DESIGNING AND DETAILING OF BUILDING SYSTEMS. MECHANICS IN CIVIL ENGINEERING

Fluctuations of the membrane with piecewise smooth contour and mixed boundary conditions

Vestnik MGSU 11/2015
  • Algazin Sergey Dmitrievich - Ishlinsky Institute for Problems in Mechanics of the Russian Academy of Sciences (IPMekh RAN) leading research worker, chief research worker, Ishlinsky Institute for Problems in Mechanics of the Russian Academy of Sciences (IPMekh RAN), 101-1 Prospekt Vernadskogo str., Moscow, 119526, Russian Federation; This e-mail address is being protected from spambots. You need JavaScript enabled to view it .

Pages 29-37

The eigenvalue problem for the two-dimensional operator Laplace is classical in mathematics and physics. However, computing methods for calculation of eigenvalues have still many problems, especially in applications to acoustic and electromagnetic wave guides. The investigated below two-dimensional spectral for the Laplace operator have been previously considered by the author only in smooth areas. The solutions of these tasks (eigen functions) are infinitely differentiated or. even analytical and therefore in order to create effective algorithms it is necessary to consider this enormous a priori information. Traditional methods of finite differences and finite elements almost do not practically use the information on smoothness of the decision, i.e. these are methods with saturation. The term “saturation” was entered by K.I. Babenko. Using the method of computing experiment the author investigates the task about fluctuations of the membrane with the piecewise smooth contour for two-dimensional area, obtained by conformal representation of the square. It is shown that eigen functions are infinitely differentiated. Therefore, numerical algorithms without saturation are applicable. In article the calculation algorithm of eigenvalues in this two-dimensional area is developed, which allows determining up to 10 natural frequencies with the accuracy acceptable for practice on the grid 10×10.

DOI: 10.22227/1997-0935.2015.11.29-37

References
  1. Algazin S.D. Chislennye algoritmy klassicheskoy matematicheskoy fiziki [Numerical Algorithms of Classical Mathematical Physics]. Moscow, Dialog-MIFI Publ., 2010, 240 p. (In Russian)
  2. Babenko K.I. Osnovy chislennogo analiza [Fundamentals of Numerical Analysis]. 2nd edition, revised and enlarged. Moscow; Izhevsk, RKhD Publ., 2002, 847 p. (In Russian)
  3. Algazin S.D., Babenko K.I., Kosorukov A.L. O chislennom reshenii zadachi na sobstvennye znacheniya [On the Numerical Solution of the Task on Eigenvalues]. Moscow, 1975, 57 p. (Preprint. IPM; no. 108, 1975). (In Russian)
  4. Algazin S.D. Vychislenie sobstvennykh chisel i sobstvennykh funktsiy operatora Laplasa (Lap123) [Calculation of Eigenvalues and Eigenfunctions of Laplace Operator]. SVIDETEL’’STVO o gosudarstvennoy registratsii programmy dlya EVM № 2012617739. Zaregistrirovana v Reestre programm dlya EVM [Certificate on State Registration of the Computer Program № 2012617739. Registered in Software Registration Book]. August 27, 2012, 18 p. (In Russian)
  5. Kuttler J.R., Sigillito V.G. Eigenvalues of the Laplacian in Two Dimensions. SIAM Review. Apr. 1984, vol. 26, no. 2, pp. 163—193. DOI: http://dx.doi.org/10.1137/1026033

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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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