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