### #20 The scaling law of shaking water off-I

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In this brief series, we will discuss the scaling law of the frequency of shaking water off in this episode, and talk about the energy budget of removing extra water.
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Dogs and various animals shake off water when they get soaking wet. However, how does this act make them dry and why should they do so? Is there any difference between little mice and big tigers in their shaking behavior?

The authors of the suggested reading found that the shaking frequencies of dogs of similar sizes are roughly the same (about 4.5Hz). The shaking frequencies of mice are also roughly a constant (about 18Hz), which is much faster than that of dogs. To reveal the scaling properties, they collected videos of various animals shaking water off their bodies, ranging from mice to bears, from Youtube and BBC channel. They discovered that there is a power law relationship between shaking frequencies and body mass, and its power is -0.22. That is, f ~ M^(-0.22).

Before we dive into the physics, let's consider why these animals should actively shake off water rather than wait until it evaporates, just like somebody does after taking shower. The authors believe that it comes from the problem of energy budget. The heat of evaporation of water, λ, is rather large, measuring 2257kJ/kg. Previous experiments showed that our body must contribute 60% of all energy to evaporate the water, which cost a lot. Is it more efficient to shake it off? We will estimate the energy budget in the following episode.

Let's consider how to build a physical model for shaking water off. The researchers observed the motion of water droplet under high-speed photography and hypothesized that the eccentric force must exceed the surface tension to shake water off. To estimate the mass of water droplet, assume that the radius of the water droplet is comparable to the radius of wetted hair bunches R0, the distance from the body surface to the rotational axis R (which is estimated as half of the chest diameter), the shaking angular frequency ω, and the surface tension of water σ. The mass of water droplet shaken off could be estimated to be:

m = 2πσR0/(Rω^2) * F
in which F is a correction term. The researchers estimated F to be 0.4 by isolated fur.

We could then predict the scaling law from the above equation. Obviously, σ and F do not vary with the size of animals. The authors also assumed that R0 does not change with the size of animals. Finally, they assumed that the minimal size of the water droplet, m, that could be shaken off by the animal is the same. Then we could derive the scaling law ω~R^(-1/2). What about the scaling law between chest radius R and body mass M?

We have discussed the elastic similarity hypothesis proposed by Thomas A. McMahon in episode 3. According to the hypothesis, the length of a supportive structure must vary with the 2/3 power of its diameter, or L ~ D^(2/3). Since M ~ LD^2, it's fair to conclude that D ~ M^(3/8), or R~ M^(3/8). So the shaking angular frequency is proportional to the -3/16 power of body mass, or ω ~ M^(-3/16), which is very close to the observed power -0.22.
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