[Aleksey_at_cfd]

Please look at this thread : air dancer maximum height problem

I am willing to pay for help. I want to do this by small steps.

The first step is very simple, analytical, like this

I will start from a simple model in laminar case: Hagen-Poisueille solution in a pipe. That has a pressure decreasing along the axis. The weight sustained at a certain height must be balanced by the total wall stress acting at a that height.
Actually, when the flow starts transition, you have vortical structures in the pipe that will cause the fluctuation of the pressure.

You have an analytical solution for the velocity, you can compute the stresses at the wall and integrate them along the surface up to the height until you get a force that sustains the weight.

The next steps should be using some CFD IDE to model the same for turbulent flow, introduce that "vortical structures", take into account the features of the walls, etc.