Skip to main content
Log in

Viscoelastic fluid flow in a prismatic channel of square cross-section with reference to the example of rubber mixtures

  • Published:
Fluid Dynamics Aims and scope Submit manuscript

Abstract

Flow in a prismatic channel of the melts of the SKI-3 and SKMS30-ARKM-15-based rubber mixtures widely used in the chemical industry is numerically investigated. The description of these media, which exhibit viscoelstic properties when being processed, requires a particular approach that would take their rheological behavior and various anomalies into account. Amidst many rheological equations governing flows of rheologically complex media the equation that would ensure not only the good reliability of the results but also the feasibility in its practical use should be selected. The special features of viscoelastic fluids clearly manifest themselves in the flow in a prismatic channel of a noncircular cross-section. Secondary flows characteristic of the viscoelastic flows in such channels force the fluid particles to move in spiral trajectories along the channel. In the numerical calculations the Phan-Thien–Tanner (PTT) rheological model is used; its parameters are obtained on the basis of experimental data. The calculations are performed using the COSMOL Multiphysics software complex. the method of solution was tested against the analytical solution of the PTT fluid flow in a round tube which was compared with the numerical solution.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. G. Schramm, A Practical Approach to Rheology and Rheometry, Gebrüder HAAKE GmbH, Karlsruhe (2000).

    Google Scholar 

  2. G. Oldroyde, “On the Formulation of Rheological Equations of State,” Proc. Roy. Soc. London 200, 523 (1950).

    Article  ADS  MathSciNet  Google Scholar 

  3. J.L. White and A. Metzner, “Rheological Equations from Molecular Network Theories,” J. Appl. Polymer Sci. No. 7, 1867 (1963).

    Article  Google Scholar 

  4. N. Phan-Thien and R.I. Tanner, “A New Constitutive Equation Derived from Network Theory,” J. Non-Newtonian Fluid Mech. No. 2, 353 (1977).

    Article  MATH  Google Scholar 

  5. H. Giesekus, “A Simple Constitutive Equation for Polymer Fluids Based on the Concept of Deformation Dependent Tensorial Mobility,” J. Non-Newtonian Fluid Mech. 11, 69 (1982).

    Article  MATH  Google Scholar 

  6. A.I. Leonov, “Nonequilibrium Thermodynamics and Rheology of Viscoelastic Polymer Melts,” Rheol. Acta 15, 85 (1976).

    Article  MATH  Google Scholar 

  7. A.I. Leonov and A.N. Prokunin, Nonlinear Phenomena in Flows of Viscoelastic Polymer Fluids, Chapman & Hall, New York (1994).

    Book  Google Scholar 

  8. M. Doi and S.F. Edwards, The Theory of Polymer Dynamics, Clarendon Press (1988).

    Google Scholar 

  9. R.B. Bird, C.F. Curtiss, R.C. Armstrong, and O. Hassager, Dynamics of Polymeric Liquids. Vol. 2. Kinetic Theory, Wiley, New York (1987).

  10. T.C.B. McLeish and R.G. Larson, “Molecular Constitutive Equations for a Class of Branched Polymers: the Pom-Pom Polymer,” J. Rheol. 42, 81 (1998).

    Article  ADS  Google Scholar 

  11. R.G. Larson, The Structure and Rheology of Complex Fluids (Topics in Chemical Engineering), Oxford Univ. Press, Oxford (1999).

    Google Scholar 

  12. O.S. Carneiro, J.M. Nobrega, F.T. Pinho, and P.J. Oliveira, “Computer Aided Rheological Design of Extrusion Dies for Profiles,” J. Mater. Process. Tech. 11, 75 (2001).

    Article  Google Scholar 

  13. A.E. Green and R.S. Rivlin, “Steady Flow of Non-Newtonian Fluids through Tubes,” Quart. J. Appl. Math. 14, 299 (1956).

    MathSciNet  MATH  Google Scholar 

  14. P. Townsend, K. Walters, and D.M. Waterhouse, “Secondary Flows in Pipes of Square Cross-Section and the Measurement of the Second Normal Stress Difference,” J. Non-Newtonian Fluid Mech. No. 1, 107 (1976).

    Article  MATH  Google Scholar 

  15. B. Debbaut, T. Avalosse, J. Dooley, “On the Development of Secondary Motions in Straight Channels Induced by the Second Normal Stress Difference,” J. Non-Newtonian Fluid Mech. 69, 255 (1997).

    Article  Google Scholar 

  16. B. Debbaut and J. Dooley, “Secondary Motions in Straight and Tapered Channels. Experiments and Three-Dimensional Finite Element Simulation with a Multimode Differential Viscoelastic Model,” J. Rheol. 43, 1525 (1999).

    Article  ADS  Google Scholar 

  17. A.G. Dodson, P. Townsend, and K. Walters, “Non-Newtonian Flow in Pipes of Non-Circular Cross-Section,” J. Computers Fluids 19, 317 (1974).

    Article  MATH  Google Scholar 

  18. G.M. Danilova-Volkonskaya and R.V. Torner, “Method of Calculation of Rheological and Relaxation Characteristics of Polymer Material Melts from the Data of Capillary Viscosimetry,” Plasticheskie Massy No. 5, 46 (2002).

    Google Scholar 

  19. B.S. Petukhov, Heat Transfer and Drag in Laminar Flows of Fluids in Tubes [in Russian], Energiya, Moscow (1967).

    Google Scholar 

  20. D.O.A. Cruz, F.T. Pinho, and P.J. Oliveira, “Analytical Solutions for Fully Developed Laminar Flow od Some Viscoelastic Fluids with a Newtonian Solvent Contribution,” J. Non-Newtonian Fluid Mech. 132, 28 (2005).

    Article  MATH  Google Scholar 

  21. D.V. Anan’ev, E.K. Vachagina, A.I. Kadyirov, A.A. Kainova, and G.T. Osipov, “Determination of Existence Conditions for the Solution with a Weak Discontinuity for Simplest Viscoelastic Fluid Flows,” Fluid Dynamics 49 (5), 576 (2014).

    Article  MATH  Google Scholar 

  22. S.-C. Xue, N. Phan-Thien, and R.I. Tanner, “Numerical Study of Secondary Flows of Viscoelastic Fluid in Straight Pipes by an Implicit Finite Volume Method,” J. Non-Newtonian Fluid Mech. 59, 191 (1995).

    Article  Google Scholar 

  23. P. Yue, J. Dooley, and J.J. Feng, “A General Criterion for Viscoelastic Secondary Flow in Pipes of Noncircular Cross Section,” J. Rheol. 52, 315 (2008).

    Article  ADS  Google Scholar 

  24. H. Giesekus, “Sekundarstromungen in viskoelastiken Flussigkeiten bei stationarer und periodischer Bewegung,” Rheol. Acta No. 4, 85 (1965).

    Article  Google Scholar 

  25. V. Semjonow, “Sekundarstromungen hochpolymerer Schmelzen in einem Rohr von elliptischen Querschnitt,” Rheol. Acta No. 6, 171 (1967).

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to E. K. Vachagina.

Additional information

Original Russian Text © E.K. Vachagina, A.I. Kadyirov, A.A. Kainova, G.R. Khalitova, 2016, published in Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, 2016, Vol. 51, No. 1, pp. 9–17.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Vachagina, E.K., Kadyirov, A.I., Kainova, A.A. et al. Viscoelastic fluid flow in a prismatic channel of square cross-section with reference to the example of rubber mixtures. Fluid Dyn 51, 8–17 (2016). https://doi.org/10.1134/S0015462816010026

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0015462816010026

Keywords

Navigation