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Differential Models of Rheologically Nonstationary Fluids

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Journal of Engineering Physics and Thermophysics Aims and scope

An analysis has been made of basic differential models of rheologically nonstationary (viscoelastic) fluids as well as of their development and interrelation. The considered models cover a rich variety of viscoelastic media: polymer solutions and melts, natural formations (glaciers), and others. Among these models, a key role is played by the Maxwell upper convection model: it provides a theoretical basis for experimental determination of the dynamic characteristics of viscoelastic fluids and for development of new rheological models. It has been shown that to improve the reliability of results obtained with the aid of a complex rheological model, it is expedient to ensure a possibility of reducing it to the existing models and thus finding analytical solutions for a number of the simplest flows. Examples of use of the models in question when results of rheometric investigations are approximated and viscoelastic-fluid flows are calculated have been given. Special emphasis has been placed on an analysis of the correspondence of the derived solutions to the physical essence of the described processes, and also of the correctness of interpretation of some results or others.

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References

  1. W. L. Wilkinson, Non-Newtonian Fluids [Russian translation], Mir, Moscow (1964).

    Google Scholar 

  2. A. Pichler, A. Pozderović, and J. Pavlović, Influence of sugars, modified starches, and hydrocolloids additions on the rheological properties of raspberry cream filling, Czech J. Food Sci., 30, No. 3, 227–235 (2012).

    Article  Google Scholar 

  3. V. Yu. Konyukhov, Polymers and Colloidal Systems [in Russian], Moskovskii Gos. Univ. Pechati, Moscow (1999).

    Google Scholar 

  4. P. Coussot, Mudflow Rheology and Dynamics, IAHR Monograph Series, Balkema Publishers, Rotterdam (1997).

    MATH  Google Scholar 

  5. E. Mitsoulis, Flow of viscoplastic materials: Models and computations, in: Rheol. Rev.: The British Society of Rheology, 135–178 (2007).

  6. I. Fridtjov, Rheology and Non-Newtonian Fluids, Springer, Cham, Switzerland (2014).

    MATH  Google Scholar 

  7. A. V. Gnoevoi, D. M. Klimov, and V. M. Chesnokov, Foundations of the Theory of Flows of Bingham Media [in Russian], Fizmatlit, Moscow (2004).

    Google Scholar 

  8. E. A. Kirsanov and V. N. Matveenko, Non-Newtonian Behavior of Structured Systems [in Russian], Tekhnosfera, Moscow (2016).

    Google Scholar 

  9. G. V. Vinogradov and A. Ya. Malkin, The Rheology of Polymers [in Russian], Khimiya, Moscow (1977).

    Google Scholar 

  10. M. Renardy, Mathematical analysis of viscoelastic flows, Annu. Rev. Fluid Mech., 21, No. 1, 21–36 (1989).

    Article  MathSciNet  MATH  Google Scholar 

  11. J. G. Oldroyd, On the formulation of rheological equations of state, Proc. Roy. Soc. London, Ser. A, A200, 523–541 (1950).

    MathSciNet  MATH  Google Scholar 

  12. M. Renardy, Mathematical analysis of viscoelastic fluids, in: C. M. Dafermos and M. Pocorny (Eds.), Handbook of Differential Equations: Evolutionary Equations, Vol. 4, Elsevier, Amsterdam (2008), Chap. 5, pp. 229–266.

  13. G. Schramm, A Practical Approach to Rheology and Rheometry, Gebruder HAAKE GmbH, Karlsruhe (1998).

    Google Scholar 

  14. T. G. Mezger, The Rheology Handbook, Vincentz Network, Hannover (2014).

    Google Scholar 

  15. G. M. Bartenev and S. Ya. Frenkel’, Polymer Physics [in Russian], Khimiya, Leningrad (1990).

    Google Scholar 

  16. T. Kajiwara, G. Barakos, and E. Mitsoulis, Rheological characterization of polymer solutions and melts with an integral constitutive equation, Int. J. Polym. Anal. Charact., 1, No. 3, 201–215 (1995).

    Article  Google Scholar 

  17. S. G. Hatzikiriakos and E. Mitsoulis, Capillary extrusion flow of a fluoropolymer melt, Int. J. Mater. Form., 6, No. 1, 29–40 (2013).

