Table of Contents

## Introduction

Viscosity is the resistance of a fluid to flow. When a fluid is flowing, the molecules comprising it experience friction due to the molecular interaction among them. It means that different fluids with different viscosities flow at different speeds when the same force is applied. Another way to look at it is that the force required to induce movement of the fluid will be larger for more viscous fluids. Honey or corn syrup has a viscosity in the range of 2000-3000 cP (*centipoise*), while water only around 1 cP. Clearly water flows more easily, since it has much lower viscosity.

In other words, the shear stress developed in the fluids is a function of viscosity. The fluids in which the shear stress is proportional to the shear rate, are called Newtonian fluids and the constant of proportionality is called viscosity:

(1)

**So why is this important?**

Viscosity is key for any process involving fluid flow. It will determine, for example – How long a reservoir will take to fill. Viscosity will also determine how much energy is required to pump out process fluid. Viscosity can also help you determine the structure of molecules such as proteins in a solution or the force required to inject a new drug formulation into a patient.

The unit of viscosity is newton-second per square meter (), which is usually expressed as pascal-second () in SI units. In lubrication practice, the earlier mentioned cP unit is widely used, .

**Dynamic vs Kinematic Viscosity: What is the difference?**

1. **Dynamic viscosity:** Dynamic viscosity is also known as absolute viscosity. The SI physical unit of dynamic viscosity (*μ*) is the Pascal-second (Pa s), identical to 1 kg m−1 s−1. The physical unit for dynamic viscosity in the centimetre gram second system of units (cgs) is the *poise* (P). It is more commonly expressed, particularly in ASTM standards, as centipoise (cP).

2. **Kinematic viscosity:** Kinematic viscosity is the ratio of dynamic viscosity and the density, so . Kinematic viscosity has SI units of . The physical unit for kinematic viscosity is the *stokes* (St), named after George Stokes. However, due to practicality reasons typically it is expressed in terms of *centistokes* (cS or cSt); 1 stokes = 100 centistokes = 1 cm2 s−1 = 0.0001 m2 s−1. Kinematic viscosity is typically used and reported in industry.

## Table of Viscosities of Common Lubricants

Dynamic Viscosity Measured at Atmospheric Pressures, mPas | Pressure-Viscosity Coefficient, 1/Gpa | |||||
---|---|---|---|---|---|---|

