Physical Approach to Solving the Mathematical Navier-Stokes Problem

Instead of modeling based on using an infinitesimal fluid element, which is treated as a continuous medium, we consider fluid flow in a fluid system as a model gas flow in a model gas system identical to the fluid system. We assigned to the model gas properties, which differ from the properties typically assigned to the ideal gas. In our approach, we mimic the movement of each particle/molecule composing the model gas and then gather that movement into macro quantities characterizing the fluid flow. We formulated integro-differential balance equations for mass, momentum, and energy applied to any non-moving point in three-dimensional space occupied by the model gas and operable from the continuum through the rarefied to the ballistic flow regimes. In parallel, we worked toward understanding how our derived integro-differential equations of the balance may relate to the existing Navier-Stokes equations. Since Navier-Stokes equations are limited to the continuum and collision-dominated flow regimes, we reduced the derived mass and momentum integro-differential balance equations by if during the period between sequential collisions, the relative change of any property value or any parameter characterizing the model gas is insignificant. Then by applying the method of vector differentiation, we converted the mass and momentum integro-differential balance equations into corresponding vector differential balance equations. We were surprised to observe that our converted differential forms look identically to the Navier-Stokes equations. This finding has led us to the conclusion that, in the collision-dominated flow regime, the formulated integro-differential forms of the balance are exact implicit solutions for corresponding Navier-Stokes equations. This finding also suggests that the proposed matching vector integro-differential forms of the mass, momentum, and energy balance, which can be solved by computer-implemented methods with no difficulty, may represent a physical problem described by the Navier-Stokes equations. We also provided five additional validation tests demonstrating the feasibility of the proposed method.