Transcript BE105_22_low_reynolds_numbers

Lecture #22: Low Reynolds number
Re = u L / n
• Forces linearly proportional to velocity
• Flow reversible
• Boundary layers large
Drag
u
S
Drag = ½ CD r S u2
CD = 2 Drag / r S u2
Re =
L
100
intermediate
r u2 S
muS/L
high (laminar)
=uL/n
high (turbulent)
10
low
CD
CD is not behaving
like a constant
1
0.1
0.1
1
106
10
Reynolds number
low Re:
small things
slow speeds
high viscosity
Consider:
Drag
u
IF Re << 1
Drag = 6 p
aS
mua
“Stokes’ Law”
6 p m u a = ½ CD r S u2
Let S = frontal area = p a2
Let Re = u (2a) / n
Then:
CD = 24/Re
CD= 24/Re
100
intermediate
high (laminar)
high (turbulent)
10
low
CD
1
0.1
0.1
1
106
10
Reynolds number
George
Stokes
Passive locomotion at low Re, e.g. pollen
What is descent velocity of pollen?
Drag = mg = 6 p
mua
terminal velocity, u = mg / 6 p m a
u = 25 mm/sec
Re = 0.1
Slow descent increases dispersal, more time
To be carried laterally by the wind.
Locomotion at low Reynolds numbers:
lateral
undulation
But, reversibility of flow means that lateral undulations cannot generate thrust!
Two basic strategies for Low Reynolds number locomotion:
1) Cilia
high drag on
power stroke,
less drag on
recovery
power stroke
recovery stroke
METACHRONY
power stroke
recovery stroke
distance
boundary
layer
effects
fluid velocity
2. Flagella (two kinds)
a) Eukaryotic flagella
(time lapse)
b) prokaryotic flagella
Drag on body
is 6 p
What is drag on tail?
mua
u
uT
q
uN
L
d
uN = u cos q
uT = u sin q
 4p 
drag N  mul  2l 1 
 ln( d )  2 
 2p 
dragT  mul  2l 1 
 ln( d )  2 
dragN  CN mul; CN  3.3
What is drag on cylinder
normal and
tangent to flow?
dragT  CT mul; CT  2.2
dragN
dragT
What are forces in direction of motion:
FForward
q
‘body’
drag
FLateral
F forward  drag N sin q  dragT cosq
Flateral  drag N cosq  dragT sin q
F forward  mul (C N  CT ) sin q cosq
Flateral  mul (CN cos2 q  CT cos2 q )
• Forward thrust adds along length of flagellum
• Forward thrust is proportional to viscosity
• Forward thrust maximal at q =45 deg.
• Production of thrust
relies on difference of CN and CT
Thrust must offset drag
on ‘head’, given by Stokes’ Law.
• Lateral forces cancel over length
• Lateral forces reduce efficiency
Boundary layers
Laminar
flow over
solid surface
Velocity, u =
u oo (mean stream flow)
u=0
(no slip condition)
boundary
layer
solid surface
uinf
 5
y
x
flat plate with upstream edge
xn
u
flow through
cylinder array
high Reynolds number
(small boundary layers)
low Reynolds number
(large boundary layers)
Size
of boundary
layer increase
with viscosity,
decreases with
Velocity.
Flow slows
between
hairs.
Hairy legs and wings