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Newton’s law of viscosity does not predict the shear stress in all fluids.

Fluids are classified as Newtonian or non-Newtonian, depending upon the relation between shear stress and the rate of shearing strain. In Newtonian fluids, the relation is linear, as shown in Figure 7.3.

In non-Newtonianfluids, the shear stress depends upon the rate of shear strain. While fluids deform continu-ously under the action of shear stress, plastics will sustain a shear stress

h

h

Figure 7.2 Velocity and shear stress profiles for flow between two parallel plates.

y

Element at

time t Element at

time t + t Figure 7.1 Deformation of afluid element.

1The derivation of velocity profiles is discussed in Chapter 8.

Ideal plastic Real plastic

Pseudo plastic Newtonian fluid

Dilatant Yield

stress

Rate of strain Figure 7.3 Stress rate-of-strain relation for Newtonian and non-Newtonianfluids.

before deformation occurs. The“ideal plastic” has a linear stress rate-of-strain relation for stresses greater than the yield stress. Thixotropic substances such as printer’s ink have a resistance to deformation that depends upon deformation rate and time.

In a Newtonianfluid there is a linear relationship between shear stress and shear strain.

Non-Newtonianfluids are fluids that do not obey Newton’s law of viscosity, and as a result shear stress depends on the rate of shear strain in a nonlinear fashion. Examples of non-Newtonianfluids are toothpaste, honey, paint, ketchup, and blood.

As shown in Figure 7.3, there are several types of non-Newtonianfluids. Pseudoplastics are materials that decrease in viscosity with increased shear strain and are called“shear thinning” fluids, since the more the fluid is sheared, the less viscous it becomes. Common examples of shear thinning materials are hair gel, plasma, syrup, and latex paint.

Dilatants are non-Newtonianfluids that exhibit an increase in viscosity with shear stress and are called“shear-thickening” fluids, since the more the fluid is sheared, the more viscous it becomes. Common examples of shear-thickeningfluids include silly putty, quicksand, and the mixture of cornstarch and water.

A third common type of non-Newtonianfluids are designated viscoelastic fluids that return, either partially or fully, to their original shape after the applied shear stress is released.

There are numerous models used to describe and characterize these non-Newtonianfluids.

A Bingham plastic is a material, like toothpaste, mayonnaise, and ketchup, that requires a finite yield stress before it begins to flow and can be described by the following equation:

t ˆ mdv

dy t0 (7-5)

where thet0is the yield stress. Whent < t0the material is a rigid plastic, and whent > t0, thefluid behaves more like a Newtonian fluid.

The Ostwald-De Waele model or power law model is another commonly used model to describe non-Newtonian fluids where the so-called apparent viscosity is a function of the shear rate raised to a power,

t ˆ mdv dy

 

^{n 1}dv

dy (7-6)

where m and n are constants that are characteristic of the fluid. Power law fluids are classified based on the value of n:

nˆ 1 fluid is Newtonian and m ˆ m

n< 1 fluid is pseudoplastic or shear thinning n> 1 fluid is a dilatants or shear thickening

It can be seen that when nˆ 1 that the power law model reduces to Newton’s law of viscosity (7-4).

The No-Slip Condition

Although the substances above differ in their stress rate-of-strain relations, they are similar in their action at a boundary. In both Newtonian and non-Newtonianfluids, the layer of fluid adjacent to the boundary has zero velocity relative to the boundary. When the boundary is a stationary wall, the layer offluid next to the wall is at rest. If the boundary or wall is moving, the layer offluid moves at the velocity of the boundary, hence the name no-slip (boundary) condition. The no-slip condition is the result of experimental observation and fails when the fluid no longer can be treated as a continuum.

7.2 Non-Newtonian Fluids ◀ 87

The no-slip condition is a result of the viscous nature of thefluid. In flow situations in which the viscous effects are neglected—the so-called inviscid flows—only the component of the velocity normal to the boundary is zero.

▶ 7.3

VISCOSITY

The viscosity of afluid is a measure of its resistance to deformation rate. Tar and molasses are examples of highly viscous fluids; air and water, which are the subject of frequent engineering interest, are examples of fluids with relatively low viscosities. An under-standing of the existence of the viscosity requires an examination of the motion offluid on a molecular basis.

The molecular motion of gases can be described more simply than that of liquids. The mechanism by which a gas resists deformation may be illustrated by examination of the motion of the molecules on a microscopic basis. Consider the control volume shown in Figure 7.4.

