HYDRODYNAMIC MODELS OF A UNIFORM GRANULAR MEDIUM H2O.AL-3.5 C-13 DATING OF FOSSILS

This paper uses a fluid-mechanical model of a granular medium to calcu late the hydrodynamic modes of a spatially uniform basic state. These modes are the granular analogs of the heat, sound, and shear modes of the standard fluid. Attention is focused on the possibility of an unst able mode that might result in the spontaneous development of inhomoge neities in density. Two cases are considered: the cooling medium which loses energy without replenishment, and the heated medium which reach es a steady state when an energy source balances the loss of energy th rough particle collisions. The spatially uniform state of the cooling granular medium is unstable. Two modes, analogous to the shear and hea t conduction modes of a standard fluid, are unstable at long wavelengt hs. The growth of these modes is algebraic, rather than exponential, i n time. The shear mode does not involve the formation of density inhom ogeneities, but the heat mode does. At long wavelengths the heat mode can be visualized by imagining a converging velocity increasing the de nsity of particles in a certain region. The increased collisional diss ipation of granular thermal energy reduces the pressure, and prevents it from reversing the convergent velocities, so the condensation is no t checked. The stability of the heated granular medium depends on the energy source. If the energy source selectively deposits energy in hot regions of a disturbance, the diffusion and collisional damping can b e overwhelmed, and the disturbance grows exponentially. The standard f luid (completely elastic particles) can be recovered as a special case of the heated granular medium. In all cases, waves analogous to the s ound and heat conduction modes are present. In some cases, a second ty pe of sound wave is present at long wavelengths with the peculiar prop erty of being damped more quickly for more elastic particles.