Abstract
The time-dependent mathematical model describing the vortex motion of an incompressible polymeric liquid is discussed. In the steady-state case certain particular solutions are found. In the case of the steady-state pressure along the axis of cylinder, a version of deriving this model for both fixed and free boundaries is given.
Similar content being viewed by others
References
J. G. Oldroyd, “On the Formulation of Rheological Equations of State,” Proc. R. Soc. 200, No. 1063, 523–541 (1950).
A. I. Leonov and A. N. Prokunin, Nonlinear Phenomena in Flows of Viscoelastic Polymer Fluids (Chapman and Hall, New York, 1994).
P. G. De Gennes, Scaling Concepts in Polymer Physics (Cornell University Press, 1979).
M. Doi and S. F. Edwards, The Theory of Polymer Dynamics (Oxford, Clarendon Press, 1986).
R. B. Bird, C. F. Curtiss, R. C. Armstrong, and O. Hassager, Dynamics of Polymeric Liquids, Vol. 2 (Wiley, New York, 1987).
H. C. Ottinger, “A Thermodynamically Admissible Reptation Model for Fast Flows of Entangled Polymers,” J. Rheol. 43, No. 6, 1461–1493 (1999).
P. E. Rouse, “A Theory of the Linear Viscoelastic Properties of Dilute Solutions of Cooling Polymers,” J. Chem. Phys. 21, No. 7, 1272–1280 (1953).
T. C. B. McLeish and R.G. Larson, “Molecular Constitutive Equations for a Class of Branched Polymers: the Pom-Pom Polymer,” J. Rheol. 42, No. 1, 81–110 (1998).
W. M. H. Verbeeten, G.W.M. Peters, and F. P. T. Baaijens, “Differential Constitutive Equations for Polymer Melt: the Extended Pom-PomModel,” J. Rheol. 45, No. 4, 821–841 (2001).
V. N. Pokrovskii, Statistical Mechanics of Dilute Suspensions (Nauka, Moscow, 1978) [in Russian].
Yu. A. Altukhov, A. S. Gusev, and G. V. Pyshnograi, Introduction to Mesoscopic Theory of Flowable Polymeric Systems (AltGPA, Barnaul, 2012) [in Russian].
G. V. Pyshnograi, A. S. Gusev, and V. N. Pokrovskii, “Constitutive Equations for Weakly Entangled Linear Polymers,” J. Non-Newtonian Fluid Mech. 163, No. 1, 17–28 (2009).
J. L. Smith, “An Experimental Study of the Vortex in the Cyclone Separator,” Trans. ASME, J. Basic Engineerin. 84, No. 4, 602–608 (1962).
G. A. Wan and C. C. Chang, “Measurement of the Velocity Field in a Simulated Tornado-Like Vortex Using a Three-Dimensional Velocity Probe,” J. Atmospher. Sci. 29, No. 1, 116–127 (1972).
A. Yu. Gubin, “Stability of a Vortex Motion of an Viscous Incompressible Fluid,” Sib. Zh. Ind. Mat. VII, No 2(18), 40–53 (2004).
O. N. Ul’yanov, “One Class of Viscous Fluid Flows,” in: Dynamics of Liquid and Gas. Collected Works. Proceedings of IMM of the Ural Branch of the Russian Academy of Sciences [in Russian]. 9, No. 2, 129–136 (2003).
N. E. Golovicheva, G. V. Pyshnograi, and V. I. Popov, “Generalization of the Poiseuille Law on the Basis of the Constitutive Rheological Relation for Polymeric Liquids,” Zh. Prikl. Mekhan. Tekhn. Fiz. 40, No. 5, 158–163 (1999).
M. Abramowitz and I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (National Bureau of Standards, Washington, D.C., 1964; Nauka, Moscow, 1979).
E. Kamke, Differentialgleichungen. Lösungmethoden und Lösungen, Vol. 1 (Leipzig, 1959; Nauka, Moscow, 1971).
N. N. Yanenko, Fractional Step Method for Solving the Multidimensional Problems of Mathematical Physics (Nauka, Novosibirsk, 1967) [in Russian].
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © A.M. Blokhin, R.E. Semenko, 2018, published in Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, 2018, No. 2, pp. 3–15.
Rights and permissions
About this article
Cite this article
Blokhin, A.M., Semenko, R.E. Vortex Motion of an Incompressible Polymer Liquid in the Cylindrical Near-Axial Zone. Fluid Dyn 53, 177–188 (2018). https://doi.org/10.1134/S0015462818020040
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0015462818020040