Solvability of a Boundary Value Problem for One-Dimensional Motion of a Granular Matter

Granular material is one of the most common in nature. Recently, there has been a significant amount of interest in the granular materials, such as powders or sand. It is challenging to describe the motion of such kind of macroscopic size matter, because it behaves like solid, fluid or gas. Size division, pattern formation, avalanches, packing and convection - these are just a few examples from a wide range of observed phenomena that occur during the motion process of granular matter. Most of these processes are based by flow and no wonder that much effort has been devoted to obtain a hydrodynamic description in which granular materials are treated as a continuous medium. During the research of vertically shaken granular medium motion in an opened container it was found that experimental and numerical results are explained by hydrodynamic theory. This paper investigates the motion of a granular matter for a shallow, vertically shaken bed. We assume the Leidenfrost state as initial and the matter resembles a fluid heated up from below.A one-dimensional isothermal and a non-isothermal problem of granular material motion,simulatedby a hydrodynamic model are considered, and the material is assumed to be a continuous medium. The local temporal solvability of isothermal and non-isothermal initial boundary-value problem in Sobolev’s spaces is proved. This work presents a numerical study of the one-dimensional model of granular medium flow. The feature of this model is consideration of the Navier-Stokes equations accounting an interpolation form of VanderWaals constitutive equation for pressure. A finite-difference approximation for numerical solution is proposed