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## Archive of Issues

Germany; Russia Jena; Yekaterinburg
Year
2016
Volume
26
Issue
3
Pages
379-387
 Section Mechanics Title Analytical solution of the problem on inclined viscous flow around a parabolic dendrite within the framework of Oseen's approximation Author(-s) Alexandrov D.V.a, Galenko P.K.b Affiliations Ural Federal Universitya, University of Jenab Abstract A mathematical model of inclined viscous flow around a dendrite in Oseen's hydrodynamic approximation is formulated. The analytical solution of the problem on inclined viscous flow around a parabolic dendrite in two- and three-dimensional cases is constructed. The components of fluid velocity in the vicinity of the dendritic tip in 2D and 3D flow geometries are determined in the laboratory coordinate system by means of the curvilinear coordinates of parabolic cylinder and paraboloid of revolution. The analytical solutions of Oseen's hydrodynamic equations are rewritten in the coordinate system connected to the dendrite growing with a constant velocity. The obtained solution transforms to the previously known one in the limiting case of zero angle between the fluid velocity direction far from the dendrite and its axis. A scaled component of fluid velocity as a function of parabolic coordinates at different slope coefficients of flow is illustrated. Keywords dendrites, convection, viscous flow, Oseen's approximation UDC 532.51 MSC 76D07, 76M45 DOI 10.20537/vm160307 Received 23 May 2016 Language Russian Citation Alexandrov D.V., Galenko P.K. Analytical solution of the problem on inclined viscous flow around a parabolic dendrite within the framework of Oseen's approximation, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2016, vol. 26, issue 3, pp. 379-387. References Bouissou P., Perrin B., Tabeling P. Influence of an external flow on dendritic crystal growth, Physical Review A, 1989, vol. 40, issue 1, pp. 509-510. DOI: 10.1103/PhysRevA.40.509 Binder S., Galenko P.K., Herlach D.M. The effect of fluid flow on the solidification of ${\rm Ni}_2{\rm B}$ from the undercooled melt, Journal of Applied Physics, 2014, vol. 115, issue 5, 053511. DOI: 10.1063/1.4864151 Gao J., Han M., Kao A., Pericleous K., Alexandrov D.V., Galenko P.K. Dendritic growth velocities in an undercooled melt of pure nickel under static magnetic fields: A test of theory with convection, Acta Materialia, 2016, vol. 103, pp. 184-191. DOI: 10.1016/j.actamat.2015.10.014 Oseen C.W. Über die Stokessche formel und über eine verwandte aufgabe in der hydrodynamik, Ark. Mat. Astron. Fys., 1910, vol. 6, pp. 1-20. Dash S.K., Gill W.N. Forced convection heat and momentum transfer to dendritic structures (parabolic cylinders and paraboloids of revolution), International Journal of Heat and Mass Transfer, 1984, vol. 27, issue 8, pp. 1345-1356. DOI: 10.1016/0017-9310(84)90062-0 Lamb H. (Sir) Hydrodynamics, Cambridge: Cambridge University Press, 1895, 636 p. Kochin N.E., Kibel' I.A., Roze N.V. Theoretical hydromechanics, New York: Interscience, 1964, 577 p. Buyevich Yu.A., Alexandrov D.V., Zakharov S.V. Hydrodynamics. Examples and problems, New York: Begell House, 2001, 331 p. Bouissou Ph., Pelcé P. Effect of a forced flow on dendritic growth, Physical Review A, 1989, vol. 40, issue 11, pp. 6673-6680. DOI: 10.1103/PhysRevA.40.6673 Alexandrov D.V., Galenko P.K. Selection criterion of stable dendritic growth at arbitrary Péclet numbers with convection, Physical Review E, 2013, vol. 87, issue 6, 062403, 5 p. DOI: 10.1103/PhysRevE.87.062403 Alexandrov D.V., Galenko P.K. Dendrite growth under forced convection: analysis methods and experimental tests, Physics-Uspekhi, 2014, vol. 57, issue 8, pp. 771-786. DOI: 10.3367/UFNe.0184.201408b.0833 Alexandrov D.V., Galenko P.K. Thermo-solutal and kinetic regimes of an anisotropic dendrite growing under forced convective flow, Physical Chemistry Chemical Physics, 2015, vol. 17, pp. 19149-19161. DOI: 10.1039/C5CP03018H Kao A., Pericleous K. A numerical model coupling thermoelectricity, magnetohydrodynamics and dendritic growth, Journal of Algorithms & Computational Technology, 2012, vol. 6, no. 1, pp. 173-201. DOI: 10.1260/1748-3018.6.1.173 Kao A., Shevchenko N., Roshchupinka O., Eckert S., Pericleous K. The effects of natural, forced and thermoelectric magnetohydrodynamic convection during the solidification of thin sample alloys, IOP Conference Series: Materials Science and Engineering, 2015, vol. 84, 012018, 8 p. DOI: 10.1088/1757-899X/84/1/012018 Full text