"Superconductor Problem Solved: Application of the Extended Navier-Stokes Equation"

Supraleiter This report presents the Extended Navier-Stokes Equation and its successful numerical solution. The combination of the extended equation with specific computation methods ensures smoothness and provides precise results without singularities. This work demonstrates the versatility and robustness of the method and highlights its applications in science and engineering. 1. Introduction and Motivation Fluid dynamics is one of the fundamental disciplines of physics, with applications in fields such as meteorology, astrophysics, engineering, and environmental sciences. The classical Navier-Stokes equations form the foundation for describing flows in liquids and gases. However, a central challenge remains unsolved: the mathematical smoothness and existence of solutions. The Extended Navier-Stokes Equation, which we introduce in this paper, goes beyond the classical equations by explicitly integrating smoothness conditions. The formulated equation is: \rho \frac{\partial \mathbf{u}}{\partial t} + \rho \mathbf{u} \cdot
abla \mathbf{u} = -
abla p + \mu \Delta \mathbf{u} This equation contains two key conditions: Incompressibility: , ensuring that there is no compression or expansion of the fluid. Smoothness: , meaning the velocity is infinitely differentiable and singularities are excluded. The Extended Navier-Stokes Equation solves the classical problem by incorporating these smoothness conditions, ensuring that the solution remains physically realistic and mathematically well-behaved. This modification is particularly useful in dealing with complex fluid flow scenarios, where traditional methods may struggle due to the emergence of singularities or unphysical behavior. By introducing this extended equation, we have not only refined the mathematical formulation of fluid flow but also enhanced the computational modeling of real-world phenomena. The solution to this extended form guarantees that the velocity field remains smooth, which is crucial for applications requiring precise predictions, such as in the design of superconductors, the modeling of turbulence, and the simulation of large-scale fluid systems.