Many rivers in sub-zero winter temperatures don't freeze ([photo][1]).
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][2], ${v^2 \over 2}+gz+{p\over\rho}=p_0$, the inclination which causes loss of potential energy causes the flow speed to increase.
 - According to the expression for adiabatic [stagnation temperature][3], $T_0 = T + \frac{v^2}{2C_p}$, 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 flow speed the only reason some rivers don't freeze?
 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 the only thing that prevents the water from freezing is the inclination?

  [1]: http://ravenshadow.lmd.net/Images/Jan05Downtown003.jpg
  [2]: https://en.wikipedia.org/wiki/Bernoulli%27s_principle
  [3]: https://en.wikipedia.org/wiki/Stagnation_temperature