I would guess it's in part because of the viscosity of the ink. That would explain why the effect is more seen when it's colder. I don't know how doable it is, but you could try filling an ink cartridge with ink used for fountain pens, which is typically less viscous. You might get blotches of ink, but my guess is you won't get dry strokes. So maybe ink manufacturers used easy-flowing ink in ballpoints at first, but then saw it flowed too easily, and made more viscous inks. But this is speculative.
I think the answer to the question is that there is no mechanism by which angular momentum is transferred to the water from outside, after you stop the initial swirling. Rather, the vortex takes angular momentum from the draining water and
transfers it to that remaining in the bottle. I give a fuller explanation below -
it talks about water draining from a bath but the principles are the same.
Three basic physical principles apply:
First, angular momentum is conserved: at any time during the draining of the
bath the total angular momentum of the water (inside and outside the bath)
stays the same.
Second, the only energy input to the system is that available from gravity
as the water drains.
Third, the water is viscous, so that any rotating volume of water within the
bath will tend to transfer its angular momentum to the body of water as a whole.
The presence of the vortex tube with a water/air surface shows that the force
experienced by the water at that interface points radially outward and must be balanced by the normal pressure of the water in the bath. As water exits the bottom of the vortex, pressure from the body of bath water outside pushes more water inward towards the vortex centre.
Conservation of angular momentum means that as this water moves in, its angular
velocity and consequent rotational energy increase. At the same time viscosity
acts to decrease the shear in water velocities, transferring angular momentum
from the fast rotating water to the main body of water away from the vortex.
The two effects combine to build rotational energy near the vortex but
transfer angular momentum away from it.
What appears to the watcher to be the adding of angular momentum to the water is,
if this model is right, rather its transfer from one part of the water volume to
another. The liquid that falls out of the bottom end of the vortex has high angular energy but low angular momentum. Most of its original angular momentum remains in the liquid still in the vessel; as the amount of liquid in the vessel decreases so its angular speed increases.
The increase in angular energy is fuelled by the decrease in gravitational potential energy as the water falls down the vortex.
Some numbers. If the air tube at the centre of the vortex in a draining sink has a radius of 3mm, and the gradient of the vortex tube surface is 9, then the angular velocity of the water must be:
$\omega^2 \cdot 3\times 10^{-3} = 9g$
that is, $\omega=171$ rad/s; quite rapid. Conversely, a speck of water rotating with the same angular momentum at a distance of 20cm from the vortex centre would have
$\omega=3.9\times 10^{-2}$ rad/s (2.2 deg/s); almost too slow to notice. Though the two specks
of water have the same angular momentum, their rotational energies differ by a
factor of 4325.
Best Answer
Assuming you start with a full bottle of water, when you tip the bottle upside down, a 'partial vacuum' (ie below atmospheric pressure) is created at top of the bottle as the water pours out the bottom. Atmospheric air then 'bubbles through' the mouth of the bottle to compensate. This slows down the flow of water through the mouth of the bottle. Each time air is 'bubbling' its way into the mouth, it impedes the flow of water out the bottle.
Once some air gets into the bottle, this air can 'expand' to let some water out of the mouth, until the air pressure inside drops sufficiently low that atmospheric air can 'bubble through' the mouth again. The more air that is inside the bottle, the more this air can 'expand' before bubbling through. That's why the water pours out faster as the bottle gets emptier.
When you swirl the water, the vortex forms a 'gap' in the centre of the mouth of the bottle through which air can flow freely from atmosphere into the bottle. As long as this continuous 'gap' of air is maintained from the mouth of the bottle to the 'top' of the water inside the bottle, there is no 'partial vacuum' above the water, so it doesn't 'bubble' and the water can pour out more evenly & hence faster.