Generalized incompressible fluid dynamical system interpolating between the Navier-Stokes and Burgers equations in two dimensions

TL;DR

This paper introduces a continuum of generalized 2D incompressible fluid equations interpolating between Burgers and Navier-Stokes, analyzing their properties both theoretically and numerically to understand flow regularity and integrability.

Contribution

It proposes a novel parameterized system connecting Burgers and Navier-Stokes equations, enabling comparison of their properties and a perturbative approach to approximate Navier-Stokes solutions.

Findings

Flow regularity deteriorates as the angle parameter increases.

Burgers equations are integrable via the heat kernel.

Numerical solutions align with perturbative predictions for short times.

Abstract

We propose a set of generalized incompressible fluid dynamical equations, which interpolates between the Burgers and Navier-Stokes equations in two dimensions and study their properties theoretically and numerically. It is well-known that under the assumption of potential flows the multi-dimensional Burgers equations are integrable in the sense they can be reduced to the heat equation, via the so-called Cole-Hopf linearization. On the other hand, it is believed that the Navier-Stokes equations do not possess such a nice property. Take, for example, the 2D Navier-Stokes equations and rotate the velocity gradient by 90 degrees, we then obtain a system which is equivalent to the Burgers equations. Based on this observation, we introduce a system of generalized incompressible fluid dynamical equations by rotating velocity gradient through a continuous angle parameter. That way we can compare properties of an integrable system with those of non-integrable ones by relating them through a continuous parameter. Using direct numerical experiments we show how the flow properties change (actually, deteriorate) when we increase the angle parameter $\alpha$ from 0 to $\pi/2$. It should be noted that the case associated with least regularity, namely the Burgers equations ($\alpha=\pi/2$), is integrable via the heat kernel. We also formalize a perturbative treatment of the problem, which in principle yields the solution of the Navier-Stokes equations on the basis of that of the Burgers equations. The principal variations around the Burgers equations are computed numerically and are shown to agree well with the results of direct numerical simulations for some time.