me 3560 presentation chi

ME3560 – Fluid Mechanics

Chapter I. Introduction

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ME 3560 Fluid Mechanics


• Fluid Mechanics is the science that deals with the behavior of fluids atrest or in motion, and the interaction of fluids with solids or other fluidsat the boundaries.



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1.1 Brief History of Fluid Mechanics• One of the first engineering problems was the supply of water to citiesfor domestic use and for the irrigation of crops.

•Roman aqueducts are a good example of water systems constructed atthe beginning of civilization.

•Roman aqueducts inSegovia, Spain, builtaround the 1st

Century A.D.



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•It has been found that from283 to 133 BC the Hellenistic city ofPergamon (Turkey) built a series of pressurized led and clay pipelines,up to 45 kmlong that operated at a pressure exceeding 1.7 Mpa (180 mof head).

•The earliest contribution theory to fluid mechanics was made byArchimedes (285–212 BC). He formulated and applied the buoyancyprinciple.

•During the Middle Ages the application of fluid machinery expanded,piston pumps were used for dewatering mines, water and wind millswere perfected to grind grains and forge metal.



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•The figure shows a mine hoist powered by areversible water wheel (Georgius Agricola 1556).

•The development of fluid systems and machinescontinued during the Renaissance. The scientificmethod was perfected and adopted throughout Europe.

• Simon Stevin (1548–1617), Galileo Galilei (1562–1642), EdmeMariotte (1620–1684), and Evangelista Torricelli (1608–1647) wereamong the first to apply the scientific method to investigate hydrostaticpressure distributions and vacuums.

•Blaise Pascal integrated and refined the work developed by thesescientists.



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• The Italian monk Benedetto Castelli (1577–1644) was the first personto publish a statement of the continuity principle for fluids.

•Sir Isaac Newton (1643–1727) applied his laws to fluids and exploredfluid inertia and resistance, free jets, and viscosity.

•The Swiss engineer Daniel Bernoulli (1700–1782) and his associateLeonard Euler (1707–1789) built upon Newton’s studies and defined theenergy and momentumequations.

•Bernoulli published in 1738 his treatiseHydrodynamica. This may beconsidered the first fluid mechanics text.

•Jean d’Alembert (1717–1789) developed the idea of velocity andacceleration components, a differential expression of continuity , and his“paradox” of zero resistance to steady uniformmotion.



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• Fluid mechanics theory up through the end of the 18th century had littleimpact on engineering since fluid properties and parameters were poorlyquantified.

•The French school of engineering led by Riche de Prony (1755–1839)along with Ecole Polytechnic and the Ecole Ponts et Chaussees were thefirst to incorporate calculus and scientific theory to the engineeringcurriculum. This brought a change to the engineering theory by making itmore practical and capable of solving real world problems.

•Scientists such as Antonie Chezy (1718–1798), Louis Navier (1785–1836), Gaspar Coriolis (1792–1843), Henry Darcy (1803–1858)contributed to fluid engineering and theory and were students and/orinstructors at these schools



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• By the mid 19th century the advances in fluid mechanics were comingfrom different fronts:•Jean Poiseuille (1799–1869) had accurately measured flowin capillarytubesfor multiple fluids.•Gotthilf Hagen (1797–1884) had differentiated between laminar andturbulent flowin pipes.•Osborn Reynolds (1842–1912) continued Hagen’s work and developedthe dimensionless number that bears his name.

•In parallel work Louis Navier and George Stokes (1819–1903)completed the general equations of fluid motion with friction (Navier–Stokes equations).

•James Francis (1815–1892) and Lester Pelton (1829–1908) pioneeredwork in turbines.•Clemens Herschel (1842–1930) invented the Venturi meter



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• Irish and English engineers such as WilliamThompson, Lord Kelvin(1824–1907), WilliamStrutt, Lord Rayleigh (1842 –1919), and SirHorace Lamb (1849–1934) investigated problems such as dimensionalanalysis, irrotational flow, vortex motion, cavitation, and waves.

•At the dawn of the 20th century the Wright brothers (Wilbur, 1867–1912; Orville, 1871–1948) through application of theory andexperimentation perfected the airplane.

