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Physics 106 Lecture 13

Fluid MechanicsSJ 7th Ed.: Chap 14.1 to 14.5

• What is a fluid?• Pressure• Pressure varies with depth• Pascal’s principle• Methods for measuring pressure• Buoyant forces • Archimedes principle• Fluid dynamics assumptionsy p• An ideal fluid• Continuity Equation• Bernoulli’s Equation

What is a fluid?

Solids: strong intermolecular forcesdefinite volume and shaperigid crystal lattices, as if atoms on stiff springsdeforms elastically (strain) due to moderate stress (pressure) in any direction

Fluids: substances that can “flow”no definite shape molecules are randomly arranged, held by weak cohesive intermolecular forces and by the walls of a containerliquids and gases are both fluids

Liquids: definite volume but no definite shapeoften almost incompressible under pressure (from all sides)can not resist tension or shearing (crosswise) stress

Fluid Statics - fluids at rest (mechanical equilibrium) Fluid Dynamics – fluid flow (continuity, energy conservation)

can not resist tension or shearing (crosswise) stressno long range ordering but near neighbor molecules can be held weakly together

Gases: neither volume nor shape are fixedmolecules move independently of each othercomparatively easy to compress: density depends on temperature and pressure

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Mass and Density

• Density is mass per unit volume at a point:

Vm or

Vm

≡ρΔΔ

≡ρ • scalar• units are kg/m3, gm/cm3..• ρwater= 1000 kg/m3= 1.0 gm/cm3

• Volume and density vary with temperature - slightly in liquids• The average molecular spacing in gases is much greater than in liquids.

Force & Pressure

• The pressure P on a “small” area ΔA is the ratio of the magnitude of the net force to the area

A/FP or AF P =

ΔΔ

=n̂PΔA

• Pressure is a scalar while force is a vector• The direction of the force producing a pressure is perpendicular

to some area of interest

n̂APAPF Δ=Δ=ΔnPΔA

• At a point in a fluid (in mechanical equilibrium) the pressure is the same inequilibrium) the pressure is the same in any directionh

Pressure units:1 Pascal (Pa) = 1 Newton/m2 (SI)1 PSI (Pound/sq. in) = 6894 Pa.1 milli-bar = 100 Pa.

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Forces/Stresses in Fluids

• Fluids do not allow shearing stresses or tensile stresses (pressures)

Tension Shear Compression

• The only stress that can be exerted on an object submerged in a static fluid is one that tends to compress the object from all sides

• The force exerted by a static fluid on an object is always• The force exerted by a static fluid on an object is always perpendicular to the surfaces of the object

Question: Why can you push a pin easily into a potato, say, using very little force, but your finger alone can not push into the skin even if you push very hard?

A sensor to measure absolute pressure

• The spring is calibrated by a known force• Vacuum is inside• The force due to fluid pressure presses on the top of the piston and

compresses the spring until the spring force and F are equal.• The force the fluid exerts on the piston is then measured

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Pressure in a fluid varies with depth

Fluid is in static equilibrium The net force on the shaded volume = 0

• Incompressible liquid - constant density ρ

• Horizontal surface areas = A y1 F1

y=0

P1Horizontal surface areas A• Forces on the shaded region:

– Weight of shaded fluid: Mg– Downward force on top: F1 =P1A– Upward force on bottom: F2 = P2A

y2 Mg

F2 P2

h

MgAPAP F 2y −−==∑ 10

AhVM ρ=Δρ=

• In terms of density, the mass of the shaded fluid is:

The extra pressure at extra depth h is:

ghPPP ρ=−=Δ 12 yyh 21 −≡ghAAP AP 2 ρ+≡∴ 1

Pressure relative to the surface of a liquid

ghPP h ρ+= 0

air

P0

h

Example: The pressure at depth h is:

P is the local atmospheric (or ambient)

All points at the same depth are at the same pressure; otherwise, the fluid could not be in equilibriumThe pressure at depth h does not depend

Ph

liquid

P0 is the local atmospheric (or ambient) pressurePh is the absolute pressure at depth hThe difference is called the gauge pressure

P di ti l The pressure at depth h does not depend on the shape of the container holding the fluid

P0

Ph

Preceding equations also hold approximately for gases such as air if the density does not vary much across h

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Atmospheric pressure and units conversions

• P0 is the atmospheric pressure if the liquid is open to the atmosphere.

