## Stokes' law

**Introduction: **Stokes’ law describes the motion of a small spherical object in a viscous fluid. The particle is subjected to the downward directed force Fg of gravity and the buoyant force FA, also known as the force exerted by Archimedes’ law. The both forces are:

F_{g}= (4/3)r^{3}ρ_{p}g ….. (1a)

F_{A}= (4/3)r^{3}ρ_{f}g, …… (1b)

Where r is the radius (the Stokes radius, see More Info) and g is the acceleration due to gravity (9.8 m/s^{2}). The resulting force is:

F_{r}= F_{g}─ FA = (4/3)r_{3}g(ρ _{p} ─ ρ_{f}.) ……. (2)

The resulting force is directed downward when the specific density of the particle ρ_{p}is higher than that of the fluid ρf (e.g. a glass bead) and upward when it is smaller (a polystyrene sphere). Here it is supposed that the former case holds, and so the sphere will move downward.

As soon as the sphere starts moving there is a third force, the frictional force Ff of the fluid. Its direction is opposite to the direction of motion. The total resulting force is:

F_{tot} = F_{r}, ─ F_{f }..….. (3)

As long as Ftot is positive, the velocity increases. However, F_{f} is dependent on the velocity. Over a large range of velocities the frictional force is proportional to the velocity (v):

F_{f}= 6πrηv ……(4)

Where η is the dynamic fluid viscosity. Expression (4) is Stokes’ law.

After some time the velocity does not increase anymore but becomes constant. Then equilibrium is reached. In other words, F_{r}is canceled by F_{f}and so F_{tot} = 0. From now on the particle has a constant velocity. The equilibrium or setting velocity v_{s}can be calculated from (2), (3) and (4) with F_{tot}= F_{r}, ─ F_{f } = 0. The result is:

v_{s}= (2/9)r_{2}g(ρ_{p}─ ρ_{f})/η, …….(5)

The proportionality of vs with r_{2 }means that doubling the radius gives a reduction of a factor of four for the setting time.

Equation (6) only holds under ideal conditions, such as a very large fluid medium, a very smooth surface of the sphere and a small radius.

**Application: **The law has many applications in science, e.g. in earth science where measurement of the setting time gives the radius of soil particles.

Blood cells: In medicine, a well-known application is the precipitation of blood cells. After setting, on the bottom of a hematocrit tube are the red cells, the erythrocytes, since they are large and have the highest density. In the middle are the white cells, the leucocytes despite their often larger volume. However, they are less dense and especially less smooth, which slows their speed of setting. On top, and hardly visible, is a thin band of the much smaller platelets, the thrombocytes. The relative height of the band (cylinder) with the red cells is the hematocrit. Although the red cells are not spherical and the medium is not large at all (a narrow hematocrit micro tube) the process still behaves rather well according to the law of Stokes .Red cells can clotter to money rolls, which sett faster (of importance for the hematologist).

The equivalent radius of a red blood cell can be calculated from the various parameters: hematocrit (assume 0.45 L/L), setting velocity (0.003 m/hour = 0.83∙10-6 m/s), density of red cells (1120 (kg/m^{3}) and plasma (1000 (kg/m^{3}). Everything filled out in eq. (5) yields 3.5 μm. Actually the red cell is disk-shaped with a radius of about 3.75 μm and a thickness of 2 μm.

**Centrifugation:**

An important application is the process of centrifugation of a biochemical sample. The centrifuge is used to shorten substantially the setting time. In this way proteins and even smaller particles can be harvested, such as radio nucleotides (enrichment of certain isotopes of uranium in an ultracentrifuge). With centrifugation, the same equations hold, but the force of gravity g should be replaced by the centrifugal acceleration a. : a = 4π^{2}f^{2}R, where f is the number of rotations/s and R the radius of the centrifuge (the distance of the bottom of the tube to the center). In biochemistry, a can easily reach 104 g and in physics even 106 g.