Viscosity is a measure of a fluid’s resistance to flow and is caused by the internal resistance to motion. A liquid with low viscosity, like water, flows easily compared to one with high viscosity, like honey. In general, viscosity for a liquid decreases as temperature increases. Honey, for example, is very viscous if cold and would ooze from the jar if you tried to pour it. Heat the honey, though, and it becomes easy to pour.

If you imagine layers of fluid-like sheets of paper stacked in a sheaf, the force per unit area, applied along the length of each sheet (call this the *x*-direction), to move one layer past another is the *shear stress* (*F*/*A*). The change of fluid velocity with distance perpendicular to the applied force (call this the *y*-direction) is the velocity gradient (Δ_v_/Δ_y_) or *shear rate*.

## Dynamic Viscosity

*Dynamic viscosity*, also known as *absolute viscosity* or simply *viscosity* (or *η*), is the ratio of shear stress to shear rate and has units in the SI system of pascal-seconds (Pa-s):

*η* = (*F*/*A*) / (Δ_v_/Δ_y_)

In CGS terms, viscosity has units of poise (P), where 1 g-cm^{-1}-s^{-1} = 1 P = 0.1 Pa-s. The poise is inconveniently large for most fluids, however, so the centipoise (cP) is more commonly used.

Dynamic viscosity is often measured with a rotational viscometer, which uses a spinning probe immersed in a sample of liquid. Depending on the nature of the test, the probe may be a cylinder, a sphere, a disk or some other shape. Viscosity is determined by the force needed to rotate the probe at a chosen speed. Because viscosity varies with temperature, any meaningful description of viscosity must also include temperature.

## Newtonian and Non-Newtonian Fluids

A *Newtonian fluid*, like water, has a constant viscosity despite the amount of shear stress. Non-Newtonian fluids have viscosities that vary with the magnitude of shear stress.

Paint, ketchup and mayonnaise are non-Newtonian fluids that undergo stress-thinning, which means their viscosities decrease with stirring. In contrast, a mixture of cornstarch and water, sometimes called *oobleck* and used for some slime recipes, exhibits stress-thickening – you can slowly push your hand into oobleck, but it becomes rigid if struck quickly.

## Kinematic Viscosity

Unlike dynamic viscosity, *kinematic viscosity* measures the resistance to flow under only the influence of gravity and is primarily used for Newtonian fluids. Kinematic viscosity *υ* is the ratio of dynamic viscosity *η* and the mass density *ρ* of the liquid:

*υ* = *η*/*ρ*

Kinematic viscosity has CGS units of cm^{2}/s, called *stokes* (St), named after Irish mathematician Sir George Gabriel Stokes, who did much work in fluid mechanics. Because of the range of values for most fluids, kinematic viscosity is more conveniently expressed in centistokes (cSt), where 1 cSt = 0.01 St.

## Measuring Saybolt Seconds Universal (SSU)

In the United States, kinematic viscosity is often described as Saybolt viscosity, expressed as Saybolt Universal Seconds (SUS) or Saybolt Seconds Universal (SSU). Saybolt viscosity, also called SSU viscosity, is measured with a Saybolt viscometer, which heats the liquid to a desired temperature and pours it through a calibrated orifice.

Standard test temperatures are 77, 100, 122 and 210 degrees Fahrenheit (25.0, 37.8, 50.0 and 98.9 degrees Celsius). The time in seconds for 60 mL of the liquid to flow through the orifice is the SSU.

## Converting Saybolt Universal Seconds to Centistokes

In its document ASTM D2161-19, the American Society for Testing and Materials has defined how to convert between kinematic viscosity *υ* in centistokes and SSU *t* in seconds. The method is derived empirically, and consequently it is somewhat complex. In simplified form, converting SSU to cSt uses the following equations for different ranges of SSU viscosity:

*υ* = 0.226_t_ − 195/*t* (32 < *t* < 100)

*υ* = 0.220_t_ − 135/*t* (t > 100)

Converting kinematic viscosity to SSU requires the inverse relationship, which can be obtained by using the solution to the quadratic equation:

*t* = [-b + SQRT(b^{2} - 4ac) ] / [2a]

Where: b = -*υ* in centistokes (cSt)

a = 0.226 and c = -195 (1 ≥ *υ* ≥ 20.63)

a = 0.220 and c = -135 (20.63 > *υ* > 52)

## Example: Converting Centistokes to SSU Viscosity

If the kinematic viscosity *υ* = 30 centistokes:

- Determine terms a, b and c: a = 0.226, b = -30 and c = -195.
- Calculate b
^{2}= 900.00. - Calculate 4ac = 4 × 0.226 × -195 = -176.28.
- Calculate 2a = 2 × 0.226 = 0.452.
- Calculate SQRT(b
^{2}-4ac) = √1076.28 = 32.81. - Calculate SSU viscosity t = [-b + SQRT(b
^{2}-4ac)] / [2a] = 138.9 seconds.

References

Resources

Tips

- There are websites on-line that calculate estimated converted values of Centistokes and Saybolt Universal seconds that can be easily found through search engines like Google.

About the Author

H. L. M. Lee is a writer, electronics engineer and owner of a small high-tech company. He also produces web content and marketing materials, and has taught physics for students taking the Medical College Admissions Test. In addition, he has written numerous scripts for engineering and physics videos for JoVE, the Journal of Visualized Experiments. H.L.M Lee earned his undergraduate engineering degree at UCLA and has two graduate degrees from the Massachusetts Institute of Technology. More information about him and his work may be found on his web site at https://www.hlmlee.com/