Hollo echos my thoughts.
On a nice evening, I take the door off (to be at one with the world) and tour around at a speed more a kin to endurance than range or race. I have had a (fortunately few) occasions, where staying up and orbiting was preferrable to landing right away. Waiting for the clearing of an obstructed runway, and float plane landing area come to mind. On some occasions, where I have a very large tailwind at high altitude, flight at endurance speed is the most efficient, as it becomes about the same speed as flight for best range.
I also participate in public service searches, where flight for endurance is the most appropriate. I have had to interrupt my search to go for fuel, and that's a little embarrassing!
A few additional comments:
Some engine installations are better than others in cooling, but I would expect all engine installations to cool adequately indefinately, during flight at endurance speed. This characteristic would be covered during the climb cooling test at the time of certification for the aircraft. The climb speed would be best rate, which would be very close to the speed for endurance, and the climb at full power and leaned (if appropriate). Presuming design compliance has been shown in this phase of flight, flying at the same speed in level flight with the power pulled well back should have no difficulty cooling.
Turning the prop slowly has merit, but not "over square" unless the engine/propeller design specifically approves this type of operation. Over square operation will rapidly intrduce the risk of detonation, increasing with increased power. Further, it will increase propeller blade strain. We don't think of that very often, but I have been reminded by observations and resulting limitations on a test propeller imposed by the propeller manufacturer, for an aircraft we are flight testing right now (engine change STC). The engine involved (a well know brand in the 200 HP range) is limited with other propellers, so as to have "avoid operation" speed ranges. It looks like this engine/propeller into airframe installation in this case may be similarly limited. Time and testing will tell. We are presently considering the climb cooling performance of this installation.
The mixture setting of an engine will have nearly nothing to do with the speed at which the plane will fly. As the original question was asked about speed, as opposed to fuel flow, I suggest that mixture settings would not be an element of the answer. Mixture is a distinct subject, which will forever be the basis for many discussions. Fiddle with mixture as you will. In a carburetted engine (any older 172) you will never achieve the ideal mixture into all cylinders. There will probably be a power setting at which you come close, but this is not published, varies considerably engine to engine, and can only be determined with careful experimentation with a scanner. For my 150, the preferred RPM for equal EGT for the four cylinders is 2450, If I run more or less power, I'll be leaning only to the leanest cylinder (which changes), and thus wasting gas. Thus, If I fly at the speed for best endurance, two of the cylinders are running un-necessarily rich, and wasting gas. I'm just betting that the amount conserved by the overall reduced power, is better than the amount wasted by the uneven fuel flow. Truth is, I really don't care that much! Fuel injected are a vast improvement over this situation, though still not perfect. Fuel injected dilikes Mogas, and my engine and I like Mogas! In the past, we have converted injected engines to carburettors so we can run Mogas, but I thread drift......
Pilot DAR