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FUNDAMENTALS OFFLUID MECHANICS

Chapter 2 Fluids at Rest- Pressure and its Effect

Be sure to BUY THE TEXT because everything important cannot be delivered in class.

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MAIN TOPICS Pressure at a Point Basic Equation for Pressure Field Pressure variation in a Fluid at Rest Standard AtmosphereMeasurement of PressureManometryMechanical and Electronic Pressure Measuring Devices Hydrostatic Force on a Plane Surface Pressure Prism Hydrostatic Force on a Curved Surface Buoyancy, Floating, and Stability Rigid-Body Motion

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Pressure at a Point 1/4

Pressure ? Indicating the normal force per unit area at a given point

acting on a given plane within the fluid mass of interest.How the pressure at a point varies with the orientation of

the plane passing through the point ?

Pressure direction

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Pressure at a Point 2/4

Consider the free-body diagram within a fluid mass. In which there are no shear stress (Ex 유체가정지해있을때), the

only external forces acting on the wedge are due to the pressure and the weight.

Forces:1. Body force: gravity (중력)2. Surface force: pressure, friction (shear stress)

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Pressure at a Point 3/4

The equation of motion (Newton’s second law, F=ma) in the y and z direction are,

+=−>−=−

=−

=−=

=−=

∑

∑

zSzzSz

ySy

zSzZ

ySyy

a2δzρ

2δzγPPδxδya

2δxδyδzρ

2δxδyδzγ-δxδyPδxδyP

a2δyρPPδxδz

a2

δxδyδzρ2

δxδyδzγ-δxδscosθPδxδyPF

a2

δxδyδzρδxδssinθPδxδzPF

( )2δyρaPP

2δzγρaPP ySyZSZ =−+=−

θδδθδδ sinszcossy ==

δx=0、δy=0、δz=0 SyZ PPP ==

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Pressure at a Point 4/4

The pressure at a point in a fluid at rest, or in motion, is independent of the direction as long as there are no shear stresses present.

The result is known as Pascal’s law named in honor ofBlaise Pascal (1623-1662).

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Basic Equation for Pressure Field

To obtain an basic equation for pressure field in a static fluid.

Apply Newton’s second law to a differential fluid mass

∑ = aδmFδThere are two types of

forces acting on the mass of fluid: surface force and body force. Vm ρδ=δ

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Body Force on Element

( )kzyxVkg

Vkgjgig

VgmgF

zyx

B

δδγδρδ

ρδ

ρδδδ

−=−=

++=

==

Vm ρδ=δ

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Surface Forces 1/4

No shear stresses, the only surface force is the pressure force.

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Surface Forces 2/4

The pressure at the left face

The pressure at the right face

The pressure force in y direction

δxδyδzypδxδz

2dy

yppδxδz

2dy

yppδFy

∂∂

−=

∂∂

+−

∂∂

−=

( )2

dyypp

2dy

yppyy

yppp LL ∂

∂−=

−

∂∂

+=−∂∂

+=

( )2

dyypp

2dy

yppyy

yppp RR ∂

∂+=

∂∂

+=−∂∂

+=

Taylor series expansion

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Surface Forces 3/4

The pressure force in x direction

The pressure force in z direction

δxδyδzxpδyδz

2dx

xppδyδz

2dx

xppδFx

∂∂

−=

∂∂

+−

∂∂

−=

δxδyδzzpδxδy

2dz

zppδxδy

2dz

zppδFz

∂∂

−=

∂∂

+−

∂∂

−=

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Surface Forces 4/4

The net surface forces acting on the element

δxδyδzkzpj

ypi

xpkδFjδFiδFFδ ZYXs

∂∂

+∂∂

+∂∂

−=++=

kzpj

ypi

xpppgrad

∂∂

+∂∂

+∂∂

=∇=

δxδyδzδxδyδz)( Fδ s ppgrad −∇=−=

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General Equation of Motion

zyxakzyxzyxp δδδρδδγδδδδ =−∇−

BS FFF

δδδ +=

akp ργ =−∇−

The general equation of motion for a fluid in which there are no shear stresses

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Pressure Variation in a Fluid at Rest

For a fluid at rest a=0 agpakp ρρργ =+−∇>−==−∇− 0

directionz...0gzp

directiony...0gyp

directionx...0gxp

z

y

x

−=+∂∂

−

−=+∂∂

−

−=+∂∂

−

ρ

ρ

ρ

γ−=∂∂

=∂∂

=∂∂

zP

0yP

0xP

gg,0g,0g

z

yx

−=

==

0 <−=−= γρgdzdp

Pressure decreases as height increases.