•In 1904, Ludwig Prandtl (1875–1953) showed that fluid flows can bederived into a layer near the walls, theboundary layer, where the frictioneffects are significant and an outer layer where such effects are negligibleand the simplified Euler and Bernoulli equations are applicable.



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• By the mid 20th century the existing theories were adequate for thetasks at hand and the fluid properties and parameters were well defined,thus supporting a enormous expansion of the aeronautical, chemical,industrial and water resources sector.

• In the late 20th century, Fluid Mechanics research was dominated by thedevelopment of the digital computer. The ability to solve large complexproblems, such as global climate modeling or to optimize the design of aturbine blade.

•The principles that we will study in this curse apply to flows rangingfrom very small to extremely large scales.



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1.2 Definition of a Fluid• A fluid is defined as a substance that deforms continuously when actedon by a shearing stress of any magnitude.

•When common solids such as steel or other metals are acted on by ashearing stress, they will initially deform(usually a very smalldeformation), but they will not continuously deform(flow).•Common fluids such as water, oil, and air satisfy the definition of afluid—that is, they will flowwhen acted on by a shearing stress.•Some materials, such as slurries, tar, putty, toothpaste are not easilyclassified since they will behave as a solid if the applied shearing stressis small, but if the stress exceeds some critical value, the substance willflow. The study of such materials is called rheology.



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•To describe the behavior of fluids at rest or in motion, we consider theaverage, or macroscopic, value of the quantity of interest.

•The average is evaluated over a small volume containing a largenumber of molecules.

•The volume is small compared with the physical dimensions of thesystem of interest, but large compared with the average distancebetween molecules.

•For gases at normal pressures and temperatures, the spacing is on theorder of 10−6 mm. For gases, the number of molecules per cubicmillimeter is on the order of 1018.

•For liquids it is on the order of 10−7 mm. For liquids, the number ofmolecules per cubic millimeter is on the order of 1021.



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1.3 The Non–Slip Condition• Fluid flow is often confined by solid surfaces, and it is important tounderstand howthe presence of solid surfaces affects fluid flow.

•.Consider the flowof a fluid in a stationary pipe or over a solid surfacethat is nonporous. Experimental observation indicates that a fluid inmotion comes to a complete stop at the surface and assumes zerovelocity relative to that surface.

•That is, a fluid in direct contact with a solid “sticks” to the surface dueto viscous effects and there is no slip.

•This is known as thenon–slip condition.

• The fluid property responsible for thenon–slip condition is theviscosity.



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• A fluid layer adjacent to a moving surface has the same velocity as thesurface

•A consequence of the non–slip condition is that all velocity profilesmust have zero values with respect to the surface at the points ofcontact.

•Another consequence of the non–slip condition is thesurface drag,which is the force a fluid exerts on a surface in the direction of the flow.



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1.4 Classification of Fluid Flows• There is a wide variety of fluid flowproblems and it is convenient toclassify thembased on some common characteristics to group them.

Viscous versus Inviscid Regions of Flow•When two fluid layers move relative to each other, a friction forcedevelops between themand the slower layer tries to slowdown thefaster layer.

•This internal resistance to flowis quantified by theviscosity. Theviscosity is caused by cohesive forces between the molecules in liquidsand by molecular collisions in gases. There is no fluid with zeroviscosity.

•Flows in which the viscous effects are important are calledviscousflows.



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• In many flows of practical interest, there are regions where the viscousforces are negligibly small compared to inertial of pressure forces.

• Neglecting the viscous effects in suchinviscid flow regions greatlysimplifies the analysis without much loss in accuracy.

• The development of viscous and inviscidregions of flowas a result of inserting a flatplate parallel to a fluid streamof uniformvelocity is shown in the picture.

•The fluid sticks to the plate on both sidesdue to the non – slip condition.

• Two zones are present, a viscous flowregion (boundary layer) and aninviscid flow region.



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Internal versus External Flow•A fluid flow is internal or external depending on whether the fluid isforced to flowin a confined channel or over a surface.

