Ball immersed into a fluid

Question

A plastic ball is pushed inside the tank and then released, can anyone plot velocity vs time and velocity vs height as long as ball is completely inside the water? For simplicity we can assume no viscous forces

Answer 1

If we ignore viscous forces:

at the full immersion, the buoyancy is the weight of water the sphere of the ball has displaced.

$$F_{B}=V_{ball}*1000kg/m^3= \frac{4}{3}\pi R^3$$

And the ball will emerge with an acceleration of $\alpha= F/m$

As the ball partially emerges from the water there will be a growing cap of the ball standing above the water until the ball is fully ejected.

The cap volume is

$$V_{cap}=\frac{1}{3}\pi h^2(3R-h)$$

The buoyancy force, Fb is

$$F_{b}= F_{Bfull-Bcap}= \frac{4}{3}\pi R^3-\frac{1}{3}\pi h^2(3R-h)$$

• m= mass of the ball
• R= radius of the ball
• h= height of the cap measured from the top of the ball to water level

I would slice the ball into say 20 sections and write an excel formula using the above EQ to calculate the force and speed at any given height, h.

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Answer 2

Even ignoring friction, this is a significant challenge. It relies on a decent understanding of the added mass effect, and the ability to compute the added mass.

If we ignore the ball for a minute and just imagine the fluid collapsing around an air bubble, which for some reason remains spherical all the while, we have an intuitive solution for the asymptote based on the idea that all things accelerate at the same rate in a gravity field. So regardless of the density of the fluid (and therefore the magnitude of static buoyancy force), if the ball is very light, the acceleration will be the same regardless of the fluid density because the fluid will fall around the bubble identically regardless of the fluid's density. The only complication now is to figure out how a ball with nontrivial density relative to the fluid affects the result. Obviously, if the ball has the same density as the fluid, the acceleration is zero. The interesting part is what happens between the two. Accounting for added mass effects is hugely important in any fluid-body dynamic system.

The added mass perspective uses the frame of reference of the fluid and computes an added mass term to add to the mass of the body that is a function of the fluid properties and the body geometry. This allows a quasi-static approach to the acceleration terms of the body in the fluid. So $$Force = (body\ mass + added\ mass) * acceleration$$ In the case of the air bubble above, the added mass is up to 400 times larger than the mass in the bubble.

So where does it come from? well imagine we take a ping pong ball and move it through a basin of still water. The water has to accelerate out of the way and move around the ball. There is a field of fluid with nonzero velocity where there used to be still water, and it takes energy to initiate that. Compute the velocity field induced around the body when the body moves at a speed of 1. Compute the kinetic energy in that induced flow. Compute the mass of fluid moving at a speed of 1 that would have the same kinetic energy as the induced flow field. That is the added mass.

So now you can begin. Here's a link for added mass associated with spherical bodies - http://web.mit.edu/2.016/www/handouts/Added_Mass_Derivation_050916.pdf