.06.004
[3] Keller, H. B.: Numerical solution of bifurcation and non-linear eigenvalue problems, Applications of bifurcation theory, 359-384 (1977) · Zbl 0581.65043
[4] Beyn, W. J.; Doedel, E. J.: Stability and multiplicity of solutions to discretizations of nonlinear ordinary-differential equations, SIAM J. Sci. stat. Comp. 2, 107-120 (1981) · Zbl 0466.65049 · doi:10.1137/0902009
[5] Parker, T. S.; Chua, L. O.: Practical numerical algorithms for chaotic systems, (1989) · Zbl 0692.58001
[6] Seydel, R.: Practical bifurcation and stability analysis: from equilibrium to chaos, (1994) · Zbl 0806.34028
[7] Govaerts, J. F. W.: Numerical methods for bifurcations of dynamical equilibria, (2000) · Zbl 0935.37054 · doi:10.1137/1.9780898719543
[8] , IMA volumes in mathematics and its applications 119 (2000)
[9] Dhooge, A.; Govaerts, W.; Kuznetsov, Y. A.: MATCONT: a Matlab package for numerical bifurcation analysis of odes, ACM trans. Math. softw. 29, 141-164 (2003) · Zbl 1070.65574 · doi:10.1145/779359.779362
[10] Doedel, E. J.; Govaerts, W.; Kuznetsov, Y. A.; Dhooge, A.: Numerical continuation of branch points of equilibria and periodic orbits, Int. J. Bifurcation chaos 15, 841-860 (2005) · Zbl 1081.37054 · doi:10.1142/S0218127405012491
[11] Govaerts, J. F. W.; Kuznetsov, Y. A.; Dhooge, A.: Numerical continuation of bifurcations of limit cycles in Matlab, SIAM J. Sci. comp. 27, 231-252 (2005) · Zbl 1087.65118 · doi:10.1137/030600746
[12] Kelley, C. T.: Iterative methods for linear and nonlinear equations, (1995) · Zbl 0832.65046
[13] Shroff, G. M.; Keller, H. B.: Stabilization of unstable procedures — the recursive projection method, SIAM J. Numer. anal. 30, 1099-1120 (1993) · Zbl 0789.65037 · doi:10.1137/0730057
[14] Siettos, C. I.; Pantelides, C. C.; Kevrekidis, I. G.: Enabling dynamic process simulators to perform alternative tasks: timestepper based toolkit for computer-aided analysis, Ind. chem. Ind. res. 42, 6795-6801 (2004)
[15] Fyrillas, M.; Georgiou, G. C.: Linear stability diagrams of the shear flow of an Oldroyd-B fluid with slip along the fixed wall, Rheol. acta 37, 61-67 (1998)
[16] Brasseur, E.; Fyrillas, M. M.; Georgiou, G. C.; Crochet, M. J.: The time-dependent extrudate-swell problem of an Oldroyd-B fluid with slip along the wall, J. rheol. 42, 549-566 (1998)
[17] Fyrillas, M. M.; Georgiou, G. C.; Vlassopoulos, D.; Hatzikiriakos, S. G.: A mechanism for extrusion instabilities in polymer melts, Polymer eng. Sci. 39, 2498-2504 (1999)
[18] Georgiou, G.: Stick-slip instability, Polymer processing instabilities: control and understanding, 161-206 (2004)
[19] Taliadorou, E.; Georgiou, G.; Alexandrou, A. N.: A two-dimensional numerical study of the stick-slip extrusion instability, J. non-Newtonian fluid mech. 146, 30-44 (2007) · Zbl 1127.76023 · doi:10.1016/j.jnnfm.2006.11.005
[20] Dubbeldam, J. L. A.; Molenaar, J.: Dynamics of the spurt instability in polymer extrusion, J. non-Newtonian fluid mech. 112, 217-235 (2003) · Zbl 1065.76541 · doi:10.1016/S0377-0257(03)00101-0
[21] Shore, J. D.; Ronis, D.; Piché, L.; Grant, M.: Model for melt fracture instabilities in the capillary flow of polymer melts, Phys. rev. Lett. 77, 655-658 (1996)
[22] Shore, J. D.; Ronis, D.; Piché, L.; Grant, M.: Theory of melt fracture instabilities in the capillary flow of polymer melts, Phys. rev. E 55, 2976-2992 (1997)
[23] Shore, J. D.; Ronis, D.; Piché, L.; Grant, M.: Sharkskin texturing instabilities in the flow of polymer melts, Physica A 239, 350-357 (1997)
[24] Black, W. B.; Graham, M. D.: Wall-slip and polymer melt flow instability, Phys. rev. Lett. 77, 956-959 (1996)
[25] Black, W. B.; Graham, M. D.: Effect of wall slip on the stability of viscoelastic plane shear flow, Phys. fluids 11, 1749-1756 (1999) · Zbl 1147.76328 · doi:10.1063/1.870040
[26] Lust, K.; Roose, D.; Spence, A.; Champneys, A. R.: An adaptive Newton-Picard algorithm with subspace iteration for computing periodic solutions, SIAM J. Sci. comp. 19, 1188-1209 (1998) · Zbl 0915.65088 · doi:10.1137/S1064827594277673
[27] Theodoropoulos, K.; Qian, Y. -H.; Kevrekidis, I. G.: Coarse stability and bifurcation analysis using timesteppers: a reaction-diffusion example, Pnas 97, 9840 (2002) · Zbl 1064.65121 · doi:10.1073/pnas.97.18.9840
[28] Kelley, C. T.; Kevrekidis, I. G.; Qiao, L.: Newton – Krylov solvers for timesteppers, SIAM J. Sci. comp. 19, 1188-1209 (2004)
[29] Möller, J.; Runborg, O.; Kevrekidis, P. G.; Lust, K.; Kevrekidis, I. G.: Effective equations for discrete systems: a timestepper based approach, Int. J. Bifurcation chaos 15, 975-996 (2005) · Zbl 1140.93422 · doi:10.1142/S0218127405012399
[30] Kevrekidis, I. G.; Gear, C. W.; Hyman, J. M.; Kevrekidis, P. G.; Runborg, O.; Theodoropoulos, C.: Equation-free multiscale computation: enabling microscopic simulators to perform system-level tasks, Commun. math. Sci. 1, 715-762 (2003) · Zbl 1086.65066
[31] Cybenko, G.: Just in time learning and estimation, Identification, adaptation, learning, 423-434 (1996) · Zbl 0854.68085
[32] Saad, Y.: Numerical methods for large eigenvalue problems, (1992) · Zbl 0991.65039
[33] Christodoulou, K. N.; Scriven, L. E.: Finding leading modes of a viscous free surface low: an asymmetric generalized problem, J. sci. Comp. 3, 355 (1998) · Zbl 0677.65032 · doi:10.1007/BF01065178
[34] Georgiou, G. C.; Crochet, M. J.: Compressible viscous flow in slits, with slip at the wall, J. rheol. 38, 639-654 (1994)
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