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

Flows occur in a saturated porous fluid medium in many environmental, industrial, and biological processes. The transport of substances from surface water to groundwater is a major issue. The mathematical model is based on the interpenetrating model of two-phase media (Rahmatullin model). The proposed equations allow for a uniform analysis of the flow of a liquid in and out of the porous zone. The Navier-Stokes equation is obtained in the liquid region in this case. The equations in the porous region are very similar to the Brinkman model. There is no need to set boundary conditions in the separation area (such as Beavers – Joseph – Saffman) when describing the flow from the perspective of a single equation for the entire region. If the porous region’s energy calculation is used, cross-border conditions occur. The order of the systems of equations in each field is different in this case. The energy estimate for the Rahmatullin equation is derived using energy inequalities based on the proposed model. The difference scheme of Rahmatullin’s equation is constructed, and the constructed scheme’s stability is determined.

Author (s) Details

U. Dalabaev
The University of World Economy and Diplomacy, Buyuk Ipak Yuli Str., Tashkent, Uzbekistan.

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