• Atmospheric pressure varies locally due to altitude, temperature, motion of air masses, other factors.

• Sea level atmospheric pressure P0 = 1.00 atm= 1.01325 x 105 Pa = 101.325 kPa = 1013.25 mb (millibars)= 29.9213” Hg = 760.00 mmHg ~ 760.00 Torr= 14.696 psi (pounds per square inch)

Pascal (Pa)

bar (bar) atmosphere(atm)

torr(Torr)

pound-force persquare inch (psi)

1 Pa ≡ 1 N/m2 10−5 9 8692×10−6 7 5006×10−3 145 04×10−61 Pa ≡ 1 N/m 10 9.8692×10 7.5006×10 145.04×10

1 bar 100,000 ≡ 106 dyn/cm2

0.98692 750.06 14.5037744

1 atm 101,325 1.01325 ≡ 1 atm 760 14.696

1 torr 133.322 1.3332×10−3 1.3158×10−3 ≡ 1 Torr; ≈ 1 mmHg

19.337×10−3

1 psi 6.894×103 68.948×10−3 68.046×10−3 51.715 ≡ 1 lbf/in2

Pressure Measurement: Barometer

• Invented by Torricelli (1608-47)• Measures atmospheric pressure P0 as it varies

with the weather• The closed end is nearly a vacuum (P = 0)• One standard atm = 1.013 x 105 Pa.

near-vacuum

Mercury (Hg) ghP Hgρ=0

One 1 atm = 760 mm of Hg= 29.92 inches of Hg

m 0.760 )m/s .)(m/kg10(13.6

Pa 101.013 g

Ph 23

5

Hg

=×

×=

ρ=

80930

How high is the Mercury column?

m 10.34 )m/s .)(m/kg10(1.0

Pa 101.013 g

Ph 23

5

water=

××

=ρ

=8093

0

How high would a water column be?

Height limit for a suction pump

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Pressure Measurements: Manometer

• Measures the pressure of the gas in the closed vessel

• One end of the U-shaped tube is open to the atmosphere at pressure P0

• The other end is connected to theThe other end is connected to the pressure PA to be measured

• Points A and B are at the same elevation.• The pressure PA supports a liquid

column of height h• The Pressure of the gas is:

ghPP P BA ρ+== 0 What effect would increasing P have on

Absolute vs. Gauge Pressure• The gauge pressure is PA – P0= ρgh

– What you measure in your tires• Gauge pressure can be positive or negative.• PA is the absolute pressure – you have to add the ambient pressure P0 to

the gauge pressure to find it.

increasing P0 have on the column height?

Example: Blood Pressure

• Normal Blood pressure is: 120 (systolic = peak) over 70 (diastolic = resting) – in mmHg.

• What do these numbers mean?• How much pressure do they correspond to in Pa. and psi?

Recall: 760.00 mmHg = 1.01325 x 105 Pa =14.696 psi

(101,325/760.00) = 133.322 Pa./mmHg

(101,325/14.696) = 6894. Pa./psi

Systolic pressure of 120= 16,000 Pa.= 2 32 psi

Mercury Sphygmomanometer

Aneroid Sphygmomanometer

= 2.32 psi

Diastolic pressure of 70= 9333 Pa.= 1.354 psi

Gauge or absolute?

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Pascal’s PrincipleA change in the pressure applied to an enclosed incompressible fluid is transmitted undiminished to every point of the fluid and to the walls of the container.

ΔFΔP

• The pressure in a fluid depends on depth h and on the value of P0 at the surface

• All points at the same depth have the same pressure.

