New conception in continuum theory of constitutive equation for anisotropic crystalline polymer liquids

Author(s)
Shifang Han

ABSTRACT

A new continuum theory of the constitutive equation of co-rotational derivative type is developed for anisotropic viscoelastic fluid—liquid crystalline (LC) polymers. A new concept of simple anisotropic fluid is introduced. On the basis of principles of anisotropic simple fluid, stress behaviour is described by velocity gradient tensor and spin tensor instead of the velocity gradient tensor in the classic Leslie—Ericksen continuum theory. Analyzing rheological nature of the fluid and using tensor analysis a general form of the constitutive equ- ation of co-rotational type is established for the fluid. A special term of high order in the equation is introduced by author to describe the sp- ecial change of the normal stress differences which is considered as a result of director tumbling by Larson et al. Analyzing the experimental results by Larson et al., a principle of Non- oscillatory normal stress is introduced which leads to simplification of the problem with relaxation times. The special behaviour of non- symmetry of the shear stress is predicted by using the present model for LC polymer liquids. Two shear stresses in shear flow of LC polymer liquids may lead to vortex and rotation flow, i.e. director tumbling in the flow. The first and second normal stress differences are calculated by the model special behaviour of which is in agree- ment with experiments. In the research, the com- putational symbolic manipulation such as computer software Maple is used. For the anisotropic viscoelastic fluid the constitutive equation theory is of important fundamental significance.

A new continuum theory of the constitutive equation of co-rotational derivative type is developed for anisotropic viscoelastic fluid—liquid crystalline (LC) polymers. A new concept of simple anisotropic fluid is introduced. On the basis of principles of anisotropic simple fluid, stress behaviour is described by velocity gradient tensor and spin tensor instead of the velocity gradient tensor in the classic Leslie—Ericksen continuum theory. Analyzing rheological nature of the fluid and using tensor analysis a general form of the constitutive equ- ation of co-rotational type is established for the fluid. A special term of high order in the equation is introduced by author to describe the sp- ecial change of the normal stress differences which is considered as a result of director tumbling by Larson et al. Analyzing the experimental results by Larson et al., a principle of Non- oscillatory normal stress is introduced which leads to simplification of the problem with relaxation times. The special behaviour of non- symmetry of the shear stress is predicted by using the present model for LC polymer liquids. Two shear stresses in shear flow of LC polymer liquids may lead to vortex and rotation flow, i.e. director tumbling in the flow. The first and second normal stress differences are calculated by the model special behaviour of which is in agree- ment with experiments. In the research, the com- putational symbolic manipulation such as computer software Maple is used. For the anisotropic viscoelastic fluid the constitutive equation theory is of important fundamental significance.

Cite this paper

Han, S. (2010) New conception in continuum theory of constitutive equation for anisotropic crystalline polymer liquids.*Natural Science*, **2**, 948-958. doi: 10.4236/ns.2010.29116.

Han, S. (2010) New conception in continuum theory of constitutive equation for anisotropic crystalline polymer liquids.

References

[1] Han, S.F. (2000) Constitutive equation and computational analytical theory of nonNewtonian Fluids, in Chinese. Science Press, Beijing.

[2] Han, S.F. (2008) Continuum mechanics of anisotropic nonNewtonian fluids—Rheology of liquid crystalline polymer, in Chinese. Science Press, Beijing.

[3] Onogi, S. and Asada T. (1980) Rheology and rheooptics of polymer liquid crystal. In: Astria, G., Marrucci, G. and Nicolai, Eds., Rheology, Plenum, New York, 127147.

[4] Baek, S.G., Magda, J.J. and Larson, R.G. (1993) Rh eological differences among liquidcrystalline polymers I. The first and second normal stress differences of PBG solutions. Journal of Rheology, 37(6), 12011224.

[5] Baek, S.G., Magda, J.J., Larson, R.G. and Hudson, S.D. (1994) Rheological differences among liquidcrystalline polymers II. T Disappearance of negative N1 in densely packed lyotropic and thermotropes. Journal of Rheology, 38(5), 14731503.

[6] Huang, C.M., Magda, J.J. and Larson, R.G. (1999)The effect of temperature and concentration on N1 and tumbling in a liquid crystal polymer. Journal of Rheology, 43(1), 3150.

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

[8] Ericksen, J.L. (1960) Anisotropic fluids. Archive for Rational Mechanics and Analysis, 4(1), 231237.

[9] Ericksen, J.L. (1961) Conservation Laws for Liquid Crystals. Transactions of the Society of Rheology, 5(1), 2334.

[10] Leslie, F.M. (1979) Theory of flow phenomena in liquid crystals. In: Brown, G.H., Ed., Advances in Liquid Crystals, Academic New York, Vol. 1, pp. 181.

[11] Chandrasekhar, S. (1977) Liquid crystals. Cambridge University Press, Cambridge.

[12] Smith, G.F. and Rivlin, R.S. (1957) The anisotropic tensors. Quarterly of Applied Mathematics, Vol. 15, No. 3, 308314.

[13] Green, A.E. (1964) Anisotropic simple fluid. Proceedings of the Royal Society of London A, 279(1379), 437445.

[14] Green, A.E. (1964) A continuum theory of anisotropic fluids. Proceedings of the Cambridge Philosophical Society, 60, 123128.

[15] Volkov, V.S. and Kulichikhin, V.G., (1990) Anisotropic viscoelasticity of liquid crystalline polymers. Journal of Rheology, 34(3), 281293.

