Friday, December 2, 2016

Sharks, Blood, and Diffusion

Every year I get very excited for Shark Week, and make my family watch the specials with me. Sharks are known for having incredible senses of smell, and it is a common misconception that sharks can smell a drop of blood in the water from miles away. Though still impressive, sharks can actually only detect blood from a distance of about a quarter of a mile. Unlike humans, sharks have separate openings for breathing and smelling. They are able to detect scents as particles (such as blood cells) suspended in water are pulled into their nasal cavities. These particles reach sharks by two different methods: diffusion and with help from currents. As we have studied Brownian Motion, I was interested to see how long it would take a shark to recognize blood by diffusion alone from a quarter of a mile away. For the purpose of simplicity, we will only consider sharks detecting red blood cells (which carry hemoglobin, a chemical to which sharks are very sensitive).
Given that the diffusion constant of red blood cells (at 20 °C) D = 6.8 x 10-14 m2/s

¼ mile = ~402m

As this is three dimensional motion, the time for the average particle’s displacement to be ¼ mile is given by

<r2 >= 6Dt

<402m2> =6(6.8 x 10-14 m2/s)t

t=3.96 x 1017 s = 6.60 1015 minutes for the average red blood cell to travel ¼ mile

However, sharks also require a concentration of blood of one part per million (.001 kg/m3) in order to be able to detect it. I was interested to see how long it would take for a shark to detect the red blood cells from all of the blood in a human body when that blood is spilled on one side of a massive tank ¼ mile long and 2883 m deep (the max depth sharks are found).

To do this we can use Fick’s law

Δm/Δt = J= DA[(C1-C2)/ Δx]

There are ~4 x 1013 red blood cells in the body at any given time, which weigh about 1.968 x10^-13 kg each. So, the total mass transferring is about 8 kg. The red blood cells are originally at a concentration of 1600 kg/m3.

 8 kg/ Δt = (6.8 x 10-14 m2/s)(2883m x 402m)[( 1600 kg/m3 - .001 kg/m3)/402m]
t = 2.6 x 107 seconds = 4.3 x 105 minutes
Obviously diffusion alone is extremely slow, and currents are very helpful to sharks when they are searching for food. 



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