Any research on Veritasium's hydrodynamic levitation? So this Youtube video has been around for a while now Veritasium Hydrodynamic levitation! (2017).
Also these questions Fluid Mechanics explanation of an object levitated next to an air jet and Explain hydrodynamic Levitation may be related.
Description
To avoid rot links and people that do not want to see the video, I'll try to explain the phenomena.
You have a sphere (radius of about various centimetres) and a water jet. The water jet is vertical (jet velocity goes against gravity), and the jet touches the sphere almost tangentially. The ball not only rotates due to the water jet but also stays at constant height from the ground. The angular velocity can be defined in the direction of the cross product between the position of the contact point between the jet and the sphere (taken the origin in the center of the sphere) and the direction of the jet velocity. Some water drops fly away at the top of the sphere away from the jet. The whole system seems stationary, as long as the water flow is constant, the height and frequency seem to also stay constant.
Suggestions
Derek clearly states it is not about Bernoulli principle, as in this case the ball rotation is important.
There seems to be two explanations "water adhere to the sphere and is ejected conserving horizontal momentum" (Newtonian effect?) and the Magnus effect (ME). The later is not so obvious to me as the system is at rest with respect to still air and there are two fluids, water and air, (does ME mean air pushes the ball horizontally?).
Note that another explanation is Coandă effect (CE) is the typical hairdryer ping-pong ball levitation demo which is explained by Bernoulli's principle and works for small balls as air can surround the ball. So I am not sure if CE works as a satisfactory answer.
Question
Has any recent paper/videos discussed what happens if you do the same experiment with hydrophobic spheres? Can it be predicted with laminar flow theories or does it need turbulence? Would this phenomena occur without air for example?
Too many questions have been posed, but not so many responses on the internet yet.
I just wanted to know if any standard hydrodynamic formalism has been developed to model this phenomena or at least if there has been some experimental characterisation of the phenomena (any study of the link between friction, density, frequency, height, radius, and so on).
 A: This is because of Friction, Coanda effect followed by Magnus effect.
In the middle, he also showed how the water followed the curve of the ball. This is mainly because of 'Coanda effect'. 
From Wikipedia,

The tendency of a jet of fluid emerging from an orifice to follow an adjacent flat or curved surface and to entrain fluid from the surroundings so that a region of lower pressure develops.

Here, the water is trying to entrain fluid from surroundings. When it comes near to a surface, it cannot pull air from surroundings which makes a low-pressure region between surface and jet. This is because, 'when something is emptied from its place, then something should fill the void to keep the balance.'
Firstly, the ball starts to rotate because of the friction between water and ball surface which is just like a Tesla turbine. As the ball is being hit by the water on one side (not centre), it will push the ball to the other side because it comes in the way of the water. Once the ball starts rotating, the fluid following the surface don't adhere to the surface much longer and drift apart tangentially. 
This is where the 'Magnus effect' kicks in. The magnus effect creates a force perpendicular to the jet direction. This force pushes back the ball to remain in contact with the jet on one side. So, the weight of the rotating ball is born by the jet completely. 
The magnus force is proportional to the speed of rotation of the ball which is proportional to the velocity of the jet of water. 
So, the ball levitates in air as long as the jet discharge is kept constant.
A: I am unconvinced by his (simplistic) use of Bernoulli's law to try to explain this effect. Clearly, the water adheres to the ball due to viscous effects and surface tension, where Bernoulli cannot be used because it assumes that viscous forces are negligible.
Instead this is an application of the Magnus effect which explains how a spinning ball curves away from its principal flight path as observed e.g. in ballsports such as baseball. Due to the rotation of the ball, it drags air faster around one side, creating a difference in pressure that moves it in the direction of the lower-pressure side.
