Thermoviscous Fluids

  • L. R. Rakotomanana
Part of the Progress in Mathematical Physics book series (PMP, volume 31)


Intuitively, a fluid is a material that has neither a preferred configuration nor a ȁCnatural state.” The nonexistence of such a natural configuration implies that every simple fluid, in the sense Noll gives it, has an isotropic symmetry. In the framework of a nonsimple material, the definition of fluid should be revisited. For the sake of clarity, let us give an overview of the different fluid-like continua with or without singularity. The basic method for deriving the governing equations lies within the choice of bases and affine connections. This allows us to propose a classification of the main possible models.


Stream Function Singularity Distribution Torsion Tensor Affine Connection Dissipation Potential 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Among Noll’s simple materials, e.g., [146], a simple fluid is distinguished by the invariance condition of stress for any unimodular transformation of the referential configuration.Google Scholar
  2. 2.
    Projection of a volume form on an orthonormal basis is equal to unity. However its objective time derivative is not null. The basis is in fact embedded in the continuum during only an infinitesimal lapse of time to remain parallel to the spatial axes at every instant (such a method goes back to Euler, e.g., [67]).Google Scholar
  3. 3.
    Reynolds in 1883-1894 identified laminar and turbulent flows by using the criterion for the onset of turbulence in terms of the Reynolds number. Furthermore, he decomposed the flow into a mean part and fluctuating parts and therefore identified additional stresses due to turbulence: “Reynolds stresses.”Google Scholar
  4. 4.
    For another point of view, e.g., Batra [9] has suggested a further dependence of the constitutive dual variables on the rate of temperature and has defined an equivalent temperature variable θ* (θ, ζθ) ), in addition to the metric, the temperature, and their gradients. A theory of nonsimple material thermomechanics has thus been developed based on the constitutive laws \( \Im = \hat \Im \left( {g,\theta ,\nabla g,\nabla \theta ,\zeta \theta } \right). \). Without going into details, it has been shown mainly that either the derivative \( \frac{{\partial \theta ^ * }} {{\partial \zeta _\theta }} = 0 \) and then thermal disturbances propagate with infinite speed, and constitutive laws writes \( \Im = \hat \Im \left( {g,\theta ,\nabla g,\nabla \theta ,\zeta \theta } \right). \) ) (nonsimple materials), or \( \frac{{\partial \theta ^ * }} {{\partial \zeta _\theta }} \ne 0 \) and thus finite speed of thermal disturbances occur. But the Coleman and Noll method imposes independence from higher metric gradients to give \( \Im = \hat \Im \left( {g,\theta ,\nabla \theta ,\zeta \theta } \right). \) This result may bring new insight for the analysisGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2004

Authors and Affiliations

  • L. R. Rakotomanana
    • 1
  1. 1.Institut Mathématique de Rennes Campus de BeaulieuUniversité de Rennes 1Rennes CedexFrance

Personalised recommendations