Viscosity_Measurement.pdf

G.E. Leblanc, et. al.. "Viscosity Measurement."

Viscosity Measurement

G.E. Leblanc

McMaster University

R.A. Secco

30.1

Shear Viscosity

Newtonian and Non-Newtonian Fluids • Dimensions and

Units of Viscosity • Viscometer Types • Capillary

Viscometers • Falling Body Methods • Oscillating Method •

Ultrasonic Methods

The University of Western Ontario

M. Kostic

Northern Illinois University

30.1Shear Viscosity

. Physical systems and applications as diverse as

ﬂuid ﬂow in pipes, the ﬂow of blood, lubrication of engine parts, the dynamics of raindrops, volcanic

eruptions, planetary and stellar magnetic ﬁeld generation, to name just a few, all involve ﬂuid ﬂow and

are controlled to some degree by ﬂuid viscosity.

An important mechanical property of ﬂuids is

viscosity

is deﬁned as the internal friction of a ﬂuid. The

microscopic nature of internal friction in a ﬂuid is analogous to the macroscopic concept of mechanical

friction in the system of an object moving on a stationary planar surface. Energy must be supplied (1) to

overcome the inertial state of the interlocked object and plane caused by surface roughness, and (2) to

initiate and sustain motion of the object over the plane. In a ﬂuid, energy must be supplied (1) to create

viscous ﬂow units by breaking bonds between atoms and molecules, and (2) to cause the ﬂow units to

move relative to one another. The resistance of a ﬂuid to the creation and motion of ﬂow units is due

to the viscosity of the ﬂuid, which only manifests itself when motion in the ﬂuid is set up. Since viscosity

involves the transport of mass with a certain velocity, the viscous response is called a

Viscosity

momentum transport

process

. The velocity of ﬂow units within the ﬂuid will vary, depending on location. Consider a liquid

between two closely spaced parallel plates as shown in Figure 30.1 . A force,

, applied to the top plate

causes the ﬂuid adjacent to the upper plate to be dragged in the direction of

. The applied force is

communicated to neighboring layers of ﬂuid below, each coupled to the driving layer above, but with

diminishing magnitude. This results in the progressive decrease in velocity of each ﬂuid layer, as shown

by the decreasing velocity vector in Figure 30.1, away from the upper plate. In this system, the applied

force is called a

shear

(when applied over an area it is called a

shear stress

), and the resulting deformation

. The

mathematical expression describing the viscous response of the system to the shear stress is simply:

rate of the ﬂuid, as illustrated by the

velocity gradient

, is called the

shear strain rate

(30.1)

where

, the shear stress, is the force per unit area exerted on the upper plate in the

-direction (and

hence is equal to the force per unit area exerted by the ﬂuid on the upper plate in the

-direction under

the assumption of a no-slip boundary layer at the ﬂuid–upper plate interface);

is the gradient of

the

-velocity in the

-direction in the ﬂuid; and

is the

coefﬁcient of viscosity

. In this case, because one

is concerned with a shear force that produces the ﬂuid motion,

is more speciﬁcally called the

shear

System for deﬁning Newtonian viscosity. When the upper plate is subjected to a force, the ﬂuid

between the plates is dragged in the direction of the force with a velocity of each layer that diminishes away from

the upper plate. The reducing velocity eventually reaches zero at the lower plate boundary.

FIGURE 30.1

. In ﬂuid mechanics, diffusion of momentum is a more useful description of viscosity

where the motion of a ﬂuid is considered without reference to force. This type of viscosity is called the

dynamic viscosity

kinematic viscosity

and is derived by dividing dynamic viscosity by

, the mass density:

(30.2)

(i.e., layered or sheet-like) or

streamline ﬂow as depicted in Figure 30.1, and it refers to the molecular viscosity or

The deﬁnition of viscosity by Equation 30.1 is valid only for

laminar

The molecular viscosity is a property of the material that depends microscopically on bond strengths,

and is characterized macroscopically as the ﬂuid’s resistance to ﬂow. When the ﬂow is turbulent, the

diffusion of momentum is comprised of viscous contributions from the motion, sometimes called the

intrinsic viscosity

eddy viscosity

, in addition to the intrinsic viscosity. Viscosities of turbulent systems can be as high as 10

times greater than viscosities of laminar systems, depending on the Reynolds number.

