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.
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G. Schramm, A Practical Approach to Rheology and Rheometry, Gebrüder HAAKE GmbH, Karlsruhe (2000).
G. Oldroyde, “On the Formulation of Rheological Equations of State,” Proc. Roy. Soc. London 200, 523 (1950).
J.L. White and A. Metzner, “Rheological Equations from Molecular Network Theories,” J. Appl. Polymer Sci. No. 7, 1867 (1963).
N. Phan-Thien and R.I. Tanner, “A New Constitutive Equation Derived from Network Theory,” J. Non-Newtonian Fluid Mech. No. 2, 353 (1977).
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).
A.I. Leonov, “Nonequilibrium Thermodynamics and Rheology of Viscoelastic Polymer Melts,” Rheol. Acta 15, 85 (1976).
A.I. Leonov and A.N. Prokunin, Nonlinear Phenomena in Flows of Viscoelastic Polymer Fluids, Chapman & Hall, New York (1994).
M. Doi and S.F. Edwards, The Theory of Polymer Dynamics, Clarendon Press (1988).
R.B. Bird, C.F. Curtiss, R.C. Armstrong, and O. Hassager, Dynamics of Polymeric Liquids. Vol. 2. Kinetic Theory, Wiley, New York (1987).
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).
R.G. Larson, The Structure and Rheology of Complex Fluids (Topics in Chemical Engineering), Oxford Univ. Press, Oxford (1999).
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).
A.E. Green and R.S. Rivlin, “Steady Flow of Non-Newtonian Fluids through Tubes,” Quart. J. Appl. Math. 14, 299 (1956).
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).
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).
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).
A.G. Dodson, P. Townsend, and K. Walters, “Non-Newtonian Flow in Pipes of Non-Circular Cross-Section,” J. Computers Fluids 19, 317 (1974).
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).
B.S. Petukhov, Heat Transfer and Drag in Laminar Flows of Fluids in Tubes [in Russian], Energiya, Moscow (1967).
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).
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).
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).
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).
H. Giesekus, “Sekundarstromungen in viskoelastiken Flussigkeiten bei stationarer und periodischer Bewegung,” Rheol. Acta No. 4, 85 (1965).
V. Semjonow, “Sekundarstromungen hochpolymerer Schmelzen in einem Rohr von elliptischen Querschnitt,” Rheol. Acta No. 6, 171 (1967).
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.
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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
- viscoelastic fluids
- Phan-Thien–Tanner model
- secondary flows
- rubber mixtures
- prismatic channel