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E.A. Brujan Fundamentals of Fluid Mechanics

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1. PROPERTIES OF FLUIDS1.1. Density, specific weight, specific volume, and specific gravity

The density of a fluid is its mass per unit volume, while the specific weight is its weightper unit volume. In SI units, density will be in kilograms per cubic meter, while specific gravity will

have the units of force per unit volume or newton per cubic meter.

Density and specific weight of a fluid are related as follows:

gg

== or . (1.1)

It should be noted that density is absolute since it depends on mass which is independent of

location. Specific weight, on the other hand, is not absolute for it depends on the value of

gravitational accelerationgwhich varies with location, primarily latitude and elevation above mean

sea level. The density of water at 20oC is 1000 kg/m3, while the density of air at the same

temperature and at a pressure of 1 bar is 1.29 kg/m3.

Specific volume v is the volume occupied by a unit mass of a fluid. It is commonly applied togases and is usually expressed in cubic meter per kilogram. Specific volume is the reciprocal of

density.

Specific gravity s of a liquid is the ratio of its density to that of pure water at a standard

temperature. In the metric system the density of water at 4oC is 1 g/cm3 and hence the specific

gravity (which is dimensionless) has the same numerical value for a liquid in that system as its

density expressed in g/cm3.

1.2. Compressibility of liquidsCompressibility is the property of a fluid to change its volume with pressure. The

compressibility of a liquid is inversely proportional to its volume modulus of elasticity, also known

as the bulk modulus. This modulus is defined as

( )dpdvvdvdpvEv == , (1.2)

where v is the specific volume andp the unit pressure. As v/dv is a dimensionless ratio, the units of

Ev and p are the same. The bulk modulus is analogous to the modulus of elasticity for solids;

however, for fluids it is defined on a volume basis rather than in terms of the familiar one-

dimensional stress-strain relation for solid bodies.

The volume modulus of mild steel is about 170,000 MN/m2. Taking a typical value for the

volume modulus of cold water to be 2,200 MN/m2 it is seen that water is about 80 times as

compressible as steel. The compressibility of liquids covers a wide range. Mercury, for example, is

approximately 8% as compressible as water, while the compressibility of nitric acid is nearly six

times greater than that of water.

The quantity reciprocal to bulk modulus is called volume compressibility defined as:

pv

v

=0

, (1.3)

where v0 is the initial volume, v is the change in volume, and p is the change in pressure.

1.3. Surface tensionThe mechanical model of a liquid surface is that of a skin under tension. Any given patch of

the surface thus experiences an outward force tangential to the surface on the perimeter. The force

per unit length of an interfacial perimeter acting perpendicular to the perimeter is defined as the

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surface tension, . The interface between two immiscible liquids, such as oil and water, also has atension associated with it, which is generally referred to as the interfacial tension.

The interface between a solid and a fluid also has a surface tension associated with it. Figure

shows a liquid droplet at rest on a solid surface surrounded by air. The region of contact between

the gas, liquid, and solid is termed contact line. The liquid-gas surface meets the solid surface with

an angle measured through the liquid, which is known as the contact angle. The system shown in

Figure 1.1(a) has a smaller contact angle that shown in Figure 1.1(b). The smaller the contact angle,the better the liquid is said to wet the solid surface. For = 0, the liquid is said to be perfectlywetting.

When the system illustrated in Figure 1.2 is in static mechanical equilibrium, the contact line

between the three interfaces is motionless, meaning that the net force on the line is zero. Forces

acting on the contact line arise from the surface tensions of the converging solid-gas, solid-liquid,

and liquid-gas interfaces, denoted by SG,SL, andLG, respectively (Figure 1.2). The condition ofzero net force along the direction tangent to the solid surface gives the following relationship

between the surface tensions and contact angle :

cosLGSLSG

+= . (1.4)

This is known as Youngs equation.

Figure 1.1. Illustration of contact angle and wetting. The liquid in (a) wets better the solid than that in (b).

