44
3 Mechanical Aspects of Biosystems
forces is the Archimedes buoyancy force. In the plane of rotation, this will be
B = ρ F ω 2 rV , where ρ F is the fluid density and ω 2 r is the centripetal acceleration
of the cell when at radius r in the centrifuge. If the suspended cell is spherical (with
radius a and density ρ c ), and its radial speed is sufficiently small, the viscous drag
is well approximated by Stokes’ law, 26
F η = 6πηav ,
(3.6)
where η is the fluid viscosity and v is the radial speed of the cell. (See Fig. 3.6.)
For other shapes, the viscous force is still proportional to the speed, at least for
relatively slow radial speeds typical of cells and macromolecules in a centrifuge.
(The proportionality works well even for the viscous drag on a small particle moving
in a gas, i.e. F v = b v. In this case, Einstein proved that b = k B T /D, with D the
gas diffusion constant.)
The particles initially suspended in a liquid and then placed in a centrifuge may
accelerate radially, but after a short time, will reach a ‘terminal’ velocity, when the
forces on the particles balance. Newton’s 2nd law in the horizontal plane then gives
6πηav = (ρ c − ρ F )[(4/3)π a
3
] ω
2 r .
(3.7)
(The expression in the brackets is the volume of the cell V = (4/3)π a 3 . In the
vertical plane, 6πηav u = (ρ c −ρ F )[(4/3)π a 3 ] g, where v u is the upwardly directed
velocity. Since ω 2 r >> g, v u << v.)
Optical centrifuges send focused light from the centrifuge chamber to the axis
of rotation, and then out along the axis to an observer. In this way, sedimentation
speeds can be directly observed and measured, as well as other static and dynamic
reactions to high g.
3.3.5 Drag on Body
As we have indicated, for any object not moving radially outward too fast in the
liquid, the viscous drag is proportional to the velocity, so the relation in Eq. (3.7)
generalizes to
b v = (ρ c − ρ F )V ω
2 r .
(3.8)
26 Stokes’ law works well here provided the Reynolds number associated with the fluid flow
satisfies R e ≡ ρv/η < 0.2. Empirical formulae for the range 0.2 < R e < 10 5 exist. (See
references in M.D. Mikhailov and A.P. Silva Freire, The drag coefficient of a sphere, Powder
Technology 237, 432–435 (2013).) An example given by D.J. Dunn is F η = [2/5 + 24/R e +
6/(1 + R
1/2
e )][ρ f v 2 /2 · πa 2 ].
3 Mechanical Aspects of Biosystems
forces is the Archimedes buoyancy force. In the plane of rotation, this will be
B = ρ F ω 2 rV , where ρ F is the fluid density and ω 2 r is the centripetal acceleration
of the cell when at radius r in the centrifuge. If the suspended cell is spherical (with
radius a and density ρ c ), and its radial speed is sufficiently small, the viscous drag
is well approximated by Stokes’ law, 26
F η = 6πηav ,
(3.6)
where η is the fluid viscosity and v is the radial speed of the cell. (See Fig. 3.6.)
For other shapes, the viscous force is still proportional to the speed, at least for
relatively slow radial speeds typical of cells and macromolecules in a centrifuge.
(The proportionality works well even for the viscous drag on a small particle moving
in a gas, i.e. F v = b v. In this case, Einstein proved that b = k B T /D, with D the
gas diffusion constant.)
The particles initially suspended in a liquid and then placed in a centrifuge may
accelerate radially, but after a short time, will reach a ‘terminal’ velocity, when the
forces on the particles balance. Newton’s 2nd law in the horizontal plane then gives
6πηav = (ρ c − ρ F )[(4/3)π a
3
] ω
2 r .
(3.7)
(The expression in the brackets is the volume of the cell V = (4/3)π a 3 . In the
vertical plane, 6πηav u = (ρ c −ρ F )[(4/3)π a 3 ] g, where v u is the upwardly directed
velocity. Since ω 2 r >> g, v u << v.)
Optical centrifuges send focused light from the centrifuge chamber to the axis
of rotation, and then out along the axis to an observer. In this way, sedimentation
speeds can be directly observed and measured, as well as other static and dynamic
reactions to high g.
3.3.5 Drag on Body
As we have indicated, for any object not moving radially outward too fast in the
liquid, the viscous drag is proportional to the velocity, so the relation in Eq. (3.7)
generalizes to
b v = (ρ c − ρ F )V ω
2 r .
(3.8)
26 Stokes’ law works well here provided the Reynolds number associated with the fluid flow
satisfies R e ≡ ρv/η < 0.2. Empirical formulae for the range 0.2 < R e < 10 5 exist. (See
references in M.D. Mikhailov and A.P. Silva Freire, The drag coefficient of a sphere, Powder
Technology 237, 432–435 (2013).) An example given by D.J. Dunn is F η = [2/5 + 24/R e +
6/(1 + R
1/2
e )][ρ f v 2 /2 · πa 2 ].