3-dimensional physically consistent diffusion in anisotropic media with memory

Caputo, Michele

Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 9 (1998) no. 2, pp. 131-143

Résumé

Some data on the flow of fluids exhibit properties which may not be interpreted with the classic theory of propagation of pressure and of fluids [21] based on the classic D’Arcy’s law which states that the flux is proportional to the pressure gradient. In order to obtain a better representation of the flow and of the pressure of fluids the law of D’Arcy is here modified introducing a memory formalisms operating on the flow as well as on the pressure gradient which implies a filtering of the pressure gradient without singularities; the properties of the filtering are also described. We shall also modify the second constitutive equation of diffusion, which relates the density variations of the fluid to its pressure variations, by introducing the rheology of the fluid also represented by derivatives of fractional order operating on the pressure as well as on the density. Moreover the medium will be considered anisotropic. We shall obtain the diffusion equation with these conditions in an anisotropic medium and find the Green function for a point source.