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Pressure-Height Relation

The basic pressure-height relation of static fluid :

Restriction:Static fluid.Gravity is the only body force.The z axis is vertical and upward.

Integrated to determine the pressure distribution in a static fluid with appropriateboundary conditions.

How the specific weight varies with z?

γρ −=−= gdzdp

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Pressure in Incompressible Fluid

A fluid with constant density is called an incompressible fluid.

∫∫ −= 1

2

2

1

z

z

p

pdzdp γ

p1 － p2 = γ(z2－z1)=γhp1=γh +p2

h= z2－z1，h is the depth of fluid measured downward from the location of p2.

This type of pressure distribution is called a hydrostatic distribution.

γρ −=−= gdzdp

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Pressure Head in Static Fluid 1/2

The pressure difference between two points in a fluid at rest: p1 － p2 = γ(z2－z1)=γh

γ−

= 21 pph h is called the pressure head and is interpreted as the height of a column of fluid of specific weight γ required to give a pressure difference p1-p2.

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Pressure Head in Static Fluid 2/2

The pressure p at any depth h below the free surface is given by p = γh + po

The pressure in a homogeneous, incompressible fluid at restdepends on the depth of the fluid relative to some reference plane, and it is not influenced by the size or shape of the tank or container in which the fluid is held.

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Example 2.1 Pressure-Depth Relationship

Because of a leak in a buried gasoline storage tank, water has seeped in to the depth shown in Figure E2.1. If the specific gravity of the gasoline is SG=0.68. Determine the pressure at the gasoline-water interface and at the bottom of the tank. Express the pressure in units of lb/ft2, lb/in2, and as pressure head in feet of water.

Determine the pressure at the gasoline-water interface and at the bottom of the tank

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Example 2.1 Solution1/2

( )20

03

0OH1

ft/lbp721

p)ft17)(ft/lb4.62)(68.0(phSGp2

+=

+=+= γ

The pressure at the interface is

po =0

ft6.11ft/lb4.62ft/lb721p

in/lb01.5ft/in144ft/lb721ft/lb721p

3

2

OH

1

222

22

1

2

==γ

===

po is the pressure at the free surface of the gasoline.

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Example 2.1 Solution2/2

ft6.14ft/lb4.62ft/lb908p

in/lb31.6ft/in144ft/lb908p

ft/lb908ft/lb721)ft3)(ft/lb4.62(php

3

2

OH

2

222

2

2

2231OHOH2

2

22

==γ

==

=+=+γ=

The pressure at the tank bottom

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Transmission of Fluid Pressure

The required equality of pressure at equal elevations throughout a system is important for the operation ofhydraulic jacks, lifts, and presses, as well as hydraulic controls on aircraft and other type of heavy machinery.(유압기계)

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222211 F

AAFpAFpAF =⇒==

The transmission of fluid pressure throughout a stationary fluid is the principle upon which many hydraulic devices are based.

F2:F1 = A2:A1

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Pressure In Compressible Fluid 1/2

For compressible fluid, ρ=ρ(P,T) , how to determine the pressure variation?

The density must be expressed as a function of one of the other variable in the equation.

For example: Determine the pressure variation in the ideal gas (ex. air at normal conditions).

RTgp

dzdpRTp −=⇒= ρ

−−=>−

−−=−== ∫∫

0

1212

0

12

01

2 expln 2

1

2

1 Tzz

Rgpp

Tzz

Rg

Tdz

Rg

pp

pdp z

z

p

p

γρ −=−= gdzdp

Constant Temperature To

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Pressure In Compressible Fluid 2/2

T=T0=constant

−−

=o

1212 RT

)zz(gexppp

T=Ta-βz

Rg

aa

Rg

aa

aa

a

aza

p

p

z

a

z

a

p

p

a

TTp

Tzpp

Tz

Rg

pp

TzT

RgzT

Rgp

zTdz

Rg

zTRgdz

pdp

dzzTR

pgdzRTpggdzdp

a

a

βββ

ββ

ββ

ββ

ββ

ββ

βρ

//

0

00

)()1(

)1ln(ln

)(ln)ln(ln

)()(

)(

=−=⇒

−=

−=−=

−−

=−

−=

−−=−=−=

∫∫∫pa is the absolute pressure at z=0

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Example 2.2 Incompressible and Isothermal Pressure-Depth Variations The Empire State Building in New York City, one of the tallest

building in the world, rises to a height of approximately 1250ft. Estimate the ratio of the pressure at the top of the building to the pressure at its base, assuming the air to be at a common temperature of 59°F. Compare this result with that obtained by assuming the air to be incompressible with γ=0.0765fb/ft3 at 14.7psi (abs).

