On the Stabilizing Effect of Convection in 3D Incompressible Flows

We investigate the stabilizing effect of convection in 3D incompressible Euler and NavierStokes equations. The convection term is the main source of nonlinearity for these equations. It is often considered destabilizing although it conserves energy due to the incompressibility condition. In this paper, we show that the convection term together with the incompressibility condition actually has a surprising stabilizing effect. We demonstrate this by constructing a new 3D model which is derived for axisymmetric flows with swirl using a set of new variables. This model preserves almost all the properties of the full 3D Euler or Navier-Stokes equations except for the convection term which is neglected in our model. If we add the convection term back to our model, we would recover the full Navier-Stokes equations. We will present numerical evidence which seems to support that the 3D model may develop a potential finite time singularity. We will also analyze the mechanism that leads to these singular events in the new 3D model and how the convection term in the full Euler and Navier-Stokes equations destroys such a mechanism, thus preventing the singularity from forming in a finite time. Keyword: Finite time singularities, 3D Navier-Stokes equations, stabilizing effect of convection.