Reference Work Entry

Encyclopedia of Tribology

pp 2416-2420

# Newton’s Law of Viscosity, Newtonian and Non-Newtonian Fluids

• Herman F. GeorgeAffiliated withApplied Science, The Lubrizol Corporation Email author
• , Farrukh QureshiAffiliated withApplied Science, The Lubrizol Corporation

## Definition

Viscosity is the physical property that characterizes the flow resistance of simple fluids. Newton’s law of viscosity defines the relationship between the shear stress and shear rate of a fluid subjected to a mechanical stress. The ratio of shear stress to shear rate is a constant, for a given temperature and pressure, and is defined as the viscosity or coefficient of viscosity. Newtonian fluids obey Newton’s law of viscosity. The viscosity is independent of the shear rate.

Non-Newtonian fluids do not follow Newton’s law and, thus, their viscosity (ratio of shear stress to shear rate) is not constant and is dependent on the shear rate.

Dynamic viscosity is the coefficient of viscosity as defined in Newton’s law of viscosity. Kinematic viscosity is the dynamic viscosity divided by the density.

## Scientific Fundamentals

Newton’s law of viscosity can be developed by considering a fluid at rest between two parallel plates (Fig. 1). A horizontal force is applied to the top plate, causing the fluid between the plates to shear. A velocity gradient will be developed between the top layer of fluid and the bottom layer of fluid. Newton postulated that the stress applied to the top plate (force/area) is proportional to the velocity gradient (velocity/gap between the plates). The velocity gradient is the shear rate and thus Newton’s postulate is
\eqalign{ {\text{Shear stress}}/{\text{shear rate}} =\, & {\text{constant}} \cr &= {\text{coefficient of viscosity}}.}
Newton defined this property of viscosity in his 1687 work, “Philosophie Principia Mathematica”:

The resistance which arises from the lack of slipperiness of the parts of the liquid, other things being equal, is proportional to the velocity with which the parts of the liquid are separated from one another

A more mechanical approach can also be applied to the system of Fig. 1. Once the force is applied to the top plate, the top plate will move a certain distance over a certain period of time as shown. The stress applied is the force per unit area. The strain on the fluid is the distance moved by the top plate divided by the gap between the two plates (or the film thickness). The rate of strain is the strain divided by time and is defined as the shear rate. Newton’s postulate in this system is expressed as the stress is proportional to the rate of strain (shear rate). The proportionality constant is the coefficient of viscosity. Thus, Newton’s law becomes
\eqalign{ {\text{Viscosity}} & =\, {\text{stress}}/\left( {{\text{strain}}/{\text{time}}} \right){\text{ where }}\left( {{\text{strain}}/{\text{time}}} \right) \cr &= {\text{shear rate}} }
$$\eta = \tau /(\epsilon /t)$$
Newton’s law of viscosity pertains to purely viscous flow of a material. Highly elastic (solid-like) systems, such as mechanical springs, are governed by Hooke’s law:
$${\text{Stiffness}} = {\text{stress}}/{\text{strain}}$$
$$k = \tau /\epsilon$$

The stress is proportional to the strain. These two laws set the boundary conditions for flow behavior of fluids. Fully viscous behavior is governed by Newton and fully elastic (solid-like) behavior by Hooke. Many fluids (e.g., industrial greases) will exhibit characteristics of both viscous and elastic behavior and are commonly referred to as viscoelastic fluids. Viscoelastic fluids are considered non-Newtonian fluids (257).

### Units of Viscosity

The units of viscosity can be obtained from the definition.
• In the SI system:

• Stress = F/A = newton/m2 = Pascal (Pa)

• Strain − distance moved/gap = m/m = dimensionless

• Rate of strain = shear rate = strain/time = dimensionless/s = 1/s (s−1)

• Viscosity = stress/shear rate =Pa/ (1/s) = Pa-s

• In the cgs system:

• Stress = F/A = dyne/cm2

• Strain − distance moved/gap = cm/cm = dimensionless

• Rate of strain = shear rate = strain/time = dimensionless/s = 1/s (s−1)

• Viscosity = stress/shear rate =dyne/cm2/(1/s) = dyne s/cm2 = Poise

The common units used in lubrication are milli Pa-sec (mPa-s) and centipoise (cP).

