Lecture #22: Low Reynolds number Re = u L / Forces linearly proportional to velocity Flow reversible Boundary layers large

low intermediatehigh (laminar)high (turbulent) Drag u Drag = ½ C D  S u 2 C D = 2 Drag /  S u 2 Re =  u 2 S  u S / L = u L / L S CDCD Reynolds number low Re: small things slow speeds high viscosity C D is not behaving like a constant

IF Re << 1 Drag = 6  u a “Stokes’ Law” Drag u S a low intermediatehigh (laminar)high (turbulent) CDCD Reynolds number Consider: 6  u a = ½ C D  S u 2 Let S = frontal area =  a 2 Let Re = u (2a) / Then: C D = 24/Re George Stokes

What is descent velocity of pollen? Drag = mg = 6  u a terminal velocity, u = mg / 6  a u = 25 mm/sec Re = 0.1 Slow descent increases dispersal, more time To be carried laterally by the wind. Passive locomotion at low Re, e.g. pollen

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 power stroke recovery stroke high drag on power stroke, less drag on recovery power stroke recovery stroke METACHRONY

distance fluid velocity boundary layer effects

2. Flagella (two kinds) a) Eukaryotic flagella (time lapse) traveling wave b) prokaryotic flagella

Drag on body is 6  u a What is drag on tail? What is drag on cylinder normal and tangent to flow? u N = u cos  u T = u sin  u uNuN uTuT L d 

What are forces in direction of motion: 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 C N and C T Lateral forces cancel over length Lateral forces reduce efficiency F Forward F Lateral drag T  Thrust must offset drag on ‘head’, given by Stokes’ Law. ‘body’ drag drag N

Boundary layers solid surface Velocity, u = u (mean stream flow) o o u = 0 (no slip condition) boundary layer Laminar flow over solid surface u inf x y flat plate with upstream edge

Size of boundary layer increase with viscosity, decreases with Velocity. Flow slows between hairs. low Reynolds number (large boundary layers) high Reynolds number (small boundary layers) flow through cylinder array

Hairy legs and wings