1. Lecture #23: Internal Flows

2. 1 cell cellular sheet cellular bilayer bilayered canister ecto- derm endo- derm one way gut mouth anus cephalization mesoderm Body Plan Evolution

3. heart lung/gill bodyintestine Basic circulatory circuit convection In dedicated plumbing diffusion in dedicated exchangers

4. Convection vs. Diffusion x C1C1 C2C2 S Fick’s Law: C= concentration in mass/volume D = diffusion coefficient Units = L 2 /T Basic strategy of circulatory systems: Pluming uses bulk flow (convection) to move fluids to capillary beds where diffusion can take place over short distances. Relative importance of bulk flow to diffusion given by Peclet number:

5. Problems with gas exchange: consider simple gas exchanger: water blood DIFFUSION distance equilibrium partial pressure (0 2 ) driving force convection

6. Problems with gas exchange: consider countercurrent gas exchanger: water blood distance partial pressure (0 2 ) driving force

8. What about lungs? distance partial pressure (0 2 ) blood air

9. Birds have more efficient system

11. What determines flow in pipes? x r a P1P1 P2P2 L If Re < 2000 (i.e. laminar flow): flow ~ pressure gradient flow ~ 1 / viscosity parabolic flow distribution What is maximum flow velocity? At center of pipe, r=0:

12. What determines flux through pipe? Flux (Q) = velocity x area: = Hagen-Poiseuille equation Flux through a system: proportional to pressure gradient inversely proportional to viscosity has fourth order dependence on diameter

13. distance pressure flow velocity heartlungintestinebody heart lung intestine body Pressure is lost (drops) across network of pipes. 10% of our total metabolic cost! 5% of our total Weight in blood!

14. Problems with blood Blood is not a ‘Newtonian’ fluid, Mostly because of red blood cells. From Hagen-Poiseuille Equation: ‘Resistance’ Blood is very viscous due to red blood cells % hematocrit viscosity carrying capacity 0 2 carried/unit cost optimum at 58%

15. Thoughts about plumbing: Consider simple branch point: S0S0 S1S1 S1S1 If S 1 = 2 S 2 then velocity is same in all branches; flux is ½ the original value. a0a0 a2a2 If a 0 = 2 a 1 then 16 times the pressure is required in small pipe for same flux! Consider change in diameter: Circulatory systems cannot compensate with large trunks – Blood volume would become too large.

16. Murray’s Law: what is geometry of branching network? 1) Cost to pump = Q x pressure gradient, or 2) Cost to make new pipe Total cost 3) Find optimum as a function of diameter:

17. if then a0a0 a1a1 a2a2 a.k.a. Murray’s Law For simple symmetrical branching case: Mass flux ~ cube of vessel diameter But, by law of continuity, Q0Q0 Q1Q1 Q2Q2 thus More generally……

19. How does a growing vascular network ‘know’ to follow Murray’s Law? x r a du/dr Shear stress at wall,  =  du/dr It can be shown that:But by Murray’s Law: So with r = a (at wall): Thus, shear stress at wall is constant in network obeying Murray’s Law. Algorithm could be: ‘Grow vessel until shear stress reaches certain value.’