Many rivers in sub-zero winter temperatures don't freeze (photo). For simplicity let's discuss a (small) water pipe/canal with uniform cross section, inclined by an angle $\theta$.

It seems that:

• According to the heat equation, $\frac{\partial T}{\partial t} - \alpha \nabla^2 T=0$, the temperature of the water in the canal supposed to change (drop) to meet equilibrium conditions with the environment.
• According to the incompressible Bernoulli's equation, ${v^2 \over 2}+gz+{p\over\rho}=p_0$, the inclination which causes loss of potential energy causes the flow speed to increase.
• According to the expression for adiabatic stagnation temperature, $T_0 = T + \frac{v^2}{2C_p}$, the reduction of the stagnation temperature (due to heat loss), will cause the flow velocity to decrease.

So:

1. What would eventually happen to the running water in the canal?
2. Is the only thing that prevents the water from freezing is the inclination?
3. Is there a known threshold for flow speed as a function of environment temperature ratio to prevent freezing?
4. What's the velocity profile in the canal?
5. Is flow speed the only reason some rivers don't freeze?