  This work shows how the early stages of perturbation growth in a
viscosity-stratified flow are different from those in a constant-viscosity
flow, and how nonlinearity is a crucial ingredient. We derive the
viscosity-varying adjoint Navier-Stokes equations, where gradients in viscosity
force both the adjoint momentum and the adjoint scalar (here temperature). By
the technique of direct-adjoint looping, we obtain the nonlinear optimal
perturbation which maximises the perturbation kinetic energy of the nonlinear
system. While we study three-dimensional plane Poiseuille (channel) flow with
the walls at different temperatures, and a temperature-dependent viscosity, our
findings are general for any flow with viscosity variations near walls. The Orr
and modified lift-up mechanisms are in operation at low and high perturbation
amplitudes respectively at our subcritical Reynolds number. The nonlinear
optimal perturbation contains more energy on the hot (less-viscous) side, with
a stronger initial lift-up. However, as the flow evolves, the important
dynamics shifts to the cold (more-viscous) side, where wide high-speed streaks
of low viscosity grow and persist, and strengthen the inflectional quality of
the velocity profile. We provide a physical description of this process, and
show that the evolution of the linear optimal perturbation misses most of the
physics. The Prandtl number does not qualitatively affect the findings at these
times. The study of nonlinear optimal perturbations is still in its infancy,
and viscosity variations are ubiquitous. We hope that this first work on
nonlinear optimal perturbation with viscosity variations will lead to wider
studies on transition to turbulence in these flows.
