An eigen-representation of the Navier–Stokes equations

Abstract

In this paper we demonstrate that the incompressible Navier–Stokes equations can be formulated as a system of quadratic polynomial equations with a regular, tractable structure. This is first shown to be possible by using the combination matrix of Cheung and Zaki (2014) on the time-harmonic Navier–Stokes with periodic boundaries. We also show that the initial value problem can be rewritten in a similar fashion using the Laguerre polynomial basis and the appropriate version of the combination matrix. The solution to both these formulations, when using a finite number of terms in the series expansion, can be found through the null space of the Macaulay matrix and leads to an eigenvalue problem. We also provide two examples which illustrate the methods discussed in this work. The approach is demonstrated on a nonlinear, univariate model differential equation, and also on the two dimensional Taylor Green vortex problem.