\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

Energy stability for thermo-viscous fluids with a fading memory heat flux

Abstract Related Papers Cited by
  • In this work we consider the thermal convection problem in arbitrary bounded domains of a three-dimensional space for incompressible viscous fluids, with a fading memory constitutive equation for the heat flux. With the help of a recently proposed free energy, expressed in terms of a minimal state functional for such a system, we prove an existence and uniqueness theorem for the linearized problem. Then, assuming some restrictions on the Rayleigh number, we also prove exponential decay of solutions.
    Mathematics Subject Classification: Primary: 45K05, 35Q79; Secondary: 80A17, 76D03.

    Citation:

    \begin{equation} \\ \end{equation}
  • [1]

    G. Amendola, Free energies for incompressible viscoelastic fluids, Quart. Appl. Math., 68 (2010), 349-374.doi: 10.1090/S0033-569X-10-01185-3.

    [2]

    G. Amendola and M. Fabrizio, Thermal convection in a simple fluid with fading memory, J. Math. Anal. Appl., 366 (2010), 444-459.doi: 10.1016/j.jmaa.2009.11.043.

    [3]

    G. Amendola, M. Fabrizio and J. M. Golden, Thermodynamics of Materials with Memory: Theory and Applications, Springer, New York, 2012.doi: 10.1007/978-1-4614-1692-0.

    [4]

    G. Amendola, M. Fabrizio and A. Manes, On energy stability for a thermal convection in viscous fluids with memory, Palestine Journal of Mathematics, 2 (2013), 144-158.

    [5]

    C. M. Dafermos, Contraction semigroups and trend to equilibrium in continuous mechanics, in Applications of Methods of Functional Analysis to Problems in Mechanics, Lectures Notes in Mathematics, 503, Springer-Verlag, Berlin-Heidelberg, 1976, 295-306.doi: 10.1007/BFb0088765.

    [6]

    R. Datko, Extending a theorem of A. M. Lyapunov to Hilbert space, J. Math. Anal. Appl., 32 (1970), 610-616.doi: 10.1016/0022-247X(70)90283-0.

    [7]

    L. Deseri, M. Fabrizio and J. M. Golden, The concept of a minimal state in viscoelasticity: New free energies and applications to $PDE_S$, Arch. Rational Mech. Anal., 181 (2006), 43-96.doi: 10.1007/s00205-005-0406-1.

    [8]

    C. R. Doering, B. Eckhardt and J. Schumacher, Failure of energy stability in Oldroyd-B fluids at arbitrarily low Reynolds numbers, J. Non-Newtonian Fluid Mech., 135 (2006), 92-96.doi: 10.1016/j.jnnfm.2006.01.005.

    [9]

    M. Fabrizio, C. Giorgi and A. Morro, Free energies and dissipation properties for systems with memory, Arch. Rational Mech. Anal., 125 (1994), 341-373.doi: 10.1007/BF00375062.

    [10]

    M. Fabrizio and B. Lazzari, On asymptotic stability for linear viscoelastic fluids, Diff. Integral Equat., 6 (1993), 491-505.

    [11]

    A. Lozinski and R. G. Owens, An energy estimate for the Oldroyd-B model: Theory and applications, J. Non-Newtonian Fluid Mech., 112 (2003), 161-176.doi: 10.1016/S0377-0257(03)00096-X.

    [12]

    A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Lectures Notes in Mathematics, 10, University of Maryland, 1974.

    [13]

    L. Preziosi and S. Rionero, Energy stability of steady shear flows of a viscoelastic fluid, Int. J. Eng. Sci., 27 (1989), 1167-1181.doi: 10.1016/0020-7225(89)90096-7.

    [14]

    M. Slemrod, An energy stability method for simple fluids, Arch. Rational Mech. Anal., 68 (1978), 1-18.doi: 10.1007/BF00276175.

    [15]

    B. Straughan, The Energy Method, Stability, and Non Linear Convection, $2^{nd}$ edition, Springer-Verlag, New York, 2004.doi: 10.1007/978-0-387-21740-6.

  • 加载中
SHARE

Article Metrics

HTML views() PDF downloads(161) Cited by(0)

Access History

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return