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

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

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?

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

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 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?

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?