Abstract: In a magnetohydrodynamic (MHD) driven fluid cell, a plane non-parallel flow in a square domain satisfying a free-slip boundary condition is examined. The energy dissipation of the flow is controlled by the viscosity and linear friction. The latter arises from the influence of the Hartmann bottom boundary layer in a three-dimensional (3D) MHD experiment in a square bottomed cell. The basic flow in this fluid system is a square eddy flow exhibiting a network of $N^2$ vortices rotating alternately in clockwise and anticlockwise directions. When $N$ is odd, the instability of the flow gives rise to secondary steady-state flows and secondary time-periodic flows, exhibiting similar characteristics to those observed when $N=3$. For this reason, this study focuses on the instability of the square eddy flow of nine vortices. It is shown that there exist eight bi-critical values corresponding to the existence of eight neutral eigenfunction spaces. Especially, there exist non-real neutral eigenfunctions, which produce secondary time-periodic flows exhibiting vortices merging in an oscillatory manner. This Hopf bifurcation phenomenon has not been observed in earlier investigations.

Key words: two-dimensional (2D) Navier-Stokes equation, non-parallel square vortex flow, primary bifurcation, secondary steady-state flow, secondary time-periodic flow