Friday, August 19, 2005

Science discussion topic

It's been a while since we've had a good old-fashioned physics debate, so let's have one now. Would a swimmer swim faster in H2O or D2O? Assume a 400m Olympic race.

7 Comments:

Blogger Justin said...

Isn't it obvious that the times would be slower? D2O is denser and more viscous than H2O, which would seem to be a double whammy. A little bit more density might help keep the swimmers relatively higher in the water, reducing their profile, but I doubt that would be significant at the Olympic level, since they are already about as streamlined as you can get. The extra enery required to move through a more viscous and more dense aquatic environment would certainly overwhelm whatever advantage being a bit higher in the water provides.

8/20/2005 01:33:00 PM  
Blogger Vincent said...

Well, if you think that one is too easy, what about H2-18O vs. D2-16O?

8/21/2005 05:08:00 PM  
Blogger Justin said...

Any links to the density and viscosity of those two compounds?

8/22/2005 01:21:00 PM  
Blogger Qian said...

I thought I read somewhere that some physicists proved that swimming in water and swimming in molasses would be about the same speed? Something about the swimmer being able to apply more force against the fluid which cancels out the higher viscosity. Not completely impossible, but sounds a bit rum to me.

8/22/2005 03:04:00 PM  
Blogger Eric said...

Justin: Fish are streamlined, people are not. Density of ordinary water at some reasonable T & P is about 1g/cm^3, so D_2O^16 and H_2O^18 must be about 20/18 g/cm^3. Since the nuclear structure won't affect the fluid properties much, all three should have the same chemical properties. Viscosity for normal water is about 0.01 Poisse at normal T & P. Care to revise your opinion? If so, don't read any further...

Qian: Molasses have a much higher viscosity and somewhat higher density. Surely


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The flippant explanation... Try swiming in water as faster you can, then try running on ground. The reason running is faster is not due to the different viscosities of dirt and water, but rather their different densities. You sink into water, not dirt. So I propose that swimming D_2O is faster.

The more scientific explanation... Assuming a distance of 400m, a time of 3m41s, a cross-sectional area of 15x50cm^2, a density of 1g/cm^3 and a viscosity of 0.01 Poise, I compute a Reynolds number of 5.6e5. Since turbulence sets in around a Reynolds number of 1e4, even someone swimming at a tenth the speed of an olympic swimmer will be in the turbulent regime (not stokes drag regime) in either H_2O or D_2O. In the turbulent regime drag is approximately proportional to density*speed^2*cross-sectional-area. I would guess that a swimmer's speed is limited by a Pmax, maximum human mechanical power output (order of magnitude estimate of 100W if you just keep living or 500W is approximately what Lance Armstrong can _sustain_ on a bike). By neglecting air resistance we can equate:
Pmax ~ density * speed^2 * cross-sectional-area-underwater,
so we find that the swimmer's speed is inversely proportional to the square root of both the fluid's density and cross-sectional-area-underwater.

Now, we need to figure out how much the cross-sectional-area-underwater changes as we change the density of the fluid for swimming. Let's assume that swimmers are an elipsoid of uniform density, rho_h, with semi-major axes (r_1 < r_2 < r_3) and that they are in an infinitely large swimming pool containing water with density rho_w. The swimmer's total mass is rho_h*4pi/3*r_1*r_2*r_3 which we can equate with the displaced fluid mass of rho_w*4pi/3*pi*h*r_2*r_3*g(h,r_1,r_2,r_3,shape), where h is the height that the water level comes up on the swimmer and g is a function that depends on the shape. Most things cancel except the ratio of densities, and the function g. If the swimmer is an ellipsoid, then this would contain some elliptic function and then the ratio of heights in the two fluids would be greater than the ratio of densities, so the swimmer would be faster in the D_2O.

However, not much... If the swimmer were a rectangular prism, then this g funciton would be a constant (well a tophat) and the ratio of heights in the two fluids would cancel the ratio of densities. So perhaps there are higher order effects necessary to consider for a swimming brick that is nearly neutrally boyant.

Note that in order for the cross sectional area under water to change more rapidly than the density, you would need to orient your body in a way which was unstable to perturbations (e.g., pointed as if you were standing up in the swimming pool), so you'd have to expend more energy to maintain your orientation (at least assuming uniform density... If you had a nonuniform density (e.g., a mushroom shaped body with a more dense step and less dense umbrella), then you could avoid this stability constraint. Alternatively, if your surface area to mass is high enough you could make use of surface tension (e.g., insects) and significant chance the relevantn forces.)

A couple of caveats... I have assumed the key is maintaining speed in the face of dissipation, not accelerating. For accelerating, it probably requires more power to move the denser water. However, based on the given 400m race and that the swimers dive in at a good clip and get a good push off the wall at each lap, I'm assuming that the acceleration stage is a relatively small fraction of the race.

Finally, since the speed up in D_2O is due to the shape of the body, and the swimmer could in principle change their shape slightly (e.g., shoulder positioning, angle of attack, closeness of hands/legs to body), so perhaps I could be wrong if those human effects are more significant.

As for D_2O^16 vs H_2O^18, that's tough. The speeds must be very nearly the same. If I had to guess, I'd guess that the electronic structure of D_2O^16 is very slightly more compact allowing a very slightly higher density. But doing that kind of quantum mechanics in my head is highly error prone.

8/29/2005 03:05:00 PM  
Blogger Vincent said...

Now that's what I call a physics debate. Methinks Eric has done fluid dynamics more recently than the rest of us.

8/29/2005 11:39:00 PM  
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