Interior Structural Bifurcation of 2D Symmetric Incompressible Flows

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.