Lecture #23: Internal Flows

1 cell cellular sheetcellular bilayer

bilayered canister

ecto-derm

endo-derm

one way gut

mouthanus

cephalization

mesoderm

Body Plan Evolution

heart

lung/gill

body intestine

Basic circulatory circuit

convectionIn dedicatedplumbing

diffusionin dedicatedexchangers

Convection vs. Diffusionx

C1 C2S

x

CCDS

time

massJ 12

Fick’s Law:

C= concentration in mass/volumeD = diffusion coefficientUnits = L2/T

Basic strategy of circulatory systems:Pluming uses bulk flow (convection) to move fluids to capillary beds where diffusioncan take place over short distances.

Relative importance of bulk flow to diffusion given by Peclet number:

D

ulPe

Problems with gas exchange:

consider simple gas exchanger:

water

bloodDIFFUSION

x

CCDSJ 12

distance

equilibriumpartial

pressure(02)

drivingforce

convection

Problems with gas exchange:

consider countercurrent gas exchanger:

water

blood

x

CCDSJ 12

distance

partialpressure

(02)

drivingforce

What about lungs? x

CCDSJ 12

distance

partialpressure

(02)

blood

air

Birds have more efficient system

Birds have more efficient system

What determines flow in pipes?

x

r

aP1 P2

L

If Re < 2000 (i.e. laminar flow):

4)(

)(22 ra

L

Prux

• flow ~ pressure gradient• flow ~ 1 / viscosity• parabolic flow distribution

4)(

max2a

L

Pux

What is maximum flow velocity?

At center of pipe, r=0:

8)(

4

42

21

2 a

L

Pa

a

L

PQ

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

distance

pre

ssu

reflo

w velo

city

heart lung intestine body

heartlung

intestinebody

Pressureis lost (drops)across networkof pipes.

10% of our totalmetabolic cost!

5% of our totalWeight in blood!

Problems with blood

Blood is not a ‘Newtonian’ fluid,Mostly because of red blood cells.

4

8

a

L

Q

P

From Hagen-Poiseuille Equation:

‘Resistance’

Blood is very viscous due to red blood cells

% hematocrit

visc

osity

carrying capacity

02 carried/unit cost

optimumat 58%

Thoughts about plumbing:

Consider simple branch point:

S0

S1

S1

If S1 = 2 S2 then velocity is same in all branches; flux is ½ the original value.

a0a2

If a0 = 2 a1 then 16 times thepressure 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.

Murray’s Law: what is geometry of branching network?

1) Cost to pump = Q x pressure gradient, or

4

28

a

QQ

L

P

2) Cost to make new pipe2aM

Total cost 2

4

28aM

a

Q

3) Find optimum as a function of diameter:

)8

( 24

2

aMa

Qa da

dopt

6/13/1 )

16(M

Qaopt

3/1~Qaopt3kaQif then

a0

a1

a2

32

31

30 aaa

a.k.a. Murray’s Law

01

01

01

26.0

63.0

79.0

uu

SS

aa

For simple symmetrical branching case:

Mass flux ~ cube ofvessel diameter

But, by law of continuity,

Q0

Q1

Q2

210 QQQ thus

333

32

31

30 ... naaaaa

More generally……

How does a growing vascular network ‘know’ to follow Murray’s Law?

x

r

adu/dr

Shear stress at wall, = du/dr

It can be shown that:

3

4

r

Q

But by Murray’s Law:

3kaQ

So with r = a (at wall):

k4

Thus, shear stress at wall is constant in network obeying Murray’s Law.Algorithm could be: ‘Grow vessel until shear stress reaches certain value.’