Abstract
We aim to derive the incompressible Navier–Stokes equations from classical mechanics. We define Lagrange and Euler coordinates and the mass density within the framework of measure theory. This yields a mathematical statement that expresses the mass conservation principle, which allows to derive the mass conservation equation. We introduce the incompressible flows and focus on their kinematic, starting with the deformation tensor and the vorticity and then the local deformations of a ball of fluid in an incompressible flow by standard ODEs. We introduce the fluid motion equation for Newtonian fluids through appropriate measures, based on the fundamental law of classical mechanics and the expression of the stress tensor in terms of the deformation tensor. The mass conservation equation coupled to the fluid motion equation yields the incompressible Navier–Stokes equations. This chapter ends with a comprehensive list of boundary conditions associated with the Navier–Stokes equations.
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Chacón Rebollo, T., Lewandowski, R. (2014). Incompressible Navier–Stokes Equations. In: Mathematical and Numerical Foundations of Turbulence Models and Applications. Modeling and Simulation in Science, Engineering and Technology. Birkhäuser, New York, NY. https://doi.org/10.1007/978-1-4939-0455-6_2
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DOI: https://doi.org/10.1007/978-1-4939-0455-6_2
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