Example: open container

ghPP h ρ+= 0

p0

ph

Add i t f A ith l d b ll it & i ht W• Add piston of area A with lead balls on it & weight W. Pressure at surface increases by ΔP = W/A

gh P P exth ρ+=

• Pressure at every other point in the fluid (Pascal’s law), increases by the same amount, including all locations at depth h.

PPP ext Δ+= 0

Ph

Pascal’s Law Device - Hydraulic pressA small input force generates a large output force

• Assume the working fluid is incompressible• Neglect the (small here) effect of height on pressure

• The volume of liquid pushed down on th l ft l th l h d

xA xA 21 Δ=Δ 21

• Assume no loss of energy in the fluid no friction etc

the left equals the volume pushed up on the right, so:

AA

xx

1

2

2

1=ΔΔ

∴

xF Work xF Work 2 21 Δ==Δ= 211

the fluid, no friction, etc.

Other hydraulic lever devices using Pascal’s Law:

• Squeezing a toothpaste tube• Hydraulic brakes• Hydraulic jacks• Forklifts, backhoes

AA

xx

FF

2

1

1

2

1

2=ΔΔ

=∴

mechanical advantage

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Archimedes Principle C. 287 – 212 BC• Greek mathematician, physicist and engineer• Computed π and volumes of solids• Inventor of catapults, levers, screws, etc.• Discovered nature of buoyant force – Eureka!

Why do ships float and sometimes sink?Why do objects weigh less when submerged in a fluid?Why do objects weigh less when submerged in a fluid?

ball of liquidin equilibrium

hollow ball same

upward force

identical pressures at every point

• An object immersed in a fluid feels an upward buoyant force that equals the weight of the fluid displaced by the object. Archimedes’s Principle

– The fluid pressure increases with depth and exerts forces that are the same whether the submerged object is there or not.

– Buoyant forces do not depend on the composition of submerged objects. – Buoyant forces depend on the density of the liquid and g.

Archimedes’s principle - submerged cube

• The extra pressure at the lower surface compared to the top is:

fltb t ghPPP ρ=−=Δ

A cube that may be hollow or made of some material is submerged in a fluidDoes it float up or sink down in the liquid?

• The pressure at the top of the cube causes a downward force of Ptop A

• The larger pressure at the bottom of the cube causes an upward force of Pbot A

• The upward buoyant force B is the weight of the fluid displaced by the cube:

fltopbot ghPPP ρ=−=Δ

gMgVghAPA B flfl fl =ρρ=Δ= =

• The weight of the actual cube is:gV gM F cubecubeg ρ==

Similarly for irregularly shaped objects

ρcube is the average density

The cube rises if B > Fg (ρfl > ρcube)The cube sinks if Fg> B (ρcube > ρfl)

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Archimedes's Principle: totally submerged object

Object – any shape - is totally submerged in a fluid of density ρfluid

The upward buoyant force is the weight of displaced fluid:

fluiduidfl gV B ρ=The downward gravitational force on the

The direction of the motion of an object

The downward gravitational force on the object is:

objectobjectobjectg gV gM F ρ==The volume of fluid displaced and the object’s volume are equal for a totally submerged object. The net force is:

objectobjectuidfl[ gnet V g ] FBF ρ−ρ=−=

The direction of the motion of an object in a fluid is determined only by the densities of the fluid and the object– If the density of the object is less than

the density of the fluid, the unsupported object accelerates upward

– If the density of the object is more than the density of the fluid, the unsupported object sinks

The apparent weight is the external force needed to restore equilibrium, i.e.

netgapparent F B F W −=−=

Archimedes’s Principle: floating object

At equilibrium the upward buoyant force is balanced by the downward weight of the object:

An object sinks or rises in the fluid until it reaches equilibrium. The fluid displaced is a fraction of the object’s volume.

ggnet F B 0 FBF =⇒=−=

The volume of fluid displaced Vfluid corresponds to the portion of the object’s volume below the fluid level and is always less than the object’s volume.Equate:

objectobjectgfluiduidfl gVFgV B ρ==ρ=

Solving:

objectobjectfluiduidfl VV ρ=ρ

VV

uidfl

object

object

fluid ρ

ρ=

Objects float when their average density is less than the density of the fluid they are in.

