Fundamentals of Fluids Mechanics_ Brujan

8/13/2019 Fundamentals of Fluids Mechanics_ Brujan

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E.A. Brujan Fundamentals of Fluid Mechanics

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1. PROPERTIES OF FLUIDS

1.1. Density, specif ic weight, specif ic volume, and specif ic gravityThe density of a fluid is its mass per unit volume, while the specific weight is its weight

per unit volume. In SI units, density will be in kilograms per cubic meter, while specific gravity willhave the units of force per unit volume or newton per cubic meter.

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

g g

== or . (1.1)

It should be noted that density is absolute since it depends on mass which is independent oflocation. Specific weight, on the other hand, is not absolute for it depends on the value ofgravitational acceleration g which varies with location, primarily latitude and elevation above meansea level. The density of water at 20 oC is 1000 kg/m 3, while the density of air at the sametemperature and at a pressure of 1 bar is 1.29 kg/m 3.

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 ofdensity.

Specific gravity s of a liquid is the ratio of its density to that of pure water at a standardtemperature. In the metric system the density of water at 4 oC is 1 g/cm 3 and hence the specificgravity (which is dimensionless) has the same numerical value for a liquid in that system as itsdensity expressed in g/cm 3.

1.2. Compressibi li ty of l iqu idsCompressibility 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

( )dpdvvdvdpv E v == , (1.2)

where v is the specific volume and p the unit pressure. As v/dv is a dimensionless ratio, the units of E v 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/m 2. Taking a typical value for thevolume modulus of cold water to be 2,200 MN/m 2 it is seen that water is about 80 times ascompressible as steel. The compressibility of liquids covers a wide range. Mercury, for example, isapproximately 8% as compressible as water, while the compressibility of nitric acid is nearly sixtimes greater than that of water.

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

pvv

=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. Figureshows a liquid droplet at rest on a solid surface surrounded by air. The region of contact betweenthe gas, liquid, and solid is termed contact line . The liquid-gas surface meets the solid surface withan angle measured through the liquid, which is known as the contact angle . The system shown inFigure 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. Forcesacting on the contact line arise from the surface tensions of the converging solid-gas, solid-liquid,and liquid-gas interfaces, denoted by SG, SL, and LG, 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 :

cos LGSLSG

+= . (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 beunderstood by considering the forces acting on a curved section of surface as illustrated in Figure1.3( a). Because of the curvature, the surface tension forces pull the surface toward the concave sideof the surface. For mechanical equilibrium, the pressure must then be greater on the concave side ofthe surface. Figure 1.3( b) shows a saddle-shaped section of surface in which surface tension forcesoppose 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 R R

p p B A , (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 andadhesion, both of which are forms of molecular attraction. Cohesion enables a liquid to resist tensilestress, while adhesion enables it to adhere to another body. Capillarity is due to both cohesion andadhesion. When the former is of less effect than the latter, the liquid will wet a solid surface withwhich it is in contact and rise at the point of contact; if cohesion predominates, the liquid surfacewill 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 glasswith a contact angle of less than 90 o, then the liquid is drawn up into the tube. The surface tension isdirectly proportional to the height of rise, h, of the liquid in the tube relative to the flat liquidsurface 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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v E c = . (1.11)

This is the velocity of a pressure or sound wave, commonly referred to as the sound speed. Forincompressible liquids c . For normal air at 0 oC, the sound velocity is 314 m/s while at 20 oC,

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

C is about 1496 m/s.

Figure 1.5. Illustrative example of sound velocity

1.5. Viscosity An ideal fluid may 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 relativeto another. These friction forces are due to a property of the fluid called viscosity . The frictionforces in fluid flow result from the cohesion and momentum interchange between the molecules inthe 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 thicknessY between them. The lower plate is stationary, and the upper plate can be set in motion by a force F resulting in velocity U . The movement of the upper plane first sets the immediately adjacent layerof liquid molecules into motion; this layer transmits the action to the subsequent layers underneathit because of the intermolecular forces between the liquid molecules. In a steady state, the velocitiesof these layers range from U (the layer closest to the moving plate) to 0 (the layer closes to thestationary plate). The applied force acts on an area, A, of the liquid surface (surface force), inducinga 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 Y is not too great or the velocity U too high, the velocity gradient will be astraight line. Experiment has shown that for a large class of fluids

Y AU

F ~ . (1.12)

It may be seen from similar triangles in Figure 1.5 that U /Y can 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

dydu

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 withrate of deformation is said to be a Newtonian fluid and can be represented by a straight line inFigure 1.7. The slope of this line is determined by the viscosity. The ideal fluid, with no viscosity, isrepresented by the horizontal axis, while the true elastic solid is represented by the vertical axis. A

plastic which sustains a certain amount of stress before suffering a plastic flow can be shown by astraight 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. Formost of these fluids, the shear viscosity decreases with increasing shear rate, and we call them

shear-thinning fluids . 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 athigh shear rates in order to keep the dissipated energy small. An example is given in Figure 1.8 forthe case of two polymer aqueous solutions. If the shear viscosity increases with shear rate, we speakof shear-thickening fluids . Note that this notation is not unique, and shear-thinning fluids are oftencalled 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 orshear rate. In the metric system the dimensions of dynamic viscosity are

sPam

s Ns

N/m of dimensions

of dimensions of Dimensions 21

2

==== dydu

A widely used unit for viscosity in the metric system is the poise (P). The poise = 0.1 Ns/m 2.The centipoise (cP) (= 0.01 P = mNs/m 2) is frequently a more convenient unit. It has a furtheradvantage that the viscosity of water at 20 oC is 1 cP. Thus the values of the viscosity in centipoisesis an indication of the viscosity of the fluid relative to that of water at 20 oC.The dynamic viscosity of water can be calculated with

( ) ( ) ( ) = = =

5

0

6

0

****0 11exp

i j

ji

ij T H T , (1.15)

where H ij is given by

i \ j 0 1 2 3 4 5 60 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

=

= iiiT

H T T , (1.16)

with H i = (1, 0.978, 0.58, -0.202), T * = T / T c, c /* = , T c = 647.3 K, and c = 317.76 kg/m 3.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 m 2/s while in the metric system the common units arecm 2/s, also called the stoke (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 isordinarily 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 thefollowing 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 section1.5.1 we can define the extensional viscosity as

dr du =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 astrong 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 whenthe stress is removed), with very high values (~10 4) possible in extreme cases (see, for example,Figure 1.10). This observation has emphasized the potential importance of extensional viscosity insuch areas as fibre spinning, lubrication, drag reduction and enhanced oil recovery. It must be notedhere 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 andconsistent way through a suitable defined extensional viscosity and strain rate.

Figure 1.9. Illustrative example of an Figure 1.10. Extensional viscosity of twoextensional flow polymer aqueous solutions

1.5.3. Engler viscosity The Engler viscosity is defined by the formula

w1EV = , (1.19)

where 1 is the time during which 200 cm 3 of the fluid under investigation flow through the gaugedorifice of a viscometer at a given temperature, T , and w is the time during which 200 cm 3 ofdistilled water flow through it at 20 oC. Kinematic viscosity can be determined from the Englerviscosity 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 stressrelaxation , or relaxation for short. If the fluid is suddenly stressed and then the stress is maintainedconstant 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 arecalled features of viscoelasticity.