, depending on the

type of strain involved. Shear viscosity is a measure of resistance to isochoric ﬂow in a shear ﬁeld, whereas

volume viscosity is a measure of resistance to volumetric ﬂow in a three-dimensional stress ﬁeld. For

most liquids, including hydrogen bonded, weakly associated or unassociated, and polymeric liquids as

well as liquid metals,

Molecular viscosity

is separated into

shear viscosity

and bulk or

volume viscosity

1, suggesting that shear and structural viscous mechanisms are closely related

[1].

The shear viscosity of most liquids decreases with temperature and increases with pressure, which is

opposite to the corresponding responses for gases. An increase in temperature usually causes expansion

and a corresponding reduction in liquid bond strength, which in turn reduces the internal friction.

Pressure causes a decrease in volume and a corresponding increase in bond strength, which in turn

enhances the internal friction. For most situations, including engineering applications, temperature

effects dominate the antagonistic effects of pressure. However, in the context of planetary interiors where

the effects of pressure cannot be ignored, pressure controls the viscosity to the extent that, depending

on composition, it can cause fundamental changes in the molecular structure of the ﬂuid that can result

in an anomalous viscosity decrease with increasing pressure [2].

Newtonian and Non-Newtonian Fluids

Equation 30.1 is known as Newton’s law of viscosity and it formulates Sir Isaac Newton’s deﬁnition of

the viscous behavior of a class of ﬂuids now called Newtonian ﬂuids.

FIGURE 30.2

Flow curves illustrating Newtonian and non-Newtonian ﬂuid behavior.

If the viscosity throughout the ﬂuid is independent of strain rate, then the ﬂuid is said to be a

Newtonian

ﬂuid

. The constant of proportionality is called the coefﬁcient of viscosity, and a plot of stress vs. strain

rate for Newtonian ﬂuids yields a straight line with a slope of

, as shown by the solid line ﬂow curve

in Figure 30.2 . Examples of Newtonian ﬂuids are pure, single-phase, unassociated gases, liquids, and

solutions of low molecular weight such as water. There is, however, a large group of ﬂuids for which the

viscosity is dependent on the strain rate. Such ﬂuids are said to be non-Newtonian ﬂuids and their study

is called

. In differentiating between Newtonian and non-Newtonian behavior, it is helpful to

consider the time scale (as well as the normal stress differences and phase shift in dynamic testing)

involved in the process of a liquid responding to a shear perturbation. The velocity gradient,

rheology

, in

the ﬂuid is equal to the shear strain rate,

, and therefore the time scale related to the applied shear

perturbation about the equilibrium state is

, where

–1

. A second time scale,

called the

relaxation

time

characterizes the rate at which the relaxation of the strain in the ﬂuid can be accomplished and is

related to the time it takes for a typical ﬂow unit to move a distance equivalent to its mean diameter.

For Newtonian water,

are rare in practice, the

time required for adjustment of the shear perturbation in water is much less than the shear perturbation

period (i.e.,

~ 10

–12

s and, because shear rates greater than 10

–1

). However, for non-Newtonian macromolecular liquids like polymeric liquids, for

colloidal and ﬁber suspensions, and for pastes and emulsions, the long response times of large viscous

ﬂow units can easily make

. An example of a non-Newtonian ﬂuid is liquid elemental sulfur, in

which long chains (polymers) of up to 100,000 sulfur atoms form ﬂow units that are easily entangled,

which bind the liquid in a “rigid-like” network. Another example of a well-known non-Newtonian ﬂuid

is tomato ketchup.

