So this Youtube video has been around for a while now Veritasium Hydrodynamic levitation! .
Also these questions Fluid Mechanics explanation of an object levitated next to an air jet and Explain hydrodynamic Levitation may be related.
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 and a water jet. The water jet is vertical (velocity goes against gravity), and 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 (origin in the center of the sphere) and the direction of the jet. 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 the same the height and frequency seem to stay constant.
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" and "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 means air pushes the ball horizontally?).
Note that Coandă effect (another usual solution) is the typical hairdryer ping-pong ball levitation demo which is explained by Bernoulli's principle so I am not sure if it works as a satisfactory answer.
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 characterization of the phenomena (any study of the link between friction, density, frequency, height, radius, and so on).