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Part of the book series: CISM International Centre for Mechanical Sciences ((CISM,volume 344))

Abstract

The aim of these lectures is to study the mathematical properties of the equations governing the motion of a viscous, incompressible second-grade fluid, such as existence, uniqueness of classical solutions and stability of steady-state flows.

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References

  1. Adams R.A. (1975), Sobolev spaces, Academic Press, New York.

    MATH  Google Scholar 

  2. Agmon, S., Douglis, A. & L. Nirenberg (1959) Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions I, Comm. Pure Appl. Math., 12, 623–727.

    Article  MathSciNet  MATH  Google Scholar 

  3. Cattabriga L. (1961), Su un problema ai contorno relativo al sistema di equazioni di Stokes , Rend. Sem. Mat. Padova 31, 308–340.

    MathSciNet  MATH  Google Scholar 

  4. Coscia V. & G.P. Galdi (1994) Existence, uniqueness and stability of regular steady motions of a second-grade fluid, Int. J. Non-Linear Mechanics, 29 (4), 493–506.

    Article  MathSciNet  MATH  Google Scholar 

  5. Coleman B.D. & H. Markovitz (1964), Normal stress effects in second-order fluids, J. App. Phys. 35, 1–48.

    Article  MathSciNet  MATH  Google Scholar 

  6. Dunn J.E. & R.L. Fosdick (1974)y Thermodynamics, stability and boundedness of fluids of complexity 2 and fluids of second grade, Arch. Rational Mech. Anal. 56, 191–252.

    Article  MathSciNet  MATH  Google Scholar 

  7. Dunn J.E. & K.R. Rajagopal (1994), Fluids of differential type: critical review and thermodynamic analysis, to appear in Int. J1. Engng Sci.

    Google Scholar 

  8. Fosdick R.L. & K.R. Rajagopal (1978), Anomalous features in the model of second order fluids, Arch. Rational Mech. Anal. 70, 145–152.

    MathSciNet  Google Scholar 

  9. Foias C. & R. Temam (1978), Remarques sur les équations de Navier-Stokes stationnaires et les phénomènes successifs de bifurcation, Ann. Scuola Norm. Sup. Pisa, S. IV, 5, 29–63

    MathSciNet  MATH  Google Scholar 

  10. Friedman A. (1969), Partial differential equations, Holt, Rinehart & Winston Inc., New York.

    MATH  Google Scholar 

  11. Galdi G.P. (1994), An introduction to the mathematical theory of the Navier-Stokes equations, Vol. I: Linearized steady Problems, Springer Tracts in Natural Philosophy, Springer Verlag, Heidelberg.

    Google Scholar 

  12. Galdi G.P. (1994), An introduction to the mathematical theory of the Navier-Stokes equations, Vol. II: Nonlinear steady problems, Springer Tracts in Natural Philosophy, Springer Verlag, Heidelberg.

    Google Scholar 

  13. Galdi G.P., M. Grobbelaar-Van Dalsen & N. Sauer (1993), Existence and uniqueness of classical solutions of the equations of motion for second grade fluids, Arch. Rational Mech. Anal, 124, 221–237.

    Article  MathSciNet  MATH  Google Scholar 

  14. Galdi G.P., M. Grobbelaar-Van Dalsen & N. Sauer (1993), Existence and Uniqueness of Solutions of the Equations of a Fluid of Second Grade with Non-homogenous Boundary Conditions, to appear on Int. J. Non-Linear Mech.

    Google Scholar 

  15. Galdi G.P., M. Padula & K.R. Rajagopal (1990), On the conditional stability of the rest state of a fluid of second grade in unbounded domains, Arch. Rational Mech. Anal. 109, 173–182.

    Article  MathSciNet  MATH  Google Scholar 

  16. Galdi G.P. & K.R. Rajagopal (1994), On the slow of a body in a second-grade fluid, to appear on Arch. Rational Mech. Anal.

    Google Scholar 

  17. Galdi G.P. & A. Sequeira (1994), Further existence results for classical solutions of the equations of a second-grade Fluid, to appear on Arch. Rational Mech. Anal.

    Google Scholar 

  18. Rajagopal K.R. (1992), Flow of viscoelastic fluids between rotating plates, Theor. and Comput. Fluid Dyn., 3, 185–216.

    Article  MATH  Google Scholar 

  19. Rajagopal K.R. & P.N. Kaloni (1989), Some remarks on boundary conditions for lows of fluids of the differential type, in: Continuum Mechanics and its Applications, Hemisphere Press.

    Google Scholar 

  20. Rivlin R.S. & J.L. Ericksen (1955), Stress-deformation relations for isotropic materials, J. Rational Mech. Anal. 4, 323–425.

    MathSciNet  MATH  Google Scholar 

  21. Simader C.G. & H. Sohr (1993) The weak and strong Dirichlet problem for A in L q in bounded and exterior domains, Pitman Research Notes in Mathematics, in press.

    Google Scholar 

  22. Temam R. (1975), On the Euler equation of incompressible perfect fluids, J. Funct. Anal. 20, 32–43.

    Article  MathSciNet  MATH  Google Scholar 

  23. Temam R. (1986), Remarks on the Euler equation, In: Proc. Symp. Pure Math. 45, 429–430.

    Article  MathSciNet  Google Scholar 

  24. Truesdell C. & W. Noll (1965), The nonlinear field theories of Mechanics, Handbuch der Physik HI/3, Springer-Verlag, Heidelberg.

    Google Scholar 

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© 1995 Springer-Verlag Wien

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Galdi, G.P. (1995). Mathematical Theory of Second-Grade Fluids. In: Galdi, G.P. (eds) Stability and Wave Propagation in Fluids and Solids. CISM International Centre for Mechanical Sciences, vol 344. Springer, Vienna. https://doi.org/10.1007/978-3-7091-3004-9_3

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  • DOI: https://doi.org/10.1007/978-3-7091-3004-9_3

  • Publisher Name: Springer, Vienna

  • Print ISBN: 978-3-211-82687-4

  • Online ISBN: 978-3-7091-3004-9

  • eBook Packages: Springer Book Archive

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