### Different Sizes, Different Forces, Different Problems

```Different Sizes, Different
Forces, Different Problems
Diffusion
• A random walk
• Steps of “mean free
path” length
• Random direction
after collision
http://www.geocities.com/piratord/browni/Difus.html
Statistics of Diffusion
But it’s random…
• Distance dependent
on t^(1/2)
– Distances larger than
a cell are inefficient to
diffuse over
• While any one particle
is unpredictable, an
ensemble is
• Diffusion smoothes
over concentration
2 2
z (t )  ( v  )t
3
2
Diffusion across a membrane
• Mass/time proportional to:
– Diffusion Coefficient, D
– Area S of the slab
– Concentration of the gradient across the slab
m
 m1 V1  m2 V2 
 DS

t
x


m
 C 
  DS 

t
 x 
Diffusion Values
http://www.nanomedicine.com/NMI/Tables/3.3.jpg
Convection
• Movement though
smooth currents
• Behavior determined
through complicated
fluid dynamics
Reynolds Number
• A measure of
viscosity versus
inertia
– ρ is density
– μ is viscosity
– L is a characeristic
length
– V is the relative velocity
of the fluid relative to the
object or sides
lv
Re 

Spermatozoa ~ 1e−2
Blood flow in brain ~ 1e2
Blood flow in aorta ~ 1e3
Onset of turbulent flow ~ 2.3e35.0e4 for pipe flow to 10^6 for
boundary layers
Typical pitch in Major League
Baseball ~ 2e5
Person swimming ~ 4e6
Blue Whale ~ 3e8
A large ship (RMS Queen Elizabeth
2) ~ 5e9
http://www.aanda
.org/index.php?o
ption=article&acc
ess=standard&Ite
mid=129&url=/art
icles/aa/full/2006/
19/aa449905/aa449905.right.html
Low Reynolds Number Regime
• Small organisms with
little mass to break
surface tension
• Cannot stroke and
glide
Purcell – “It helps to imagine under what conditions a man would be
swimming at, say, the same Reynolds number as his own sperm. Well, you
put him in a swimming pool that is full of molasses, and then you forbid him
to move any part of his body faster than one centimeter per minute. Now
imagine yourself in that condition: you’re in the swimming pool in molasses,
and now you can only move like the hands of a clock. If under those ground
rules you were able to move a few meters in a couple of weeks, you may
qualify as a low Reynolds number swimmer.
High Reynolds Number
• Turbulent,
irreversible flow
• Fast forward
pushes dominate
slow backwards
pushes
Surface Tension
• In water, attractive force
between molecules
• On Surface, attractive
force in, no force out
• Liquids minimize surfaces
• Order l
– Cross sectional areas go as
l2
– But, Distances get further
apart as things get bigger
Gravity
• Order l3
• Cross sectional
areas go as l2
• Gravity become
increasingly
important to big
things
Conclusions
Size Matters!
```