Section 5.3.4 discusses vascular networks. It states that large organisms, unlike bacteria, cannot rely on diffusion to feed themselves. This is due to the relation which describes whether "stirring" (anything other than diffusion) is favourable - eqn 5.16: v > D/d where v is the velocity of "stirring", D is the diffusion constant and d is the distance. In larger organisms the ratio D/d is small (due to large increase in d).
The section started by modelling the vascular system as a simplified situation with steady, (zero acceleration) laminar (frictional forces dominate) flow of a Newtonian fluid through a straight cylindrical pipe of radius R and derived a function for the velocity as a function as the radial distance from the centre of the pipe, r. Unfortunately I couldn't follow the derivation but the section ended up with the Hagen-Poiseuille relation which describes the volume flow rate: Q = (pi*R^4)*p/8L*eta, where L is the length of the pipe and eta is the viscosity. The general form Q = p/Z, where p is pressure and Z = 8*eta*L/(pi*R^4) is the hydrodynamic resistance. This lets us realise an important point: that due to the 1/R^4 term the resistance decreases rapidly with radius. This can explain why the vascular system of humans is not just a straight cylindrical pipe with radius R, it can dilate quickly and thus increase the volume flow rate.
Just as a side-note this blog is not terribly good at presenting equations in a readable format. I think it would be useful for a physics blog if we had some sort of TEX function.
ReplyDeleteI agree. I don't like writing equations on this blog because it looks so bad. I was almost tempted to write a post in latex, then post it as a document somehow.
ReplyDeleteI third that motion. Calvin, I'd show you how to do this on here, but it would look messy. Since you're not going to be at class this week, find me sometime and I'll go through it in person with you.
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