lecture #22: low reynolds number re = u l / forces linearly proportional to velocity flow reversible...
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Lecture #22: Low Reynolds number
Re = u L / • Forces linearly proportional to velocity• Flow reversible• Boundary layers large
low
intermediate high (laminar) high (turbulent)
DraguDrag = ½ CD S u2
CD = 2 Drag / S u2
Re = u2 S
u S / L= u L /
L
S
CD
Reynolds number10 106
100
10
1
0.1
0.1 1
low Re:small thingsslow speedshigh viscosity
CD is not behaving like a constant
IF Re << 1
Drag = 6 u a
“Stokes’ Law”
Dragu
Sa
low
intermediate high (laminar) high (turbulent)
CD
Reynolds number10 106
100
10
1
0.1
0.1 1
Consider:
6 u a = ½ CD S u2
Let S = frontal area = a2
Let Re = u (2a) / Then:CD = 24/Re
CD= 24/Re
GeorgeStokes
What is descent velocity of pollen?
Drag = mg = 6 u a
terminal velocity, u = mg / 6 au = 25 mm/sec Re = 0.1
Slow descent increases dispersal, more timeTo be carried laterally by the wind.
Passive locomotion at low Re, e.g. pollen
Locomotion at low Reynolds numbers:
lateralundulation
But, reversibility of flow means that lateral undulations cannot generate thrust!
Two basic strategies for Low Reynolds number locomotion:
1) Cilia
power stroke
recovery stroke
high drag on power stroke,less drag on
recovery
power stroke recovery stroke
METACHRONY
distance
fluid velocity
boundarylayereffects
2. Flagella (two kinds)
a) Eukaryotic flagella
(time lapse)
traveling wave
b) prokaryotic flagella
Drag on body
is 6 u aWhat is drag on tail?
What is drag on cylinder normal and
tangent to flow?
212 )ln(
4
dlN uldrag
212 )ln(
2
dlT uldrag
uN = u cos uT = u sin
u
uN
uT
L
d
3.3; NNN CulCdrag
2.2; TTT CulCdrag
What are forces in direction of motion:
cossin)(
cossin
TNforward
TNforward
CCulF
dragdragF
• Forward thrust adds along length of flagellum• Forward thrust is proportional to viscosity• Forward thrust maximal at =45 deg.• Production of thrust relies on difference of CN and CT
)coscos(
sincos22
TNlateral
TNlateral
CCulF
dragdragF
• Lateral forces cancel over length • Lateral forces reduce efficiency
FForward
FLateral
dragT
Thrust must offset drag on ‘head’, given by Stokes’ Law.
‘body’drag
dragN
Boundary layers
solid surface
Velocity, u =u (mean stream flow)oo
u = 0(no slip condition) boundary
layer
Laminarflow oversolid surface
uinf
x
y
flat plate with upstream edge
u
x 5
Sizeof boundarylayer increasewith viscosity,decreases withVelocity.
Flow slowsbetweenhairs.
low Reynolds number(large boundary layers)
high Reynolds number(small boundary layers)
flow through cylinder array
Hairy legs and wings