On existence of general solution of the Navier-Stokes equations for 3D non-stationary incompressible flow

TL;DR

This paper presents a novel approach to solving the 3D non-stationary incompressible Navier-Stokes equations by decomposing the flow into irrotational and solenoidal parts, and proves the existence of a general solution.

Contribution

It introduces a new decomposition method for Navier-Stokes equations, separating flow into curl-free and divergence-free components, and demonstrates the existence of a general solution.

Findings

Existence of a general solution is linked to solving a PDE-system of linear equations.

Flow velocity decomposed into irrotational and solenoidal components.

Final solution is a sum of curl-free and variable curl components.

Abstract

A new presentation of general solution of Navier-Stokes equations is considered here. We consider equations of motion for 3-dimensional non-stationary incompressible flow. The field of flow velocity as well as the equation of momentum should be split to the sum of two components: an irrotational (curl-free) one, and a solenoidal (divergence-free) one. The obviously irrotational (curl-free) part of equation of momentum used for obtaining of the components of pressure gradient. As a term of such an equation, we used the irrotational (curl-free) vector field of flow velocity, which is given by the proper potential (besides, the continuity equation determines such a potential as a harmonic function). The other part of equation of momentum could also be split to the sum of 2 equations: - with zero curl for the field of flow velocity (viscous-free), and the proper Eq. with viscous effects but variable curl. A solenoidal Eq. with viscous effects is represented by the proper Heat equation for each component of flow velocity with variable curl. Non-viscous case is presented by the PDE-system of 3 linear differential equations (in regard to the time-parameter), depending on the components of solution of the above Heat Eq. for the components of flow velocity with variable curl. So, the existence of the general solution of Navier-Stokes equations is proved to be the question of existence of the proper solution for such a PDE-system of linear equations. Final solution is proved to be the sum of 2 components: - an irrotational (curl-free) one and a solenoidal (variable curl) components.