Advertisement

Steady Stokes Flow in Domains with Unbounded Boundaries

  • Giovanni P. Galdi
Part of the Springer Tracts in Natural Philosophy book series (STPHI, volume 38)

Abstract

So far, with the exception of the half-space, we have considered flows occurring in domains with a compact boundary. Nevertheless, from the point of view of the applications it is very important to consider flows in domains Ω having an unbounded boundary, such as channels or pipes of possibly varying cross section. In studying these problems, however, due to the particular geometry of the region of flow, completely new features, which we are going to explain, appear. To this end, assume Ω to be an unbounded domain of ℝ n with m > 1 “exits” to infinity, of the type (see Section III.4.3)
$$\Omega = \bigcup\limits_{i = 0}^m {{\Omega _i}} ,$$
where Ω0 is a smooth compact subset of Ω while Ω i , i = 1,..., m, are disjoint domains which, in possibly different coordinate systems (depending on Ω i ) have the form
$${\Omega _i} = \left\{ {x \in {R^n}:{x_n} > 0,x' \equiv \left( {{x_1},...,{x_{n - 1}}} \right) \in \sum\nolimits_i {{x_n}} } \right\}$$

Keywords

Weak Solution Pressure Field Sobolev Inequality Decay Estimate Stoke Flow 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliography

  1. Amick, C.J., 1977, Steady Solutions of the Navier-Stokes Equations in Unbounded Channels and Pipes, Ann. Scuola Norm. Pisa, (4) 4, 473–513 [1.3, Introduction to VI]MathSciNetMATHGoogle Scholar
  2. Amick, C.J., 1978, Properties of Steady Solutions of the Navier-Stokes Equations for Certain Unbounded Channels and Pipes, Nonlin. Anal., Theory, Meth. Appl, 2, 689–720 [1.3, Introduction to VI]MathSciNetMATHCrossRefGoogle Scholar
  3. Horgan, C.O., and Wheeler, L.T., 1978, Spatial Decay Estimates for the Navier-Stokes Equations with Application to the Problem of Entry Flow, SI AM J. Appl Math., 35, 97–116 [1.3, Notes for III]MathSciNetMATHGoogle Scholar
  4. Ladyzhenskaya, O.A., and Solonnikov, V.A., 1980, Determination of Solutions of Boundary Value Problems for Steady-State Stokes and Navier-Stokes Equations in Domains Having an Unbounded Dirich-let Integral, Zap. Nauchn. Sem. Leningrad Otdel. Mat. Inst. Steklov (LOMI), 96, 117–160; English Transl.: J.Soviet Math., 21, 1983, 728–761 [1.3, Introduction to VI, VI. 1, VI.2, VI.3]Google Scholar
  5. Horgan, C.O., 1978, Plane Steady Flows and Energy Estimates for the Navier-Stokes Equations, Arch. Rational Mech. Anal., 68, 359–381 [Notes for VI]MathSciNetMATHGoogle Scholar
  6. Ames, K.A., and Payne, L.E., 1989, Decay Estimate in Steady Pipe Flow, SI AM J. Math. Anal., 20, 789–815 [Notes for VI]MathSciNetMATHGoogle Scholar
  7. Padula, M., and Pileckas, K., 1992, Steady Flow of a Viscous Ideal Gas in Domains with Noncompact Boundaries: Existence and Asymptotic Behavior in a Pipe, Ann. Mat. Pura Appl., in Press [Notes for VI]Google Scholar
  8. Heywood, J.G., 1976, On Uniqueness Questions in the Theory of Viscous Flow, Acta Math., 136, 61–102 [Introduction to III, III.4, III.5, Notes for III, Introduction to VI, VI.3, VI.4]MathSciNetMATHCrossRefGoogle Scholar
  9. Solonnikov, V.A., and Pileckas, K., 1977, On Certain Spaces of Solenoidal Vectors and on the Solvability of a Boundary-Value Problem for the System of Navier-Stokes Equations in Domains with Noncompact Boundaries, Zap. Nauch. Sem. Len. Otdel. Mat. Inst. Steklov (LOMI), 73, 136–151; English Transl.: J. Soviet Math., 34, 1986, 2101–2111 [III.4]Google Scholar
  10. Fraenkel, L.E., 1973, On a Theory of Laminar Flow in Channels of a Certain Class, Proc. Cambridge Phil Soc., 73, 361–390 [Notes for VI]MathSciNetMATHCrossRefGoogle Scholar
  11. IosiF’jan, G.A., 1978, An Analogue of Saint-Venant’s Principle and M e Uniqueness of the Solutions of the First Boundary Value Prob for Stokes’ System in Domains with Noncompact Boundaries, Dokl. Akad. Nauk SSSR, 242, 36–39; English Transl.: Soviet Math Dokl., 19, 1048–1052 [Notes for VI]Google Scholar
  12. Pileckas, K., 1980a, On Unique Solvability of Boundary Value Problems for the Stokes System of Equations in Domains with Noncompact Boundaries, Trudy Mat. Inst. Steklov, 147, 115–123; English TVansl.: Proc. Steklov Math Inst., 147, 1981, 117–126 [Notes for VI]Google Scholar
  13. Pileckas, K., 1980b, Three-Dimensional Solenoidal Vectors,Zap. Nauch. Sem. Len. Otdel. Mat. Inst. Steklov (LOMI), 96, 237–239; English TVansl.: J. Soviet Math. 21, 1983, 821–823 [Notes for III]Google Scholar
  14. Gilbarg, D., and Weinberger, H.F., 1978, Asymptotic Properties of Steady Plane Solutions of the Navier-Stokes Equations with Bounded Dirichlet Integral, Ann. Scuola Norm. Sup. Pisa,, (4), 5, 381–404 [1. 2, Notes for II, Notes for VI]Google Scholar
  15. Pileckas, K., 1980a, On Unique Solvability of Boundary Value Problems for the Stokes System of Equations in Domains with Noncompact Boundaries, Trudy Mat. Inst. Steklov, 147, 115–123; English TVansl.: Proc. Steklov Math Inst., 147, 1981, 117–126 [Notes for VI]MathSciNetGoogle Scholar
  16. Galdi, G.P., and Sohr, H., 1992, Existence, Uniqueness and Asymp-totics of Solutions to the Stationary Navier-Stokes Equations in Certain Domains with Noncompact Boundaries, Preprint #5, Istituto di Ingegneria dell’Università di Ferrara [1.3, Notes for VI]Google Scholar

Copyright information

© Springer Science+Business Media New York 1994

Authors and Affiliations

  • Giovanni P. Galdi
    • 1
  1. 1.Istituto di IngegneriaUniversità di FerraraFerraraItaly

Personalised recommendations