On the generalized method lines applied to the time-independent incompressible Navier-Stokes system

TL;DR

This paper develops a generalized method of lines approach to solve the time-independent incompressible Navier-Stokes equations, particularly when external forces are gradients, using fixed point theory for validation.

Contribution

It introduces a linear system for special Navier-Stokes cases and applies the generalized method of lines with fixed point theorems to approximate solutions.

Findings

Linear system solution for gradient external forces

Approximate solutions via generalized method of lines

Validation through Banach fixed point theorem

Abstract

In the first part of this article, we obtain a linear system whose the solution solves the time-independent incompressible Navier-Stokes system for the special case in which the external forces vector is a gradient. In a second step we develop approximate solutions, also for the time independent incompressible Navier-Stokes system, through the generalized method of lines. We recall that for such a method, the domain of the partial differential equation in question is discretized in lines and the concerning solution is written on these lines as functions of the boundary conditions and boundary shape. Finally, we emphasize these last main results are established through applications of the Banach fixed point theorem.