The top of the control volume is enlarged to show that even though the top of the element is a streamline of the flow, individual molecules cross this plane. The paths of the molecules between collisions are represented by the random arrows. Because the top of the control volume is a streamline, the net molecularflux across this surface must be zero;

hence, the upward molecularflux must equal the downward molecular flux. The molecules that cross the control surface in an upward direction have average velocities in the x direction corresponding to their points of origin. Denoting the y coordinate of the top of the control surface as y0, we shall write the x-directional average velocity of the upward molecularflux as vxj_y , where the minus sign signifies that the average velocity is evaluated at some point below y₀. The x-directional momentum carried across the top of the control surface is then mvxjy per molecule, where m is the mass of the molecule. If Z molecules cross the plane per unit time, then the net x-directional momentumflux will be

∑^Z

nˆ1mn…vxjy vxjy‡† (7-7)

y

x Figure 7.4 Molecular motion at the surface of a control volume.

Theflux of x-directional momentum on a molecular scale appears as a shear stress when thefluid is observed on a macroscopic scale. The relation between the molecular momentum flux and the shear stress may be seen from the control-volume expression for linear momentum

Thefirst term on the right-hand side of equation (5-4) is the momentum flux. When a control volume is analyzed on a molecular basis, this term includes both the macroscopic and molecular momentumfluxes. If the molecular portion of the total momentum flux is to be treated as a force, it must be placed on the left-hand side of equation (5-4). Thus, the molecular momentumflux term changes sign. Denoting the negative of the molecular momentumflux as t, we have

t ˆ ∑^Z

nˆ1mn…vxjy vxjy‡† (7-8) We shall treat shear stress exclusively as a force per unit area.

The bracketed term, …vxjy vxjy‡† in equation (7-8), may be evaluated by noting that vxjy ˆ vxjy₀ …dvx=dyjy₀†d; where y ˆ y0 d. Using a similar expression for y ‡, we obtain, for the shear stress,

t ˆ 2 ∑

In the above expression d is the y component of the distance between molecular collisions. Borrowing from the kinetic theory of gases, the concept of the mean free path,l, as the average distance between collisions, and also noting from the same source thatd ˆ 2/

3l, we obtain, for a pure gas, as the shear stress.

Comparing equation (7-9) with Newton’s law of viscosity, we see that m ˆ4

3mlZ (7-10)

The kinetic theory gives Zˆ NC=4, where

Nˆ molecules per unit volume Cˆ average random molecular velocity and thus

2In order of increasing complexity, the expressions for mean free path are presented in R. Resnick and D. Halliday, Physics, Part I, Wiley New York, 1966, Chapter 24, and E. H. Kennard, Kinetic Theory of Gases, McGraw-Hill Book Company, New York, 1938, Chapter 2.

7.3 Viscosity ◀ 89

where d is the molecular diameter andk is the Boltzmann constant, we have

Equation (7-11) indicates thatm is independent of pressure for a gas. This has been shown, experimentally, to be essentially true for pressures up to approximately 10 atmo-spheres. Experimental evidence indicates that at low temperatures the viscosity varies more rapidly than ffiffiffiffiffi

T:

p The constant-diameter rigid-sphere model for the gas molecule is respon-sible for the less-than-adequate viscosity-temperature relation. Even though the preceding development was somewhat crude in that an indefinite property, the molecular diameter, was introduced, the interpretation of the viscosity of a gas being due to the microscopic momentumflux is a valuable result and should not be overlooked. It is also important to note that equation (7-11) expresses the viscosity entirely in terms offluid properties.

A more realistic molecular model utilizing a force field rather than the rigid-sphere approach will yield a viscosity-temperature relationship much more consistent with experimental data than the ffiffiffiffi

pT

result. The most acceptable expression for nonpolar molecules is based upon the Lennard–Jones potential energy function. This function and the development leading to the viscosity expression will not be included here. The interested reader may refer to Hirschfelder, Curtiss, and Bird³ for the details of this approach. The expression for viscosity of a pure gas that results is

m ˆ 2:6693  10 ⁶ ffiffiffiffiffiffiffiffi pMT s²Wm

(7-12) wherem is the viscosity, in pascal-seconds; T is absolute temperature, in K; M is the molecular weight;s is the “collision diameter,” a Lennard–Jones parameter, in Å (Angstroms); Wmis the

“collision integral,” a Lennard–Jones parameter that varies in a relatively slow manner with the dimensionless temperaturekT/e ; k is the Boltzmann constant, 1.38?10 ¹⁶ergs/K ; ande is the characteristic energy of interaction between molecules. Values ofs and e for various gases are given in Appendix K, and a table ofWmversus kT/e is also included in Appendix K.