The unit of kinematic viscosity (discussed below) is the centistokes (cSt) and is related by
$${\text{cSt}} = {\text{cP}}/{\text{density}}$$
Table 1 provides approximate viscosities of some common fluids at ambient conditions.
Newton’s Law of Viscosity, Newtonian and Non-Newtonian Fluids, Table 1

Approximate viscosities of some common fluids at ambient conditions

Fluid

Viscosity, mPa-s

Air

.001

Water

1

Milk

10

Olive oil

100

Engine oil

1,000

Honey

10,000

## Key Applications

### Newtonian Fluids

The behavior of a Newtonian fluid is shown in Fig. 2. A plot of stress versus shear rate yields a straight line with the slope equal to the viscosity. A viscosity versus shear rate plot yields a horizontal line as the viscosity is independent of shear rate. All gases obey Newton’s law and are Newtonian fluids. Mineral oils and blends of mineral oils are also generally Newtonian in flow behavior. The viscosity of a Newtonian fluid, at a given temperature and pressure, can be determined with a single measurement at any shear rate.

In general, most additives used in the lubrication industry would affect the absolute value of the viscosity of the formulation but not the Newtonian behavior (454,348). Common dispersants and detergents used in engine oil formulations, for example, will raise the viscosity of the formulation due to their own higher viscosities. However, the formulation will still obey Newton’s law and follow Newtonian fluid behavior.

### Non-Newtonian Fluids

Many fluids do not obey Newton’s law of viscosity (349). The viscosity will vary with shear rate and a single measurement is not sufficient to characterize the flow properties of the fluid. Other factors that may affect flow properties include pressure and temperature.

In general, for non-Newtonian fluids there is a general characteristic shape to the viscosity versus shear rate curve, as shown in Fig. 3, regardless of the fluid. The difference between fluids will reside in the scale of the two axes and at what shear rates the two Newtonian regions appear. There is a low shear Newtonian region followed by a region where the viscosity decreases with shear rate (shear thinning or power law region). This is followed by a high shear Newtonian region. Lubricating oils are commonly tested for viscosity in the low shear Newtonian regime. The shear thinning region for engine oils does not appear until the shear rate approaches 500,000–1,000,000 s−1. The engine oil formulations will appear Newtonian in behavior unless measured at very high shear rates (328,330).

Polymeric additives are the most common cause of non-Newtonian behavior in fully formulated lubricating oils (131,164). The major factors affecting this flow behavior are the molecular weight and concentration of the polymer (625). Polymer morphology is a minor factor affecting the non-Newtonian behavior. The higher the molecular weight and the higher the concentration, the more pronounced the non-Newtonian behavior of the fluid. These effects relate to new, unused lubrication formulations.

As oils age in service and become used, other factors can cause them to take on more non-Newtonian characteristics even at the lower shear rates. These factors include contamination particles, soot/sludge from incomplete combustion, and oxidation by products and particles. Some of the factors that can determine the degree of non–Newtonian behavior include particle size, shape and distribution, volume fraction of particles, electrostatic charges on particles, and steric effects, among others. The more the particles can interact with one another, the greater the likelihood of non-Newtonian behavior.

### Non-Newtonian Fluid Models

Figure 4 shows various types of non-Newtonian behavior displayed on a shear stress versus shear rate curve. Many empirical models have been developed to describe the non-Newtonian behavior of fluids. Some of the common models are discussed below.

(Note: Figure 4 refers to shear thinning as power law, whereas shear thickening also can be described by power law. The figure can also show typical thixotropic behavior.)