Figure 1.2. Surface tension forces acting on the contact line

The surface tension of a droplet causes an increase in pressure in the droplet. This can be

understood by considering the forces acting on a curved section of surface as illustrated in Figure

1.3(a). Because of the curvature, the surface tension forces pull the surface toward the concave side

of the surface. For mechanical equilibrium, the pressure must then be greater on the concave side of

the surface. Figure 1.3(b) shows a saddle-shaped section of surface in which surface tension forces

oppose each other, thus reducing or eliminating the required pressure difference across the surface.

The mean curvature of a two-dimensional surface is specified in terms of the two principal radii of

curvature, R1 and R2, which are measured in perpendicular directions. A detailed mechanicalanalysis of curved tensile surfaces shows that the pressure change across the surface is directly

proportional to the surface tension and to the mean curvature of the surface:

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+=

21

11

RRpp BA , (1.5)

where the quantity in brackets is twice the mean curvature. The sign of the radius of curvature is

positive if its center lies in phase A and negative if it lies in phase B. This equation is known as theYoung-Laplace equation, and the pressure change across the interface is termed the Laplace

pressure.

Figure 1.3. Mechanics of curved surfaces that have principal radii of curvature of: (a) the same sign, and (b)

the opposite sign

We conclude this section with the introduction of capillarity. Liquids have cohesion and

adhesion, both of which are forms of molecular attraction. Cohesion enables a liquid to resist tensile

stress, while adhesion enables it to adhere to another body. Capillarity is due to both cohesion and

adhesion. When the former is of less effect than the latter, the liquid will wet a solid surface with

which it is in contact and rise at the point of contact; if cohesion predominates, the liquid surface

will be depressed at the point of contact. For example, capillarity makes water rise in a glass tube,

while mercury is depressed below the true value. Capillary rise in a tube is depicted in Figure 1.4.

Figure 1.4. Illustrative example of capillarity

If a glass capillary tube is brought into contact with a liquid surface, and if the liquid wets the glass

with a contact angle of less than 90o, then the liquid is drawn up into the tube. The surface tension is

directly proportional to the height of rise, h, of the liquid in the tube relative to the flat liquid

surface in the larger container. By applying the Young-Laplace equation to the meniscus in thecapillary tube, the following relationship is obtained:

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=

gbh

cos2, (1.6)

where b is the radius of curvature at the center of the meniscus and is the density differencebetween liquid and gas. For small capillary tubes, b is well approximated by the radius of the tube

itself, r, assuming that the contact angle of the liquid on the tube is zero. For larger tubes, the value

ofh must be corrected for gravitational deformation of the meniscus.

Surface tension decreases slightly with increasing temperature. Room-temperature organic

liquids typically have surface tensions in the range of 20 mN/m to 40 mN/m, while pure water has a

value of 72 mN/m at 25oC. The surface tension of water at the atmospheric pressure can be

calculated with

=

c

c

c

c

T

TT

T

TT625,012358,0

256,1

, (1.7)

where Tis temperature and Tc = 647.3 K.Surface tension effects are generally negligible in most engineering situations; however,

they may be important in problems involving capillary rise, the motion of drops and bubbles, the

breakup of liquid jets, and in hydraulic model studies where the model is small.

1.4.Sound velocity

Consider a fluid at rest in a rigid pipe of cross-sectional area A (Figure 1.5). Suppose a

piston at one end is suddenly moved with a velocity Vfor a time dt. This will produce an increase in

pressure which will travel through the fluid with velocity c. While the piston moves the distance

Vdt, the wave front will move the distance cdt. During this time the piston will displace a mass offluid AVdt, and during this same time the increase in pressure dp will increase the density of the

portion between 1 and 2 by d. Equating the mass displaced by the piston to the gain in massbetween 1 and 2 due to the increased density, AVdt=Acdtd, from which

d

Vc = . (1.8)

From mechanics the impulse of a force equals the resulting increase in momentum. The

impulse of the force produced by the piston is Adtdp. The mass Acdtis initially at rest, but asthe pressure wave travels through it, each element of it will have its velocity increased to V, so that

at the end of the time dtthe entire mass up to section 2 will have the velocity V. Hence the increase

in momentum is AcVdt. ThusAdtdp = AcVdt, and

=

V

dpc , (1.9)

or

ddp

c =2 . (1.10)

In section 1.2 the volume modulus of elasticity is defined asEv = (v/dv)dp. Since = 1/v,v = 1 = constant, and thus dv + vd = 0, and v/dv = +/d. HenceEv = dp/d. Therefore, wefinally have

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vEc = . (1.11)

This is the velocity of a pressure or sound wave, commonly referred to as the sound speed. For

incompressible liquids c . For normal air at 0oC, the sound velocity is 314 m/s while at 20oC,

the velocity is 340 m/s. The sound speed in water at 24

o

C is about 1496 m/s.

