Abstract
Our modeling fluid flow, especially if the fluid is referred to as a gas, is established on mimicking each particle/molecule’s movement and then gathering that movement into macro quantities characterizing the fluid flow. It has resulted in discovering a new principle of the property (mass, momentum, and energy) balance in space. We named it the Ballistic Principle of the Property Balance in Space as described earlier in our publications. This paper uses a different scheme of defining a net rate of total property efflux than our original paper. Using this scheme, we formulated integro-differential forms of mass balance and momentum balance equations adapted to the incompressible fluid flow (gas flow with a mass-flow velocity less than 0.3 Ma) at the non-uniform temperature in the infinite gas space. We also investigated the analytical behavior of the integro-differential equations in the region bounding the point of singularity by applying the Taylor series expansion method to transform the integro-differential mass and momentum balance equations into the corresponding vector differential equations. Then we compared them with the Navier-Stokes equations of mass and momentum conservation for an incompressible fluid. We were surprised to find that the Navier-Stokes momentum balance equation does not describe the fluid flow adequately. Particularly, it does not consider the momentum associated with the part of velocity acquired by each gas particle during its free path traveling in the body force field. Also, the Navier-Stokes momentum balance equation is silent about the influence of the temperature non-uniformity on the momentum balance. Finally, we have demonstrated that the Navier-Stokes equations are not applicable to govern fluid flow on R3 × [0, ∞).
Highlights
Navier-Stokes equations are designed to govern the motion of fluids, including gases
We investigated the analytical behavior of the integro-differential equations in the region bounding the point of singularity by applying the Taylor series expansion method to transform the integro-differential mass and momentum balance equations into the corresponding vector differential equations
We may conclude the following: 1) We derived integro-differential forms of the mass balance and momentum balance equations adapted to the incompressible non-isothermal gas flow in the infinite gas space
Summary
Navier-Stokes equations are designed to govern the motion of fluids, including gases. One can expect that the term 1 ρ ∇p should not appear in the Navier-Stokes momentum balance equation if it is expanded to describe the fluid flow in the infinite space Vc = V∞ = 3 ×[0, ∞) and the ambient pressure p∞ is constant at infinity. We proposed a new approach in the modeling of fluid dynamics problems It recognizes that each gas particle moves with a probability between two points in space occupied by the gas and follows a ballistic trajectory governed by a law of motion in free space. This paper provides further insights into the newly discovered BPPBS and its application to formulate governing mass and momentum balance equations for incompressible gas flow in integro-differential and differential forms. Recognizing the unconventional nature of our approach and in the interests of a better understanding of our method, we provide a short introduction to the methodology adapted to the incompressible gas flow at the non-uniform temperature in general
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