A new constitutive theory for extrusion-extensional flow of anisotropic liquid crystalline polymer fluid

Author(s)
Shifang Han

ABSTRACT

A new continuum theory of the constitutive equation of co-rotational derivative type was developed by the author for anisotropic viscoelastic fluid－liquid crystalline (LC) polymers (S.F. Han, 2008, 2010) . This paper is a continuation of the recent publication [1] to study extrusion-extensional flow of the fluid. A new concept of simple anisotropic fluid is introduced. On the basis of anisotropic simple fluid, stress behavior is described by velocity gradient tensor F and spin tensor W instead of the velocity gradient tensor D in the classic Leslie?Ericksen continuum theory. A special form of the constitutive equation of the co-rotational type is established for the fluid. Using the special form of the constitutive equation in components a computational analytical theory of the extrusion-extensional flow is developed for the LC polymer liquids - anisotropic viscoelastic fluid. Application of the constitutive theory to the flow is successful in predicting bifurcation of elongational viscosity and contraction of extrudate for LC polymer liquids–anisotropic viscoelastic fluid. The contraction of extrudate of LC polymer liquids may be associated with the stored elastic energy conversion into that necessary for bifurcation of elongational viscosity in extrusion extensional flow of the fluid.

A new continuum theory of the constitutive equation of co-rotational derivative type was developed by the author for anisotropic viscoelastic fluid－liquid crystalline (LC) polymers (S.F. Han, 2008, 2010) . This paper is a continuation of the recent publication [1] to study extrusion-extensional flow of the fluid. A new concept of simple anisotropic fluid is introduced. On the basis of anisotropic simple fluid, stress behavior is described by velocity gradient tensor F and spin tensor W instead of the velocity gradient tensor D in the classic Leslie?Ericksen continuum theory. A special form of the constitutive equation of the co-rotational type is established for the fluid. Using the special form of the constitutive equation in components a computational analytical theory of the extrusion-extensional flow is developed for the LC polymer liquids - anisotropic viscoelastic fluid. Application of the constitutive theory to the flow is successful in predicting bifurcation of elongational viscosity and contraction of extrudate for LC polymer liquids–anisotropic viscoelastic fluid. The contraction of extrudate of LC polymer liquids may be associated with the stored elastic energy conversion into that necessary for bifurcation of elongational viscosity in extrusion extensional flow of the fluid.

Cite this paper

nullHan, S. (2011) A new constitutive theory for extrusion-extensional flow of anisotropic liquid crystalline polymer fluid.*Natural Science*, **3**, 307-318. doi: 10.4236/ns.2011.34040.

nullHan, S. (2011) A new constitutive theory for extrusion-extensional flow of anisotropic liquid crystalline polymer fluid.

References

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[2] Han Shifang, (2000), Constitutive equation and computational analytical theory of Non-Newtonian Fluids, Beijing: Science Press, in Chinese.

[3] Han Shifang, (2008), Continuum mechanics of anisotropic non-Newtonian fluids — Rheology of liquid crystalline polymer, Beijing: Science Press, in Chinese.

[4] Wagner M.H. A, Th.Ixner, and K. Geiger, A note on the melt strength of liquid crystalline polymer, J. Rheol. 1997, 41(5) 1087-1093

[5] F.P. Mantia and A. Valenza, (1989) ,Shear and nonisothermal elongational characterization of a liquid crystalline polymer, Polymer Engineering and Science, V. 29 No. 10 625-631

[6] Gotsis, A.D. and Odriozola, M.A., (2000), Extensional viscosity of a thermotropic liquid crystalline polymer J.Rheol., 44(5): 1205-1225.

[7] Baek S.-G., Magda J. J. and Larson R.G., (1993), Rheo- logical differences among liquid-crystalline polymers I. The first and second normal stress differences of PBG solutions J. of Rheology.

[8] Baek S.-G.,Magda J. J. ,Larson R.G. and Hudson S. D, (1994),Rheological differences among liquid-crystalline polymers II. T Disappearance of negative N1 in densely packed lyotropic and thermotropes J. of Rheology, 38(5) 1473-1503

[9] Huang C.M., J. J. Magda , R.G. Larson, (1999), The effect of temperature and concentration on N1 and tumbling in a liquid crystal polymer, J. Rheol. 43 No 1, p31-50

[10] Doi,M. and Edwards S.F., (1986), The Theory of Polymer Dynamics, Oxford London.

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[13] Leslie F.M., (1979), Theory of Flow Phenomena in Liquid Crystals, in Advances in Liquid Crystals ed. by G.H. Brown Academic New York , Vo. 1 p. 1

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[15] Smith G.F. & Rivlin R.S. (1957), The anisotropic tensors, Quart. Appl. Math. 15, 308-314.

[16] Green A.E., (1964), Anisotropic simple fluid, Proc. Roy. Soc. Londonser, A 279, p 437-445.

[17] Green A.E., (1964), A continuum theory of anisotropic fluids, Proc. Camb. Phil. Soc., 60 p 123-128.

[18] Volkov V.S. and Kulichikhin V.G., (1990), Anisotropic Viscoelasticity of Liquid Crystalline Polymers, J. Rheol., 34(3) 281-293.

[19] Volkov V.S.and Kulichikhin V.G., (2000), Non-symmetric viscoelasticity of anisotropic polymer liquids, J. Rheol., 39(3) 360-370.

[20] R.G. Larson, (1993), Roll-cell instability in shearing flows of nematic polymers, J. of Rheology, 39, 2, Mar/ Apr., 175-197.

