On the stick-slip flow from slit and cylindrical dies of a Phan-Thien and Tanner fluid model. II. Linear stability analysis

Physics of Fluids

Volumn 25, Issue 9, 2013, Pages

  Karapetsas, George ^a   Tsamopoulos, John ^b  

^a UNIVERSITY OF THESSALY   (Greece)

^b UNIVERSITY OF PATRAS   (Greece)

Author keywords

[No Author keywords available]

Indexed keywords

DIES; EXTRUSION; HOPF BIFURCATION; PERIODIC STRUCTURES; REYNOLDS EQUATION; REYNOLDS NUMBER; SLIP FORMING; STICK-SLIP; TENSILE STRENGTH; VISCOELASTICITY;

EXTRUSION PROCESS; FLOW INSTABILITIES; PHAN-THIEN TANNER MODELS; RHEOLOGICAL PARAMETER; SPATIALLY PERIODIC STRUCTURE; VIS-COELASTIC FLUIDS; VISCOELASTIC MODELING; WEISSENBERG NUMBER;

LINEAR STABILITY ANALYSIS;

EID: 84885064301     PISSN: 10706631     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.4821805     Document Type: Article

References (58)

1. Petrie C.J. S. Denn M.M. Instabilities in polymer processing. AIChE J. 1976, 22:209-236. 10.1002/aic.690220202
2. Denn M.M. Issues in viscoelastic fluid mechanics. Annu. Rev. Fluid Mech. 1990, 22:13-34. 10.1146/annurev.fl.22.010190.000305
3. Denn M.M. Extrusion instabilities and wall slip. Annu. Rev. Fluid Mech. 2001, 33:265-287. 10.1146/annurev.fluid.33.1.265
4. Larson R.G. Instabilities in viscoelastic flows. Rheol. Acta 1992, 31:213-263. 10.1007/BF00366504
5. Graham M.D. The sharkskin instability of polymer melt flows. Chaos 1999, 9:154-163. 10.1063/1.166386
6. Hatzikiriakos S.G. Migler K. Polymer Processing Instabilities: Control and Understanding 2005, and (Marcel Dekker, New York).
7. Koopmans R. den Doelder J. Molenaar J. Polymer Melt Fracture 2011, and (CRC Press, Boca Raton).
8. Ramamurthy A.V. Wall slip in viscous fluids and influence of materials of construction. J. Rheol. 1986, 30:337-357. 10.1122/1.549852
9. Kalika D.S. Denn M.M. Wall slip and extrudate distortion in linear low-density polyethylene. J. Rheol. 1987, 31:815-834. 10.1122/1.549942
10. Piau J.M. El Kissi N. Tremblay B. Influence of upstream instabilities and wall slip on melt fracture and sharkskin phenomena during silicones extrusion trough orifice dies. J. Non-Newtonian Fluid Mech. 1990, 34:145-180. 10.1016/0377-0257(90)80016-S
11. El Kissi N. Piau J.M. Stability phenomena during polymer melt extrusion. J. Rheol. 1994, 38:1447-1463. 10.1122/1.550552
12. El Kissi N. Piau J.M. Toussaint F. Sharkskin and cracking of polymer melt extrudates. J. Non-Newtonian Fluid Mech. 1997, 68:271-290. 10.1016/S0377-0257(96)01507-8
13. Pomar G. Muller S.J. Denn M.M. Extrudate distortions in linear low-density polyethylene solutions and melt. J. Non-Newtonian Fluid Mech. 1994, 54:143-151. 10.1016/0377-0257(94)80019-7
14. Venet C. Vergnes B. Experimental characterizations of sharkskin in polyethylenes. J. Rheol. 1997, 41:873-892. 10.1122/1.550837
15. Cogswell F.N. Stretching flow instabilities at the exits of extrusion dies. J. Non-Newtonian Fluid Mech. 1977, 2:37-47. 10.1016/0377-0257(77)80031-1
16. Georgiou G.C. Crochet M.J. Time-dependent compressible extrudate-swell problem with slip at the wall. J. Rheol. 1994, 38:1745-1755. 10.1122/1.550524
17. 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. 1998, 42:549-566. 10.1122/1.550959
18. Migler K.B. Son Y. Qiao F. Flynn K. Extensional deformation, cohesive failure and boundary conditions during sharkskin melt fracture. J. Rheol. 2002, 46:383-400. 10.1122/1.1445186
