# How does the viscosity of a fluid affect the angular speed and resultant force of a cylinder that is rolling?

*This is not a homework question - I'm taking the iB and need to understand how liquid properties affects certain variables of a cylinder for my Extended Essay.

So basically, here's the scenario: There's a cylinder that's rolling down a ramp with a known liquid of known viscosity inside it.

I want to know how changing the viscosity of the liquid will affect the angular speed and resultant force and therefore the translational speed of the cylinder at the end of the ramp.

In my experiment, I used a PVC pipe of radius 10cm and inside it is half filled with water. Moreover, I vary the viscosity by adding corn syrup to the water and finding the viscosity using this method: http://www.wikihow.com/Measure-Viscosity

Moreover, I changed the volume of the liquid inside for each viscosity. I found a positive linear relationship between volume and translational speed. As volume increases, speed increases. I also do not understand why this happens. I've attempted to explain it using moment of inertia and conservation of energy but it didn't work because all the mass is cancelled in the equation. I'm guessing it's due to slosh dynamics but I can't seem to find much information on it.

TL;DR I want to know how liquid viscosity and liquid volume affects speed.

• What does "I changed the volume of the liquid inside for each viscosity" mean? Did you start with an empty cylinder and added more and more liquid until full? Because I would expect stable results for empty and full case, and unstable/fluctuating results for the half-full cases. – Steeven May 4 '17 at 7:13
• I started with an empty PVC pipe until half filled – IntJ May 4 '17 at 15:30
• Note that you would expect a full solid cylinder to roll down a ramp faster than a hollow cylinder; see here. Note also that you would expect a cylinder filled with zero-viscosity fluid to act basically like a hollow cylinder, since the fluid wouldn't rotate with the cylinder; and a cylinder filled with a high-viscosity fluid would act like a solid cylinder, since the fluid would rotate perfectly with the cylinder. But the half-filled cylinder (which you're asking about here) is much trickier. – Michael Seifert Dec 19 '18 at 16:06

An empty cylinder has a fixed moment-of-inertia. A full cylinder as well, assuming an even mixture. The motion should be perfectly predictable.

But when containing a mixed phase of liquid/gas, the moment-of-inertia $I$ is not fixed. The liquid can move around and since $I=\int r^2\;dm$ depends on distance, this splashing around will severally change $I$.

Such changes in $I$ will change the angular speed $\omega$ during the motion, since $K=\frac 12 I \omega^2$ must be constant.

Furthermore there is the issue of inertia, and with a mixed-phase content, the cylinder might not move syncronized with the content. What I mean is that if the cylinder suddenly stops, the liquid has a split-second more to continue forward before reaching the cylinder wall. The cylinder might thus stop in a bumpy manner and not evenly.

The viscosity only changes the liquids ease of moving around inside the cylinder. The higher the viscosity, the slower are the fluctuations in $I$ due to the content displacing itself. In other words, I would expect higher viscosity to make the motion more stable with less sudden changes by evening out the bumps and changes in $I$.

• Won't changing the viscosity change the friction force between the fluid and the inside walls of the cylinder and thus change the speed? As it affects the inertia of the moving cylinder – IntJ May 4 '17 at 15:31

This is an old question, but I think it is kind of interesting. I think the two issues (how does speed depend on volume of fluid, and how does it depend on viscosity) are somewhat separate. The OP did not actually say what happens experimentally if the cylinder is full and the viscosity is varied (I think that is the more interesting question). OP claims that speed increases with filling (for fixed viscosity?).

1. Variation with fluid volume: My guess is that the main effect is the following. For an empty cylinder the velocity is reduced compared to a point particle or a solid ball because some of the potential energy is converted to rotational kinetic energy, not translational energy. This is a standard intro physics problem, and there are many movies on YouTube that demonstrate the effect. I would expect that if some fluid is added there may be some sloshing, but mostly the fluid does not acquire vorticity, and more potential energy is converted to translational motion. This is what the OP claims to observe.

2. Variation with viscosity, filled cylinder: In a filled cyinder the fluid (because of viscosity) will try to come into co-rotation with the cylinder, and mimic a solid cylinder. Note that a solid cylinder rolls faster than a hollow cylinder, because of the effect described in 1). However, this cannot happen instantaneously. The time scale is determined by vorticity diffusion (see, for example, here), and given by $$\tau \sim \frac{R^2}{\nu}$$ where $$R$$ is the radius of the cylinder, and $$\nu$$ is the kinematic viscosity of the fluid. During the time of order $$\tau$$ the fluid is not co-rotating yet, and I would expect the cylinder to roll faster (this means low viscosity should lead to faster rolling). Note that there is also an effect from viscous heating of the fluid (proportional to $$\nu$$). This effect has the same sign, but I would expect to be smaller. It should be possible to make this quantitative (using the solution linked to above, but this would require some effort).