[21] Han Shifang, (2001), Constitutive equation of liquid crystalline polymer - anisotropic viscoelastic fluid Beijing: Acta Mechanica Sinica, No 5, p 588-600, in Chinese

[22] Han Shifang, (2004), Constitutive equation of co-rotational derivative type for anisotropic viscoelastic fluid Beijing: Acta Mechanica Sinica, No: 2 p 46-53.

[23] Han Shifang, (2007), An unsymmetric constitutive equation for anisotropic viscoelastic fluid, Beijing: Acta Me- chanica Sinica, No: 2 p 46-53.

[24] Han Shifang, (2008), Research advances of un-symme- tric constitutive theory of anisotropic viscoelastic liquids and its hydrodynamic Behavior, J. of central South Univ. Technol. v. 15 Suppl. 1. p 1-4 Springer.

[25] R. L. Tanner (1985), Engineering Rheology, Clarendon Press. Oxford.

[26] Zahorski S., (1982), Mechanics of viscoelastic fluids, Martinus Nijhoff Publishers, The Hague/Boston/London.

[27] Truesdell,C. (1951),A new definition of a fluid II. The Maxwell fluid. J. Math. Pure Appl. (9)30, 115-158.

[1] Han Shifang (2010) New conception in continuum theory of constitutive equation for anisotropic crystalline polymer liquids, Natural Science 2(9) p 948-958

[2] Han Shifang, (2000), Constitutive equation and computational analytical theory of Non-Newtonian Fluids, Beijing: Science Press, in Chinese.

[3] Han Shifang, (2008), Continuum mechanics of anisotropic non-Newtonian fluids — Rheology of liquid crystalline polymer, Beijing: Science Press, in Chinese.

[4] Wagner M.H. A, Th.Ixner, and K. Geiger, A note on the melt strength of liquid crystalline polymer, J. Rheol. 1997, 41(5) 1087-1093

[5] F.P. Mantia and A. Valenza, (1989) ,Shear and nonisothermal elongational characterization of a liquid crystalline polymer, Polymer Engineering and Science, V. 29 No. 10 625-631

[6] Gotsis, A.D. and Odriozola, M.A., (2000), Extensional viscosity of a thermotropic liquid crystalline polymer J.Rheol., 44(5): 1205-1225.

[7] Baek S.-G., Magda J. J. and Larson R.G., (1993), Rheo- logical differences among liquid-crystalline polymers I. The first and second normal stress differences of PBG solutions J. of Rheology.

[8] Baek S.-G.,Magda J. J. ,Larson R.G. and Hudson S. D, (1994),Rheological differences among liquid-crystalline polymers II. T Disappearance of negative N1 in densely packed lyotropic and thermotropes J. of Rheology, 38(5) 1473-1503

[9] Huang C.M., J. J. Magda , R.G. Larson, (1999), The effect of temperature and concentration on N1 and tumbling in a liquid crystal polymer, J. Rheol. 43 No 1, p31-50

[10] Doi,M. and Edwards S.F., (1986), The Theory of Polymer Dynamics, Oxford London.

[11] Ericksen J.L. (1960) ,Anisotropic fluids, Arch. Rational Mech. Anal. 4, p231-237.

[12] Ericksen J.L.,(1961)Conversation laws for liquid crystalls. Transaction of Society of Rheology, 5(1), 23-34.

[13] Leslie F.M., (1979), Theory of Flow Phenomena in Liquid Crystals, in Advances in Liquid Crystals ed. by G.H. Brown Academic New York , Vo. 1 p. 1

[14] Chandrasekhar S., Liquid Crystals, (1977), Cambridge Univ. Press.

[15] Smith G.F. & Rivlin R.S. (1957), The anisotropic tensors, Quart. Appl. Math. 15, 308-314.

[16] Green A.E., (1964), Anisotropic simple fluid, Proc. Roy. Soc. Londonser, A 279, p 437-445.

[17] Green A.E., (1964), A continuum theory of anisotropic fluids, Proc. Camb. Phil. Soc., 60 p 123-128.

[18] Volkov V.S. and Kulichikhin V.G., (1990), Anisotropic Viscoelasticity of Liquid Crystalline Polymers, J. Rheol., 34(3) 281-293.

[19] Volkov V.S.and Kulichikhin V.G., (2000), Non-symmetric viscoelasticity of anisotropic polymer liquids, J. Rheol., 39(3) 360-370.

[20] R.G. Larson, (1993), Roll-cell instability in shearing flows of nematic polymers, J. of Rheology, 39, 2, Mar/ Apr., 175-197.

[21] Han Shifang, (2001), Constitutive equation of liquid crystalline polymer - anisotropic viscoelastic fluid Beijing: Acta Mechanica Sinica, No 5, p 588-600, in Chinese

[22] Han Shifang, (2004), Constitutive equation of co-rotational derivative type for anisotropic viscoelastic fluid Beijing: Acta Mechanica Sinica, No: 2 p 46-53.

[23] Han Shifang, (2007), An unsymmetric constitutive equation for anisotropic viscoelastic fluid, Beijing: Acta Me- chanica Sinica, No: 2 p 46-53.

[24] Han Shifang, (2008), Research advances of un-symme- tric constitutive theory of anisotropic viscoelastic liquids and its hydrodynamic Behavior, J. of central South Univ. Technol. v. 15 Suppl. 1. p 1-4 Springer.

[25] R. L. Tanner (1985), Engineering Rheology, Clarendon Press. Oxford.

[26] Zahorski S., (1982), Mechanics of viscoelastic fluids, Martinus Nijhoff Publishers, The Hague/Boston/London.

[27] Truesdell,C. (1951),A new definition of a fluid II. The Maxwell fluid. J. Math. Pure Appl. (9)30, 115-158.