On the propagation of small perturbations in viscous compressible fluid
Object of this paper is to prove that the paradox of the instantaneous propagation of small perturbations in the flow of a compressible viscous fluid is removed if one takes the relation between the stress tensor and the deformation rate tensor not the one given by the classical theory but that obtained considering the dependence on the time of the distribution function φ of the molecular velocities. The result is derived either determining the momentum transfer due to the thermal molecular motion in analogous way as that of C. Cattaneo in the problem of the heat transfer, or assuming a form of Ф on the ground of invariant considerations according to the method used by M. M. Brillouin. The general motion equations corresponding to the constitutive equation so obtained are written and applied to the case of slow motion and initial value problem (Cauchy problem); the motion equations in this case can be reduced to a system of three partial differential equations of the first order, that results to be totally hyperbolic. The characteristic lines are deduced as well as the variation laws of all the physical quantities along them, and it is shown how the solution can be obtained by an iterative method.
KeywordsConstitutive Equation Small Perturbation Viscous Fluid Characteristic Line Instantaneous Propagation
Unable to display preview. Download preview PDF.
- Duhem, P.: Récherches sur l’élasticité. Paris: Gauthier-Villars 1906.Google Scholar
- Levi-Civita, T.: Caractéristiques des Systèmes différentiels et Propagation des ondes. Paris: Librairie Félix Alcan 1932.Google Scholar
- Possio, C.: L’influenza della viscosita’ e della conducibilita’ termica sulla propagazione del suono R. Accademia delle Scienze di Torino, Vol. 78. 1942.Google Scholar
- Cattaneo, C.: Seminario Matematico e Fisico dell’Università di Modena. Modena: Società Tipografica Modenese 1948.Google Scholar
- Brillouin, M. M.: Théorie Moleculaire des Gas. Diffusion de Mouvement et de l’Energie. Annales de Chémie et de Physique. Vol. 20 (XXV). 1900.Google Scholar
- Rocard, Y.: L’Hydrodynamique et la Théorie cinétique des gas. Paris: Gauthier-Villars 1932.Google Scholar
- Courant, R.: Partial differential equations. New York—London: Interscience Publishers, John Wiley and Sons 1962.Google Scholar
- Lorentz, H. A.: Annalen der Physik. Vol. 12. 1881.Google Scholar