Shear Stress in Laminar Flow

I

n the analysis offluidflow thus far, shear stress has been mentioned, but it has not been related to thefluid orflow properties. We shall now investigate this relation for laminar flow. The shear stress acting on afluid depends upon the type offlow that exists. In the so-called laminarflow, thefluidflows in smooth layers or lamina, and the shear stress is the result of the (nonobservable) microscopic action of the molecules. Turbulentflow is characterized by the large-scale, observablefluctuations influid andflow properties, and the shear stress is the result of thesefluctuations. The criteria for laminar and turbulent flows and the shear stress in turbulentflow will be discussed in Chapter 12.

▶ 7.1

NEWTON’S VISCOSITY RELATION

In a solid, the resistance todeformationis the modulus of elasticity. The shear modulus of an elastic solid is given by

shear modulusˆshear stress

shear strain (7-1)

Just as the shear modulus of an elastic solid is a property of the solid relating shear stress and shear strain, there exists a relation similar to (7-1), which relates the shear stress in a parallel, laminarflow to a property of thefluid. This relation is Newton’s law of viscosity:

viscosityˆ shear stress

rate of shear strain (7-2)

Thus, the viscosity is the property of afluid to resist therateat which deformation takes place when thefluid is acted upon by shear forces. As a property of thefluid, the viscosity depends upon the temperature, composition, and pressure of thefluid, but is independent of the rate of shear strain.

The rate of deformation in a simpleflow is illustrated in Figure 7.1. Theflow parallel to thexaxis will deform the element if the velocity at the top of the element is different than the velocity at the bottom.

The rate of shear strain at a point is defined as dd=dt. From Figure 7.1, it may be seen that dd

dt ˆ lim

Dx;Dy;Dt®0

djt‡Dt djt

Dt

ˆ lim

Dx;Dy;Dt®0

fp=2 arctan‰…vjy‡Dy vjy†Dt=DyŠg p=2 Dt

(7-3)

In the limit, dd=dtˆdv=dyˆrate of shear strain.

85

Combining equations (7-2) and (7-3) and denoting the viscosity bym, we may write Newton’s law of viscosity as

tˆmdv

dy (7-4)

The velocity profile and shear stress variation in afluidflowing between two parallel plates is illustrated in Figure 7.2. The velocity profile¹in this case is parabolic; as the shear stress is proportional to the derivative of the velocity, the shear stress varies in a linear manner.

▶ 7.2

NON-NEWTONIAN FLUIDS 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 forflow 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 arefluids 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 thefluid 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 thefluid 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 toflow and can be described by the following equation:

tˆmdv

dyt0 (7-5)

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

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,

tˆmdv 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 ofn:

nˆ1 fluid is Newtonian andmˆm

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

It can be seen that when nˆ1 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 offluid 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. Inflow situations in which the viscous effects are neglected—the so-called inviscidflows—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 molecularflux. 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 theycoordinate of the top of the control surface as y0, we shall write the x-directional average velocity of the upward molecularflux asvxj_y , where the minus sign signifies that the average velocity is evaluated at some point belowy₀. Thex-directional momentum carried across the top of the control surface is thenmvxjy per molecule, wheremis the mass of the molecule. IfZmolecules cross the plane per unit time, then the netx-directional momentumflux will be

∑^Z

nˆ1

mn…vxjy vxjy‡† (7-7)

y

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

Theflux ofx-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

∑Fˆ Z Z

c:s:rv…v_?n†dA‡@

@t Z Z Z

c:v:rvdV (5-4)

Thefirst term on the right-hand side of equation (5-4) is the momentumflux. 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 momentumflux 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 ast, we have

tˆ ∑^Z

nˆ1

mn…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 ˆvxjy₀ …dvx=dyjy₀†d;where y ˆy₀ d. Using a similar expression fory‡, we obtain, for the shear stress,

tˆ2 ∑^Z

nˆ1

mn

dvx

dy

y₀

dn

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 thatdˆ2/

3l, we obtain, for a pure gas,

tˆ4

3mlZdvx

dy

y₀

(7-9) as the shear stress.