The ratio of densities equals the fraction of the object’s volume that is below the surface

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Archimedes’s Principle: King’s Crown

King to Archimedes: “Is the crown made of pure gold?”• The crown’s density ρcrown = mcrown/ Vcrown should = ρgold = 19,300 kg/m3

• Weigh the crown in air to find mcrown• But: how to find Vcrown??

• The Buoyant force equals the difference in scale readings...

Solution:• Weigh crown in air to find mcrown• Weigh it again fully submerged in a fluid• The apparent weight T2 is:

B F T g2 −=

crownfl2g gV TF B ρ=−=• solve:

gTF

Vfl

2gcrown ρ

−=

TFF

flg

gcrown ρ

−=ρ∴

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Example: What fraction of an iceberg is underwater?

VV

VV

uidfl

object

object

fluid total

underwater ρ

ρ==

displaced seawaterApply Archimedes’ Principle

glacial fresh water ice

3iceobject kg/m . 3109170 ×=ρ=ρ

3seawaterfluid kg/m . 310031 ×=ρ=ρ

From table:

%. V

V t t l

underwater 0389=∴

Water expands when it freezes. If not.......ponds, lakes, seas freeze to the bottom in winter

Vtotal

What if iceberg is in a freshwater lake?3

freshwaterfluid kg/m . 310001 ×=ρ=ρ

1.7% V

V total

underwater 9=

Floating objects are more buoyant in saltwaterFreshwater tends to float on top of seawater...

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Water has special properties

Maximum density at 4o C.• ρwater= 1000 kg/m3= 1.0 gm/cm3

• turnover of ponds and lakes• ice floats, so bodies of water do not freeze to the bottom

Fluids’ Flow is affected by their viscosity

• Viscosity measures the internal friction in a fluid.

Low viscosity• gases

Medium viscosity• water• other fluids that pour and flow easily

High viscosity• honey• oil and grease• glass

• Viscous forces depend on the resistance that two adjacent layers of fluid have to relative motion.

• Part of the kinetic energy of a fluid is converted to internal energy, analogous to friction for sliding surfaces

y g

Ideal Fluids – four approximations to simplify the analysis of fluid flow:

Th fl id i i i l f i i i l d• The fluid is nonviscous – internal friction is neglected• The flow is laminar (steady, streamline flow) – all particles

passing through a point have the same velocity at any time.• The fluid is incompressible – the density remains constant• The flow is irrotational – the fluid has no angular momentum

about any point. A small paddle wheel placed anywhere does not feel a torque and rotate

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Laminar Flow - Streamlines

Steady state flow – at any point the velocity of particles is constantEach particle of the fluid follows a smooth path called a streamline. Particles’ velocities are tangent to streamlines (field lines).The paths of different particles never cross each otherEvery given fluid particle arriving at a given point has the same velocity

velocity field lines

Every given fluid particle arriving at a given point has the same velocity

Turbulent flow

An irregular flow characterized by small whirlpool-like regionsTurbulent flow occurs when the particles go above some critical (Mach) speedEnergy is dissipated, more work must be done to maintain flow

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Flow of an ideal fluid through a short section of pipe

Constant density and velocity within volume element dVIncompressible fluid means dρ/dt = 0

Mass flow rate = amount of mass crossing area A per unit time= a “current”sometimes called a “mass flux”

length dx

velocity v

cross-section area A

Av dtdxA

)V(dtd

dtdM I rateflow massmass

ρ=ρ=

ρ===

AdxdV cylinder in fluid fo volume ==

length dxAv I rateflow massmass ρ=≡

Av I rateflow volumevol =≡

v J areaflow/unit massmass ρ=≡

Equation of Continuity: conservation of mass• An ideal fluid is moving through a pipe of nonuniform diameter• The particles move along streamlines in steady-state flow• The mass entering at point 1 cannot disappear or collect in the pipe• The mass that crosses A1 in some time interval is the same as the mass that

crosses A2 in the same time interval.

vAvA outflowmassinflowmass ρ==ρ=

ρ1

ρ2

222111 vAvA outflow massinflow mass ρ==ρ=• The fluid is incompressible so:

ttancons a =ρ=ρ 21

vA vA 2211 =∴

• This is called the equation of continuityfor an incompressible fluid ρ1

p• The product of the area and the fluid

speed (volume flux) at all points along a pipe is constant.

