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12 Internal Flows in Marine Organisms
rate at which energy is used up by the blood vessel due to metabolism (Fung,
1997):
(12.15)
in which Q is the pipe discharge in unit time (m 3 /s), 6.p is the pressure drop
over a pipe length l, and K is the proportionality constant. For a given vessel
length, l, and a flow rate, Q, there is an optimal pipe radius, a, at which the
cost function C F is minimum. This condition is satisfied when:
a(CF) = O.
aa
Thus, the radius a should be:
(
1611 ) 1/6
_ _ t'"_
Ql/3
a -
2
.
7rK
Now, from Eq. (12.14), the conservation of mass becomes:
(12.16)
(12.17)
(12.18)
which is known as Murray's law. After substituting Eq. (12.17) into Eq. (12.15)
we obtain a minimum value of the cost function:
(12.19)
The minimum value of C F for bifurcated blood vessels, as in Fig. 12.3, can be
found by the variation of the lengths h, hand l3, and the radii aI, a2 and a3,
in such a way that the following condition is satisfied:
(12.20)
The result is (Fung, 1997):
(12.21)
(12.22)
Equations (12.18), (12.21) and (12.22) form the necessary conditions for optimal bifurcation pattern of blood vessels.