30°C | 60°C | 100°C | 30°C | 60°C | 100°C | |

High VI oils | ||||||

Light machine oil | 38 | 12.1 | 5.3 | - | 18.4 | 13.4 |

Heavy machine oil | 153 | 34 | 9.1 | 23.7 | 20.5 | 15.8 |

Heavy machine oil | 250 | 50.5 | 12.6 | 25 | 21.3 | 17.6 |

Cylinder oil | 810 | 135 | 26.8 | 34 | 28 | 22 |

Medium VI oils | ||||||

Spindle oil | 18.6 | 6.3 | 2.4 | 20 | 16 | 13 |

Light machine oil | 45 | 12 | 3.9 | 28 | 20 | 16 |

Medicinal white oil | 107 | 23.3 | 6.4 | 29.6 | 22.8 | 17.8 |

Heavy machine oil | 122 | 26.3 | 7.3 | 27 | 21.6 | 17.5 |

Heavy machine oil | 171 | 31 | 7.5 | 28 | 23 | 18 |

Low VI oils | ||||||

Spindle oil | 30.7 | 8.6 | 3.1 | 25.7 | 20.3 | 15.4 |

Heavy machine oil | 165 | 30 | 6.8 | 33 | 23.8 | 16 |

Heavy machine oil | 310 | 44.2 | 9.4 | 34.6 | 26.3 | 19.5 |

Cylinder oil | 2000 | 180 | 24 | 41.5 | 29.4 | 25 |

Other fluids and lubricants | ||||||

Water | 0.8 | 0.47 | 0.28 | 0 | 0 | 0 |

Ethylene oxide- | ||||||

propylene oxide copolymer | 204 | 62.5 | 22.5 | 17.6 | 14.3 | 12.2 |

Castor oil | 360 | 80 | 18 | 15.9 | 14.4 | 12.3 |

Di(2-ethylhexyl) phthalate | 43.5 | 11.6 | 4.05 | 20.8 | 16.6 | 13.5 |

Glycerol (glycerine) | 535 | 73 | 13.9 | 5.9 | 5.5 | 3.6 |

Polypropylene glycol 750 | 82.3 | - | 17.8 | - | ||

Polypropylene glycol 1500 | 177 | - | - | 17.4 | - | - |

Tri-arylphosphate ester | 25.5 | - | - | 31.6 | - | - |

## Viscosity-Temperature Relation for Water

It is interesting to know that the viscosity of liquids decreases rapidly with an increase in temperature, and the viscosity of gases increases with an increase in temperature. Thus, upon heating, liquids flow more easily, whereas gases flow more sluggishly. For example, the viscosities of water at 27 °C (81 °F) is 0.85 × 10^{−3 }pascal-second and at 77 °C (171 °F) is and 0.36 × 10^{−3} pascal-second, respectively, but those of air at the same temperatures are 1.85 × 10^{−5} and 2.08 × 10^{−5} pascal-second.

Viscosity of water as a function of temperature can be obtained according to Vogel-Fulcher-Tammann equation:

(2)

where is the dynamic viscosity in , . is given in kelvins.

Here is a simple calculator that allows to calculate the viscosity of water as a function of temperature according to the equation above:

## Viscosity-Temperature Relations for Lubricating Oils: Walther equation

For lubricating oils, viscosity drops with temperature rapidly. There are a number of models exist to describe the relation between viscosity and temperature. These models are usually fits obtained using experimental data. The known equations are due to Reynolds, Vogel, but the most commonly used one is known as Walther equation. This equation is the standard used it ASTM, ISO and DIN charts. Here is the Walther equation:

(3)

Here is the kinematic viscosity given in , is temperature in kelvins, is a constant for a particular fluid. For lubrication oils, it can be chosen as 0.7 (see the reference). The constants and are calculated from the kinematic viscosities at 2 temperatures ( and ):

(4)

The lubricant manufacturers typically specify kinematic viscosities at 40C and 100C, which are then used in the equation above (but don’t forget to translate to kelvins). Here is a simple calculator for calculating the viscosity of an oil according to the Walther equation:

## Viscosity-Pressure-Temperature Relations for Lubricating Oils

In lubricated contacts the contact pressure is typically in the range of 0.1-2 GPa. Viscosity of lubricants depends strongly on the pressure. Thus, for the calculation of the oil film thickness and lubricant friction in lubricated contacts, it is necessary to know the viscosity at the given pressure and temperature. There are several models available, here the most commonly used ones are presented.

### Barus equation

This equation is a very simple one and therefore very common. It links the viscosity of the oil to the pressure as follows:

(5)

where is the viscosity at ambient pressure, is the pressure-viscosity coefficient. For lubricating oils, to .

This equation is known to be inaccurate for high pressure (>1 GPa), predicting a larger increase rate with increase of pressure than it really is. Here is a simple calculator for calculating the viscosity of an oil at the given pressure:

### Roelands equation

A more accurate relation that also includes temperature dependence, is by Roelands:

(6)

Here

- – temperature at which was measured (in kelvins)
- – viscosity-temperature index, commonly
- – pressure-viscosity index, commonly
- are given in in this equation.

The pressure-viscosity coefficient used in the Barus equation above is related to the pressure-viscosity index in the Roelands equation as follows:

(7)

## Viscosity Units Conversion

Multiply by | |||||
---|---|---|---|---|---|

Convert from | Convert to | ||||

Poiseuille, (Pa s) | Poise, (dyne s/cm2), (g/cm s) | centiPoise | kg/m h | kgf s/m2 | |