Mechanical models are often used to discuss the viscoelastic behaviour of fluids. Figure1.11 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 supposedto produce a velocity proportional to the load at any instant. Thus, if F is the force acting in a springand u is its extension, then F = . If the force F acts on a dashpot, it will produce a velocity ofdeflection u&, and u F &= . 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:

F F

u +=

&&

. (1.21)

Furthermore, if the force is suddenly applied at the instant of time t = 0, the spring will be suddenlydeformed 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 theelastic 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 ofthe Maxwell linear fluid


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where the dynamic viscosity is constant. Note that, for human blood, 0 is very small: of theorder of 0.05 dyn/cm 2, 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 ofred cells in saline plus albumin has a zero yield stress; a suspension of red cells in plasmacontaining fibrinogen has a finite yield stress.


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difference between the pressure of the fluid to which they are connected and that of the surroundingair.

If the pressure is below that of the atmosphere, it is called vacuum and its gage value is theamount by which it is below that of the atmosphere. Gage pressures are positive if they are abovethat of the atmosphere and negative if they are below that of the atmosphere (vacuum) (Figure 2.2).

2.3. Barometer The absolute pressure of the atmosphere is measured by the barometer. If a tube such as in

Figure 2.3 has its lower end immersed in a liquid which is exposed to the atmospheric pressure, andif air is exhausted from the tube, the liquid will rise in it. If the air was completely exhausted, theonly pressure on the surface of the liquid in the tube would then be that of its own vapour pressureand the liquid would reached its maximum height.

From concepts developed in section 2.1 the pressure at O within the tube and at a at thesurface of the liquid outside the tube must be the same; that is pO = pa. We may write

vapour atm ph p += (2.7)

and if the vapour pressure on the surface of the liquid in the tube were negligible, then we shouldhave:

h p atm = (2.8)

The liquid employed for barometers is usually mercury, because its density is sufficientlylarge to enable a reasonably short tube to be used, and also because its vapour pressure is negligiblysmall at ordinary temperatures. If some other liquid was used, the tube necessarily would be so highas to be inconvenient and its vapour pressure at ordinary temperatures would be appreciable; hencea nearly perfect vacuum at the top of the column would not be attainable.

The sea-level atmospheric pressure may be expressed as follows:

101.3 kN/m 2, abs (1,013 mbars, abs) , 760 mm Hg , 10.3 m H 2O

2.4. M easur ement of pressur e There are many ways by which pressure in a fluid may be measured. Some of them are

discussed below.

Figure 2.3. Illustrative example of barometer


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Figure 2.4. Piezometer column Figure 2.5. A simple manometer

2.4.1. Piezometer column A piezometer column is a simple device for measuring moderate pressures of liquids. It consists ofa tube (Figure 2.4) in which the liquid can freely rise without overflowing. The pressure is given by

p = h. To reduce capillary error the tube should be at least 12 mm in diameter.If the pressure of a flowing liquid is to be measured, special precautions should be taken in

making the connection. The hole must be drilled absolutely normal to the interior surface of thewall, and the surface roughness near the hole must be removed. Also, the hole should be small,

preferable not larger than 3 mm.

2.4.2. Simple manometer Since the open piezometer tube is cumbersome for use with liquids under high pressure and cannot

be used with gases, the simple manometer or U-tube of Figure 2.5 is a convenient device formeasuring pressure. To determine the pressure in A we may write:

C B p p = (2.9)where

z p p A B = 1 and y p p atmC += 2 (2.10 a,b )and, therefore,

y z p p atm A ++= 21 (2.11)

Although mercury is generally used as the measuring fluid in the simple manometer, otherliquids can be used. As the density of the measuring fluid approaches that of the fluid whose

pressure is being measured, the reading becomes larger for a given pressure, thus increasing the

accuracy of the instrument, provided the density is accurately known.

2.4.3. Differential manometer In many cases only the difference between two pressures is desired and for this purpose differentialmanometers, such as shown in Figure 2.6, may be used. For the geometry illustrated in the left-handside of Figure 2.6, the measuring fluid is of greater density than that of the fluid whose pressuredifference is involved. We may write

DC p p = (2.12)where

A AC z p p = 1 and y z p p B B D 21 += (2.13 a,b )and thus

y z p z p B B A A 211 += where y z z B A = (2.14)


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or( ) y p p B A = 12 (2.15)

and finally

y p p B A

= 1

1

2

1

(2.16)

Figure 2.6. A differential manometer

This equation is applicable only if A and B are at the same elevation. If their elevations aredifferent, an elevation-difference term must be added to this equation. The differential manometer,when used with a heavy liquid such as mercury, is suitable for measuring large pressure differences.For a small pressure difference a light fluid, such as oil, or even air, may be used, and in this casethe manometer is arranged as in the right-hand side of Figure 2.6. Naturally, the fluid must be one

that will not mix with the fluid in A or B. By the same method of analysis as the preceding, it may be shown that for the simple case with identical liquids in A or B, and with both A and B at the sameelevation

y p p B A

=

1

2

1

1

(2.17)

where the ratio 2 / 1 has a value less than 1. As the density of the measuring fluid approaches thatof the fluid being measured, (1 - 2 / 1) approaches zero and larger values of y are obtained for small

pressure differences, thus increasing the sensitivity of the gage. Once again, this equation must bemodified if A and B are not at the same elevation.

2.5. F lui d masses subj ected to acceleration Under certain conditions there may be no relative motion between the particles of a fluid

mass yet the mass itself may be in motion. If a body of fluid in a tank is transported at a uniformvelocity, the conditions are those of ordinary fluid statics. But if it is subjected to acceleration,special treatment is required.

Consider the case of a liquid mass in an open tank moving horizontally with a linearacceleration a x, as shown in Figure 2.7( a). A free body diagram (Figure 2.7( b)) of a small particle(mass m) of liquid on the surface indicates that the forces exerted by the surrounding fluid on the

particle are F z = W and F x = m ax; the latter is required to produce acceleration ax of the particle.Equal and opposite to these forces are F x and F z of Figure 2.7( a), the forces exerted by the particleon the surrounding fluid. The resultant of these forces is F . The liquid surface must be at rightangles to F , for it were not, the particle would not maintain its relative position in the liquid. Hence


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g a x= tan . The liquid surface and all planes of equal hydrostatic pressure must be inclined atangle with the horizontal.

In Figure 2.7( c) is shown a free-body diagram of an elemental cube of the liquid at thecenter of which the pressure is p. Applying the equation of motion in the x-direction

= x x am F (2.18) xa z y x g

z y x

x p

p z y x

x p

p =

+

22 (2.19)

which reduces to

x x aa g x p ==

(2.20)

In the vertical direction

=

z z am F (2.21)

022

=

+

z y x y x z

z p

p y x z

z p

p

(2.22)

Figure 2.7. A fluid mass subjected to horizontal acceleration


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which yields

g z p ==

(2.23)

Thus for the case where the fluid is subjected to a horizontal acceleration a x

( )2/122 g an

p x +=

(2.24)

where n is the direction at right angles to and outward from the liquid surface.In the foregoing discussion the acceleration of the fluid mass was at right angles to the

direction of the gravitational acceleration. When the acceleration of the fluid mass is in some otherdirection, the same general approach can be used. For a fluid subject to both horizontal and verticalacceleration a x and a z, it can be shown that

( )[ ]2/1222 g aan p

z x ++= (2.25)

When ax = a z = 0, this equation reduces to = n p , which is essentially the same as the basichydrostatic equation. The last equation indicates that, if fluid in a container is subjected to anupward acceleration, there will be an increase of pressure within the fluid. A downward accelerationresults in a decrease in pressure.