    Article  Google Scholar 

  18. M. A. Barrera, J. F. Vega, M. Aguilar, and J. Martinez-Salazar, Melt flow index on high molecular weight polyethylene: A comparative study of experiments and simulation, J. Mater. Process. Technol., 174, 171–177 (2006).

    Article  Google Scholar 

  19. J. R. Funk, G. W. Hall, R. Crandall, and W. D. Pilkey, Linear and quasi-linear viscoelastic characterization of ankle ligaments, J. Biomech. Eng., 122, No. 2, 15–22 (2000).

    Article  Google Scholar 

  20. J. A. Shashkina, O. E. Philippova, Y. D. Zaroslov, A. R. Khokhlov, T. A. Pryakhina, and I. V. Blagodatskikh, Rheology of viscoelastic solutions of cationic surfactant. Effect of added associating polymer, Langmuir, 21, No. 4, 1524–1530 (2005).

    Article  Google Scholar 

  21. B. Yesilata, C. Clasen, and G. H. McKinley, Nonlinear shear and extensional flow dynamics of wormlike surfactant solutions, J. Non-Newtonian Fluid Mech., 133, 73–90 (2006).

    Article  Google Scholar 

  22. H. Jeffreys, The Earth: Its Origin, History, and Physical Constitution, 6th edn., Cambridge University Press, London (1976).

    MATH  Google Scholar 

  23. A. D. Polyanin and A. V. Vyazmin, Decomposition of three-dimensional linearized equations for Maxwell and Oldroyd viscoelastic fluids and their generalizations, Theor. Found. Chem. Eng., 47, No. 4, 321–329 (2013).

    Article  Google Scholar 

  24. S. B. Pillapakkam, P. Singh, D. Blackmore, and N. Aubry, Transient and steady state of a rising bubble in a viscoelastic fluid, J. Fluid Mech., 589, 215–252 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  25. O. M. Sokovnin, N. V. Zagoskina, and S. N. Zagoskin, Hydrodynamics of motion of spherical particles, droplets and bubbles in a non-Newtonian liquid: Experimental studies, Theor. Found. Chem. Eng., 47, No. 4, 356–367 (2013).

    Article  Google Scholar 

  26. D. Funfschilling and H. Z. Li, Flow of non-Newtonian fluids around bubbles: PIV measurements and birefringence visualization, Chem. Eng. Sci., 56, 1137–1141 (2001).

    Article  Google Scholar 

  27. J. R. Herrera-Velarde, R. Zenit, D. Chehata, and B. Mena, The flow of non-Newtonian fluids around bubbles and its connection to the jump discontinuity, J. Non-Newtonian Fluid Mech., 111, Nos. 2–3, 199–209 (2003).

  28. B. D. Coleman and W. Noll, An approximation theorem for functionals with applications in continuum mechanics, Arch. Ration. Mech. Anal., 6, No. 5, 355–370 (1960).

    Article  MathSciNet  MATH  Google Scholar 

  29. B. D. Coleman and W. Noll, Foundation of linear viscoelasticity, Rev. Mod. Phys., 33, No. 2, 239–249 (1961).

    Article  MathSciNet  MATH  Google Scholar 

  30. D. D. Joseph, Instability of the rest state of fluids of arbitrary grade greater than one, Arch. Ration. Mech. Anal., 75, No. 3, 251–256 (1981).

    Article  MathSciNet  MATH  Google Scholar 

  31. J. Dunwoody, On the stability of the rest state of fluids of complexity n, Proc. Roy. Soc. London, Ser. A, A378, 437–444 (1981).

    Article  MathSciNet  MATH  Google Scholar 

  32. R. S. Rivlin and J. L. Ericksen, Stress deformation relations for isotropic materials, J. Ration. Mech. Anal., 4, 323–425 (1955).

    MathSciNet  MATH  Google Scholar 

  33. J. E. Dunn and R. L. Fosdick, Thermodynamics, stability and boundedness of fluids of complexity 2 and fluids of second grade, Arch. Ration. Mech. Anal., 56, No. 3, 191–252 (1974).

    Article  MathSciNet  MATH  Google Scholar 

  34. R. L. Fosdick and K. R. Rajagopal, Anomalous features in the model of "second order fluids," Arch. Ration. Mech. Anal., 70, No. 2, 145–152 (1979).

    Article  MathSciNet  MATH  Google Scholar 

  35. K. R. Rajagopal, T. Y. Na, and A. S. Gupta, Flow of a viscoelastic fluid over a stretching sheet, Rheol. Acta, 23, No. 2, 213–215 (1984).

