What happens to running water in sub-zero environment? 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:


*

*What would eventually happen to the running water in the canal?

*Is the inclination the only thing that prevents the water from freezing?

*Is there a known threshold for flow speed as a function of environment temperature ratio to prevent freezing?

*What's the velocity profile in the canal?

*Is flow speed the only reason some rivers don't freeze?

 A: It should be noted that the Water freezes on surface, cause the density of the Water is highest at 4 degrees, which makes the 0-degree water to float on the top.
1. What would eventually happen to the running water in the canal?
Typically the running water simply is mixed, which means that the whole water mass must be cooled (and even the earth below it) to zero before it starts to freeze. And if this happens, then the river starts to freezes from bottom or where ever it simply founds a point to build chrystals. 
The term is "Frazil ice".
2. Is the inclination the only thing that prevents the water from freezing?
No. If the water doesn't find points to build chrystal, it just becomes supercooled. You can actually cool a still distilled water down to - 48 Celcius and still have it liquid.
3. Is there a known threshold for flow speed as a function of environment temperature ratio to prevent freezing?
Froude number is also here practical; FR<1 typically doens't freeze, and Fr>1 freezes.
The air temperature doesn't make so much of a difference.
4. What's the velocity profile in the canal?
"Typical", I would say. Means the highest velocity is near the surface in the middle of the flow.
5. Is flow speed the only reason some rivers don't freeze?
The velocity is not so relevant as Froude Number. Many other aspects are also explained already. 