Figure 2.8. Force on a plane area

2.6. F orce on pl ane area If the pressure is uniformly distributed over an area, the force is equal to the pressure times

the area, and the point of application of the force is at the centroid of the area. This is the case ofgases where the pressure variation with vertical distance is very small because of the low specificweight. In the case of liquids the distribution of pressure is not uniform; hence further analysis arenecessary.

In Figure 2.8 let MN be the trace of a plane area making an angle with the horizontal. Tothe right is the projection of this area upon a vertical plane. Let h be the variable depth to any pointand y be the corresponding distance from OX , the intersection of the plane produced and the freesurface.


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Consider an element of area so chosen that the pressure is uniform over it. Such an elementis a horizontal strip. If x denotes the width of the area at any depth, then dA = x dy. As p = h and h = y sin , the force dF on a horizontal strip is

dA ydAhdA pdF === sin (2.26)

The pressure distribution over the area forms a pressure prism, the volume of which is equal to thetotal force acting on the area.

Integrating the preceding expression,

== sinsin A y ydA F c (2.27)where yc is the distance to the centroid of the wetted area of the wall A. If the vertical depth of thecentroid is denoted by hc, then hc = yc sin and

Ah F c = (2.28)

Thus the total force acting on any plane area submerged in a liquid is found by multiplyingthe specific weight of the liquid by the product of the area and the depth of its centroid. The valueof F is independent of the angle of inclination of the plane so long as the depth of its centroid isunchanged (For a plane submerged as in Figure 2.8, it is obvious that this equation applies to oneside only. As the pressures are there identical on the two sides, the net force is zero. In most

practical cases where the thickness of the plane is not negligible, the pressures on the two sides arenot the same).

Since h is the pressure at the centroid, another statement is that the total force on any planearea submerged in a liquid is the product of the area and the pressure at its centroid.

The point of application of the resultant force on an area is called the center of pressure .Taking OX in Figure 2.8 as an axis of moments, the moment of an elementary force y sin dA is:

dA ydF y = sin2 (2.29)

and if y p denotes the distance to the center of pressure,

o p I dA y F y == sinsin 2 (2.30)where I o is the moment of inertia of the plane area about an axis through O.

If the preceding expression is divided by the value of F given by Ah F c = the result is

A y I

A y I

yc

o

c

o p ==

sin

sin (2.31)

That is, the distance of the center of pressure from the axis where the plane or plane producedintersects the liquid surface is obtained by dividing the moment of inertia of the area A about thesurface axis by its static moment about the same axis.

This may also be expressed in another form, by noting that

cco I A y I += 2 (2.32)

where I c is the moment of inertia of an area about its centroidal axis. Thus


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A y I

y A y

I y A y

c

cc

c

cc p +=

+=2

(2.33)

From this equation it may be seen that the location of the center of pressure is independent

of the angle ; that is, the plane may be rotated about axis OX without affecting the location of thecenter of pressure. Also, it may be seen that the center of pressure is always below the centroid andthat, as the depth of immersion is increased, the center of pressure approaches the centroid.

The lateral location of the center of pressure may be determined by considering the area to be made up of a series of elemental horizontal strips. The center of pressure for each strip would beat the midpoint of the strip. Since the moment of the resultant force F , must be equal to the momentof the distributed force system about any axis, say, the y axis

= dA p x F X p p (2.34)where X p is the lateral distance from the selected y axis to the center of pressure of the resultantforce F , and x p is the lateral distance to the center of any elemental horizontal strip of area dA onwhich the pressure is p.

2.7. F orce on cur ved sur face On any curved area such as MN in Figure 2.8, the forces upon the various elements are

different in direction and magnitude, and an algebraic summation is impossible. But for anynonplanar area the resultant forces in certain directions can be found.

Figure 2.9. Hydrostatic forces on curved surfaces

2.7.1. Horizontal force on curved surface Any irregular curved area (Figure 2.9) may be projected upon a vertical plane whose trace is MN .The projecting elements, which are all horizontal, enclose a volume whose ends are the vertical

plane MN and the irregular surface MN . This volume of liquid is in static equilibrium. Acting onthe vertical projection MN is a force F , and the horizontal force on the irregular area is F x.Gravity W is vertical, and the lateral forces on all the projecting elements are normal to theseelements and hence normal to F . Thus the only horizontal forces on MNMN are F and F x, andtherefore F x = F .

Hence the horizontal force in any given direction upon any area is equal to the force uponthe projection of that area upon a vertical plane normal to the given direction .


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2.7.2. Vertical force on curved surface The vertical force on a curved area, such as MN in Figure 2.9, can be found by considering thevolume of liquid enclosed by the area and vertical elements extending to the free surface. Thisvolume of liquid is in static equilibrium. Disregarding the pressure on the free surface, the onlyvertical forces are gravity W and F z, the vertical force on the irregular area. The forces on thevertical elements are normal to these and hence are horizontal. Therefore, F z = W .

Hence the vertical force upon any area is equal to the weight of the volume of fluidextending above that area to the free surface . The line of action of F z must be the same as that ofW ; that is, it must pass through the center of gravity of the volume.

2.7.3. A particular case: horizontal and vertical forces on a cylindrical surface The horizontal force is equal to the force of pressure of the fluid on the vertical projection of thecylindrical surface vert c x Ah g F = where hc is the depth of submergence of the centre of massand Avert is the wetted area of the wall, while the vertical component of the pressure force is equal tothe force of gravity acting in the volume V of fluid bounded from above by the free surface of thefluid, and from below by the curviliniar solid surface under consideration, and from the sides, bythe vertical surface drawn through the perimeter of the walls V g F

z = . The total force of

pressure acting on a cylindrical surface is ( ) 2/122 z x F F F += . The direction of the total force of pressure is determined by the angle formed by the vector F with the horizontal plane

x z F F = tan .For the case illustrated in Figure 2.10 we have

Figure 2.10. Horizontal and vertical forces on a cylindrical surface

2.8. Buoyancy of submerged and floating bodies The body DHCK immersed in the fluid in Figure 2.11 is acted upon by gravity and pressures

of the surrounding fluid. On its upper surface the vertical component of the force is F z and is equalto the weight of the volume of fluid ABCHD . In similar manner the vertical component of force on

the undersurface is z F and is equal to the weight of the volume of fluid ABCKD . The difference between these two volumes is the volume of the body DHCK .

The buoyant force of a fluid is denoted by F B, and it is vertically upward and equal to

z z F F , which is equal to the weight of the volume of fluid DHCK . That is, the buoyant force on

any body is equal to the weight of fluid displaced. If the body is in equilibrium, W = F B, which means that the densities of body and

fluid are equal. If W > F B, the body will sink.

Horizontal force RL R F x 2= , L is the length of the cylindrical surface

Vertical force:

( ) L RV V V F OBAO AO A ABO A z 2'' 21

===

(directed upwards)


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25

If W < F B, the body will rise until its density and that of the fluid are equal, as inthe case of a balloon in the air or, in the case of a free surface, the body will riseuntil the weight of the displaced liquid equals the weight of the body. Hence afloating body displaces a volume of liquid equivalent to its weight. If the body isless compressible than the fluid, there is a definite level at which it will reachequilibrium. If it is more compressible than the fluid, it will rise indefinitely,

provided the fluid has no definite limit of height, as in the case of earthsatmosphere.

I ll ustrative example. Determine whether a gold nugget contains an impurity if its weight in air isknown to be G0 = 9.65 N and in water Gw = 9.15 N. The density of pure gold is g = 19.3 103 kg/m 3.

The gold nugget contains no impurity if its density g is equal to the density of gold g. Thedensity of the nugget is

nn gV

G0= .