19. McLeish T.C. B. Ball R.C. A molecular approach to the spurt effect in polymer melt flow. J. Polym. Sci. 1986, 24:1735. 10.1002/polb.1986.090240809
20. Georgiou G.C. Vlassopoulos D. On the stability of the simple shear flow of a Johnson-Segalman fluid. J. Non-Newtonian Fluid Mech. 1998, 75:77-97. 10.1016/S0377-0257(97)00078-5
21. Adewale K.E. P. Leonov A.I. On modelling spurt flows of polymers. J. Non-Newtonian Fluid Mech. 1993, 49:133-138. 10.1016/0377-0257(93)85027-8
22. Shaqfeh E.S. G. Purely elastic instabilities in viscometric flows. Annu. Rev. Fluid Mech. 1996, 28:129-185. 10.1146/annurev.fl.28.010196.001021
23. Black W.B. Graham M.D. Effect of wall slip on the stability of viscoelastic plane shear flow. Phys. Fluids 1999, 11:1749-1756. 10.1063/1.870040
24. Karapetsas G. Tsamopoulos J. On the stick-slip flow from slit and cylindrical dies of a Phan-Thien and Tanner fluid model. I. Steady state. Phys. Fluids 2009, 21:123101. 10.1063/1.3271495
25. Natarajan R. Acrivos A. The instability of the steady flow past sphere and disks. J. Fluid Mech. 1993, 254:323-344. 10.1017/S0022112093002150
26. Sureshkumar R. Beris A.N. Linear stability analysis of viscoelastic Poiseuille flow using an Arnoldi-based orthogonalization algorithm. J. Non-Newtonian Fluid Mech. 1995, 56:151-182. 10.1016/0377-0257(94)01279-Q
27. Carvalho M.S. Scriven L.E. Three-dimensional stability analysis of free surface flows: Application to forward deformable roll coating. J. Comput. Phys. 1999, 151:534-562. 10.1006/jcph.1999.6195
28. Grillet A.M. Bogaerds A.C. B. Peters G.W. M. Baaijens F.P. T. Stability analysis of constitutive equations for polymer melts in viscometric flows. J. Non-Newtonian Fluid Mech. 2002, 103:221-250. 10.1016/S0377-0257(02)00005-8
29. Smith M.D. Armstrong R.C. Brown R.A. Sureshkumar R. Finite element analysis of two-dimensional viscoelastic flows to three-dimensional perturbations. J. Non-Newtonian Fluid Mech. 2000, 93:203-244. 10.1016/S0377-0257(00)00124-5
30. Karapetsas G. Matar O.K. Valluri P. Sefiane K. Convective rolls and hydrothermal waves in evaporating sessile drops. Langmuir 2012, 28:11433. 10.1021/la3019088
31. Phan-Thien N. Tanner R.I. A new constitutive equation derived from network theory. J. Non-Newtonian Fluid Mech. 1977, 2:353-365. 10.1016/0377-0257(77)80021-9
32. Phan-Thien N. A nonlinear network viscoelastic model. J. Rheol. 1978, 22:259-283. 10.1122/1.549481
33. Brown R.A. Szady M.J. Northey P.J. Armstrong R.C. On the numerical stability of mixed finite-element methods for viscoelastic flows governed by differential constitutive equations. Theor. Comput. Fluid Dyn. 1993, 5:77-106. 10.1007/BF00311812
34. Saad Y. Variations of Arnoldi's method for computing eigenelements of large unsymmetric matrices. Linear Algebr. Appl. 1980, 34:269-295. 10.1016/0024-3795(80)90169-X
35. Lehoucq R.B. Sorensen D.C. Yang C. ARPACK User's Guide: Solution of Large Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods 1998, and (SIAM, Philadelphia, PA).
36. Natarajan R. An Arnoldi-based iterative scheme for nonsymmetric matrix pencils arising in finite element stability problems. J. Comput. Phys. 1992, 100:128-142. 10.1016/0021-9991(92)90315-P
37. Schenk O. Gärtner K. Solving unsymmetric sparse systems of linear equations with PARDISO. Future Gener. Comput. Syst. 2004, 20:475-487. 10.1016/j.future.2003.07.011
