Numerical Resolution of the Navier-Stokes Equations in 3D Using the Finite Volume Method: Application to the Millennium Problem
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Barack Ndenga
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Abstract
We present a robust numerical framework for solving the three-dimensional incompressible
Navier–Stokes equations using the finite volume method. Our Python-based implementation
employs explicit time integration, pressure correction via a Poisson solver, and advanced 3D
visualization tools—including vortex identification and particle tracking. The simulations capture
the formation, evolution, and dissipation of vortex structures, with a monotonic decay of kinetic
energy consistent with the physics of viscous incompressible flows. While this work does not
constitute a formal proof, our results provide new insights into the regularity and energy
properties of solutions, directly addressing the Clay Mathematics Institute’s Millennium Problem.
All code and visualization tools are openly available to ensure full reproducibility and to foster
further research on the existence and smoothness of Navier–Stokes solutions in three
dimensions.While the numerical methods employed are well established, this work distinguishes
itself by providing a fully open-source, Python-based 3D framework—complete with advanced
visualization, detailed documentation, and explicit orientation towards the Millennium Problem.
Description
My work presents a robust, open-source, and visually interpretable Python framework for solving 3D incompressible Navier-Stokes equations using the finite volume method (FVM