Figure 1.5. Illustrative example of sound velocity

1.5.Viscosity

An ideal fluidmay be defined as one in which there is no friction. Thus the forces acting on

any internal section of the fluid are purely pressure forces, even during motion. In a real fluid,

shearing (tangential) and extensional forces always come into play whenever motion takes place,

thus given rise to fluid friction, because these forces oppose the movement of one particle relative

to another. These friction forces are due to a property of the fluid called viscosity. The friction

forces in fluid flow result from the cohesion and momentum interchange between the molecules in

the fluid.

1.5.1. Shear viscosity

The shear viscosity of a fluid is a measure of its resistance to tangential or sheardeformation. To better understand the concept of shear viscosity we assume the following model

(Figure 1.6): two solid parallel plates are set on the top of each other with a liquid film of thickness

Y between them. The lower plate is stationary, and the upper plate can be set in motion by a forceF

resulting in velocity U. The movement of the upper plane first sets the immediately adjacent layer

of liquid molecules into motion; this layer transmits the action to the subsequent layers underneath

it because of the intermolecular forces between the liquid molecules. In a steady state, the velocities

of these layers range from U (the layer closest to the moving plate) to 0 (the layer closes to the

stationary plate). The applied force acts on an area,A, of the liquid surface (surface force), inducing

a so-called shear stress (F/A). The displacement of liquid at the top plate, x, relative to thethickness of the film is called shear strain (x/L), and the shear strain per unit time is called theshear rate (U/Y).

Figure 1.6. Illustrative example of shear viscosity

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If the distance Yis not too great or the velocity Utoo high, the velocity gradient will be a

straight line. Experiment has shown that for a large class of fluids

Y

AUF~ . (1.12)

It may be seen from similar triangles in Figure 1.5 that U/Ycan be replaced by the velocity gradientdu/dy. If a constant of proportionality is now introduced, the shearing stress between any two thinsheets of fluid may be expressed by

dy

du

Y

U

A

F === , (1.13)

which is called Newtons equation of viscosity. In transposed form it serves to define the

proportionality constant

dydu

= , (1.14)

which is called the dynamic coefficient of viscosity. The term du/dy = & is called the shear rate.

Figure 1.7. Rheological behaviour of materials Figure 1.8. Shear-thinning behaviour of two

polymer aqueous solutions

A fluid for which the constant of proportionality (i.e., the viscosity) does not change with

rate of deformation is said to be a Newtonian fluid and can be represented by a straight line in

Figure 1.7. The slope of this line is determined by the viscosity. The ideal fluid, with no viscosity, is

represented by the horizontal axis, while the true elastic solid is represented by the vertical axis. Aplastic which sustains a certain amount of stress before suffering a plastic flow can be shown by a

straight line intersecting the vertical axis at the yield stress. There are certain non-Newtonian fluids

in which varies with the shear rate. Typical representatives of non-Newtonian fluids are liquidswhich are formed either partly or wholly of macromolecules (polymers), and two phase materials,

like, for example, high concentration suspensions of solid particles in a liquid carrier solution. For

most of these fluids, the shear viscosity decreases with increasing shear rate, and we call them

shear-thinningfluids. Here the shear viscosity can decreases by many orders of magnitude. This is a

phenomenon which is very important in the plastics industry, since the aim is to process plastics at

high shear rates in order to keep the dissipated energy small. An example is given in Figure 1.8 for

the case of two polymer aqueous solutions. If the shear viscosity increases with shear rate, we speak

ofshear-thickening fluids. Note that this notation is not unique, and shear-thinning fluids are often

called pseudoplastic, and shear-thickening fluids are called dilatant.

