Why do streamlines converge when water flows down?

[snb]

1)Can two streamlines(line of flow) converge to a single one ?

2) When water flows from the tap, as it flows down the horizontal area of cross-section decreases...An argument using the Equation of Continuity is given...but I have two doubts regarding those
i) Is water in freefall Laminar flow ?
ii) Why do streamlines come closer as water goes down ?

 

[rcgldr]

Can two streamlines(line of flow) converge to a single one?

This would mean that something changed the differening speeds of the two streamlines so they ended up the same. In reality, there's usually a gradual change in speed of flow versus cross section as opposed to infinitely thin shear boundaries between idealized streamlines.

2) When water flows from the tap, as it flows down the horizontal area of cross-section decreases ... Why do streamlines come closer as water goes down?

Mass flow across any cross section of the flow is constant (otherwise mass would be accumulating at some point). As the water falls, it's speed increases while density remains essentually constant, so the cross sectional area decreases.

i) Is water in freefall Laminar flow?

Assuming it starts off laminar from the source, it transitions from laminar flow to turbulent flow as it falls.

 

[boneh3ad]

1)Can two streamlines(line of flow) converge to a single one ?

No. The definition of streamlines ensures they cannot cross or touch. The physics implied by streamlines crossing would be that you have mass disappearing, so that is the physical intuition that should indicate that it can't happen.

2) When water flows from the tap, as it flows down the horizontal area of cross-section decreases...An argument using the Equation of Continuity is given...but I have two doubts regarding those
i) Is water in freefall Laminar flow ?

It depends on many factors. Like Randomguy88 said, it can be, It depends on whether it is laminar coming from the source and the flow rate. You can turn on your faucet such that it is laminar the entire way down or turbulent the entire way.

ii) Why do streamlines come closer as water goes down ?

Like Randomguy88 said, the water accelerates, so in order to conserve mass, as the water moves faster, it must have a smaller cross-section to conserve mass flow.

 

[snb]

Is Equation of Continuity valid for turbulent flow ?

Like Randomguy88 said, the water accelerates, so in order to conserve mass, as the water moves faster, it must have a smaller cross-section to conserve mass flow.

I wanted to know n explanation in terms of forces or something like that. I am not able to understand why should those streamlines come closer to decrease the cross-sectional area...

 

[boneh3ad]

Is Equation of Continuity valid for turbulent flow ?

Yes, as are the Navier-Stokes equations and energy equation. Regardless of whether the flow is laminar, energy, mass and momentum must still be conserved.

I wanted to know n explanation in terms of forces or something like that. I am not able to understand why should those streamlines come closer to decrease the cross-sectional area...

You are out of luck then. It has nothing to do with forces and everything to do with continuity. Think of it this way; you would start out at the top with some $v_0$ and $A_0$ and at a given point later in the fall, you would have $v_1$ and $A_1$. Now clearly $v_1 > v_0$ since the water is accelerating. Let's say the area didn't get smaller. The mass flow at the start, $\rho v_0 A$ would be less than at the lower point, $\rho v_1 A$. Clearly that can't happen because you would be creating mass from nowhere. Going back to continuity, let's use the simplified form:

$$\rho v_0 A_0 = \rho v_1 A_1$$

Density is constant, leaving us with

$$v_0 A_0 = v_1 A_1$$

Rearranging that, you can get

$$\frac{v_0}{v_1} = \frac{A_1}{A_0}$$

Since $v_1 > v_0$, you also know that $\frac{v_0}{v_1} < 1$. From our previous equation, that means that $\frac{A_1}{A_0} < 1$, which leads to the conclusion:

$$A_0 > A_1$$

The stream gets smaller.