This is something anyone could easily verify. When we open a tap slowly, water bends inwards (towards the axis) while maintaining its laminar flow. After a certain height below the opening, the flow becomes turbulent. I've approximately illustrated the shape of water near the top portion in the following diagram:
I tried to explain the above phenomenon based on my knowledge on fluid dynamics. Let us consider the following diagram:
Here, $A_1$ and $A_2$ are the areas of cross-section and $v_1$ and $v_2$ are the speeds of water molecules at two different heights (indicated by dotted red lines).
Since, the shape of water remains fairly constant and the flow is laminar, in a time interval $\Delta t$, the volume of water passing through level 1 must be equal to the volume of water passing through level 2. Mathematically, we can say:
$$A_1v_1\Delta t=A_2v_2\Delta t$$ $$A_1v_1=A_2v_2$$
Or in other words, the product of the area of cross section and the speed remains the same at all heights and this is known as the equation of continuity. Since water molecules are under the gravitation force of attraction, they are accelerated downwards. So, $v_1<v_2$. As the product of area of cross section and the speed must be a constant, $A_1>A_2$. This explains why water bends towards the axis while falling slowly from a tap.
But the above explanation fails at much lower heights above the fluctuating flow zone (where flow fluctuates from laminar to turbulent). Let us consider another diagram:
The area of cross-section remains almost constant at the intermediate heights above the red zone. It doesn't decrease in accordance to the equation of continuity. Further, my method of explanation involves a lot of assumptions and I've also neglected surface tension, viscosity etc. I'm unable to imagine how these forces would affect our results.
Is this a correct reason for "Why does water falling slowly from a tap bend inwards?" or is there any better explanation for this phenomenon?
Image Courtesy: My own work :)