•The flow of an unbounded fluid over a surface such as a plate, a wire,or a pipe isexternal flow.

•The flow in a pipe or a duct isinternal flow if the fluid is completelybounded by solid surfaces.

•The flow of liquids in a duct which is only partially filled is calledopen channel flow. Flow of rivers is an example of this type of flows.

• Internal flows are dominated by the influence of viscosity throughoutthe flow field.• In external flows the viscous effects are limited to boundary layersnear solid surfaces and to wake regions downstreamof bodies.



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Compressible versus Incompressible Flow• A flow is classified as beingcompressible or incompressible,depending on the level of variation of density during flow.

•Incompressibility is an approximation and a flowis said to beincompressible if the density remains constant, that is, the volume ofevery portion of fluid remains unchanged.

•The densities of liquids are essentially constant (incompressible).

•Gases on the other hand are highly compressible. However, gas flowcan often be considered incompressible if the density changes are under5 percent, which is usually the case whenMa < 0.3

�� =�

�=��

��



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• The speed of sound in air (roomtemperature, sea level) isc = 346 m/s.Therefore, the compressibility effects in air can be neglected at speedsunder about 100 m/s (≈ 220 mi/hr).

Laminar versus Turbulent Flow• Some flows are smooth and orderly while others are rather chaotic.

• The highly ordered fluid motion characterized by smooth layers offluid is calledlaminar.

• The highly disordered fluid motion that typically occurs at highvelocities and is characterized by velocity fluctuations is calledturbulent.

• Flow that alternates between being laminar and turbulent is calledtransitional.



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Natural (or Unforced) versus Forced Flow• A forced flow is a flow in which the fluid is forced to flowover asurface or in a pipe by external means such as pump or a fan.

• In natural flows any fluid motion is due to natural means such as thebuoyancy effect.

Steady versus Unsteady Flow• The termssteady anduniform are used frequently in engineering.

• The termsteadyimplies no change at a point with time. The oppositeof steady isunsteady.

•The termuniform implies no change with location over a specifiedregion.



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• The termsunsteady and transient are often used interchangeably,however, in fluid mechanicsunsteady applies to any flowthat is notsteady, andtransientapplies to developing flows.

One–, Two–, and Three–Dimensional Flows• The best way to describe a flowfield is through the velocitydistribution, thus the flowcan be one–, two–, or three –dimensional,depending on the number of coordinate directions required to describethe flow.

• In the most general case, a fluid flowis described by three–dimensions[V(x, y, z) or V(r, θ, z)].

• In many instances, the variation of the velocity in certain directionscan be small relative to the variation in other directions and can beignored with negligible error. Thus the flowcan be 1–D or 2–D.



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1.5 System and Control Volume•A systemis a collection of matter of fixed identity (always the sameatoms or fluid particles), which may move, flow, and interact with itssurroundings.

•A systemis a specific, identifiable quantity of matter. It may consist of arelatively large amount of mass or it may be an infinitesimal size.

•A systemmay interact with its surroundings by various means (by thetransfer of heat or the exertion of a pressure force, for example).

• A system may continually change size and shape, but italwayscontains the same mass.



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•A control volume, is a volume in space (a geometric entity, independentof mass) through which fluid may flow.

•In fluid mechanics, it is difficult to identify and keep track of a specificquantity of matter.

• In several cases, the main interest is in determining the forces puton adevice rather than in the information obtained by following a givenportion of the air (a system) as it flows along.

• For these situations it is more adequate to use the control volumeapproach.

•Identify a specific volume in space (a volume associated with the deviceof interest) and analyze the fluid flowwithin, through, or around thatvolume.



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• In general, thecontrol volumecan be a moving volume, although formost situations we will use only fixed, non-deformable control volumes.• The matter within a control volume may change with time as the fluidflows through it.• The amount of mass within the volume may change with time.•The control volume itself is a specific geometric entity, independent ofthe flowing fluid.



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•All of the laws governing the motion of a fluid are stated in their basicform in terms of a systemapproach.

•For example, “the mass of a systemremains constant,” or “the time rateof change of momentumof a systemis equal to the sumof all the forcesacting on the system.”