[16] Volkov, V.S. and Kulichikhin, V.G. (2000) Non symmetric viscoelasticity of anisotropic polymer liquids. Journal of Rheology, 39(3), 360370.

[17] Larson, R.G. (1993) Rollcell instability in shearing flo ws of nematic polymers. Journal of Rheology, 39(2), 175197.

[18] Han, S.F. (1998) Constitutive equation of Max well Oldroyd type for liquid crystalline polymer and its fluid flow. Proceedings of 3rd International Conference on Fluid Mechanics, Beijing Institute of Technology Press, Beijing, 723728.

[19] Han, S.F. (2001) Constitutive equation of liquid crystalline polymer—anisotropic viscoelastic fluid. Acta Mechanica Sinica, in Chinese, 5, 588600.

[20] Han, S.F. (2004) Constitutive equation of corotational derivative type for anisotropic viscoelastic fluid. Acta Mechanica Sinica, 2, 4653.

[21] Han, S.F. (2007) An unsymmetric constitutive equation for anisotropic viscoelastic fluid. Acta Mechanica Sinica, 2, 4653.

[22] Han, S.F. (2007) A Constitutive equation of coro tational type for liquid crystalline polymer and influe nce of orientation on material functions. Journal of Central South University Technology, 14(Suppl 1), 1418.

[23] Tanner, R.L. (1985) Engineering rheology. Clarendon Press, Oxford.

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

[25] Truesdell, C.A. (1951) New definition of a fluid II. The Maxwell fluid. Journal of Pure and Applied Mathematics, 30(9), 115158.

[1] Han, S.F. (2000) Constitutive equation and computational analytical theory of nonNewtonian Fluids, in Chinese. Science Press, Beijing.

[2] Han, S.F. (2008) Continuum mechanics of anisotropic nonNewtonian fluids—Rheology of liquid crystalline polymer, in Chinese. Science Press, Beijing.

[3] Onogi, S. and Asada T. (1980) Rheology and rheooptics of polymer liquid crystal. In: Astria, G., Marrucci, G. and Nicolai, Eds., Rheology, Plenum, New York, 127147.

[4] Baek, S.G., Magda, J.J. and Larson, R.G. (1993) Rh eological differences among liquidcrystalline polymers I. The first and second normal stress differences of PBG solutions. Journal of Rheology, 37(6), 12011224.

[5] Baek, S.G., Magda, J.J., Larson, R.G. and Hudson, S.D. (1994) Rheological differences among liquidcrystalline polymers II. T Disappearance of negative N1 in densely packed lyotropic and thermotropes. Journal of Rheology, 38(5), 14731503.

[6] Huang, C.M., Magda, J.J. and Larson, R.G. (1999)The effect of temperature and concentration on N1 and tumbling in a liquid crystal polymer. Journal of Rheology, 43(1), 3150.

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

[8] Ericksen, J.L. (1960) Anisotropic fluids. Archive for Rational Mechanics and Analysis, 4(1), 231237.

[9] Ericksen, J.L. (1961) Conservation Laws for Liquid Crystals. Transactions of the Society of Rheology, 5(1), 2334.

[10] Leslie, F.M. (1979) Theory of flow phenomena in liquid crystals. In: Brown, G.H., Ed., Advances in Liquid Crystals, Academic New York, Vol. 1, pp. 181.

[11] Chandrasekhar, S. (1977) Liquid crystals. Cambridge University Press, Cambridge.

[12] Smith, G.F. and Rivlin, R.S. (1957) The anisotropic tensors. Quarterly of Applied Mathematics, Vol. 15, No. 3, 308314.

[13] Green, A.E. (1964) Anisotropic simple fluid. Proceedings of the Royal Society of London A, 279(1379), 437445.

[14] Green, A.E. (1964) A continuum theory of anisotropic fluids. Proceedings of the Cambridge Philosophical Society, 60, 123128.

[15] Volkov, V.S. and Kulichikhin, V.G., (1990) Anisotropic viscoelasticity of liquid crystalline polymers. Journal of Rheology, 34(3), 281293.

[16] Volkov, V.S. and Kulichikhin, V.G. (2000) Non symmetric viscoelasticity of anisotropic polymer liquids. Journal of Rheology, 39(3), 360370.

[17] Larson, R.G. (1993) Rollcell instability in shearing flo ws of nematic polymers. Journal of Rheology, 39(2), 175197.

[18] Han, S.F. (1998) Constitutive equation of Max well Oldroyd type for liquid crystalline polymer and its fluid flow. Proceedings of 3rd International Conference on Fluid Mechanics, Beijing Institute of Technology Press, Beijing, 723728.

[19] Han, S.F. (2001) Constitutive equation of liquid crystalline polymer—anisotropic viscoelastic fluid. Acta Mechanica Sinica, in Chinese, 5, 588600.

[20] Han, S.F. (2004) Constitutive equation of corotational derivative type for anisotropic viscoelastic fluid. Acta Mechanica Sinica, 2, 4653.

[21] Han, S.F. (2007) An unsymmetric constitutive equation for anisotropic viscoelastic fluid. Acta Mechanica Sinica, 2, 4653.

[22] Han, S.F. (2007) A Constitutive equation of coro tational type for liquid crystalline polymer and influe nce of orientation on material functions. Journal of Central South University Technology, 14(Suppl 1), 1418.

[23] Tanner, R.L. (1985) Engineering rheology. Clarendon Press, Oxford.

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

[25] Truesdell, C.A. (1951) New definition of a fluid II. The Maxwell fluid. Journal of Pure and Applied Mathematics, 30(9), 115158.