Unable to maintain a constant temperature in the system

Thank you. I was thinking that if the wall is accelerated slowly enough to have a quasi-steady state, then use of a thermostat would remove the energy added to the flow by the wall.

that line of thinking doesn't make much sense. you *should* get a
steady state at *constant* velocity *and* when using a thermostat. if
you *change* the velocity, then also the steady state equilibrium
*will* change, regardless of the *speed* of the change. the thermostat
coupling parameter then determines how effective that removal of the
excess kinetic energy is and how far you will stray from the desired
target temperature.

I fixed the *all* to just the *flow*, that was part of the problem, but still temperature increases gradually. Also, I was of the impression that changing the damping factor would change the viscosity of the fluid as well.

yes, the stronger you couple a thermostat, the more you perturb the
dynamics. for a langevin thermostat that changes the viscosity, but
also increases the noise added to the velocities.