  Motivated by the relevance of edge state solutions as mediators of
transition, we use direct numerical simulations to study the effect of
spatially non-uniform viscosity on their energy and stability in minimal
channel flows. What we seek is a theoretical support rooted in a fully
non-linear framework that explains the modified threshold for transition to
turbulence in flows with temperature-dependent viscosity. Consistently over a
range of subcritical Reynolds numbers, we find that decreasing viscosity away
from the walls weakens the streamwise streaks and the vortical structures
responsible for their regeneration. The entire self-sustained cycle of the edge
state is maintained on a lower kinetic energy level with a smaller driving
force, compared to a flow with constant viscosity. Increasing viscosity away
from the walls has the opposite effect. In both cases, the effect is
proportional to the strength of the viscosity gradient. The results presented
highlight a local shift in the state space of the position of the edge state
relative to the laminar attractor with the consequent modulation of its basin
of attraction in the proximity of the edge state and of the surrounding
manifold. The implication is that the threshold for transition is reduced for
perturbations evolving in the neighbourhood of the edge state in case viscosity
decreases away from the walls, and vice versa.