Using the weight of displaced water, we find G0 Gw = V n w g and so,

( )33

0

0

0

0 kg/m103.19 =

== ww

w

wn GG

G

g GG g

G

The nugget contains no impurity since g n = .

Figure 2.11 . Forces acting on a submerged body


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3. KINEMATICS OF FLUID FLOW

In this chapter we deal only with velocities and acceleration and their distribution in spacewithout consideration of any forces involved.

3.1. L aminar and turbulent flow

That there are two distinctly different types of fluid flow was demonstrated by OsborneReynolds in 1883. He injected a fine, threadlike stream of colored liquid having the same density aswater at the entrance to a large glass tube through which water was flowing from a tank. A valve atthe discharge end permitted him to vary the flow. When the velocity in the tube was small, thiscolored liquid was visible as a straight line throughout the length of the tube, thus showing that the

particles of water moved in parallel straight lines. As the velocity of the water was graduallyincreased by opening the valve further, there was a point at which the flow changed. The line wouldfirst become wavy, and then at a short distance from the entrance it would break into numerousvortices beyond which the color would be uniformly diffused so that no streamlines could bedistinguished. Later observations have shown that in this latter type of flow the velocities arecontinuously subject to irregular fluctuations.

The first type is known as laminar , streamline , or viscous flow. The significance of theseterms is that the fluid appears to move by the sliding of laminations of infinitesimal thicknessrelative to adjacent layers; that the particles move in definite and observable paths or streamlines, asin Figure 3.1; and also that the flow is characteristic of a viscous fluid or is one in which viscosity

plays a significant part (Figure 1.6 of chapter 1).The second type is known as turbulent flow and is illustrated in Figure 3.2, where ( a )

represents the irregular motion of a large number of particles during a very brief time interval, while(b) shows the erratic path followed by single particle during a longer time interval. A distinguishedcharacteristic of turbulence is its irregularity, there being no definite frequency, as in wave action,and no observable pattern, as in the case of eddies.

Large eddies and swirls and irregular movements of large bodies of fluid, which can betraced to obvious sources of disturbance, do not constitute turbulence, but may be described asdisturbed flow . By contrast, turbulence may be found in what appears to be a very smoothly flowingstream and one in which there is no apparent source of disturbance. The fluctuations of velocity arecomparatively small and can often be detected only by special instrumentation.

Figure 3.2. Illustrative example of turbulent flow

At a certain instant a particle at O in Figure 3.2 b may be moving with the velocity OD , butin turbulent flow OD will vary continuously both in direction and magnitude. Fluctuations ofvelocity are accompanied by fluctuations in pressure, which is the reason why manometers attachedto a pipe in which fluid is flowing usually show pulsations. In this type of flow an individual

particle will follow a very irregular and erratic path, and no two particles may have identical or evensimilar motions. Thus a rigid mathematical treatment of turbulent flow is impossible, and insteadstatistical means of evaluation must be employed.

Figure 3.1. Illustrative example of laminar flow


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27

Figure 3.2 c . A turbulent jet of water emerging from a circular orifice into a tank of still water. The fluidfrom the orifice is made visible by mixing small amounts of a fluorescing dye and illuminating it with a thinlight sheet. The picture illustrates swirling structures of various sizes amidst an avalanche of complexity. The

boundary between the turbulent flow and the ambient is usually rather sharp and convoluted on many scales.The object of study is often an ensemble average of many such realizations. Such averages obliterate most ofthe interesting aspects seen here, and produce a smooth object that grows linearly with distance downstream.Even in such smooth objects, the averages vary along the length and width of the flow, these variations being

a measure of the spatial inhomogeneity of turbulence. The inhomogeneity is typically stronger along thesmaller dimension (the width) of the flow. The fluid velocity measured at any point in the flow is anirregular function of time. The degree of order is not as apparent in time traces as in spatial cuts, and a rangeof intermediate scales behaves like fractional Brownian motion.


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3.2. Steady fl ow and unif orm f low A steady flow is one in which all conditions at any point in a stream remain constant with

respect to time, but the conditions may be different at different points. A truly uniform flow is one inwhich the velocity is the same in both magnitude and direction at a given instant at every point inthe fluid. Both of these definitions must be modified somewhat, because true steady flow is found

only in laminar flow. In turbulent flow there are no continual fluctuations in velocity and pressure atevery point, as has been explained. But if the values fluctuate equally on both sides of a constantaverage value, the flow is called steady flow. However, a more exact definition for this case would

be mean steady flow .Steady (or unsteady) and uniform (or nonuniform) flow can exist independently of each

other, so that any of four combinations is possible. Thus the flow of liquid at a constant rate in along straight pipe of constant diameter is steady uniform flow , the flow of liquid at a constant ratethrough a conical pipe is steady nonuniform flow , while at a changing rate of flow these cases

become unsteady uniform and unsteady nonuniform flow, respectively.

3.3. Path l ines, str eamlines, and str eak l ines A path line (see Figure 3.2 b) is the trace made by a single particle over a period of time. If a

camera were to take a time exposure of a flow in which a fluid particle was colored so it wouldregister on the negative, the picture would show the course followed by the particle. This would beits path line. The path line shows the direction of the velocity of the particle at successive instants oftime.

Streamlines show the mean direction of a number of particles at the same instant of time. Ifa camera were to take a very short time exposure of a flow in which there were a large number of

particles, each particle would trace a short path, which would indicate its velocity during that briefinterval. A series of curves drawn tangent to the means of the velocity vectors are streamlines.

Path lines and streamlines are identical in the steady flow of a fluid in which there are nofluctuating velocity components, in other words, for truly steady flow. This is because particlesalways move along streamlines, since these lines show the direction of motion of every particle.Truly steady flow may be either that of an ideal frictionless fluid or that of one so viscous andmoving so slowly that no eddies are formed. This latter is the laminar type of flow, wherein thelayers of fluid slide smoothly, one upon another. In turbulent flow, however, path lines andstreamlines are not coincident, the path lines being very irregular while the streamlines areeverywhere tangent to the local mean temporal velocity.

In experimental fluid mechanics, a dye or other tracer is frequently injected into the flow totrace the motion of the fluid particles. If the flow is laminar, a ribbon of color results. This is calleda streak line , or filament line . It is an instantaneous picture of the positions of all particles in theflow which have passed through a given point (namely, the point of injection).

3.4. F low r ate and fl ow mean The quantity of fluid flowing per unit time is called the flow rate . It may be expressed in

terms of volume flow rate (in SI units cubic meters per second), or mass flow (in SI units kilograms per second). In dealing with incompressible fluids, volume flow rate is commonly used, whereasmass flow rate is more convenient with compressible fluids.

Figure 3.3 illustrates a streamline in steady flow lying in the xz plane. Element of area dA lies in the yz plane. The mean velocity at point P is u. The volume flow rate through the element ofarea dA is

( ) ( ) Ad udAudAud dQ ==== coscosAu (3.1)

where dA is the projection of dA on the plane normal to the direction of u. This indicates that thevolume flow rate is equal to the magnitude of the velocity multiplied by the flow area at right angles


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29

to the direction of the velocity. The mass flow rate may be computed by multiplying the volumeflow rate by the density of the fluid.

If the flow is turbulent, the instantaneous velocity component u along the streamline willfluctuate with time, even though the flow is nominally steady. A plot of u as a function of time isshown in Figure 3.4. The average ordinate of u over a period of time determines the temporalmean value of velocity u at point P .