    Article  Google Scholar 

  36. M. Norouzi, M. H. Kayhani, M. R. H. Nobari, and F. Talebi, Analytical investigation of viscoelastic creeping flow and heat transfer inside a curved rectangular duct, Theor. Found. Chem. Eng., 45, No. 1, 53–67 (2011).

    Article  Google Scholar 

  37. D. F. McTigue, S. L. Passman, and S. J. Jones, Normal stress effects in the creep of ice, J. Glaciol., 31, No. 108, 120–126 (1985).

    Article  Google Scholar 

  38. Ch.V. R. Murthy, K. Gowthami, K. R. Kavitha, and P. S. Kumari, Flow of a second order fluid over an inclined rigid plane, IJ-ETA-ETS, 3, No. 1, 13–17 (2010).

  39. F. Carapau, Axisymmetric motion of a generalized Rivlin–Ericksen fluid with heat-dependent normal stress coefficients, Int. J. Math. Mod. Meth. Appl. Sci., 2, No. 2, 168–175 (2008).

    Google Scholar 

  40. P. Riesen, K. Hutter, and M. Funk, A viscoelastic Rivlin–Ericksen material model applicable to glacier ice, Nonlinear Proc. Geophys., 17, No. 6, 673–684 (2010).

    Article  Google Scholar 

  41. H. Giesekus, A unified approach to a variety of constitutive models for polymer fluids based on the concept of a configuration dependent molecular mobility, Rheol. Acta, 21, No. 4, 366–375 (1982).

    Article  MATH  Google Scholar 

  42. H. Giesekus, A simple constitutive equation for polymer fluids based on the concept of deformation dependent tensorial mobility, J. Non-Newtonian Fluid Mech., 11, No. 1, 69–109 (1982).

    Article  MATH  Google Scholar 

  43. C. Wilfried, Configurational and rheological properties of cyclic polymers, J. Chem. Soc. Faraday Trans., 91, No. 16, 2525-2530 (1995).

    Article  Google Scholar 

  44. C. Rauwendaal, Polymer Extrusion, 4th edn., Hanser Publications, Munich (2001).

    Google Scholar 

  45. A. Ya. Malkin and A. I. Isaev, Rheology: Concepts, Methods, and Applications [in Russian], Professiya, St. Petersburg (2007).

    Google Scholar 

  46. M. W. Johnson and D. Segalman, A model for viscoelastic fluid behavior which allows non-affine deformation, J. Non-Newtonian Fluid Mech., 2, No. 3, 255–270 (1977).

    Article  MATH  Google Scholar 

  47. G. C. Georgiou and D. Vlassopoulos, On the stability of the simple shear flow of a Johnson–Segalman fluid, J. Non-Newtonian Fluid Mech., 75, No. 1, 77–97 (1998).

    Article  MATH  Google Scholar 

  48. K. Atalik and R. Keunings, On the occurrence of even harmonics in the shear stress response of viscoelastic fluids in large amplitude oscillatory shear, J. Non-Newtonian Fluid Mech., 122, Nos. 1–3, 107–116 (2004).

    Article  MATH  Google Scholar 

  49. P. D. Olmsted and O. Radulescu, Jonson–Segalman model with a diffusion term in cylindrical Couette flow, J. Rheol., 44, No. 2, 257–275 (2000).

    Article  Google Scholar 

  50. J. F. Berret, D. C. Roux, and G. Porte, Isotropic-to-nematic transition in wormlike micelles under shear, J. Phys. II, 4, No. 8, 1261–1279 (1994).

    Google Scholar 

  51. P. T. Callaghan, M. E. Cates, C. J. Rofe, and J. A. F. Smeulders, A study of the spurt effect in wormlike micelles using nuclear magnetic resonance microscopy, J. Phys. II, 6, No. 3, 375–393 (1996).

    Google Scholar 

  52. C. Grand, J. Arrault, and M. E. Cates, Slow transients and metastability in wormlike micelle rheology, J. Phys. II, 7, No. 8, 1071–1086 (1997).

    Google Scholar 

  53. Y. C. Hon, S. Li, and M. Huang, A meshless computational method for the shear flow of Johnson–Segalman fluid, Int. J. Comput. Meth. Eng. Sci. Mech., 6, No. 1, 59–64 (2005).