Comparing equation (7-9) with Newton’s law of viscosity, we see that mˆ4

3mlZ (7-10)

The kinetic theory givesZˆNC=4, where

Nˆmolecules per unit volume Cˆaverage random molecular velocity and thus

mˆ1

3NmlCˆrlC 3 or, using²

lˆ 1 ffiffiffi2

p pNd² and Cˆ ffiffiffiffiffiffiffiffi 8kT pm r

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

whered is the molecular diameter andkis the Boltzmann constant, we have mˆ 2

3p³⁼² ffiffiffiffiffiffiffiffiffiffi mkT p

d² (7-11)

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

mˆ2:669310 ⁶ ffiffiffiffiffiffiffiffi pMT s²Wm

(7-12) wheremis the viscosity, in pascal-seconds;Tis absolute temperature, in K;Mis the molecular weight;sis the“collision diameter,”a Lennard–Jones parameter, in Å (Angstroms);W_mis the

“collision integral,”a Lennard–Jones parameter that varies in a relatively slow manner with the dimensionless temperaturekT/e;kis the Boltzmann constant, 1.38_?10 ¹⁶ergs/K ; andeis the characteristic energy of interaction between molecules. Values ofsandefor various gases are given in Appendix K, and a table ofWmversuskT/eis 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

∑x_jfij

(7-13) wherex_i,x_jare mole-fractions of speciesiand jin the mixture, and

fijˆ 1 ffiffiffi8

p 1‡Mi

Mj

1=2

1‡ mi

mj

!1=2

Mj

Mi 1=4

2 4

3 5

2

(7-14) whereM_i, M_jare the molecular weights of speciesiandj,andmi,mjare the viscosities of speciesiandj.Note that wheniˆj, we havefijˆ1.

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 anN-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, mˆ t

dv=dy or, in dimensional form,

F=L²

…L=t†…1=L†ˆFt L² where Fˆforce,Lˆlength,tˆtime.

Using Newton’s second law of motion to relate force and mass (FˆML/t²), wefind that the dimensions of viscosity in the mass–length–time system becomeM/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 symboln. The origin of the name kinematic viscosity may be seen from the dimensions ofn:

nm 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/m²ˆ10 poiseˆ0.02089 slugs/ft _? sˆ0.02089 lbf ? s/ft²ˆ0.6720 lbm/ft _? s).

Kinematic viscosity in the metric system is expressed in (meters)²per second (1 m²/sˆ10⁴ stokesˆ10.76 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)ˆ0.001 kilogram/meter second.

1 centipoise (cP)ˆ0.001 Pascal second.

▶ 7.4

SHEAR STRESS IN MULTIDIMENSIONAL LAMINAR FLOWS OF A NEWTONIAN FLUID

Newton’s viscosity relation, discussed previously, is valid for only parallel, laminarflows.

Stokes extended the concept of viscosity to three-dimensional laminarflow. The basis of Stokes’viscosity relation is equation (7-2),

viscosityˆ shear stress

rate of shear strain (7-2)

where the shear stress and rate of shear strain are those of a three-dimensional element.

Accordingly, we must examine shear stress and strain rate for a three-dimensional body.

Shear Stress

The shear stress is a tensor quantity requiring magnitude, direction, and orientation with respect to a plane for identification. The usual method of identification of the shear stress involves a double subscript, such astxy. The tensor component,tij, is identified as follows:

t ˆ magnitude

first subscript ˆ direction of axis to which plane of action of shear stress is normal second subscriptˆ direction of action of the shear stress:

Thustxyacts on a plane normal to thexaxis (theyzplane) and acts in theydirection. In addition to the double subscript, a sense is required. The shear stresses acting on an element DxDyDz, illustrated in Figure 7.6, are indicated in the positive sense. The definition of positive shear stress can be generalized for use in other coordinate systems. A shear stress component is positive when both the vector normal to the surface of action and the shear stress act in the same direction (both positive or both negative).

