Fluid Mechanics in Minkowski Space. Modelling of Fluid Motion in Porous Materials with Anisotropic Pore Space Structure

Abstract

In the paper a fluid motion in a rigid porous medium of anisotropic pore space structure is described. Considerations are based on the new macroscopic model of saturated porous medium (Cieszko [4], [5]) in which the fluid flow through porous skeleton of anisotropic pore structure is described as a motion of the material continuum in the plane anisotropic metric space - Minkowski space - immersed in Euclidean one that is the model of the physical space. This model takes into account the fundamental fact for kinematics of fluid-saturated porous solid that pore space of permeable skeleton forms the real space for a fluid motion and its structure imposes restriction on that motion. In such approach the metric tensor of the Minkowski space is used to characterise the anisotropic structure of the skeleton pore space. It enabled one to determine the measures of any line, surface and volume elements in Minkowski and Euclidean spaces and to define the geometrical parameters characterising pore structure of porous materials: tortuosity, surface and volume porosity.The mass and linear momentum balance equations for fluid are derived and the equation for wave propagation in barotropic inviscid fluid filling orthotropic space of pores is obtained. It is shown that the velocity of the plane wave in such a medium depends on the direction of wave propagation.It worth to underline that presented description of fluid motion in the Minkowski space is a good starting point for modelling of mechanics of deformable porous solid saturated with fluid where the concept of deforming anisotropic space (Finsler space) as a model of pore space would have to be used.KeywordsSaturated porous materialsanisotropic pore structureanisotropic spacewave propagation