With reference to Figure 30.2, the more general form of Equation 30.1 also accounts for the nonlinear

response. In terms of an initial shear stress required for ﬂow to start,

(0), an initial linear term in the

•

Newtonian limit of a small range of strain rate,

g¶t

(0)/

, and a nonlinear term

(

), the shear stress

•

dependence on strain rate,

(

) can be described as:

()

() =

() +

tgt

(30.3)

•

For a Newtonian ﬂuid, the initial stress at zero shear rate is zero and the nonlinear function

(

) is zero,

•

so Equation 30.3 reduces to Equation 30.1, since

(0)/

then equals

. For a non-Newtonian ﬂuid,

•

) is nonzero. This characterizes fluids in which shear stress

increases disproportionately with strain rate, as shown in the dashed-dotted ﬂow curve in Figure 30.2,

or decreases disproportionately with strain rate, as shown in the dashed ﬂow curve in Figure 30.2. The

former type of ﬂuid behavior is known as

(0) may be zero but the nonlinear term

(

or dilatancy, and an example is a concentrated

solution of sugar in water. The latter, much more common type of ﬂuid behavior, is known as

shear thickening

shear

thinning

or pseudo-plasticity; cream, blood, most polymers, and liquid cement are all examples. Both

behaviors result from particle or molecular reorientations in the ﬂuid that increase or decreases, respec-

tively, the internal friction to shear. Non-Newtonian behavior can also arise in ﬂuids whose viscosity

changes with time of applied shear stress. The viscosity of corn starch and water increases with time

duration of stress, and this is called

. Conversely, liquids whose viscosity decreases with

time, like nondrip paints, which behave like solids until the stress applied by the paint brush for a

sufﬁciently long time causes them to ﬂow freely, are called

rheopectic behavior

Fluid deformation that is not recoverable after removal of the stress is typical of the purely viscous

response. The other extreme response to an external stress is purely elastic and is characterized by an

equilibrium deformation that is fully recovered on removal of the stress. There are an inﬁnite number

of intermediate or combined viscous/elastic responses to external stress, which are grouped under the

behavior known as

thixotropic ﬂuids

. Fluids that behave elastically in some stress range require a limiting or

yield stress before they will ﬂow as a viscous ﬂuid. A simple, empirical, constitutive equation often used

for this type of rheological behavior is of the form:

viscoelasticity

ttgh

(30.4)

where

is the

yield stress

is an

apparent viscosity

called the plastic viscosity, and the exponent

allows

for a range of non-Newtonian responses:

= 1 is pseudo-Newtonian behavior and is called a

Bingham

ﬂuid

> 1 is shear thickening behavior. Interested readers should

consult [3–9] for further information on applied rheology.

;

< 1 is shear thinning behavior; and

Dimensions and Units of Viscosity

From Equation 30.1, the dimensions of dynamic viscosity are M L

–1

and the basic SI unit is the Pascal

second (Pa·s), where 1 Pa·s = 1 N s m

–2

. The c.g.s. unit of dyn s cm

–2

is the poise

(P). The dimensions

of kinematic viscosity, from Equation 30.2, are L

–1

and the SI unit is m

–1

. For most practical situations,

this is usually too large and so the c.g.s. unit of cm

s –1 , or the stoke (St), is preferred. Table 30.1 li sts

some common ﬂuids and their shear dynamic viscosities at atmospheric pressure and 20

TABLE 30.1

Shear Dynamic Viscosity of Some Common Fluids

at 20

C and 1 atm

Fluid

Shear dynamic viscosity (Pa·s)

Air

1.8

10 –4

Water

1.0

10 –3

Mercury

1.6

10 –3

Automotive engine oil (SAE 10W30)

1.3

10 –1

Dish soap

4.0

10 –1

Corn syrup

6.0

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