For example, in Figure 7.6(a), the shear stress tyxat the top of the element acts on surfaceDxDz. The vector normal to this area is in the positiveydirection. The stresstyxacts in the positive xdirection—hence, tyx, as illustrated in Figure 7.6(a), is positive. The student may apply similar reasoning to tyx acting on the bottom of the element and conclude that tyx is also positive as illustrated.

As in the mechanics of solids, tijˆtji(see Appendix C).

Rate of Shear Strain

The rate of shear strain for a three-dimensional element may be evaluated by determining the shear strain rate in thexy, yz,andxzplanes. In thexyplane illustrated in Figure 7.7, the shear strain rate is again dd/dt; however, the element may deform in both thexand theydirections.

Hence, as the element moves from position 1 to position 2 in time Dt, dd

dt ˆ lim

Dx;Dy;Dt®0

djt‡Dt djt

Dt

ˆ lim

Dx;Dy;Dt®0

p=2 arctanf‰…vxjy‡Dy vxjy†DtŠ=Dyg Dt

arctanf‰…vyjx‡Dx vyjx†DtŠ=Dxg p=2 Dt

7.4 Shear Stress in Multidimensional Laminar Flows of a Newtonian Fluid ◀ 93

As the shear strain evaluated above is in thexyplane, it will be subscriptedxy.In the limit, ddxy/dtˆ@ vx/@y‡@vy/@x. In a similar manner, the shear strain rates in the yzand xz planes may be evaluated as

dd^yz dt ˆ@vy

@z ‡@vz

@y ddxz

dt ˆ@vy

@z ‡@vz

@y

y

x (a)

z

y x

z

Figure 7.6 Shear stress acting in a positive sense.

x y

1 2

t

Figure 7.7 Shear strain in thexyplane.

Stokes’s Viscosity Relation

(A)Shear Stress Stokes’s viscosity relation for the shear-stress components in laminar flow may now be stated with the aid of the preceding developments for rate of shear strain.

Using equation (7-2), we have, for the shear stresses written in rectangular coordinate form, t^xyˆt^yxˆm @vx

@y ‡@vy

@x

(7-15a)

tyzˆtzyˆm @vy

@z ‡@vz

@y

(7-15b) and

tzxˆtxzˆm @vz

@x ‡@vx

@z

(7-15c) (B)Normal Stress The normal stress may also be determined from a stress rate-of-strain relation; the strain rate, however, is more difficult to express than in the case of shear strain.

For this reason the development of normal stress, on the basis of a generalized Hooke’s law for an elastic medium, is included in detail in Appendix D, with only the result expressed below in equations (7-16a), (7-16b), and (7-16c).

The normal stress in rectangular coordinates written for a newtonianfluid is given by s^xxˆm 2@vx

@x 2 3r?v

P (7-16a)

syyˆm 2@vy

@y 2 3r?v

P (7-16b)

and

s^zzˆm 2@vz

@z 2 3r?v

P (7-16c)

It is to be noted that the sum of these three equations yields the previously mentioned result:

the bulk stress, sˆ …sxx‡syy‡szz†=3;is the negative of the pressure,P.

Example 1

A Newtonian oil undergoes steady shear between two horizontal parallel plates. The lower plate isfixed, and the upper plate, weighing 0.5 lbf, moves with a constant velocity of 15 ft/s. The distance between the plates is constant at 0.03 inches, and the area of the upper plate in contact with thefluid is 0.95 ft². What is the viscosity of thisfluid?

We begin with Newton’s law of viscosity and also realize that shear stress is force divided by area:

F

Aˆtyxˆmdvx

dy FdyˆmAdy 0.03 inches

Fixed bottom plate

Moving with v = 15 m/s Figure 7.8 Flow between two parallel plates where the top plate is moving and the bottom plate is stationary.

7.4 Shear Stress in Multidimensional Laminar Flows of a Newtonian Fluid ◀ 95