The power law, shear thinning, or pseudoplastic model is
$${\text{Stress }} = {\text{ constant }}^*{ }{\left( {\text{shear rate}} \right)^{\text{n}}}$$
 n = 1 This reduces to Newton’s law and the constant is the viscosity $${\text{Viscosity }} = {\text{ stress}}/{\text{shear rate}}$$ n < 1 The fluid is shear thinning with the viscosity decreasing with shear rate. This is the most common type of behavior for non-Newtonian fluids. Shear thinning is the predominant phenomena associated with polymer solution flow behavior. \eqalign{ {\text{Viscosity }} = {\text{ stress}}/{\hbox{shear rate }} = {\text{ constant}}^*{\left( {\text{shear rate}} \right)^{\text{n}}}/{\text{shear rate}} \cr { } = {\text{constant }}/{\left( {\text{shear rate}} \right)^{{{\text{n}} - {1}}}}} n > 1 The fluid is dilatant or rheopectic, with the viscosity increasing with shear rate \eqalign{ {\text{Viscosity }} = {\text{ stress}}/{\hbox{shear rate }} = {\text{ constant}}^*{\left( {\text{shear rate}} \right)^{\text{n}}}/{\text{shear rate}} \cr = {\text{ constant }}^*{\left( {\text{shear rate}} \right)^{{{\text{n}} - {1}}}}}

This is generally unusual behavior and is not observed often. Highly loaded particle systems (75+%) tend to show an increase in viscosity as the shear rate increases.

Some fluids will also show a yield stress. This yield stress is a minimum stress required to produce flow. A fluid with a yield stress will not flow as long as the applied stress is below the yield stress value. Fluid flow commences once the applied shear stress exceeds the critical yield stress. This flow commonly follows shear thinning behavior. Some fluids, however, possess a yield stress followed by Newtonian flow. These types of fluids are referred to as Bingham fluids.
$${\text{Stress}} = {\text{constant a}} + {\text{viscosity }}^*{\text{ shear rate}}$$
The Herschel–Bulkley model is
$${\text{Stress}} = {\text{constant a}} + {\text{constant }}{\left( {\text{shear rate}} \right)^{\text{n}}}$$

This describes a fluid with a yield stress (constant a) followed by a power law term. If n = 1, then this describes a Bingham fluid. This reduces to a Newtonian fluid for n = 1 and constant a (yield stress) = 0.

The cross model is a four-parameter model and describes the general non–Newtonian viscosity versus shear rate curve of Fig. 3. It includes a zero shear viscosity, infinite shear viscosity and power law, and shear thinning terms.

### Temporary and Permanent Viscosity Shear Loss

The change in viscosity with shear rate for non-Newtonian fluids (shear thinning) is termed temporary shear loss (454). It is temporary in the sense that, as the shear rate increases and decreases, the viscosity is totally recoverable. The decrease in viscosity is temporary at the high shear rates and the viscosity recovers as the shear rate is decreased. Permanent shear loss occurs when physical changes occur to the fluid, causing a permanent change in the viscosity. A polymer-containing fluid will show shear thinning character, as discussed above. If this fluid is mechanically sheared such that the polymer chains are broken, then this effectively reduces the molecular weight of the polymer. The fluid will still be non-Newtonian in behavior but the absolute value of the viscosity will be decreased by a certain amount. This permanent decrease in viscosity across shear rates is called permanent viscosity shear loss or permanent shear loss.

Thixotropic fluids show a decrease in viscosity over time, irrespective of the shear rate.

### Viscosity Measurement

Viscosity can be measured by a number of techniques that yield a single value (165). This is perfectly appropriate and valid for Newtonian fluids. Non-Newtonian fluids need to be measured under varying stress or shear rate to yield shear stress versus shear rate data to obtain the real behavior of the system. Non-Newtonian fluid viscosities must be compared at equivalent shear rates to be meaningful. The glass capillary tube is a common technique used in lubrication to measure viscosity. The stress is applied by gravitational force and thus the shear rate tends to be low for this type of measurement. As discussed above, many polymer-containing lubrication fluids are shear thinning but only at very high shear rates. The shears rates of the capillary instrument are in the range of the low shear Newtonian region and thus the fluids will behave as Newtonian fluids.

The capillary method generates viscosity termed kinematic viscosity. The units of kinematic viscosity are centistokes (cSt) and the conversion to dynamic viscosity is $${\text{cSt}} = {\text{cP}}$$ divided by density.

A variety of high-pressure capillary (271) and rotational instruments (453) are used to measure the viscosity versus shear rate behavior of non-Newtonian fluids. These include cone and plate and concentric cylinder type systems. The rotational systems use a mechanical force and operate in one of two ways: (1) the mechanical force is applied (stress) and resulting speed of rotation is measured (shear rate) or (2) speed of rotation is set and the force required to maintain speed is measured.