The difference between u and u, which may be designated as u , is called the turbulentfluctuation of this component. Thus at any instant

uuu += (3.2)

and u may be evaluated for any finite time t as ( ) = t dt ut u 01 .

The flow rate may be expressed as

v AdAuQ A

== (3.3)or

v AdAuQ Am

== (3.4)

where u is the temporal mean velocity through an infinitesimal area dA, while v is the mean , oraverage, velocity over the entire sectional area A (Note that the are A is defined by the surface atright angles to the velocity vectors). Q is the volume flow rate (m 3/s) and Qm is the mass flow rate

(kg/s). If u is known as a function of A, the foregoing may be integrated. If the flow rate has beendetermined directly by some other method, the mean velocity may be found by

Figure 3.3. A streamline in a steady flow

Figure 3.4. Fluctuating velocity at a point


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30

AQ

AQ

v m==

(3.5)

3.5. Equation of continui ty Figure 3.5 illustrates a short length of a stream tube, which may be assumed, for practical

purposes, as a collection of streamlines. Since the stream tube is bounded on all sides by streamlinesand since there can be no net velocity normal to a streamline, no fluid can leave or enter the streamtube except at the ends. The fixed volume between the two fixed sections of the stream tube isknown as the control volume and its magnitude will designated by According to

Newtonian physics ( i.e. , disregarding the possibility of converting mass to energy), mass must beconserved. If the mass of the fluid contained in the control volume of volume ( ) at time t ismass, then the mass of fluid contained in at time t + dt would be

( ) ( )dt dAudt dAut dt t 222111massmass +=+ (3.6)

But the mass contained in at t + dt can also be expressed as:

( )vol dt t t dt t

+=+

massmass (3.7)

where t is the time rate of change of the mean density of the fluid in . Equating thesetwo expressions for mass t+dt yields:

( ) ( ) )(222111 vol dt t dt dAudt dAu

= (3.8)

and

=

vol A Avol d

t dAudAu )(

21222111

(3.9)

This is the general equation of continuity for flow through regions with fixed boundaries. It statesthat the net rate of mass inflow to the control volume is equal to the rate of increase of mass withinthe control volume. This equation can be reduced to more useful forms.

For steady flow, t = 0 and

Figure 3.5. Length of stream tube as control volume


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31

=21

222111 A AdAudAu (3.10)

ormQv Av A == 222111 (3.11)

These are the continuity equations that apply to steady, compressible or incompressible flow within

rigid boundaries.If the fluid is incompressible, = constant and thus

=21

2211 A AdAudAu (3.12)

orQv Av A == 2211 (3.13)

This is the continuity equation that applies to incompressible fluids for both steady and unsteadyflow within rigid boundaries.

These equations are generally adequate for the analysis of flows in conduits with rigid boundaries, but for the consideration of flow in space, as that of air around airplane, for example, itis desirable to express the continuity equation in another, as indicated below.

3.6. Di ff erential equation of continui ty Figure 3.6 shows three coordinate axes x, y, z mutually perpendicular and fixed in space. Let

the velocity components in these three directions be u, v, w, respectively. Consider now a small parallelepiped, having sides x, y, z . In the x direction the rate of mass flow into this box throughthe left-hand face is approximately z yu , this expression becoming exact in the limit as the boxis shrunk to a point. The corresponding rate of mass flow out of the box through the right-hand faceis ( )[ ]{ } z y x xuu + . Thus the net rate of mass flow into the box in the x direction is

( )[ ] z y x xu . Similar expressions may be obtained for the y and z directions. The sum ofthe rates of mass inflow in the three directions must equal the rate of change of the mass in the box,or ( ) z y xt . Summing up, applying the limiting process, and dividing both sides of theequation by the volume of the parallelepiped, which is common to all terms, we get

( ) ( ) ( )t z

w y

v x

u=

(3.14)

which is the equation of continuity in its most general form.

Figure 3.6. An elemental volume of fluid


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32

This equation is, of course, valid regardless of whether the fluid is a real one or an ideal one. If theflow is steady, does not vary with time, but it may vary in space. Since

( ) ( ) ( ) xu xu xu += , it follows that for steady flow the equation may be written

0=

+

+

+

+

+

z w

yv

xu

z w yv xu

(3.15)

In the case of an incompressible fluid, whether the flow is steady or not, the equation of continuity becomes

0=+

+

z w

yv

xu

(3.16)

Figure 3.7. Two-dimensional flow along a curved path. ( a ) Irrotational flow. ( b) Rotational flow.

3.7. Rotational and ir rotational f low Irrotational flow may be briefly described as flow in which each element of the moving fluidsuffers no net rotation from one instant to the next, with respect to a given frame of reference. Inirrotational flow, however, a fluid element may deform as shown in Figure 3.7 a , where the axes ofthe element rotate equally toward or away from each other. As long as the algebraic averagerotation is zero, the motion is irrotational. In Figure 3.7 b is depicted an example of rotational flow.In this case there is a net rotation of the fluid element. Actually, the deformation of the element inFigure 3.7 b is less than that of Figure 3.7 a.

Figure 3.8. Deformation of a fluid element


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Let us now express the condition of irrotationality in mathematical terms. It will help torestrict the discussion at first to two-dimensional motion in the xy plane. Consider a small elementmoving as depicted in Figure 3.8 a. During a short time interval t the element moves from one

position to another and in the process it deforms as indicated. Superimposing A on A, defining an x axis along AB, and enlarging the diagram, we get Figure 3.8 b. The angle between AB and AB can be expressed as

( )[ ]t

xv

xt x xv

x B B

=

=

= (3.17)

Hence the rate of rotation of the edge of the element that was originally aligned with AB is

xv

t =

= (3.18)

Likewise

( )[ ] t yu

yt y yu

=

= / (3.19)

and the rate of rotation of the edge of the element that was originally aligned with AC is

yu

t =

= (3.20)

with the negative sign because + u is directed to the right. The rate of rotation of the element aboutthe z axis is now defined to be z , the average of and , thus

=

yu

xv

z 21

(3.21)

But the criterion we originally stipulated for irrotational flow was that the rate of rotation be zero.Therefore the flow is irrotational in the xy plane if

yu

xv

=

(3.22)

In three-dimensional flow the corresponding expressions for the components of angular-deformation rates about the x and y axes are

=

z v

yw

x 21

=

xw

z u

y 21

(3. 23 a,b )

Thus vrr

rot21= . Finally, for the general case, irrotational flow is defined to be that for which

0=== z y x

(3.24)

The primary significance of irrotational flow is that ideal (frictionless) flow is irrotational .


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3.8. Circulation To get a better understanding of the character of a flow field, we should acquaint ourselves

with the concept of circulation . Let the streamlines of Figure 3.9 represent a two-dimensional flowfield, while L represents any closed path in this field. The circulation is defined mathematically asa line integral of the velocity about a closed path. Thus

== L L

dLvd cosLv (3.25)

where v is the velocity in the flow field at the element d L of the path, and is the angle between v and the tangent to the path (in the positive direction along the path) at that point. This equation isanalogous to the common equation in mechanics for work done as a body moves along a curved

path while the force makes some angle with the path. The only difference here is the substitution ofa velocity for a force.

In a two-dimensional flow field we can write

dydx yu

xv

d

= (3.26)

The vorticity is defined as the circulation per unit of enclosed area. Thus

yu

xv

dydxd

=

= (3.27)

and we note that an irrotational flow is one for which the vorticity = 0. Similarly, if the flow isrotational, 0.