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Example 2.2 Solution1/2

( )( ) 955.0

)ft/in144)(in/lb7.14()ft1250)(ft/lb765.0(1

pzz1

pp

orzzpp

222

3

1

12

1

2

1212

=−=−γ

−=

−γ−=

For isothermal conditions

For incompressible conditions

956.0]R)46059)[(Rslug/lbft1716(

)ft1250)(s/ft2.32(exp

RT)zz(gexp

pp

2

o

12

1

2

=

°+°⋅⋅−=

−−

=

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Example 2.2 Solution2/2

Note that there is little difference between the two results. Since the pressure difference between the bottom and top of the building is small, it follows that the variation in fluid density is small and, therefore, the compressible fluid and incompressible fluid analyses yield essentially the same result.By repeating the calculation, for various values of height, h, the results shown in figure are obtained.

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Standard Atmosphere1/2 (option)

Standard atmosphere was first developed in the 1920s. The currently accepted Standard atmosphere is based on a report published in 1962 and updated in 1976.

The so-called U.S. standard atmosphere is an idealized representation of middle-latitude, year-around mean conditions of the earth’s atmosphere.

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Standard Atmosphere2/2 (option)

For example, in the troposphere, the temperature variation is of the form T = Ta – βz

where Ta is the temperature at sea level (z=0) and β is the lapse rate (the rate of change of temperature with elevation).

β

β−=

R/g

aa T

z1pp Pa is the absolute pressure at z=0.

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Measurement of Pressure: Absolute and Gage

Absolute pressure: measured with respect to vacuum.Gage pressure: measured with respect to atmospheric

pressure. atmosphereabsolutegage ppp −=

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Barometers

Mercury Barometer is used to measure atmosphere pressure:Patm＝γh +Pvapor＝γh Pvapor＝0.000023 lb / in2@68oFγ：specific weight of mercury

ghpp vaporatm ρ=−

The height of a mercury column is converted to atmosphere pressure by using

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Manometry

A standard technique for measuring pressure involves the use of liquid column in vertical or inclined tubes.

Pressure measuring devices based on this technique are called manometers. The mercury barometer is an example of one type of manometer, but there are many other configuration possible, depending on the particular application.Piezometer Tube.U-Tube manometer.Inclined-Tube manometer.

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Piezometer Tube

The fundamental equation isP = P0 + γh >> PA = γ1 h1PA : gage pressure ( P0=0)γ1 :the specific weight of the liquid in the containerh1: measured from the meniscus at the upper surface to point(1)Only suitable if the pressure in the container is greater than atmospheric pressure, and the pressure to be measured must be relatively small so the required height of the column is reasonable. The fluid in the container must be a liquid rather than a gas.

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Simple U-Tube Manometer

The fluid in the manometer is called the gage fluid.

A(1)(2)(3)OpenPA + γ1 h 1 = Popen (= Patm) +γ2h 2>> PA (gage pressure)=γ2h 2 –γ1 h 1If pipe A contains a gas

then γ1h 1≈ 0>> PA =γ2h 2

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Example 2.4 Simple U-Tube Manometer

A closed tank contains compressed air and oil (SGoil = 0.90) as is shown in Figure E2.4. A U-tube manometer using mercury (SGoil = 13.6) is connected to the tank as shown. For column heights h1 = 36 in., h2 = 6 in., and h3 = 9 in., determine the pressure reading (in psi) of the gage.

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Example 2.4 Solution1/2

221oilair1 p)hh(pp =+γ+=

As we move from level (2) to the open end, the pressure must decrease by γHgh3, and at the open end the pressure is zero. Thus, the manometer equation can be expressed as

0h)hh(phpp 3Hg21oilair3Hg21 =γ−+γ+→γ==

or 0h))(SG()hh)()(SG(p 3OHHg21OHoilair 22=γ−+γ+

The pressure at level (1) is

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Exmaple 2.4 2/2

+

+

−= tf129)lb/ft 4.62)(6.13( ft

12636)lb/ft 4.62)(9.0(p 33

air

2air lb/ft 440p =

The value for pair

So that

The pressure reading (in psi) of the gage

psi 06.3/ftin. 144

lb/ft 440p 22

2

gage ==

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Differential U-Tube Manometer