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The dimensions of dynamic viscosity are force per unit area divided by velocity gradient or

shear rate. In the metric system the dimensions of dynamic viscosity are

sPam

sN

s

N/m

ofdimensions

ofdimensionsofDimensions

21

2

=

=== dydu

A widely used unit for viscosity in the metric system is thepoise (P). The poise = 0.1 Ns/m2.

The centipoise (cP) (= 0.01 P = mNs/m2) is frequently a more convenient unit. It has a further

advantage that the viscosity of water at 20oC is 1 cP. Thus the values of the viscosity in centipoises

is an indication of the viscosity of the fluid relative to that of water at 20oC.

The dynamic viscosity of water can be calculated with

( ) ( ) ( )

=

= =

5

0

6

0

****

0 11expi j

ji

ij THT , (1.15)

whereHij is given by

i \ j 0 1 2 3 4 5 6

0 5.132 2.152 2.818 1.778 0.417 0 01 3.205 7.318 10.707 4.605 0 0.158 02 0 12.41 12.632 2.34 0 0 03 0 14.767 0 4.924 1.6 0 0.0364 7.782 0 0 0 0 0 05 1.885 0 0 0 0 0 0

and

( )

13

0*

**

0 071.55

=

= ii

i

T

H

TT , (1.16)

withHi = (1, 0.978, 0.58, -0.202), T* = T/ Tc, c /

* = , Tc = 647.3 K, and c = 317.76 kg/m3.

In many problems involving viscosity there frequently appears the value of viscosity divided

by density. This is defined as kinematic viscosity, so called because force is not involved, the onlydimensions being length and time, as in kinematics. Thus

= . (1.17)

In SI units, kinematic viscosity is measured in m2/s while in the metric system the common units are

cm2/s, also called thestoke (St). The centistoke (cSt) (0.01 St) is often a more convenient unit.

The dynamic viscosity of all fluids is practically independent of pressure for the range that is

ordinarily encountered in engineering work. The kinematic viscosity of gases varies with pressure

because of changes in density.

1.5.2.Extensional viscosity

The extensional viscosity of a fluid is a measure of its resistance to extensional or

elongational deformation. To better understand the concept of extensional viscosity we assume the

following model (Figure 1.9): a spherical bubble is collapsing in a liquid of infinite extent. The

motion is thus spherical symmetric and can be described by only one spatial coordinate, the radialdistance from the bubble center, r. The maximum velocity of the liquid is attained at the bubble

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wall while the liquid velocity is zero at infinity. By analogy with the results presented in section

1.5.1 we can define the extensional viscosity as

drdu

=E (1.18)

where E is the extensional viscosity and du/dr= & is the constant strain rate. In most applications,the extensional viscosity is presented in terms of a Trouton ratio which is defined conveniently to

be the ratio of extensional viscosity to the shear viscosity, Tr= E/. The Trouton ratio which takesthe constant value 3 for Newtonian liquids and shear-thinning inelastic liquids, is found to be a

strong function of strain rate & in many non-Newtonian elastic liquids (a material is said to beelastic if it deforms under stress (e.g., external forces), but then returns to its original shape when

the stress is removed), with very high values (~104) possible in extreme cases (see, for example,

Figure 1.10). This observation has emphasized the potential importance of extensional viscosity in

such areas as fibre spinning, lubrication, drag reduction and enhanced oil recovery. It must be noted

here that generating a purely extensional flow in the case of mobile liquids is virtually impossible.

The most that one can hope to do is to generate flows with a high extensional component and tointerpret the data in a way which captures that extensional component in a convenient and

consistent way through a suitable defined extensional viscosity and strain rate.

Figure 1.9. Illustrative example of an Figure 1.10. Extensional viscosity of two

extensional flow polymer aqueous solutions

1.5.3.Engler viscosity

TheEngler viscosity is defined by the formula

w1EV = , (1.19)

where 1 is the time during which 200 cm3 of the fluid under investigation flow through the gauged

orifice of a viscometer at a given temperature, T, and w is the time during which 200 cm3 of

distilled water flow through it at 20oC. Kinematic viscosity can be determined from the Engler

viscosity with the help of the following formula:

410EV

063.0EV073.0

= . (1.20)

For the Engler viscosity exceeding 16 Engler degrees, the formula EV104.7 6= should be used.