3.9. Veloci ty and acceleration in steady f low In a typical three-dimensional flow field the velocities are everywhere different in

magnitude and direction. Also, the velocity at any point in the field may change with time. Let usfirst consider the case where the flow is steady and thus independent of time. If the velocity of afluid particle has the components u, v, and w parallel to the x, y, and z axes, then for steady flow:

( ) z y xuu st ,,= ( ) z y xvv st ,,= ( ) z y xww st ,,=

Figure 3.9. Circulation around a closed path in atwo-dimensional velocity field


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35

Applying the chain rule of partial differentiation, the acceleration of the fluid particle for steadyflow can be expressed as

( )t d

z d z t d

yd yt d

xd x

z y xdt d

st +

+

== VVVV ,,a (3.28)

where

( )2/1222

wvu ++=V (3.29)

Noting that ut d xd = , vt d yd = , and wt d z d = , we have

z w

yv

xu st

++

= VVVa (3.30)

This equation can be written as three scalar components

( ) z u

w y

uv

x

uua

st x

+

+

= (3.31 a)

( ) z v

w yv

v xv

ua st y

++

= (3.31 b)

( ) z w

w yw

v xw

ua st z +

+

= (3.31 c)

These equation show that even though the flow is steady, the fluid may possess anacceleration by virtue of a change in velocity with change in position. Thus type of acceleration iscommonly referred to as convective acceleration .

3.10. Velocity and acceleration i n unsteady f low If the flow is unsteady, then the velocity components take the form

( )t z y xuu ,,,= ( )t z y xvv ,,,= ( )t z y xww ,,,=

Following a similar procedure to that of the preceding section results in the following set of scalarequations

z u

w yu

v xu

ut u

a x +

+

+

= (3.32a)

z v

w yv

v xv

ut v

a y

+

+

+

= (3.32b)

z w

w yw

v xw

ut w

a z +

+

+

= (3.32c)

In the above set of equations the terms containing t represent the acceleration caused bythe unsteadiness of the flow. This type of acceleration is commonly referred to as the local acceleration .

In the case of uniform flow (streamlines parallel to one another) the convective accelerationis zero and

t = Va (3.33)

while in the case of steady flow the local acceleration becomes zero.


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4. BASIC HYDRODYNAMICS

In this chapter we discuss various mathematical methods of describing the flow of fluids.The presentation here provides some notion of the possibilities of a rigorous mathematical approachto flow problem.

4.1. Th e str eam fun ction The stream function , based on the continuity principle, is a mathematical expression that

describes a flow field. In Figure 4.1 are shown two adjacent streamlines of a two-dimensional flowfield. Let ( ) y x, represent the streamline nearest the origin. Then d + is representative of thesecond streamline. Since there is no flow across a streamline, we can let be indicative of the flowcarried through the area from the origin O to the first streamline. And thus d represents the flowcarried between the two streamlines of Figure 4.1.

From continuity referring to the triangular fluid element of Figure 4.1, we see that for anincompressible fluid

yd u xd vd += . (4.1)

The total derivative d may be also express as

yd y

xd x

d +

= . (4.2)

Comparing these last equations, we note that

y

u = and

x

v = . (4.3)

Thus, if can be expressed as a function of x and y, we can find the velocity components ( u and v)at any point of a two-dimensional flow field by application of the last equations. It should be notedthat since the derivation of is based on the principle of continuity, it is necessary that continuitybe satisfied for the stream function to exist . Also, since vorticity was not considered in thederivation of , the flow need not be irrotational for the stream function to exist .

The equation of continuity

0=+

yv

xu

(4.4)

may be expressed in terms of by substituting the expressions for u and v, and doing so we get

Figure 4.1. Stream function


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37

0=

x y y x

or x y y x

=

22, (4.5)

which shows that, if = ( x,y), the derivatives taken in either order give the same result and that aflow described by a stream function automatically satisfies the continuity equation.

4.2. Velocity potenti al For two-dimensional flow the velocity potential = ( x,y) may be defined in cartesian

coordinates as

xu

= and y

v = . (4.6)

If we substitute these expressions into the continuity equation, we get

022

2

2

=

+

y x

, (4.7)

which is known as the Laplace equation.If the expressions of the velocity components are substituted into the equation of vorticity,

we get

022

=

+

=

=

=

x y y x x y y x yu

xv

. (4.8)

Since = 0, the flow is irrotational, and thus, if a velocity potential exists, the flow must be irrotational . Conversely, if the flow is rotational, the velocity potential does not exist.

The rotation of fluid particles requires the application of torque, which in turn depends onshearing forces. Such forces are possible only in a viscous fluid. In inviscid (or ideal) fluids therecan be no shears and hence no torques. Hence the flow of an ideal fluid is irrotational .

4.3. Orth ogonali ty of streamli nes and equipotenti al li nes

Figure 4.2. Orthogonality of streamlinesand equipotential lines


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38

It was found that

yd y

xd x

d +

= . (4.9)

Similarly,

yd

y

xd

x

d +

= . (4.10)

Using the expressions for the velocity components we can express these two equations as

yd u xd vd += (4.11)and

yd v xd ud = . (4.12)

Along a streamline, = constant, so d = 0, and from the first equation we get uv xd yd // = .Along an equipotential line, = constant, so d = 0, and from the second equation we get

vu xd yd // = . Geometrically, this tells us that the streamlines and equipotential lines areorthogonal, or everywhere perpendicular to each other.The equipotential lines = C i and the streamlines = K i , where C i and K i have equal

increments between adjacent lines, form a network of intersecting perpendicular lines which iscalled the flow net (Figure 4.2).

4.4. An il lu strative example A flow is defined by u = 2x and v = -2y . Find the stream function and potential function for

this flow.Check continuity:

022 ==

+

yv

xu

. (4.13)

Hence continuity is satisfied and it is possible for a stream function to exist

xdy ydxdyudxvd 22 +=+= , (4.14)

12 C xy += . (4.15)

Check to see if the flow is irrotational:

000 ==

yu

xv . (4.16)

Hence the flow is irrotational and a potential function exists

ydy xdxdyvdxud 22 +== , (4.17)

222 )( C y x ++= . (4.18)

The location of lines of equal can be found by substituting values of into the expression =2xy. Thus for = 60, x = 30/ y. In a similar fashion lines of equal potential can be plotted. Forexample, for = 60 we have x = ( y2 - 60) 0.5. The flow net depicts flow in a corner .


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4.5. Energy considerati ons in fl uid flow In this paragraph fluid flow is approached from the viewpoint of energy considerations. The

first law of thermodynamics tells us that energy can be neither created nor destroyed. Moreover, allforms of energy are equivalent. In the following sections the various forms of energy present influid flow are discussed.

4.5.1. Kinetic energy of a flowing fluid

A body of mass m when moving at a velocity V possesses a kinetic energy, 221

KE mV = .

Thus if a fluid were flowing with all particles moving at the same velocity, its kinetic energy would

be also 221

mV ; this can be written as:

( )

g

V

volume

V volume

volume

mV

2

21

21

weight

KE 222

===

.

In SI units the term g V 22 is expressed in meters.In the flow of a real fluid the velocities of different particles will usually not be the same, so

it is necessary to integrate all portions of the stream to obtain the true value of the kinetic energy. Itis convenient to express the true value in terms of the mean velocity V and a factor . Hence,

g V 2weight

KETrue 2 = .

4.5.2. Potential energy The potential energy of a particle of fluid depends on its elevation above any arbitrary datum

plane. We are usually interested only in differences in elevation and therefore the location of thedatum plane is determined solely by considerations of convenience. A fluid particle of weight W situated at a distance z above datum plane possesses a potential energy Wz . Thus, its potentialenergy per unit weight is z .

