Simulation of the stationary nonisothermal MHD flows of polymeric fluids in channels with interior heating elements

A. M. Blokhin ; B. V. Semisalov

Sibirskij žurnal industrialʹnoj matematiki, Tome 23 (2020) no. 2, pp. 17-40.

Résumé

Basing on the rheological mesoscopic Pokrovskii–Vinogradov model, the equations of nonrelativistic magneto-hydrodynamics, and the heat conduction equation with dissipative terms, we obtain a closed coupled system of nonlinear partial differential equations that describes the flow of solutions and melts of linear polymers. We take into account the rheology and induced anisotropy of polymeric fluid flow as well as mechanical, thermal, and electromagnetic impacts. The parameters of the equations are determined by mechanical tests with up-to-date materials and devices used in additive technologies (as $3D$ printing). The statement is given of the problems concerning stationary polymeric fluid flows in channels with circular and elliptical cross-sections with thin inclusions (some heating elements). We show that, for certain values of parameters, the equations can have three stationary solutions of high order of smoothness. Just these smooth solutions provide the defect-free additive manufacturing. To search for them, some algorithm is used that bases on the approximations without saturation, the collocation method, and some special relaxation method. Under study are the dependencies of the distributions of the saturation fluid velocity and temperature on the pressure gradient in the channel.

Détail

Keywords: polymeric fluid, mesoscopic model, nonisothermal MHD flow, heat dissipation, nonlinear boundary-value problem, multiplicity of solutions, method without saturation.

Bibliographie

[1] Yu. A. Altukhov, A. S. Gusev, G. V. Pyshnograi, Introduction to the Mesoscopic Theory of Flowing Polymer Systems, Izd-vo AltGPA, Barnaul, 2012 (in Russian)

[2] V. N. Pokrovskii, The Mesoscopic Theory of Polymer Dynamics, Springer-Verl., Berlin, 2010

[3] G. V. Vinogradov, V. N. Pokrovskii, Yu. G. Yanovsky, “Theory of viscoelastic behavior of concentrated polymer solutions and melts in one-molecular approximation and its experimental verification”, Rheol. Acta, 7 (1972), 258–274 | DOI

[4] L. G. Loitsyanskii, Mechanics of Liquids and Gases, Stewartson Pergamon Press, Oxford

[5] S. A. Zinovich, I. E. Golovicheva, G. V. Pyshnograi, “Influence of the molecular weight on the shear viscosity and longitudinal viscosity of linear polymers”, Prikl. Matematika i Tekhn. Fizika, 41:2 (2000), 154–160 (in Russian) | Zbl

[6] B. Y. Ahn, S. B. Walker, S. C. Slimmer et al, “Planar and three-dimensional printing of conductive inks”, J. Vis. Exp., 58 (2011), e3189 | DOI

[7] S. D. Hoath, W. K. Hsiao, G. D. Martin et al, “Oscillations of aqueous PEDOT:PSS fluid droplets and the properties of complex fluids in drop-on-demand inkjet printing”, J. Non-Newtonian Fluid Mechanics, 223 (2015), 28–36 | DOI | MR

[8] H. H. Lee, K. S. Chou, K. C. Huang, “Inkjet printing of nanosized silver colloids”, Nanotechnology, 16 (2005), 2436–2441 | DOI

[9] P. K. Kahol, J. C. Ho, Y. Y. Chen et al, “On metallic characteristics in some conducting polymers”, Synthetic Metals, 151:2 (2005), 65–72 | DOI

[10] T. H. Le, Y. Kim, H. Yoon, “Electrical and electrochemical properties of conducting polymers”, Polymers, 9:4 (2017), 150 | DOI

[11] K. Muro, M. Watanabe, T. Tamai et al, “PEDOT/PSS Nano-Particles: Synthesis and Properties”, RSC Adv., 6:90 (2016), 87147–87152 | DOI

[12] D. G. Yoon, M. G. Kang, J. B. Kim et al, “Nozzle printed-PEDOT:PSS for organic light emitting diodes with various dilution rates of ethanol”, Appl. Sci., 8:2 (2018), 203 | DOI

