Continua and Generalities About Their Equations

  • Giovanni Gallavotti
Part of the Texts and Monographs in Physics book series (TMP)


A homogeneous continuum, chemically inert, in d dimensions is described by:
  1. (a)

    a region Ω in ambient space (Ω ⊂ ℝ d ), which is the occupied volume;

  2. (b)

    a function Pρ(P) > 0, defined on Ω, giving the mass density;

  3. (c)

    a function PT(P) defining the temperature;

  4. (d)

    a function Ps(P) defining the entropy density (per unit mass);

  5. (e)

    a function Pδ(P) defining the displacement with respect to a reference configuration;

  6. (f)

    a function Pu(P) defining the velocity field;

  7. (g)

    an equation of state relating T(P), s(P), ρ(P);

  8. (h)
    a stress tensor \(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\tau } \), also denoted (τ ij ), giving the force per unit surface that the part of the continuum in contact with an ideal surface element dσ, with normal vector n, on the side of n, exercises on the part of the continuum in contact with dσ on the side opposite to n, via the formula
    $$d\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{f} = \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\tau } \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{n} d\sigma \;{(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\tau } \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{n} )_i} = \sum\nolimits_{j = 1}^d {{\tau _{ij}}} {n_j}$$
  9. (i)
    a thermal conductivity tensor \(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{k} \), giving the quantity of heat traversing the surface element dσ; in the direction of n per unit time via the formula
    $$dQ = - \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{k} \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{n} \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\partial } T\;d\sigma $$
  10. (l)

    a volume force density Pg(P);

  11. (m)

    a relation expressing the stress and conductivity tensors as functions of the observables δ, u, ρ, T, s.



Vector Field Velocity Field Euler Equation Current Line Flux Line 
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. [LL71]
    Landau, L., Lifchitz, E.: Mécanique des fluides, MIR, Moscow, 1971.Google Scholar
  2. [La32]
    Lamb, H.: Hydrodynamics, sesta edizione, Cambridge University Press, 1932.Google Scholar
  3. [Ar79]
    Arnold, V.: Metodi Matematici della Meccanica Classica, Editori Riuniti, 1979.Google Scholar
  4. [DZ94]
    Dyachenko, A.I., Zakharov, V.E.: Is free surface hydrodynamics an integrable system?, Physics Letters A190, 144–148, 1994.MathSciNetCrossRefGoogle Scholar
  5. [DLZ95]
    Dyachenko, Lvov, Y.V., A.I., Zakharov V.E: Five-wave interaction on the surface of deep fluid, Physica D, 87, 233–261, 1995.MathSciNetADSzbMATHCrossRefGoogle Scholar
  6. [CW95]
    Craig, W., Worfolk, P.: An integrable normal form for water waves in infinite depth, Physica D 84 (1995) pp. 513–531Google Scholar
  7. Craig, W.: Birkhoff normal forms for water waves, Mathematical Problems in Water Waves, Contemporary Mathematics, AMS (1996), pp. 57–74.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Giovanni Gallavotti
    • 1
  1. 1.Dipartmento di Fisica, I.N.F.N.Università degli Studi di Roma “La Sapienza”RomaItaly

Personalised recommendations