Relationship Between the Continuity Equation and the Mass Density Diffusion Equation

Belevich M. Yu.

Mathematical model of fluid mechanics includes the so-called continuity equation, which follows from the mass conservation law. In recent decades this equation repeatedly subjected to revision. However, until now the common viewpoint is not yet developed. This paper examines the expediency of modification of the continuity equation. As an argument in favor of modifications we use a simple example of a problem that has no solution in the framework of the standard approach, i.e. in case of use of the standard continuity equation in the model of the fluid. The matter concerns modeling of evolution of the mass density jump in stationary fluid under the condition of stable stratification. The absence of solution means the presence of a contradiction in the problem posing. Such a contradiction is detected and its causes and ways to overcome it are studied. The latter, as it turns out, can be achieved by proper averaging of the fluid mechanics equations that needs to be done in view of the adopted parametric description of small-scale movements. The analysis of derivation of the system of fluid mechanics equations allows understanding, what requirements should satisfy possible modifications of the continuity equation. We also analyze the averaging procedure and describe the correct averaging of equations of the fluid model, the continuity equation including and without any extra assumptions like incompressibility. Current problem adjoins to the mass density diffusion equation, suggested earlier by P.S. Lineykin. We critically assess the derivation of this equation, and discuss its possible place in the system of fluid mechanics equations.

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