Poiseuille (Pa s) | 1 | 10 | 10^3 | 3.63 10^3 | 0.102 |

Poise (dyne s/cm2) or (g/cm s) | 0.1 | 1 | 100 | 360 | 0.0102 |

centiPoise | 0.001 | 0.01 | 1 | 3.6 | 0.00012 |

kg/m h | 2.78 10^-4 | 0.00278 | 0.278 | 1 | 2.83 10^-5 |

kgf s/m2 | 9.81 | 98.1 | 9.81 10^3 | 3.53 10^4 | 1 |

lbf s/inch2 | 6.89 10^3 | 6.89 10^4 | 6.89 10^6 | 2.48 10^7 | 703 |

lbf s/ft2 | 47.9 | 479 | 4.79 10^4 | 1.72 10^5 | 0.0488 |

lbf h/ft2 | 1.72 10^5 | 1.72 10^6 | 1.72 10^8 | 6.21 10^8 | 1.76 10^4 |

lb/ft s | 1.49 | 14.9 | 1.49 10^3 | 5.36 10^3 | 0.152 |

lb/ft h | 4.13 10^-4 | 0.00413 | 0.413 | 1.49 | 4.22 10^-5 |

**Viscosity Tests**

Liquids that are resistant to flow or flow slowly have high viscosity. These are liquids like honey, sunscreen and dish soap. Liquids that are not resistant to flow or flow fast have low viscosity. These are liquids like water and vegetable oil.

**Ways to Measure Viscosity**

There are many ways to measure viscosity of liquids. Here is a list of the most common ways to measure viscosities together with videos of such operations.

**Capillary Viscometer**

A capillary viscometer is an instrument used to measure a liquid’s viscosity or thickness by measuring how long it takes the liquid to flow through a small-diameter tube or capillary. Here is a short video on the capillary viscometer operation:

**Zahn Cup**

A Zahn cup is a viscosity measurement device widely used in the paint industry. It is commonly a stainless steel cup with a tiny hole drilled in the centre of the bottom of the cup. There is also a long handle attached to the sides. There are five cup specifications, labelled Zahn cup #x, where x is the number from one through five (see table below). Large number cup sizes are used when viscosity is high, while low number cup sizes are used when viscosity is low. They are manufactured in accordance with ASTM D 4212, ASTM D1084 and ASTM D816 lubricants testing standards. Here is a quick demonstration of this test.

**Falling Sphere Viscometer**

A falling-sphere viscometer is a device used to measure the viscosity of a fluid by measuring the time required for a spherical ball to fall a certain distance under gravity through a tube filled with the fluid whose viscosity is to be determined.

**Vibrational Viscometer**

The vibrating viscometer determines viscosity by allowing a vibrating blade to vibrate in a liquid and measure the resultant viscous drag. The following formula determines the vibration amplitude when the measured fluid surrounds the vibrating shaft, and vibrational restriction occurs.

(8)

- K： constant on the driving force and the displacement
- μ： viscosity
- ρ： density
- Ea： amplitude in air
- Ex： amplitude in the measured fluid
- C : constant derived from vibration parts

The measured amplitude varies according to the fluid’s viscosity. After K is calculated using a specimen fluid for which the viscosity and density are already known, µρ can be derived by comparing Ea with Ex.

**Rotational Viscometer**

Rotational viscometers are designed to measure viscosity by analyzing the torque required to rotate a spindle submerged in a fluid at a constant speed. The continuous rotation of the spindle ensures calculations are made over time, allowing time-dependent fluid analysis. Digital models can evaluate low, medium, and high viscosity materials. Considerations for rotational viscometers should include the level of viscosity application, speed, and spindle configuration. Each consideration configuration will help hone the sensitivity and accuracy of measurement for your specific application. Additional considerations would be unit size and dimensions.

There are SAE (Society of Automotive Engineers) grades for gear oils and crankcases (engines), AGMA (American Gear Manufacturers Association) grades for gear oils, SUS (Saybolt Universal Seconds), cSt (kinematic viscosity in centistokes), and absolute viscosity.

ISO VS Viscosity Grade | Mid Point Viscosity cSt @ 40˚ C (104˚ F) | Kinemetic Viscosity Limit. cSt @ 40˚ C (104˚ F) | |

10 | 10 | 9 | 11 |

15 | 15 | 13.5 | 16.5 |

22 | 22 | 19.8 | 24.2 |

32 | 32 | 28.8 | 35.2 |

46 | 46 | 41.4 | 50.6 |

68 | 68 | 61.2 | 74.8 |

100 | 100 | 90 | 110 |

150 | 150 | 135 | 165 |

220 | 220 | 198 | 242 |

320 | 320 | 288 | 352 |

460 | 460 | 414 | 506 |

680 | 680 | 612 | 748 |

1000 | 1000 | 900 | 1100 |

1500 | 1500 | 1350 | 1650 |

2200 | 2200 | 1980 | 2420 |

3200 | 3200 | 2880 | 3520 |

4600 | 4600 | 4140 | 5060 |

6800 | 6800 | 6120 | 7480 |

References –