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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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Axial and radial velocities in the creeping flow in a pipe

Vestnik MGSU 5/2014
  • 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 127-134

The article is devoted to analytical study of transformation fields of axial and radial velocities in uneven steady creeping flow of a Newtonian fluid in the initial portion of the cylindrical channel. It is shown that the velocity field of the flow is two-dimensional and determined by the stream function. The article is a continuation of a series of papers, where normalized analytic functions of radial axial distributions in uneven steady creeping flow in a cylindrical tube with azimuthal vorticity and stream function were obtained. There is Poiseuille profile for the axial velocity in the uniform motion of a fluid at an infinite distance from the entrance of the pipe (at x = ∞), here taken equal to zero radial velocity. There is uniform distribution of the axial velocity in the cross section at the tube inlet at x = 0, at which the axial velocity is constant along the current radius. Due to the axial symmetry of the flow on the axis of the pipe (at r = 0), the radial velocities and the partial derivative of the axial velocity along the radius, corresponding to the condition of the soft function extremum, are equal to zero. The authors stated vanishing of the velocity of the fluid on the walls of the pipe (at r = R , where R - radius of the tube) due to its viscous sticking and tightness of the walls. The condition of conservation of volume flow along the tube was also accepted. All the solutions are obtained in the form of the Fourier - Bessel. It is shown that the hydraulic losses at uniform creeping flow of a Newtonian fluid correspond to Poiseuille - Hagen formula.

DOI: 10.22227/1997-0935.2014.5.127-134

References
  1. 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.
  2. Akhmetov V.K., Volshanik V.V., Zuykov A.L., Orekhov G.V. Modelirovanie i raschet kontrvikhrevykh techeniy [Modeling and Calculation of Counter Vortex Flows]. Мoscow, MGSU Publ., 2012, 252 p.
  3. Zuykov A.L. Azimutal'nyy vikhr' i funktsiya toka v polzushchem techenii v trube [Azimuthal Vorticity and Stream Function in the Creeping Flow in a Pipe]. Vestnik MGSU [Proceedings of Moscow State University of Civil Engineering]. 2014, no. 4, рp. 150—159.
  4. 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.

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