4.5.3. Internal energy Internal energy is the energy due to the motion of molecules and forces of attraction between

them. Internal energy is a function of temperature; it can expressed in terms of energy per unit massi or in terms of energy per unit of weight I . Note that i = gI .

The zero of internal energy may be taken at any arbitrary temperature, since we are usuallyconcerned only with differences. For a unit mass, i = cvT , where cv is the specific heat atconstant volume whose units are Nm/(kgK). Thus i is expressed in Nm/kg and the internal energy

I per unit of weight is expressed in meters.

4.5.4. General equation for steady flow of any fluid The first law of thermodynamics states that for steady flow the external work done on any system

plus the thermal energy transferred into or out of the system is equal to the change of energy of the system.Thus, for steady flow,

work + heat = energy.

Let us now apply the first law of thermodynamics to the fluid system defined by the fluidmass contained at time t in the control volume between sections 1 and 2 of the stream tube in Figure4.3. The control volume is fixed in position and does not move or change shape (Figure 4.3( b)). The


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40

fluid system we are dealing with consists of the fluid that was contained between sections 1 and 2 attime t . This fluid system moves to a new position during time interval t , as indicated in Figure 4.3.

Figure 4.3. Work, heat, and energy for a stream tube

During this short time interval we shall assume that the fluid moves a short distance ds1 at section 1and ds2 at section 2. In this discussion we restrict ourselves to steady flow so that 222111 ds Ads A = .In moving these short distances, work is done on the fluid system by the pressure forces p1 A1 and

p2 A2. This work is referred to as flow work . It may be expressed as

222111work flow ds A pds A p = . (4.19)

The minus sign in the second term indicates that the force and displacement are in opposite

directions.In addition to flow work , if there is a machine between section 1 and 2 there will be a shaft work . During the short time interval dt we can write

( ) M M hds Adt hdt ds

A =

==

1111

11timeweightenergy

timeweight

shaft work , (4.20)

where hM is the energy put into the flow by the machine per unit weight of flowing fluid. If themachine is a pump, hM is positive; if the machine is a turbine, hM is negative.

The heat transferred from an external source into the fluid system over time interval dt is

( ) H H Qds Adt Qdt ds

A =

=

1111

11heat , (4.21)

where QH is the energy put into the flow by the external heat source per unit weight of flowingfluid. If the heat flow is out of the fluid, the value of QH is negative.

In using the concept of the control volume, we consider a fluid system defined by the massof fluid contained in the control volume at time t . At time t + dt this same mass of fluid has movedto a new position. At that instant the energy E 2 of the fluid system equals the energy E 1 that was

possessed by the fluid mass when it was coincident with the control volume at time t plus theenergy E out that flowed out of the control volume during time interval dt minus the energy E in that flowed into the control volume during time interval dt . Thus

inout E E E E += 12 . (4.22)


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Hence the change in energy E of the fluid system under consideration during time interval dt is

inout E E E E E == 12 . (4.23)

During time interval dt the weight of fluid entering at section 1 is ( )111 ds A and for steady flowan equal weight must leave section 2 during the same time interval. Hence the energy E in whichenters at section 1 during time dt is ( ) ( ) 1211111 2 I g V z ds A ++ , while that which leaves, E out ,at section 2 is represented by a similar expression. Thus

( ) ( )

++

++==1

21

111112

22

22222 22energy I

g V

z ds A I g

V z ds A E . (4.24)

Applying the first law of thermodynamics, at the same time factoring out( ) ( )222111 ds Ads A = for steady flow and rearranging, we get

++

++=++1

21

112

22

222

2

1

1

22 I

g V

z I g

V z Qh

p p M M

, (4.25)

or

+++=++

+++

222

22

2211

1

12

11 22

I z p

g V

Qh I z p

g V

M M

. (4.26)

This equation applies to liquids, gases, and vapours, and to ideal fluids as well as to realfluids with friction. The only restriction is that it is for steady flow. In many cases this equation is

greatly shortened because certain quantities are equal and thus cancel each other, or are zero. Thus,if two points are at the same elevation, z 1 z 2 = 0. If the conduit is well insulated or if thetemperature of the fluid and that of its surroundings are practically the same, QH may be taken aszero. If there is no machine between sections 1 and 2, then the term hM drops out.

In the above equation each term has the dimension of length. Thus p/ , called pressure head ,represents the energy per unit weight stored in the fluid by virtue of the pressure under which thefluid exists; z , called elevation head , represents the potential energy per unit weight of fluid; andV 2/(2 g ), called velocity head , represents the kinetic energy per unit weight of fluid. The sum ofthese three terms is called the total head and is denoted by H , where

z p

g

V H ++= 2

2

. (4.27)

In this equation we have disregarded . In turbulent flow the value of is only a little more thanunity and thus it can be assumed equal to unity. If the flow is laminar, the velocity head is usuallyvery small compared to the other terms and thus little error is introduced if is set to 1 rather than2, its true value.

4.5.5. Equation for unsteady flow along a streamline for ideal fluid Let us consider now the frictionless unsteady flow of a fluid along a streamline. Referring to

Figure 4.3 we note p1 p, p2 ( p + dp ), z 2 z 1 dz and A1 A2 dA. We will use Newtonssecond law, and express the acceleration of the fluid element as ( ) dt dV dsdV V + . Applying = ma F to the fluid element we get for unsteady flow


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Lhdt dV

dsdV

V dsdAdAdz dpdA +

+= . (4.28)

In this equation hL represents the head loss between sections 1 and 2. Dividing through by dA gives

Lhdt dV

ds gdz VdV dp =+++

. (4.29)

The termdt dV

ds accounts for the effect of acceleration caused by the unsteadiness of the flow. This

equation may also be expressed as

Lhdt dV

g ds

dz g

V d

dp =+++2

2

. (4.30)

This equation applies to unsteady flow of both compressible and incompressible real fluids.However, an equation of state relating to p must be introduced before integration if we aredealing with a compressible fluid. For an incompressible fluid ( = constant) we can integrate fromsome section 1 to another section 2, where the distance between them is L, to obtain:

Lhdt dV

g L

z z g

V g

V p p =+++12

21

2212

22 , (4.31)

or

dt

dV

g

L z

g

V ph z

g

V p L

+

++=+

++2

222

1

211

22 . (4.32)

It should be noted here that if a pipe, for example, consists of two or more pipes in series, an

dt dV

g L

(the accelerative head ) term each pipe should appear in the equation just as there would be a

separate term for the head loss in each pipe.

4.6. M omentum and forces in fl uid f low Previously, two important fundamentals concepts of fluid mechanics were presented: the

continuity equations and the energy equation. In this paragraph a third basic concept, the impulse-momentum principle will be developed.

Consider now a flow that may be incompressible or incompressible, real or ideal, steady orunsteady. Newtons second law may be expressed as

( ) =dt md V

F . (4.33)

Thus, the sum of the external forces on a body is equal to the rate of change of momentum of that body. This equation can also be expressed as ( ) ( ) = V F md dt , i.e., impulse equals change ofmomentum, hence the terminology impulse-momentum principle used.

