Some Stable Difference Schemes to Describe the Flow in the Combined Region

In many environmental, industrial and biological processes, flows occur in a saturated porous fluid medium. The transport of substances between surface water and groundwater is a very serious problem. The basis of the mathematical model is based on the interpenetrating model (Rahmatullin model) of two-phase media. The proposed equations make it possible to study the flow of a liquid in and outside the porous region in a uniform manner. In this case, the Navier-Stokes equation is obtained in the liquid region. In the porous region, the equations are close to the Brinkman model. In connection with the description of the flow from the standpoint of a single equation for the entire region, there is no need to set boundary conditions in the separation region (such as Beavers – Joseph – Saffman). Cross-border conditions arise if the energy estimation is used for the porous region. In this case, the order of the systems of equations in each area is different. On the basis of the proposed model, the energy estimation for the equation of Rahmatullin is derived using energy inequalities. The difference scheme of the equation of Rahmatullin is constructed and the stability of the constructed scheme is obtained.