For multicomponent gas mixtures at low density, Wilke⁴has proposed this empirical formula for the viscosity of the mixture:

mmixtureˆ ∑ⁿ

iˆ1

ximi

∑xjfij

(7-13) where x_i, x_jare mole-fractions of species i and j in the mixture, and

fijˆ 1ffiffiffi

Equations (7-12), (7-13), and (7-14) are for nonpolar gases and gas mixtures at low density. For polar molecules, the preceding relation must be modified.⁵

3J. O. Hirschfelder, C. F. Curtiss, and R. B. Bird, Molecular Theory of Gases and Liquids, Wiley, New York, 1954.

4C. R. Wilke, J. Chem. Phys., 18, 517–519 (1950).

5J. O. Hirschfelder, C. F. Curtiss, and R. B. Bird, Molecular Theory of Gases and Liquids, Wiley, New York, 1954.

Although the kinetic theory of gases is well developed, and the more sophisticated models of molecular interaction accurately predict viscosity in a gas, the molecular theory of liquids is much less advanced. Hence, the major source of knowledge concerning the viscosity of liquids is experiment. The difficulties in the analytical treatment of a liquid are largely inherent in nature of the liquid itself. Whereas in gases the distance between molecules is so great that we consider gas molecules as interacting or colliding in pairs, the close spacing of molecules in a liquid results in the interaction of several molecules simultaneously. This situation is somewhat akin to an N-body gravitational problem. In spite of these difficulties, an approximate theory has been developed by Eyring, which illustrates the relation of the intermolecular forces to viscosity.⁶The viscosity of a liquid can be considered due to the restraint caused by intermolecular forces. As a liquid heats up, the molecules become more mobile. This results in less restraint from intermolecular forces.

Experimental evidence for the viscosity of liquids shows that the viscosity decreases with temperature in agreement with the concept of intermolecular adhesive forces being the controlling factor.

Units of Viscosity

The dimensions of viscosity may be obtained from Newton’s viscosity relation, m ˆ t

dv=dy or, in dimensional form,

F=L²

…L=t†…1=L†ˆFt L² where F ˆ force, L ˆ length, t ˆ time.

Using Newton’s second law of motion to relate force and mass (F ˆ ML/t²), wefind that the dimensions of viscosity in the mass–length–time system become M/Lt.

The ratio of the viscosity to the density occurs frequently in engineering problems. This ratio,m/r, is given the name kinematic viscosity and is denoted by the symbol n. The origin of the name kinematic viscosity may be seen from the dimensions ofn:

n m

r∼M=Lt M=L³ˆL²

t

The dimensions of v are those of kinematics: length and time. Either of the two names, absolute viscosity or dynamic viscosity, is frequently employed to distinguish m from the kinematic viscosity, n.

In the SI system, dynamic viscosity is expressed in pascal-seconds (1 pascal-secondˆ 1 N n _? s/m²ˆ 10 poise ˆ 0.02089 slugs/ft ? sˆ 0.02089 lbf ? s/ft²ˆ 0.6720 lbm/ft _? s).

Kinematic viscosity in the metric system is expressed in (meters)²per second (1 m²/sˆ 10⁴ stokesˆ 10.76 ft²/s).

Absolute and kinematic viscosities are shown in Figure 7.5 for three common gases and two liquids as functions of temperature. A more extensive listing is contained in Appendix I.

Table 7.1 gives viscosities of commonfluids.

6For a description of Eyring’s theory, see R. B. Bird, W. E. Stewart, and E. N. Lightfoot, Transport Phenomena, Wiley, New York, 2007, Chapter 1.

7.3 Viscosity ◀ 91

Viscosity,

Temperature, K

0 250 300 350 400 450

1 2 3 4 56 8 10 20 30 40 5060 80 100 200 300 400 500600 800 1000

CO₂ H₂ Water

Air Kerosene

105, Pa·s

Figure 7.5 Viscosity–temperature variation for some liquids and gases.

Table 7.1 Viscosities of commonfluids (at 20°C unless otherwise noted)

Fluid Viscosity (cP) at 20°C

Ethanol 1.194

Mercury 15.47

H2SO4 19.15

Water 1.0019

Air 0.018

CO₂ 0.015

Blood 2.5 (at 37°C)

SAE 40 motor oil 290

Corn oil 72

Ketchup 50,000

Peanut butter 250,000

Honey 10,000

1 centipoise (cP)ˆ 0.001 kilogram/meter second.

1 centipoise (cP)ˆ 0.001 Pascal second.

▶ 7.4

SHEAR STRESS IN MULTIDIMENSIONAL LAMINAR FLOWS