The rate of fluid volume entering one end equals the volume leaving at the other endWhere the pipe narrows (constriction), the fluid moves faster, and vice versa

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Bernoulli’s Equation – energy conservation

• An ideal fluid flows through a region where its speed, cross-sectional area, and/or altitude changes.

• The pressure in the fluid varies with these changes.• Application of energy conservation yields the Bernoulli Equation

Daniel Bernoulli, Swiss physicist, 1700 – 1782

OPTIONAL TOPIC

12 EEE W mechpressure by done work −=Δ==Δ

Pressure P1speed v1height y1

Pressure P2speed v2height y2

Volume ΔV = A1Δx1

Volume ΔV = A2Δx2

Equal volumes flow in equal timesdue to continuity equation

time ttime t + Δt

Bernoulli’s Equation – energy conservation

The fluid is incompressible. The continuity equation (no leaks) requires:

2211 xAxA V Δ=Δ=ΔThe net work done on fluid volume by external pressures:

V ]P [P dxAPdxAPdxFdxF W 212121 Δ−=−=−=Δ 221121

Vmm m 1 Δρ=≡= 2

The change in the mechanical energy:11

OPTIONAL TOPIC

Pressure P1speed v1

Pressure P2speed v

Equal volumes flow in equal timesd i i i

12212

1222

1 mgymgymvmvEE E 12 −+−=−=Δ

)yy(gvvPP 21 12212

1222

1 −ρ+−=− ρρ

Equate to the net work and divide through by the volume ΔV:

Rearrange:

gyvPgyvP 21 2222

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212

1 ρ++=ρ++ ρρ

time ttime t + Δt

speed v1height y1

speed v2height y2

Volume ΔV = A1Δx1

Volume ΔV = A2Δx2

due to continuity equation

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Bernoulli’s Equation – energy conservation

Bernoulli’s Equation is often expressed as:

gyvPgyvP 21 2222

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212

1 ρ++=ρ++ ρρ

OPTIONAL TOPIC

]y-y[gPP 121 2ρ+=

Constant gyvP =ρ++ ρ 221

For a fluid at rest Bernoulli’s Equation reduces to the earlier result for variation of pressure with depth:

For flow at constant altitude (y1 = y2) the pressure is lower where ther fluid flows faster:

vPvP 21222

1212

1ρρ +=+

Gases are compressible, but the above pressure / speed relation holds true.

Bernoulli effect causes airplane wings to work

• Streamlines flow around a moving airplane wing• Lift is the upward force on the wing from the air• Drag is the resistance• The lift depends on the speed of the airplane, the area of the wing, its

OPTIONAL TOPIC

curvature, and the angle between the wing and the horizontal

Low Pressure where flow velocity is high

Higher Pressure where flow velocity is low

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Lift – General

• In general, an object moving through a fluid experiences lift as a result of any effect that causes the fluid to change its direction as it flows past the object

• Some factors that influence lift are:

OPTIONAL TOPIC

– The shape of the object– The object’s orientation with respect to the fluid flow– Any spinning of the object– The texture of the object’s surface

Golf Ball

Th b ll i i id b k i• The ball is given a rapid backspin• The dimples increase friction

– Increases lift• It travels farther than if it was not

spinning

Atomizer

• A stream of air passes over one end of an open tube

• The other end is immersed in a liquid

OPTIONAL TOPIC

• The moving air reduces the pressure above the tube

• The fluid rises into the air stream• The liquid is dispersed into a fine

spray of droplets

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