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I discovered this bottling gallons of homebrew beer and cider.
Is it really faster when you account for the time needed to first create the whirlpool?

Conveniently, the emptying "race" in the video only starts after the whirlpool has been created!

Time it yourself by spinning the bottle right after you begin pouring, and you will notice that it is still faster. ;)
So is there a way to do this trick without your hand touching the liquid?
Yes, I do this all the time. Hold the bottle vertically with the mouth facing downwards. As the water is pouring out, flick your wrist in an upwards elliptical arc, keeping the mouth facing downwards. It only takes me one or two flicks to create the vortex, and no liquid sprays around or touches me.

It's kind of a swirling motion.

It is, we use it often (chefs), example needing to empty 12 bottles of red wine into a pot :-)

Edit: fat thumbs

I remembered seeing something along these lines a while ago, didn't realise it was as far back as 2012 though: https://www.reddit.com/r/askscience/comments/sv6kw/which_is_...

I guess it's something that many people rediscover on a semi-regular basis.

I remember seeing the trick on Tomorrow's World (a UK science TV programme) back in the 90s. Can't find a link to the particular episode though.
I was doing this in the mid eighties... think a teacher showed me it but can't remember... didn't have tv.
I'm pretty sure I saw it featured in an article in Highlights for Children during the 80s.

https://en.wikipedia.org/wiki/Highlights_for_Children

I learned it in australia many years ago from these guys: https://www.youtube.com/watch?v=DYBw7k_8EbE
Do Australians commonly say "elephant" to count seconds as the presenters do? In the U.S. it's usually "one-thousand" or "Mississippi". (I guess I can imagine that Australians wouldn't say "Mississippi"!)
I've never heard of "elephant", no. (Sydney here)
"By creating a whirlpool inside the bottle, you’ve set up a corridor of air connecting the inside of the bottle with the outside air."

It didn't appear to me that the small vortex in the video actually reached all the way down to the mouth of the bottle.

I think you are wrong.
It didn't when she still had her hand on it, but as soon as the water began to drain, the funnel/vortex immediately connected the mouth with the air pocket at the top.
I feel like this is more than just air flow. You're adding energy into the bottle by causing the water to spin in a whirlpool. It seems like that energy would also help push the water out of the bottle through path of least resistance dynamics (the open end of the bottle).
That is easily testable: just try with and without the whirlpool in a bottle that has a hole in the bottom so that the air flow is not a factor.
The counter-argument there is that after you've swirled it, each water particle has to travel a longer path to get out of the neck, because it's corkscrewing rather than going in a straight line.
Except for the fact that those particles are moving faster (velocity)...
Yeah, they're going faster on a longer path. It's not immediately obvious which factor would win.
There is a much faster way, using cavitation.. from Bill Beaty of the awesome http://amasci.com/

https://www.youtube.com/watch?v=77gWkl0ZUC8

"Ooh, cool fancy ice cubes! They look like shards of broken glass!"

"That's 'cause they are shards of broken glass."

"...on second thought, I'm not really that thirsty."