Asymptotic in Time Decay of Solutions to the Equations of a Second-Grade Fluid Filling the Whole Space

  • Giovanni P. Galdi
  • Arianna Passerini

Abstract

Second grade fluids have been introduced to describe certain nonlinear effects that can not be explained by the classical theory of Navier-Stokes for Newtonian fluids (see e.g. Truesdell and Noll, 1965; Dunn and Fosdick, 1974; Rajagopal, 1992). In this regard, the Cauchy stress tensor T for an incompressible fluid of this type, is related to the kinematical variables by
$$ {\mathbf{T}} = - \tilde{p}{\mathbf{I}} + v{{{\mathbf{A}}}_{1}} + {{\alpha }_{1}}{{{\mathbf{A}}}_{2}} + {{\alpha }_{2}}{\mathbf{A}}_{1}^{2} $$
where ̃p is the pressure, v is the viscosity, α1 and α2 the normal stress moduli, and the Rivlin-Ericksen tensors A 1 and A 2 (Rivlin and Ericksen, 1955), are defined by
$$ \begin{array}{*{20}{c}} {{{A}_{1}} = \left( {grad{\mkern 1mu} v} \right) + \left( {grad{\mkern 1mu} v} \right)T} \\ {{{A}_{2}} = \frac{{dA}}{{dt}} + {{A}_{1}}\left( {grad{\mkern 1mu} v} \right) + {{{\left( {grad{\mkern 1mu} v} \right)}}^{T}}{{A}_{1}}} \\ \end{array} $$
(v denoting the velocity and \( \frac{d}{{dt}} \) the material time derivative).

Keywords

Arianna 

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References

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Copyright information

© Springer Science+Business Media New York 1995

Authors and Affiliations

  • Giovanni P. Galdi
    • 1
  • Arianna Passerini
    • 1
  1. 1.Instituto di IngegneriaUniversità di FerraraFerraraItaly

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