(2) = (3)PA＋γ1h1= γ2h2 +γ3h3+ PB

The pressure difference isPA－ PB＝γ2h2＋γ3h3－γ1h1

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Example 2.5 U-Tube Manometer

As will be discussed in Chapter 3, the volume rate of flow, Q, through a pipe can be determined by means of a flow nozzle located in the pipes as illustrated in Figure. the nozzle creates a pressure drop, pA － pB, along the pipe which is related to the flow through the equation , where K is a constant depending on the pipe and nozzle size. The pressure drop is frequently measured with a differential U-tube manometer of the type illustrated. (a) Determine an equation for pA － pB in terms of the specific weight of the flowing fluid, γ1, the specific weight of the gage fluid, γ2, and the various heights indicated. (b) For γ1= 9.80kN/m3 , γ2 = 15.6 kN/m3 , h1 = 1.0m, and h2 = 0.5m, what is the value of the pressure drop, pA － pB?

BA ppKQ −=

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Example 2.5 Solution

(Ans)

we start at point A and move vertically upward to level (1), the pressure will decrease by γ1h1 and will be equal to pressure at (2) and (3). We can now move from (3) to (4) where the pressure has been further reduced by γ2h2 . The pressure at levels (4) and (5) are equal, and as we move from (5) to B the pressure will increase byγ1(h1 + h2)

)(hppp)hh(hhp

212BA

B2112211A

γ−γ=−⇒=+γ+γ−γ−

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Inclined-Tube Manometer (option)

To measure small pressure change, an inclined-tube manometer is frequently used:PA +γ1h1 –γ2l2sinθ –γ3h3 = PB

PA – PB =γ2l2sinθ +γ3h3 –γ1h1

If pipe A and B contain a gas thenγ3h3≒γ1h1≒0>> l2 = ( PA – PB ) /γ2 sinθ

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BUOYANCY 1/2

Buoyancy: The net vertical force acting on any body which is immersed in a liquid, or floating on its surfacedue to liquid pressure. FB

Consider a body of arbitrary shape, having a volume V, that is immersed in a fluid,

We enclose the body in a parallelepiped and draw a free-body diagram of parallelpiped with body removed as shown in (b).

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BUOYANCY 2/2

[ ]VA)hh(A)hh(FA)hh(FF

WFFF

1212B

1212

12B

−−−−=−=−

−−=

γγγ

fluids gsurroundin ofdensity theis f

fB VgFρ

ρ=

FB is the force the body is exerting on the fluid.W is the weight of the shaded fluid volume (parallelepiped minus body).A is the horizontal area of the upper (or lower) surface of the parallelepiped.

For a submerged body, the buoyancy force of the fluid is equal to the weight of displaced fluid and is directly vertically upward

A

-> Archimedes’ Principle

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Example 2.10 Buoyant Force on a Submerged Object A spherical buoy has a diameter of 1.5 m, weighs 8.50kN, and is

anchored to the seafloor with a cable as is shown in Figure a. Although the buoy normally floats on the surface, at certain times the water depth increases so that the buoy is completely immersed as illustrated. For this condition what is the tension of the cable?

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Example 2.10 Solution

WFT B −+−=0

With γ= 10.1 kN/m3 and V = πd3/6

N 10 1.785 ]m) 1.5 [(k/6)( ) N/m 10 10.1 ( F 4333B ×=×=

VFB γ=

The tension in the cable

kN35.9N10850.0N10785.1T 44 =×−×=

FB is the buoyant force acting on the buoy, W is the weight of the buoy, and T is the tension in the cable. For Equilibrium

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The Line of Action of FB and C.G.

The line of action of buoyancy force, which may be found using the method of “hydrostatic force on submerged surfaces,” acts through the centroid of the displaced volume.

The point through which the buoyant force acts is called the center of buoyancy.

C.G: The body force due to gravity on an object act through its center of gravity (C.G.).

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Stability

Stability? Stable? Unstable?A body is said to be in a stable

equilibrium position if, when displaced, it returns to its equilibrium position. Conversely, it is an unstable equilibrium position if, when displaced (even slightly), it moves to a new equilibrium position.

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Stability of Immersed Body

The location of the line of action of the buoyancy force determines stability.

While C.G. is below the center of buoyancy, a rotation from its equilibrium position will create a restoring couple formed by the weight and the buoyancy force.

If C.G. is above the center of buoyancy,…..

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Stability of Floating Body

The determination of stability depends in a complicated fashion on the particular geometry and weight distribution of the body.