Internal Friction
The sign of temperature follows from the observation that a stirred substance will come to rest due to friction with the container walls, and within the fluid. When coffee, or any other liquid, is stirred, it will spin a while after the spoon is removed. The motion will slow down because of internal friction, and finally the coffee will be at rest in the cup. The second law should describe this well- known behavior, which is observed in all viscous fluids.
With the fluid in motion, we have to account for the kinetic energy of the swirling, which must be computed by summation (i.e., integration), of the
Experience shows that over time the fluid slows down, hence the kinetic energy Ekin = [ ρ V2dV decreases over time, and will be zero in equilibrium, where the stirred substance comes to rest, V = 0; this implies
The latter inequality is compatible with the 2nd law in the form (4.30) only if the thermodynamic temperature is non-negative, T ≥ 0.
An equivalent experience is that work in transmission can be lost to friction, but not gained. Figure 4.7 shows the work and heat flows for a gearbox operating at constant temperature T , at steady state. The gearbox receives the work − W˙ in , and delivers the work W˙ out > 0. Moreover, the gearbox is in thermal contact with the environment from which it receives the heat Q˙ . The figure shows absolute values for work, the arrows indicate the direction of the work flows. The first law is straightforward to evaluate: Since the gear box operates at steady state, the energy supply must equal the energy loss. The statement of the first law can be read straight from the figure: Energy flow in (arrows pointing towards the gearbox) must be equal to energy flow out (arrows out of the gearbox), There is only a single heat flow contribution, therefore the second law becomes
Comparison between the two last equations shows, again, that the thermo- dynamic temperature must not be negative, T ≥ 0.
The lost work leaves the gearbox in form of heat 
W˙ out > 0, which is transmitted into the environment. The reason for the loss is friction within the gearbox.