Very Low Reynolds Numbers: Stokes' Drag
The equation for viscous resistance or linear drag is appropriate for objects or particles moving through a fluid at relatively slow speeds where there is no turbulence (i.e. low Reynolds number, ). Note that purely laminar flow only exists up to Re = 0.1 under this definition. In this case, the force of drag is approximately proportional to velocity, but opposite in direction. The equation for viscous resistance is:
where:
- is a constant that depends on the properties of the fluid and the dimensions of the object, and
- is the velocity of the object
When an object falls from rest, its velocity will be
which asymptotically approaches the terminal velocity . For a given, heavier objects fall faster.
For the special case of small spherical objects moving slowly through a viscous fluid (and thus at small Reynolds number), George Gabriel Stokes derived an expression for the drag constant:
where:
- is the Stokes radius of the particle, and is the fluid viscosity.
The resulting expression for the drag is known as Stokes' drag:
For example, consider a small sphere with radius = 0.5 micrometre (diameter = 1.0 µm) moving through water at a velocity of 10 µm/s. Using 10−3 Pa·s as the dynamic viscosity of water in SI units, we find a drag force of 0.09 pN. This is about the drag force that a bacterium experiences as it swims through water.
Read more about this topic: Drag (physics)
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