In most situations in physics, the effect of kinetic friction is to reduce the macroscopic kinetic energy of a system and convert it into heat, thereby increasing its temperature.
but in the case of subsonic Fanno flow, the opposite happens: temperature decreases and velocity increases. I know the mathematical theory behind this, but can someone explain how friction is playing its dissipative role over here? There is the familiar effect of entropy increase but I'm also looking for a answer to how reduction of temperature and increase in velocity can have the above effect which seems quite counter intuitive to me.
[Physics] Intuitive explanation for subsonic Fanno flow
dissipationflowfluid dynamicsthermodynamics
Best Answer
Friction still dissipates kinetic energy from the fluid. However, since we are assuming that the flow is adiabatic, heat is not transferred out of the system and is instead fed back into the fluid. To conserve total energy, this thermal energy can either increase the temperature and decrease velocity or decrease velocity and increase temperature. The choice is governed by the second law of thermodynamics: total entropy must increase.
As a thermodynamic state, entropy can written in terms of two other states. While it may be intuitive that a process that increases temperature increases the number of microstates (and therefore increases entropy), we also need to consider how the other states are affected.
For an ideal gas, the speed of sound $a$ is related to temperature by $$ a^2 = \gamma R T. $$ Using the definition of the Mach number $M = u/a$, we can get the following relationship between the flow velocity and temperature $$ \frac{\mathrm{d}T}{T} + \frac{\mathrm{d} M^2}{M^2} = \frac{\mathrm{d}u^2}{u^2}. $$
Since we are also assuming that the flow rate is constant, the continuity equation reduces to $\mathrm{d}(u \rho) = 0$. This assumption fixes the relationship between flow velocity and density $$ \frac{\mathrm{d}\rho}{\rho} + \frac{1}{2} \frac{\mathrm{d}u^2}{u^2} = 0. $$ So as the velocity increases, the density must decrease in order to maintain flow rate.
The change in entropy is then determined by the rate of temperature change and the rate of density change, both of behave differently at different Mach numbers. For subsonic flows, the combination that will result in increasing entropy is to drop temperature and density.
Let me know if this is still too hand-wavy, I can dig up some old notes and go over it step by step.