    Article  Google Scholar 

  54. J. S. Howell, Numerical Approximation of Shear-Thinning and Johnson–Segalman Viscoelastic Fluid Flows, Ph.D. Dissertation, The Graduate School of Clemson Univ., Clemson, South Carolina, USA, (2007).

  55. N. Phan-Thien and R. I. Tanner, A new constitutive equation derived from network theory, J. Non-Newtonian Fluid Mech., 2, No. 4, 353–365 (1977).

    Article  MATH  Google Scholar 

  56. N. Phan-Thien, A non-linear network viscoelastic model, J. Rheol., 22, No. 3, 259–283 (1978).

    Article  MATH  Google Scholar 

  57. M. A. Alves, F. T. Pinho, and P. J. Oliveria, Study of steady pipe and channel flows of single-mode Phan-Thien–Tanner fluid, J. Non-Newtonian Fluid Mech., 101, Nos. 1–3, 55–76 (2001).

    Article  MATH  Google Scholar 

  58. V. Ngamaramvaranggul and M. F. Webster, Simulation of pressure-tooling wire-coating flow with Phan-Thien/Tanner models, Int. J. Numer. Meth. Fluids, 38, No. 7, 677–710 (2002).

    Article  MATH  Google Scholar 

  59. J. Soulages, M. S. N. Oliveira, P. S. Sousa, et al., Investigating the stability of viscoelastic stagnation flows in T-shaped microchannels, J. Non-Newtonian Fluid Mech., 163, Nos. 1-3, 9–24 (2009).

    Article  MATH  Google Scholar 

  60. P. J. Oliveira and F. T. Pinho, Analytical solution for fully-developed channel and pipe flow of Phan-Thien–Tanner fluids, J. Fluid Mech., 387, 271–280 (1999).

    Article  MathSciNet  MATH  Google Scholar 

  61. F. T. Pinho and P. J. Oliveira, Axial annular flow of a nonlinear viscoelastic fluid — An analytical solution, J. Non-Newtonian Fluid Mech., 93, No. 2, 325–337 (2000).

    Article  MATH  Google Scholar 

  62. F. T. Pinho and P. J. Oliveira, Phan-Thien/Tanner flow in concentric annuli, in: D. M. Binding et al. (Eds.), Proc. XIII Int. Cong. Rheol., Cambridge (UK), 20–25 August 2000, British Soc. Rheol., Glasgow (2000), Vol. 2, pp. 386–388.

  63. D. O. A. Cruz, F. T. Pinho, and P. J. Oliveira, Analytical solutions for fully developed laminar flow of some viscoelastic liquids with a Newtonian solvent contribution. J. Non-Newtonian Fluid Mech., 132, Nos. 1–3, 28–35 (2005).

    Article  MATH  Google Scholar 

  64. M. Mirzazadeh, F. Rashidi, and S. H. Hashemabadi, Flow of a nonlinear viscoelastic fluid in concentric rotating cylinders with relative rotation, in: M. Behnia, W. Lin, and G. D. McBain (Eds.), Proc. 15th Austral. Fluid Mech. Conf., The University of Sydney (Sydney, Australia), 13–17 December 2004, Sydney (2004), Paper AFMC00020.

  65. S. H. Hashemabadi and S. M. Mirnajafizadeh, Analytical solution of simplified Phan-Thien/Tanner fluid between nearly parallel plates of a small inclination, J. Appl. Sci., 7, No. 9, 1271–1278 (2007).

    Article  Google Scholar 

  66. A. Ya. Malkin, The state of the art in the rheology of polymers: Achievements and challenges, Polym. Sci., Ser. A, 51, No. 1, 80–102 (2009).

    Article  Google Scholar 

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  1. Limited Liability Company “OLNI,”, 18–102 Kol′tsov Str, Kirov, 610050, Russia

    O. M. Sokovnin, N. V. Zagoskina & S. N. Zagoskin

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  1. O. M. Sokovnin
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  2. N. V. Zagoskina
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  3. S. N. Zagoskin
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Correspondence to O. M. Sokovnin.

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Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 92, No. 2, pp. 548–562, March–April, 2019.

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Sokovnin, O.M., Zagoskina, N.V. & Zagoskin, S.N. Differential Models of Rheologically Nonstationary Fluids. J Eng Phys Thermophy 92, 528–541 (2019). https://doi.org/10.1007/s10891-019-01960-4

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