fluid dynamicsnewtonian-mechanics
Recently it has become a trend to flip a bottle which is partially filled with water and land it on it's bottom. This video shows the phenomena :https://www.youtube.com/watch?v=2E7rS1q4NbU
What i would like to know is that can we model this phenomena to come up with an equation for a perfect flip of the bottle. If the phenomena does not take a shape in your mind you can ask me to illustrate it much clearly.
Surface tension is plausible : it implies that, at the level of the hole, the pressure in water will be less than 1 atm. Thus the pressure isolines will look like this:
Quantitatively, curvature $c$ will be of the order $1/(1 $mm$)$ because the hole looks about 1 mm size in your video, which with pure water would lead to a pressure difference drop of the order of 70 Pa across the interface. Then the flat interface with atmospheric pressure needs to be $c\gamma/(\rho g)$ lower than the hole, which is about 7 mm with pure water. Because you probably have some impurities, this will be less in practice. As @pwf suggests, you can use surfactants (soap) to reduce it (but that won't be a good thing for the birds!)
First up thanks to all who took an interest especially @irishphysics who stuck with the question for some time.
It turns out that the phenomena was analysed and solved by Lord Kelvin and is known as the Kelvin wave pattern. The pattern itself is the result of a spreading pressure wave which manifests itself as the curved diverging wave crests (the ones I described as bending in my original question) and a set of transverse waves that sort of follow along with (and behind) the boat.
The full (mathematical) explanation is here: (https://www.math.ubc.ca/~cass/courses/m309-01a/carmen/Mainpage.htm).
The final (modelled) wave pattern from the above link looks like this:
which for me is pretty close to my experience.
For those who suggested the depth of water had something to do with it: you were partly right in that the precise angle of the waves from the bow depends on the depth of water rather than on the speed of the boat.
There's also a good explanation of the physics here:- (http://www.prirodopolis.hr/daily_phy/pdf/speed.pdf)
Best Answer
Like the previous craze for flipping a paper/plastic cup, I think it is mostly a matter of practice to flip at the right speed to get the bottle to land on its base. When it does so, the bottle containing 1/3 water has the most stability. But many of the clips show bottles half-filled landing upright, so the amount of water is not crucial.
The larger the number of rotations before landing and the higher aspect ratio of the bottle (height/base), the more difficult it is to land the bottle successfully.
The Institute of Physics explanation points out that when optimally filled and flipped at the right speed the bottle almost stops rotating as it lands. This is compared with a diver who increases his/her moment of inertia in order to open out from the initial tight spin into a slow rotation on reaching the water.