Homogenization of a Convection–Diffusion Equation in a Thin Rod Structure



This chapter is devoted to the homogenization of a stationary convection diffusion model problem in a thin rod structure. More precisely, we study the asymptotic behavior of solutions to a boundary value problem for a convection diffusion equation defined in a thin cylinder that is the union of two nonintersecting cylinders with a junction at the origin. We suppose that in each of these cylinders the coefficients are rapidly oscillating functions that are periodic in the axial direction, and that the microstructure period is of the same order as the cylinder diameter. On the lateral boundary of the cylinder we assume the Neumann boundary condition, while at the cylinder bases the Dirichlet boundary conditions are posed.


Stationary Convection Thin Cylinder Microstructure Period Boundary Layer Function Linearize Elasticity System 


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  1. [BaPa89]
    Bakhvalov, N.S., Panasenko, G.P.: Homogenization: Averaging Processes in Periodic Media, Kluwer, Dordrecht–Boston–London (1989).Google Scholar
  2. [BLP78]
    Bensoussan, A., Lions, J. L., Papanicolaou, G.: Asymptotic Analysis for Periodic Structure, North-Holland, New York (1978).Google Scholar
  3. [GiTr98]
    Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order, Springer, Berlin (1998).Google Scholar
  4. [MuSi99]
    Murat, F., Sili, A.: Asymptotic behavior of solutions of the anisotropic heterogeneous linearized elasticity system in thin cylinders. C.R. Acad. Sci. Paris Sér. I Math., 328, 179–184 (1999).MathSciNetGoogle Scholar
  5. [KoPa92]
    Kozlova, M.V., Panasenko, G.P.: Averaging a three-dimensional problem of elasticity theory in a nonhomogeneous rod. Comput. Math. Math. Phys., 31, no. 10, 128–131 (1992).MathSciNetGoogle Scholar
  6. [KLS89]
    Krasnosel’skij, M.A., Lifshits, E.A., Sobolev, A.V.: Positive Linear Systems: The Method of Positive Operators, Heldermann, Berlin (1989).MATHGoogle Scholar
  7. [LaUr68]
    Ladyzhenskaya, O.A., Ural’tseva, N.N.: Linear and Quasilinear Elliptic Equations, Academic Press, New York (1968).MATHGoogle Scholar
  8. [Naz82]
    Nazarov, S.A.: Structure of the solutions of boundary value problems in thin regions. Vestnik Leningrad. Univ. Mat. Mekh. Astronom., 126, 65–68 (1982) (Russian).Google Scholar
  9. [Naz99]
    Nazarov, S.A.: Justification of the asymptotic theory of thin rods. Integral and pointwise estimates. J. Math. Sci., 97, 4245–4279 (1999).Google Scholar
  10. [Pa94-I]
    Panasenko, G.P.: Asymptotic analysis of bar systems. I. Russian J. Math. Phys., 2, 325–352 (1994).MathSciNetGoogle Scholar
  11. [Pa96-II]
    Panasenko, G.P.: Asymptotic analysis of bar systems. II. Russian J. Math. Phys., 4, 87–116 (1996).MathSciNetGoogle Scholar
  12. [Pa05]
    Panasenko, G.P.: Multi-Scale Modelling for Structures and Composites, Springer, Dordrecht (2005).Google Scholar
  13. [PaPi09]
    Pankratova, I., Piatnitski, A.: On the behavior at infinity of solutions to stationary convection-diffusion equation in a cylinder, DCDS-B, 11 (2009).Google Scholar
  14. [Past02]
    Pastukhova, S.: Averaging for nonlinear problems in the theory of elasticity on thin periodic structures. Dokl. Akad. Nauk, 383, 596–600 (2002) (Russian).MathSciNetGoogle Scholar
  15. [TrVi87]
    Trabucho, L., Viaño, J.M.: Derivation of generalized models for linear elastic beams by asymptotic expansion methods, in Applications of Multiple Scaling in Mechanics, Masson, Paris (1987), 302–315.Google Scholar
  16. [TuAg86]
    Tutek, Z., Aganović, I.: A justification of the one-dimensional linear model of elastic beam. Math. Methods Appl. Sci., 8, 502–515 (1986).CrossRefGoogle Scholar
  17. [Ve95]
    Veiga, M.F.: Asymptotic method applied to a beam with a variable cross section, in Asymptotic Methods for Elastic Structures, de Gruyter, Berlin (1995), 237–254.Google Scholar

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© Birkhäuser Boston, a part of Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.UniversitÉ de SaintSt. étienneFrance
  2. 2.Narvik University CollegeNarvikNorway

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