Spontaneous flow instabilities of active polar fluids in three dimensions

Active liquid crystals exhibit spontaneous flow when sufficient active stress is generated by internal molecular mechanisms. This also referred to as Fr\'{e}edericksz transition. We show this transition in three dimensions and study its dependence on the boundary conditions. Using nonlinear numerical solutions linear perturbation analysis, we find an out-of-plane under extensile for perpendicular polarity anchoring at boundary, whereas parallel permits both in-plane contractile wrinkling stress.