General Solution to 2D Steady Navier-Stokes Equation for Incompressible Flow without vorticity diffusion

TL;DR

This paper derives a comprehensive analytical solution to the 2D steady incompressible Navier-Stokes equations without vorticity diffusion, using potential functions to describe various flow types and revealing polynomial constraints on the solutions.

Contribution

It introduces a novel approach using potential functions to obtain the general solution for 2D steady Navier-Stokes flow without vorticity diffusion, extending beyond classical solutions.

Findings

Vorticity equation is a biharmonic function in potential form

Velocity and pressure fields for steady shear flow are polynomial of degree three

Flow parameters include up to four independent variables in non-unidirectional shear flows

Abstract

The study solves the general solution to 2D steady Navier-Stokes equation for incompressible flow without vorticity diffusion, which is more general than Stokes flow. In order to obtain the general solution, two potential functions are introduced to express the velocity: a vector potential describing the rotational incompressible flow and a scalar potential describing the irrotational incompressible flow. The results show that the vorticity equation expressed with potential functions is a biharmonic function, which means that the potential functions describing the flow field are polynomials of no more than fourth degree. For a steady unidirectional shear flow, the velocity and pressure fields can be described with the vector potential expressed by a polynomial of third degree. For non unidirectional two-dimensional steady shear flow, there may be four independent parameters in the two potential functions.