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1.6. Viscoelasticity

When a fluid is suddenly strained and then the strain is maintained constant afterward, the

corresponding stresses induced in the fluid decrease with time. This phenomenon is called stress

relaxation, orrelaxation for short. If the fluid is suddenly stressed and then the stress is maintained

constant afterward, the fluid continues to deform, and the phenomenon is called creep. If the fluid is

subjected to a cycling loading, the stress-strain relationship in the loading process is usuallysomewhat different from that in the unloading process, and the phenomenon is called hysteresis.

The features of hysteresis, relaxation, and creep are found in many materials. Collectively, they are

called features of viscoelasticity.

Mechanical models are often used to discuss the viscoelastic behaviour of fluids. Figure

1.11 illustrates the simplest model composed of a combination of a linear spring with spring

constant and a dashpot with coefficient of viscosity (Maxwell linear model). A linear spring issupposed to produce instantaneously a deformation proportional to the load. A dashpot is supposed

to produce a velocity proportional to the load at any instant. Thus, ifFis the force acting in a spring

and u is its extension, then F = . If the force Facts on a dashpot, it will produce a velocity ofdeflection u& , and uF &= . In the Maxwell model, the same force is transmitted from the spring tothe dashpot. This force produces a displacement F/ in the spring and a velocity F/ in thedashpot. The velocity of the spring extension is /F& if we denote a differentiation with respect totime by a dot. The total velocity is the sum of these two:

FF

u +=

&

&

. (1.21)

Furthermore, if the force is suddenly applied at the instant of time t = 0, the spring will be suddenly

deformed to u(0) = F(0)/, but the initial dashpot deflection would be zero, because there is no timeto deform. Thus the initial condition for the Maxwell model is

)0(

)0(F

u = . (1.22)

For the Maxwell fluid, the sudden application of a load induces an immediate deflection by the

elastic spring, which is followed by creep of the dashpot. On the other hand, a suddendeformation produces an immediate reaction by the spring, which is followed by stress relaxation

Figure 1.11. The mechanical model of

the Maxwell linear fluid

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according to an exponential law. The factor /= , with the dimension of time, is called therelaxation time because it characterize the rate of decay of the force.

In terms of stress and strain rate the Maxwell model reads as

e&& =+ . (1.23)

This equation is called the equation of linear Maxwell viscoelastic fluid. We can call thecharacteristic time the memory span of the fluid. As 0, we obtain the constitutive relationvalid for Newtonian fluids.

The materials mentioned until now have been pure fluids, that is materials where the

shearing forces vanish when the rate of deformation vanishes. However, we often have to deal with

substances which have a dual character. Of these substances, we shall mention here the Bingham

material, which can serve as a model for the material behaviour of paint, or more generally, for high

concentrations of solid particles in Newtonian fluids.

If the solid particles and the fluid are dielectrics, that is do not conduct electrically, then these

dispersions can take on Bingham character under a strong electric field, even if they show only pure

fluid behaviour without electric field. These electrorheological fluids, whose material behaviour

can be changed very quickly and without much effort, can find applications, for example, in the

damping of unwanted oscillations. Through appropriate measures the material can be made to self-

adjust to changing requirements and may be formed into intelligent materials, which are found

increasingly interesting. Even the behaviour of grease used as a means of lubricating ball bearings,

can be described with the Bingham model.

The constitutive equation of the Bingham material in the case of a simple shearing flow is: if

the material flows, we have for the shear stress

&+= 0 for 0 , (1.24)

otherwise the material behaves like an elastic solid, i.e. the shear stress is

G= for 0 < , (1.25)

where 0 is the yield stress and G is the shear modulus.An important constitutive equation for materials with yield stress is the Casson equation

which it is used to describe the rheology of human blood at a hematocrit content of less than 40%.

The Casson equation reads as

&+= 0 (1.26)

Figure 1.12. Behaviour of Bingham materials

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where the dynamic viscosity is constant. Note that, for human blood, 0 is very small: of the

order of 0.05 dyn/cm2, and is almost independent of the temperature in the range 10 37 oC. 0 ismarkedly influenced by the macromolecular composition of the suspending fluid. A suspension of

red cells in saline plus albumin has a zero yield stress; a suspension of red cells in plasma

containing fibrinogen has a finite yield stress.

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