38. Schenk O. Gärtner K. On fast factorization pivoting methods for symmetric indefinite systems. Electron. Trans. Numer. Anal. 2006, 23:158-179.
39. Gorodtsov V.A. Leonov A.I. On a linear instability of a plane parallel Couette flow of viscoelastic fluid. J. Appl. Math. Mech. 1967, 31:310-319. 10.1016/0021-8928(67)90156-6
40. Renardy M. Renardy Y. Linear stability of plane Couette flow of an upper convected Mawell fluid. J. Non-Newtonian Fluid Mech. 1986, 22:23-33. 10.1016/0377-0257(86)80002-7
41. Wilson H.J. Renardy M. Renardy Y. Structure of the spectrum in zero Reynolds number shear flow of the UCM and Oldroyd-B liquids. J. Non-Newtonian Fluid Mech. 1999, 80:251-268. 10.1016/S0377-0257(98)00087-1
42. Graham M.D. Effect of axial flow on viscoelastic Taylor-Couette instability. J. Fluid Mech. 1998, 360:341-374. 10.1017/S0022112098008611
43. Bogaerds A.C. B. 2009, personal communication.
44. Palmer A.S. Phillips T.N. Spectra approximation of Phan-Thien Tanner liquids. Numer. Algorithms 2005, 38:133-153. 10.1007/BF02810620
45. Keiller R.A. Numerical instability of time dependent flows. J. Non-Newtonian Fluid Mech. 1992, 43:229-246. 10.1016/0377-0257(92)80026-T
46. Papanastasiou T.C. Malamataris N. Ellwood K. A new outflow boundary condition. Int. J. Numer. Methods Fluids 1992, 14:587-608. 10.1002/fld.1650140506
47. Mackley M.R. Rutgers R.P. Gilbert D.G. Surface instabilities during the extrusion of linear low density polyethylene. J. Non-Newtonian Fluid Mech. 1998, 76:281-297. 10.1016/S0377-0257(97)00122-5
48. Vinogradov G.V. Malkin A. Rheology of Polymers 1980, and (Springer, Berlin).
49. Sornberger G. Quantin J.C. Fajolle R. Vergnes B. Agassant J.F. Experimental study of the sharkskin defect in linear low density polyethylene. J. Non-Newtonian Fluid Mech. 1987, 23:123-135. 10.1016/0377-0257(87)80014-9
50. Joo Y.L. Shaqfeh E.S. G. Viscoelastic Poiseuille flow through a curved channel: A new elastic instability. Phys. Fluids A 1991, 3:1691-1694. 10.1063/1.858223
51. Joo Y.L. Shaqfeh E.S. G. A purely elastic instability in Dean and Taylor-Dean flow. Phys. Fluids A 1992, 4:524-543. 10.1063/1.858325
52. Byars J.A. Oztekin A. Brown R.A. McKinley G.H. Spiral instabilities in the flow of highly elastic fluids between rotating parallel disks. J. Fluid Mech. 1994, 271:173-218. 10.1017/S0022112094001734
53. Ganpule H.K. Khomami B. An investigation of interfacial instabilities in the superposed channel flow of viscoelastic fluids. J. Non-Newtonian Fluid Mech. 1999, 81:27-69. 10.1016/S0377-0257(98)00088-3
54. Smith M.D. Joo Y.L. Armstrong R.C. Brown R.A. Linear stability analysis of flow of an Oldroyd-B fluid through a linear array of cylinders. J. Non-Newtonian Fluid Mech. 2003, 109:13-50. 10.1016/S0377-0257(02)00162-3
55. Drazin P.G. Reid W.H. Hydrodynamic Stability 1981, and (Cambridge University Press, Cambridge, UK).
56. Preziosi L. Chen K. Joseph D.D. Lubricated pipelining: stability of the core-annular flow. J. Fluid Mech. 1989, 201:323-356. 10.1017/S0022112089000960
57. Miller J.C. Rallison J.M. Instability of coextruded elastic liquids at high Weissenberg number. J. Non-Newtonian Fluid Mech. 2007, 143:88-106. 10.1016/j.jnnfm.2007.01.008
58. Karapetsas G. Tsamopoulos J. Steady extrusion of viscoelastic materials from an annular die. J. Non-Newtonian Fluid Mech. 2008, 154:136-152. 10.1016/j.jnnfm.2008.04.007