Brightly colored and highly spherical particles, and specifically colored and fluorescent polyethylene microspheres, are often used by scientists as highly visible tracer particles in liquids for the purposes of Particle Image Velocimetry (PIV) or fluid flow visualization.

It is important for the researches to select the correct microsphere to optimize the performance of the particle in the solution, accuracy of data collected, and success of the fluid flow experiment. For example, the experiment might require the microsphere to stay in suspension for a long time, without settling, or a desired settling velocity might need to be achieved for testing a particular instrument. For this reason, the investigator needs to understand how spherical particles behave in their particular liquid, at specified environmental conditions. This behavior of spherical particles in solution is best characterized by Stoke's law.

Stokes’s law is a mathematical equation that calculates the settling velocities of small spherical particles in a fluid medium. The law, first set forth by the British scientist Sir George G. Stokes in 1851, is derived by consideration of the forces acting on a particular particle as it sinks through a liquid column under the influence of gravity .

Stoke's law only applies with these assumptions:

• There is no other particle nearby that would affect the flow pattern.
• The motion of the particle is constant.
• The particle is spherical and rigid.
• The air velocity right at the particle surface is zero.
• The fluid is incompressible.

As you can see from the formulas below, microsphere diameter is the most critical variable for determining settling velocity. The settling velocity, and, as a result, settling time, are proportional to the diameter of the spherical particle squared. The larger the sphere diameter, the faster the particle will settle. The smaller the particle diameter, the longer it will stay suspended in the fluid.

The second most critical variable is density delta, or the difference in the density of the particle and the density of the liquid. Settling time and velocity are proportional to density delta (also known as density mismatch), which means that matching the density of the particle to the density of the liquid is very important for minimizing settling velocity and maximizing the time microspheres spend in suspension. Keep in mind that even the slightest variation in density matters, and densities of liquids sometimes vary significantly with changes in temperature, pressure, and materials added to it. Please see Density of Aqueous Solutions Reference Tables.

Settling velocity is inversely proportional to the viscosity of the fluid. Obviously, the thicker (more viscous) the fluid, the longer the settling time, the thinner (less viscous) the fluid, the faster the settling time.

The scientist designing a flow visualization experiment that includes spherical tracer particles is designing a complete system that takes into account the exact properties of the fluid medium at specific environmental conditions, matched to the diameter and density of the microspheres, and also ensuring that no external forces are affecting the behavior of the spherical particle.

Fall or Settling Velocity :
V t = gd2p - ρm)/18µ

Acceleration of Gravity :
g= 18 µ Vt /d2( ρp - ρm)

Particle Diameter :
d= v18 µ Vt /g (ρp - ρm)

Density of Medium :
ρm = ρp - 18 µ Vt/ d2

Particle Density :
ρp = 18 µ Vt /d2+ ρm

Viscosity of Medium :
µ = gd2( ρp - ρm)/18 Vt

Where,
Vt = Fall or Settling Velocity,
g = Acceleration of Gravity,
d = Particle Diameter,
ρm = Density of Medium,
ρp = Particle Density,
µ = Viscosity of Medium.

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