# Movement of a cylinder filled with water

Not long ago I was pretty bored at a dinner and I started playing with a water bottle that was not empty: I've been quite interested in its behavior when putted on its side and pushed: the bottle of course starts rolling, but its movement seems quite complex. I've been trying to obtain the equation governing this kind of system (an analytical solution would be nice too, but I have no idea if it can be found or not), but I really can't start.

Here is the model I use. I might have forgot to include some things, or maybe some assumptions are just wrong, so feel free to modify it if you think it can be improved. The following (awful) picture represents the initial situation. The only external force considered is the gravity $\vec{g}$ and it is uniform.

Modelisation of the bottle : I consider the bottle as an infinite rigid cylinder of radius $r$, so that the problem can be considered two-dimensional (side effects are neglected). The mass distribution of this cylinder / circle is uniform. When not a rest, it is rolling without slipping on the rigid horizontal floor: the point of contact between the cylinder and the floor, $A$, has always a zero velocity: $\vec{v_A} = \vec{0}$.

Modelisation of the water : Inside the cylinder there is a water (represented in blue on the picture) and air. I do not think that air really matters, so only the behavior of the water is to be studied.

The mass of the water is constant (the bottle is not leaking). We can assume that $r$ is more than a few millimeters, so that capillary effects can be neglected (as a consequence the free surface is initially horizontal). Water can be considered as an incompressible fluid, but I think that unfortunately viscosity has to be taken into account.

The movement : Now let's say we push the cylinder (or equivalently we states its acceleration) along the $x$ axis at $t=0$. What happens ? I would like to get the differential equation verified by $x_O(t)$ (the position of the center of cylinder).

I see two big issues: firstly the center of gravity of this system is always changing, and secondly it seems hard to find the force from the water acting on the cylinder. I actually really don't know how to start. How would you study this ?

• Awesome, I once studied this case for a course (long ago). Then I analyzed it one an incline, hitting a wall at the bottom. It will 'reflect' in a funny way (with a delay). Sep 16, 2014 at 18:41
• This is so cool! I just tried. Once the bottle slowed down in an oscillating way and the second time it came back to me. Sep 16, 2014 at 19:20
• You might simplify it by replacing the water with a pendulum suspended from the centerline of the bottle. Then the bottle could be replaced by a little "frictionless" cart on a horizontal track at the level of the bottle's center. Sep 17, 2014 at 19:27
• I have a feeling you're going to get radically different behavior in your bottle depending on how quickly you spin it. Nov 18, 2014 at 19:04
• (because the water will behave differently in the adiabatic limit than it will when you get waves and the like started. Nov 18, 2014 at 20:05

## 1 Answer

Great problem! Simple to state, incredibly complex to solve.

I have three suggestions:

• I'm not sure that it's a good idea to consider an infinite cylinder. It will have an infinite mass an might just sit there.

• Your question is not formulated very precisely. To study the time evolution of a dynamical system you need to fully specify it's initial conditions. In your case you should specify the speed (and position) of the cylinder as well as the state of the water. The second specification might be quite tricky. You need to specify the velocity field of the water as well as the shape of its interface with the air. Moreover you need to make sure that the velocity field is incompressible $\boldsymbol{\nabla}\cdot \boldsymbol{v}=0$, and that the physical boundary conditions are enforced at the interior edge of the cylinder. If you have a non-vanishing viscosity the velocity field must match the velocity of the cylinder when the water touches it. In conclusion, it gets really complicated.

• It might be simpler to place you cylinder on an inclined plane. Then your cylinder speeds up at first, but reaches a constant speed once the viscosity of the water is able to dissipate the gravitational potential energy as it is converted to kinetic energy. You can then look for a steady state without having to worry about initial conditions. I guess that depending on the inclination of the plane, there will be different regimes. For small inclinations the speed of the cylinder might saturate and you might find a steady state with the water surface being smooth. Then if your plane is steeper you might (???) find another steady state with turbulent water in the cylinder. Finally for a very steep plane, you will find no steady state. The water will just stick to the edge of the cylinder and roll with it as it accelerates down.

I would start with the third quasi-stationary regime (accelerating cylinder with water turning with it). It looks like it's the simplest. You could try to find out the minimum speed above which it is realised.