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1.6 Dimensions, Dimensional Homogeneity, andUnits.•The study of fluid mechanics requires to develop a systemfordescribing the fluid characteristicsqualitatively andquantitatively.

•The qualitative aspect serves to identify the nature, or type, of thecharacteristics (such as length, time, stress, and velocity).

•The quantitative aspect provides a numerical measure of thecharacteristics. The quantitative description requires both a number anda standard (unit) by which various quantities can be compared.

221 −−− === MLTFLTaLTV &&&

][2

NFs

ma

s

mV



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•All theoretically derived equations aredimensionally homogeneous,and all additive separate terms must have the same dimensions.

For example, the equation for the velocity, V, of a uniformly accelerated body is

where V0 is the initial velocity, a the acceleration, and t the time interval. In terms of dimensions the equation is

and thus this equation is dimensionally homogeneous.

atVV += 0

111 −−− += LTLTLT &



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1.6.1 Systems of Units•In addition to the qualitative description of the various quantities ofinterest, it is necessary to have a quantitative measure of any givenquantity.•We will consider three systems of units that are commonly used inengineering.- International System(SI)

Quantity Unit

Length Meter (m)

Time Second (s)

Mass Kilogram (kg)

Temperature Kelvin (K)

15.273+= CK o



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Quantity Unit

Force Newton (N)

Work Joule (J)

Power Watt (W)2

2

m/s81.9

m/s1N1J/s1W

m1N1J

1m/s1kg1N

==⋅==

⋅=⋅=

mg;gW



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-British Gravitational (BG) System.

Quantity Unit

Length Foot (ft)

Time Second (s)

Mass Slug (slug)

Temperature Rankine (oR)

Force Pound (lb)

Work lb⋅ftPower lb⋅ft/s

2

2

oo

32.2ft/s

ft/s1slug1lb

459.67FR

==⋅=

+=

mg;gW



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-English Engineering (EE) System.•In the EE system, units for forceand mass are defined independently.

Quantity Unit

Length Foot (ft)

Time Second (s)

Mass Pound mass (lbm)

Temperature Rankine (oR)

Force Pound (lb)

Work lb⋅ftPower lb⋅ft/s

2

2

oo

32.2ft/s

32.2lbm1slug

ft/s2.321lbm1lb

459.67FR

===

⋅=+=

mg;gW



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1.7 Modeling in Engineering•An engineering device can be studied eitherexperimentally oranalytically.

•The experimental approach is advantageous because it deals with theactual physical systemand the desired quantity is determined bymeasurement.

•The experimental approach is expensive, time consuming and oftenimpractical. Additionally the systemto be studied might not exist.

•On the other hand, the analytical approach (including numericalapproach) has the advantage of being fast and inexpensive.

•The results obtained are subject to the accuracy of the assumptions,approximations, and idealizations made in the analysis.



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• The description of most scientific problems involve equations thatrelate the changes in some key variables to each other.

• Usually, the smaller the increment chosen in the changing variables, themore general and accurate the description.

•Therefore, differential equations are used to investigate a wide varietyof problems in engineering and sciences.

• However, may problems can be studied without the need of usingdifferential equations.

• In this class we will learn different tools, additional to the solutionofdifferential equations to study problems in Fluid Mechanics



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1.8 Continuum• Matter is made up of atoms that are widely spaced in the gas phase.

•However, it is very convenient to disregard the atomic nature of asubstance and treat it as a continuous, homogeneous matter with noholes, that is, acontinuum.

• The continuumidealization allows the treatment of properties as pointfunctions and to assume that properties vary continuously in spacewithout discontinuities.

•This assumption is valid as long as the size of the systemconsidered islarge relative to the space between the molecules. This will be the case inall problems we analyze in this course of Fluid Mechanics



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•To describe the behavior of fluids at rest or in motion, we consider theaverage, or macroscopic, value of the quantity of interest.

•The average is evaluated over a small volume containing a largenumber of molecules.

•The volume is small compared with the physical dimensions of thesystem of interest, but large compared with the average distancebetween molecules.

