@Chet Miller Do you have some answers on this?

Google "Falling Ball Viscometry" for part 1. Even though your ball is rising (due to buoyancy), the same equations apply. For experiment 2, I don't understand the geometry.

@ChetMiller I have modified my original post by inserting a graphic representation of the geometry. Let me know...

In scenario 2, it is held stationary in the vertical direction, but can move in the horizontal direction? Why are you omitting viscous interactions with the walll/. It seems that these can be very important. won't the ball clog the exit hole if it is allowed to move horizontal?

@ChetMiller In this scenario, the sphere is kept totally stationary. It cannot move either vertically or horizontally, at least I have not thought about what would happen if the sphere moved. So what do you think?

I think that, if it can not move horizontally or vertically, its velocity is zero.

@ChetMiller What changes in the two cases where there is the sphere and where there is not?

I don't understand this question. I thought the focus was on the velocity of the sphere.

@ChetMiller The focus is on the speed of the water jet, not the speed of the sphere. That's why we don't understand each other... I would like to determine the velocity of the water flow in the two cases where there is or is not the sphere.

In scenario 2, neglecting drag by the sphere and the tube wall, the velocity is determined by the Bernoulli equation.

@ChetMiller There are no frictional forces between ball and wall, but viscous resistance acts according to the law $\vec F = - \alpha \vec V$ as in the first experiment... Also, is there an intuitive explanation of why the two velocities of the water flow are different in the two cases where there is or is not a sphere? A more detailed clarifying answer would be greatly appreciated...

Realistically, you need to consider viscous drag and the proximity of the wall in scenario 2. The ball acts as a valve, and slows the water flow. But, unless viscous drag in the region between the walll and ball is included, you will not predict the lower velocity..

@ChetMiller OK, you've convinced me... Could you show me how to derive the velocity of the water jet considering the viscous drag in the region between the sphere and the wall in the two cases where there is a sphere and there is not, and why qualitatively there is a difference in velocity in the two cases? An extended answer would be most clarifying.

I could solve this, but I feel that it's too much work for a problem you just dreamed up.

@ChetMiller Why did I 'just dream up'? Anyway, even if it's true that it would be a big job, it wouldn't be a big issue for me, I can understand that. Anytime, if you'd like to dispense some of your knowledge on this, thank you very much!

@ChetMiller I'm looking forward to your answer!