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I'm currently taking a course on compressible flow, and although I have a pretty firm grasp of the mathematics, the way that flows behave still seems confusing to me. When considering Fanno flow, I don't understand why subsonic flows are accelerated to Mach 1. I have sufficient background in biology and chemistry to be familiar with the natural tendency of systems to maximize entropy and minimize energy, but I'm failing to see how Mach 1 is the maximum entropy state. What about Mach 1 makes it have the maximum number of microstates for a flow?

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If we take a control volume as a system, even though we say the system is adiabatic, there is heat transfer. The heat transfer from the surrounding to the system is the frictional heat. Meanwhile, the system does work to the surrounding by doing frictional work and accelerating/expanding of the system.

The first law and the second law are listed below. $$dU=\delta Q -dW$$ $$dS =\frac{\delta Q}T$$

Conceptually, when the flow is subsonic, $dW$ is greater than $\delta Q$ because the work does not only include doing frictional work but also include accelerating/expanding the system. Therefore, $dU$ decreases and the temperature decreases. However, this doesn't prevent $dS$ from increasing due to increasing in volume (more microstate in space though less micro velocity (i.e. low temperature)). This will last until it can no longer accelerate/expand the system, which is Mach 1.

In the nutshell, movement is reduced but it has more space. The net is increasing in entropy but this can last forever.

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