Let us apply this equation to a body defined by the mass of fluid contained at time t in thecontrol volume of Figure 4.4. Henceforth we shall refer to this mass of fluid as the fluid system . Thecontrol volume is fixed in position; it does not move, nor does it change shape or size. At time t +


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t the fluid mass we are dealing with (i.e., the fluid system) has moved to a new position indicated by the shaded area of Figure 4.4. Let us now define some terms:

(mV )t Momentum at time t of the fluid system (coincident with the control volume attime t )

(mV )t +t Momentum at time t + t of the fluid system (coincident with the shaded area ofFigure 4.4 at time t + t )

(mV )t Momentum of the fluid mass contained within the control volume at time t

(mV )t +t Momentum of the fluid mass contained within the control volume at time t + t

(mV )out Momentum of the fluid mass that leaves the control volume during time interval t

(mV )in Momentum of the fluid mass that enters the control volume during time interval t

Figure 4.4. Control volume for general case. (left) fluid mass acted on by certain forces. (right) location ofthe fluid system at times t and t + t

At time t the momentum of the fluid system is equal to the momentum of the fluid mass

contained in the control volume at time t because the same fluid mass is involved in both cases.Thus

(mV )t = (mV )t . (4.34)

At time t + t the momentum of the fluid system is equal to the momentum of the fluid mass in thecontrol volume at t + t plus the momentum of the mass that has flowed out of the control volumeduring time interval t minus that which flowed into the control volume during time interval t .Thus

( ) ( ) ( ) ( )inout t t t t mV mV V mmV += ++ . (4.35)

The change of momentum of the fluid system is

( ) ( ) ( )t t t mV mV mV = + , (4.36)


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and thus we get( ) ( ) ( ) ( ) ( )inout t t t mV mV V mV mmV += + . (4.37)

Applying Newtons second law, dividing through by t , rearranging, and noting that thelimit of ( ) ( )dt mV d t mV = as t 0, we get

( ) ( ) ( ) ( ) ( ) ( )dt

mmdt

md md dt md

t m t t t inout

t

V V V V V V F

+==

= +

0

lim . (4.38)

The above equation states that the force acting on a fluid mass is equal to the rate of change of themomentum of the fluid mass which, in turn, is equal to the sum of the two terms on the right-handside of the equation. The first term on the right side of the equation represents the net rate ofoutflow of momentum across the control surface while the second term represents the rate ofaccumulation of momentum within the control volume. In the case of steady flow, the last term of

the equation is equal to zero and the equation is

( ) ( )dt

md md inout V V F = . (4.39)

Thus, for steady flow the force on the fluid mass is equal to the net rate of outflow of momentumacross the control surface.

It is advantageous to select a control volume such that the control surface is normal to thevelocity where it cuts the flow. Consider such a situation depicted in Figure 4.5. Also, let thevelocity be constant where it cuts across the control surface. In Figure 4.5 the fluid system we aredealing with is contained between sections 1 and 2 at time t .

This fluid system moves to a new position during time interval dt , as indicated in Figure 4.5. Duringthis short interval we will assume the fluid moves a short distance ds1 at section 1 and ds2 at section2. Also, we are restricting ourselves to steady flow. The momentum crossing the control surface atsection 1 during the interval dt is 1111 V ds A while that crossing section 2 is 2222 V ds A .

Noting that since the control surface cuts the velocity at right angles,

V = ds/dt and Q = AV , (4.40)

we get for steady flow along a stream tube

Figure 4.5. Control volume for steady flow with controlsurface cutting a constant velocity stream atright angles


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111222 V V F = QQ . (4.41)

From continuity, for steady flow,

2211 QQQ == ; (4.42)

thus we can write

( ) V V V F == QQ 12 . (4.43)

The F represents the vectorial summation of all forces acting on the fluid mass includinggravity forces, shear forces, and pressure forces including those exerted by fluid surrounding thefluid mass under consideration as well as the pressure forces exerted by the solid boundaries incontact with the fluid mass.

The corresponding scalar equations are:

( ) x x x x V QV V Q F == 12 , (4.44)

( ) y y y y V QV V Q F == 12 , (4.45)

( ) z z z x V QV V Q F == 12 . (4.46)

If the velocity is not uniform over a section, the momentum transfer across that section isgreater than that compute by using the mean velocity. Hence a correction factor must be introducedthat multiply the term V Q :

= A

dAu AV

22

1 . (4.47)

For laminar flow in a circular pipe, 34= , but for turbulent flow in circular pipes, it usuallyranges from 1.005 to 1.05.

In the succeeding sections these equations will be applied to several situations that arecommonly encountered in engineering practice. The big advantage of the impulse-momentum

principle is that one need not be concerned with the details of what is occurring within the flow,only the conditions at the end sections of the control volume govern the analysis.

4.7. F orce exerted on pr essur e conduits Consider the case of horizontal flow to the right through the reducer of Figure 4.6( a). A

free-body diagram of the forces acting on the fluid mass contained in the reducer is shown in Figure4.6( b). We shall apply the momentum-impulse equations to this fluid mass to examine the forcesthat are acting in the x direction. The forces p1 A1 and p2 A2 represent pressure forces exerted by fluidlocated just upstream and just downstream of the fluid mass under consideration. The force ( ) x F R F / represents the force exerted by the reducer on the fluid in the x direction. Neglecting shear forces atthe boundary of the reducer, the force ( ) x F R F / is the integrated effect of the normal pressure forcesthat are exerted on the fluid by the wall of the reducer. The intensity of pressure at the wall willdecrease as the diameter decreases because of the increase in velocity head.


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Figure 4.6. Forces on a reducer

Applying the momentum-impulse equations and assuming the fluid to be ideal with ( ) x F R F / directed as shown, we get

( ) ( ) == 12/2211 vvQ F A p A p F x F R x . (4.48)

In this equation each term can be evaluated independently from given flow data, except ( ) x F R F / ,which is the quantity we wish to find. Thus

( ) ( )122211/ vvQ A p A p F x F R = . (4.49)

This gives the value of the total force exerted by the reducer on the fluid in the x direction. Thisforce acts to the left as assumed in Figure 4.6. The force of the fluid on the reducer is, of course,equal and opposite to that of the reducer on the fluid. If friction were considered with the flow to theright, ( ) x F R F / would be somewhat larger.

Consideration of the weight of fluid between sections 1 and 2 in Figure 4.6 results in theconclusion that pressure are larger on the bottom half of the pipe than on the upper half. It should be

noted that it is the conditions at the end sections of the control volume that govern the analysis.What occurs within the flow between sections 1 and 2 is unimportant so far as the determination offorces is concerned.

If the fluid undergoes a change in both direction and velocity, as in the reducing pipe bendin Figure 4.7, the procedure is similar to that of the preceding case, except that it is convenient todeal with components. Assuming the flow is in a horizontal plane so that the weight can beneglected, applying the momentum-impulse equations by summing up forces acting on the fluid inthe x direction, and equating them to the change in fluid momentum in the x direction gives

( ) ( )122211/ coscos vvQ A p A p F x F B = . (4.50)

Similarly, in the y direction

( ) sinsin 222/ vQ A p F y F B += . (4.51)

In a specific case, if the numerical values determined by these equations are positive, then theassumed directions are correct. A negative value for either one merely indicates that that componentis in the direction opposite from that assumed.

Note that V F = Q is the resultant of all forces acting on the fluid, which includes the pressure forces on the two ends, while F B F / is the force exerted by the bend on the fluid. The value

of F B F / is( ) ( )2/2// y F B x F B F B F F F += , (4.52)


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and it is represented by the closing line in the force diagram shown in Figure 4.7. The total forceexerted by the fluid on the bend is equal to F B F / , but it is opposite to the direction shown in thefigure.

Figure 4.7. Forces on a reducing bend

It may be seen that such a force tends to move the portion of the pipe considered. Hence,where such c

Fundamentals of Fluids Mechanics_ Brujan

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