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HYDRAULICS. ENGINEERING HYDROLOGY. HYDRAULIC ENGINEERING

Azimuthal vorticity and stream function in the creeping flow in a pipe

Vestnik MGSU 4/2014
  • Zuykov Andrey L'vovich - Moscow State University of Civil Engineering (MGSU) Doctor of Technical Sciences, Professor, Department of Hydraulics and Water Resources, Moscow State University of Civil Engineering (MGSU), 26 Yaroslavskoye shosse, Moscow, 129337, Russian Federation; +7 (495)287-49-14, ext. 14-18; This e-mail address is being protected from spambots. You need JavaScript enabled to view it .

Pages 150-159

The article is devoted to the analytical study of the structure of steady non-uniform creeping flow in a cylindrical channel. There are many papers on the hydrodynamics of such flows, mainly related to the production of polymers. Previously we showed that the structure of steady non-uniform creeping flow in a cylindrical tube is determined by the Laplace equation relative to the azimuthal vorticity. The solution of Laplace's equation regarding the azimuthal vorticity is dedicated to the first half of the article. Fourier expansion allows us to write the azimuthal vortex in the form of two functions, the first of which depends only on the radial coordinate, and the second depends only on the axial coordinate. Fourier expansion can come to the Sturm - Liouville problem with a system of two differential equations, one of which is homogeneous Bessel equation. The radial-axial distribution of the azimuthal vorticity in the creeping flow is obtained on the basis of a rapidly convergent series of Fourier - Bessel. In the next article the radial-axial distribution of the stream function will be discussed. The solution is constructed from the Poisson equation based on the solution for the azimuthal vortex distribution. Fourier expansion can come to the Sturm - Liouville problem with a system of two differential equations, one of which is inhomogeneous Bessel equation. The inhomogeneous Bessel equation is solved through the Wronskian. The distribution of the stream function is obtained in the form of rapidly converging series of Fourier - Bessel.

DOI: 10.22227/1997-0935.2014.4.150-159

References
  1. Van Dyke M. An Album of Fluid Motion. Stanford, The Parabolic Press, 1982, 184 p.
  2. Giesekesus H. A Simple Constitutive Equation for Polymer Fluids Based on the Concept of Deformation Dependent Tensorial Mobility. Journal of Non-Newtonian Fluid Mechanics. 1982, vol. 11, pp. 69—109.
  3. Bird R.B., Armstrong R.C., Hassager O. Dynamics of Polymeric Liquids. Vol. 1 Fluid Mechanics. 2nd ed. New York, John Willey and Sons, 1987, 565 p.
  4. Snigerev B.A., Aliev K.M., Tazyukov F.Kh. Polzushchee techenie vyazkouprugoy zhidkosti so svobodnoy poverkhnost'yu v usloviyakh neizotermichnosti [Creeping Flow of Viscoelastic Fluid with a Free Surface in a Non-Isothermal]. Izvestiya Saratovskogo universiteta [Proceedings of the Saratov University]. New. Ser. Mathematics. Mechanics. Informatics. 2011, no. 3 (1), pp. 89—94.
  5. Orekhov G.V., Zuykov A.L., Volshanik V.V. Kontrvikhrevoe polzushchee techenie [Creeping Counter Vortex Flow]. Vestnik MGSU [Proceedings of Moscow State University of Civil Engineering]. 2013, no. 4, pp. 172—180.
  6. Akhmetov V.K., Volshanik V.V., Zuykov A.L., Orekhov G.V. Modelirovanie i raschet kontrvikhrevykh techeniy [Modeling and Calculation of Counter Vortex Flows]. Moscow, Moscow State University of Civil Engineering Publ., 2012, 252 p.
  7. Zuykov A.L. Raspredelenie prodol'nykh skorostey v tsirkulyatsionnom techenii [The Distribution of the Longitudinal Velocity in the Circulation Flow in the Pipe]. Vestnik MGSU [Proceedings of Moscow State University of Civil Engineering], 2009, no. 3, ðp. 200—204.
  8. Vladimirov V.S. Uravneniya matematicheskoy fiziki i spetsial'nye funktsii [The Equations of Mathematical Physics and Special Functions]. Moscow, Nauka Publ., 1988, 512 ð.
  9. Korn G.A., Korn T.M. Mathematical Handbook for Scientists and Engineers: Definitions, Theorems, and Formulas for Reference and Review. New York, General Publishing Company, 2000, 1151 p.
  10. Korenev B.G. Vvedenie v teoriyu besselevykh funktsiy [Introduction to the Theory of Bessel Functions]. Moscow, Nauka Publ., 1971, 288 ð.

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CREEPING COUNTER VORTEX FLOW

Vestnik MGSU 4/2013
  • Orekhov Genrikh Vasil’evich - Moscow State University of Civil Engineering (MGSU) Candidate of Technical Sciences, Associate Professor, Chair, Department of Hydroelectric Engineering and Use of Aquatic Resources; +7 (499) 182-99-58, Moscow State University of Civil Engineering (MGSU), 26 Yaroslavskoe shosse, Moscow, 129337, Russian Federation; This e-mail address is being protected from spambots. You need JavaScript enabled to view it .
  • Zuykov Andrey L’vovich - Moscow State University of Civil Engineering (MGSU) Doctor of Technical Sciences, Chair, Department of Hydraulics; +7(495)287-49-14, ext. 14-18, Moscow State University of Civil Engineering (MGSU), 26 Yaroslavskoe shosse, Moscow, 129337, Russian Federation; This e-mail address is being protected from spambots. You need JavaScript enabled to view it .
  • Volshanik Valeriy Valentinovich - Moscow State University of Civil Engineering (MGSU) Doctor of Technical Sciences, Professor, Professor, Department of Hydroelectric Engineering and Use of Aquatic Resource, Moscow State University of Civil Engineering (MGSU), 26 Yaroslavskoe shosse, Moscow, 129337, Russian Federation; This e-mail address is being protected from spambots. You need JavaScript enabled to view it .

Pages 172-180

The authors have performed an analytical research into one of the most complex types of heterogeneous 3D flows of fluids and gases, that is, a creeping counter vortex flow. The “creeping counter vortex flow” is the flow that is formed as a result of interaction between two or more slow concurrent co-axial circulatory longitudinal flows swirling in the opposite directions.Creeping flows are typical for numerous structural elements of machines, mechanisms, items of equipment and devices, if the flow velocity or cross dimensions of channels are small or, alternatively, if the viscosity of the fluid is high. This model designed by the coauthors, serves as the basis for the hydrodynamic theory of lubrication. If the flow velocity is small and the viscosity of the liquid media is substantial, inertial convective summands can be ignored for Navier — Stokes equations.The coauthors believe that the research into the phenomena of the creeping counter vortex flow as one of the types of heterogeneous 3D flows of fluids and gases has a strong potential in space technologies, and it may be elaborated in further research projects to be developed by the coauthors.

DOI: 10.22227/1997-0935.2013.4.172-180

References
  1. Korn G., Korn T. Spravochnik po matematike dlya nauchnykh rabotnikov i inzhenerov [Reference Book of Mathematics for Researchers and Engineers]. Moscow, Nauka Publ., 1970, 720 p.
  2. Zuykov A.L. Analiz izmeneniya profilya tangentsial’nykh skorostey v techenii za lokal’nym zavihritelem [Analysis of Changes in the Profile of Tangential Velocities of the Flow Shaped Up by the Local Swirler]. Vestnik MGSU [Proceedings of Moscow State University of Civil Engineering], 2012, no. 5, pð. 23—28.

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