[13] A. N. Papathanassiou, I. Sakellis, J. Grammatikakis et al, “Exploring electrical conductivity within mesoscopic phases of semiconducting PEDOT:PSS films by Broadband Dielectric Spectroscopy”, Appl. Phys. Lett., 103 (2013), 123304 | DOI

[14] J. Liu, X. Wang, D. Li et al, “Thermal conductivity and elastic constants of PEDOT:PSS with high electrical conductivity”, Macromolecules, 48:3 (2015), 585–591 | DOI

[15] N. Z. Khan, “Modeling and simulation of organic mem relay for estimating the coefficient of thermal expansion of PEDOT: PSS”, TechConnect briefs, 4 (2017), 120–123

[16] B. V. Semisalov, “A nonlocal algorithm for finding a solution to the poisson equation and its applications”, Zhurn. Vychisl. Matematiki i Mat. Fiziki, 54:7 (2014), 1110–1135 (in Russian) | DOI | Zbl

[17] B. V. Semisalov, Computing Code for Finding Solutions to Boundary Value Problems for Partial Differential Equations with High Accuracy and Low Computational Costs “Nonlocal Method Without Saturation”, State Registration Certificate No 2015615527 Issued on May 20, 2015

[18] B. V. Semisalov, “Fast nonlocal algorithm for solving Neumann–Dirichlet boundary value problems with an error control”, Vychisl. Metody. Programmirovanie, 17:4 (2016), 500–522 (in Russian)

[19] B. V. Semisalov, “Development and analysis of the fast pseudo spectral method for solving nonlinear Dirichlet problems”, Bull. South Ural State University. Series: Math. Modelling, Program. Comp. Software, 11:2 (2018), 123–138 | DOI | Zbl

[20] L. I. Sedov, Mechanics of Continua, v. 1, Nauka, M., 1970 (in Russian)

[21] A. B. Vatazhin, G. A. Lyubimov, S. A. Regirer, Magnetic Hydrodynamic Flows in Channels, Nauka, M., 1970

[22] Pai Shih-I, Introduction to the Theory of Compressible Flow, Literary Licensing, Whitefish, 2013

[23] A. M. Blokhin, A. S. Rudometova, “Stationary Flows of a Weakly Conducting Incompressible Polymeric Liquid Between Coaxial Cylinders”, J. Appl. Indust. Math., 11:4 (2017), 486–493 | DOI | MR | Zbl

[24] A. M. Blokhin, A. S. Rudometova, “Stationary Solutions of the Equations for Nonisothermal Electroconvection of a Weakly Conducting Incompressible Polymeric Liquid”, J. Appl. Indust. Math., 9:1 (2015), 147–156 | DOI | MR | Zbl

[25] A. I. Akhiezer, I. A. Akhiezer, Electromagnetism and Electromagnetic Waves, Vysshaya Shkola, M., 1985 (in Russian)

[26] K. Nordling, D. Osterman, Physics Handbook for Science and Engineering, Studentlitteratur, Lund, 1999 | MR

[27] Y. Shibata, “On the r-boundness for the two phase problem with phase transitions: compressiable-incompressiable”, Model Problem. Funkcialaj Ekvacioj, 59 (2016), 243–287 | DOI | MR | Zbl

[28] A. M. Blokhin, B. V. Semisalov, A. S. Shevchenko, “Stationary solutions of the equations describing nonisothermal convection of an incompressible polymeric viscous-elastic liquid”, Mat. Modelirovanie, 28:10 (2016), 3–22 (in Russian) | Zbl

[29] A. M. Blokhin, E. A. Kruglova, B. V. Semisalov, “Steady-state flow of an incompressible viscoelastic polymer fluid between two coaxial cylinders”, Comp. Math. Math. Physics, 57:7 (2017), 1181–1193 | DOI | MR | Zbl

[30] S. G. Kalashnikov, Electricity, Nauka, M., 1964 (in Russian)

[31] R. Kubo, Thermodynamics: An Advanced Course with Problems and Solutions, North Holland, Amsterdam, 1968