Interior Structural Bifurcation of 2D Symmetric Incompressible Flows

TL;DR

This paper investigates the structural bifurcation of 2D symmetric incompressible flows with interior singular points of index ±1, revealing generic bifurcation patterns like vortex pairs and saddle connections, supported by numerical evidence.

Contribution

It extends bifurcation analysis to symmetric flows with index ±1 singular points, identifying generic flow pattern bifurcations and providing numerical validation.

Findings

Flow bifurcations include vortex pairs and saddle connections.

Symmetry conditions ensure generic bifurcation scenarios.

Numerical simulations confirm theoretical predictions.

Abstract

The structural bifurcation of a 2D divergence free vector field $\mathbf{u}(\cdot, t)$ when $\mathbf{u}(\cdot, t_0)$ has an interior isolated singular point $\mathbf{x}_0$ of zero index has been studied by Ma and Wang. Although in the class of divergence free fields which undergo a local bifurcation around a singular point, the ones with index zero singular points are generic, this class excludes some important families of symmetric flows. In particular, when $\mathbf{u}(\cdot, t_0)$ is anti-symmetric with respect to $\mathbf{x}_0$, or symmetric with respect to the axis located on $\mathbf{x}_0$ and normal to the unique eigendirection of the Jacobian $D\mathbf{u}(\cdot, t_0)$, the vector field must have index 1 or -1 at the singular point. Thus we study the structural bifurcation when $\mathbf{u}(\cdot, t_0)$ has an interior isolated singular point $\mathbf{x}_0$ with index -1, 1. In particular we show that if such a vector field with its acceleration at $t_0$ both satisfy aforementioned symmetries then generically the flow will undergo a local bifurcation. Under these generic conditions, we rigorously prove the existence of flow patterns such as pairs of co-rotating vortices and double saddle connections. We also present numerical evidence of Stokes flow in a rectangular and cylindrical cavity showing that the bifurcation scenarios we present are indeed realizable.