I am having hard time proving this either right or wrong.
Let's take for example a river, does the water flowing down follow the least resistant path?
[Physics] Does water follow the path of least resistance
flowwater
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What you refer to is conservation of mass under some assumptions:
- Constant density
- A steady state flow
I'll bring us back to your equation by starting with the very fundamental mass accounting for a given fluid flow. To be comprehensive, we need to recognize that velocity isn't constant over the entire area, but we will assume that it is. Take the flow rate to be $\dot{m}$.
$$\dot{m} = \rho V A$$
Now, if we have a steady state flow along a single flow path, then this quantity will be constant over the entire path, $\dot{m}=const$. Water in the cases you are concerned about is sufficiently incompressible so $\rho = const$. This results in your conclusion that $VA$ is constant.
Gravity may or may not shift the balance from $V$ to $A$ or vice versa. It depends on if there are rigid boundaries to the flow. If you have a flow fall freely in air or flow downward in a trench (like a river) then the boundary of the fluid may change freely. If you have a pipe with a given flow area, then the velocity is fully determined from that. Anyway, there are laws that conserve other things - like energy. So in a rigid pipe flowing downward (absent friction) the pressure will increase as you go down in elevation, which results directly from gravity.
The difference between rain and water in the sink is that rain is simply falling, while water in the sink is being drawn into a center from a distance away, and the water in the sink is not perfectly still. It is rotating, if only a little bit. As it is drawn to the center, the rotation becomes more rapid.
The principle is Conservation of Angular Momentum.
Here, Ms. Kawaguti speeds up her rotation by pulling in her arms:
Best Answer
The river will not always follow the along the path of least resistance because as water flows, its momentum increases. If a river follows a bend/curve in the land, it may slosh against the bank or even go over the bank--but eventually, it will lose some of its momentum & then flow to a lower elevation or seep through the earth to a lower elevation. So as you can see the direction & momentum of water movement depends on other factors.