Fluids in Motion Supplement I

Sedimentation of a Sphere at low Re
Suppose a spherical object with density ρ 1 is sinking (or floating) within a fluid. At low Reynolds
Number, the drag force is given by the Stokes drag, so:
F d
= 6πηRv
Plugging this in and using the formula for the volume of a sphere:
4
6πη π ρ ρ
3
( )
3
Rv= R
1−
2
g
There is only a little cancellation and one may then solve for the terminal velocity of a sphere
sedimenting if the Reynolds Number is less than 1.
v =
2
9
R
2
( ρ − ρ )
1 2
η
g
Example S4 Sedimentation time for a cell in water
Suppose you have a red blood cell with an effective radius of 4 µm and density 1.15 g/cm 3
sedimenting in salt water that has a density of 1.05 g/cm 3 . How long will the cell take to
sediment a vertical distance of 5 cm?
Reasoning: Since the radius of the RBC is in the micrometer range, and we guess from
experience that the sedimentation speed is small (guess 1 mm/s) the Reynolds number = Dvρ/η ~
(2*4 x 10 -6 )(1 x 10 -3 )(1050)/(0.001) ~ 0.0084. This is very much less than 1 so we can use the
Stokes drag formula above:
2
−6
( ρ − ρ ) g ( × ) ( − )
2
2 R
4 10 1150 1050 (9.80)
1 2 2
v = =
9 η 9 (0.001)
v = ×
−6
3.5 10 m/s
So now we can use the definition of speed v = ∆x/∆t (here, v = h/t) to find the time to settle 5 cm
(= 1 x 10 -2 m):
t
h 0.05 m
= = = ×
−6
v 3.5×
10 m/s
4
1.4 10 s = 239 min = 4.0 hr