•For gases at normal pressures and temperatures, the spacing is on theorder of 10−6 mm. For gases, the number of molecules per cubicmillimeter is on the order of 1018.

•For liquids it is on the order of 10−7 mm. For liquids, the number ofmolecules per cubic millimeter is on the order of 1021.



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1.9 Measures of FluidMass andWeight1.9.1 Density•The density of a fluid is defined as its mass per unit volume.

•The specific volume of a fluid is defined as the ratio of the volumeoccupied by the volume to its mass.

•The density of liquids is assumed to be constant –incompressible fluids•The density of gases depends on the temperature and pressure of thesystem. For example, for ideal gases:

=33

,ft

slug

m

kg

V

mρ

==

lbm

ft

slug

ft

kg

m

m

Vv

333

,,1

ρ

RTp ρ=



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1.9.2 Specific Weight•Specific weight(γ) is weight per unit volume. gργ =

1.9.3 Specific Gravity

•Specific Gravity(SG) is defined as the ratio of the density of the fluidto the density of water at 4 °C (39.2 °F):ρ=1.94 slugs/ft3=1000 kg/m3.

COHCOH oo

SG4@4@ 22

γγ

ρρ ==



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1.10Viscosity•If P is applied to the upper plate, it will move continuously with avelocity,U.•The fluid in contact with the upper plate moves with a velocity,U.•The fluid in contact with the bottomfixed plate has a zero velocity.•The fluid between the two plates moves with velocityu=u(y)=Uy/b•A velocity gradient du/dy=U/b, develops in the fluid between the plates.•The experimental observation that the fluid “sticks” to the solidboundaries is referred to as theno-slip condition.



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•It can be experimentally determined that

• For a large number of fluids the relation between shear stress andvelocity gradient is linear:

y

u

b

U

A

P

∂∂→ταα

•τ shear stress in a fluid in motion•∂ u/ ∂ y. Rate of shearing strain (Velocity gradient)

y

u

∂∂= µτ •µ �Absolute (dynamic) viscosity

LT

M

LT

LLT

LM

=

⋅⋅

=1

/

1

GradientVelocity

StressShear:Dimensions

22

sPasm

kg

m

sN:Units

2⋅=

⋅=⋅

•1poise = 0.1 N⋅s/m2

Dynamic viscosity is property that relates shearing stress and fluid motion



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•Fluids for which theshearing stress islinearlyrelated to the rate ofshearing strain (also referredto as rate of angulardeformation) are designatedasNewtonian fluids.•µ = µ (T)•For gasesµ increases asTdoes.•For liquidsµ decreases asTdoes.



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•Fluids for which the shearingstress is not linearly related tothe rate of shearing strain aredesignated asnon-Newtonianfluids.

•Quite often viscosity appearsin fluid flow problemscombined with the density inthe form

ρµυ =

•ν� kinematic viscosity•The dimensions of ν are L2/T• BG units are ft2/s • SI units are m2/s.•CGS units are cm2/s = St (stoke)



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• Vapor pressure (saturation pressure) is a thermodynamic property andit is the pressure at which phase change fromliquid to gas (boiling)occurs.•Under certain circumstances in flowing fluids lowpressures can begenerated such that cavitation may occur.

1.11Vapor Pressure (pv)

http://www.youtube.com/watch?v=GpklBS3s7iU&feature=related



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1.12Compressibilityof Fluids1.12.1 Bulk Modulus•A property that is commonly used to characterize compressibility is thebulk modulus, E

ν, defined as

•The bulk modulus has dimensions of pressure,FL−2.•The units forEv are lb/in.2 (psi) and N/m2 (Pa).

ρρ // d

dp

VdV

dpEv =−=



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1.12.2 CompressionandExpansionof Gases•When gases are compressed (or expanded), the relationship betweenpressure and density depends on the nature of the process.•Isothermal process

•Isentropic Process (frictionless compression (expansion), no heat isexchanged with the surroundings)

pEp

v == cons;ρ

vpvp

vk

ccRcck

kpEp

−==

==

;/

cons;ρ



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1.12.3 Speedof Sound•The velocity at which small disturbances propagate in a fluid is calledtheacoustic velocity or thespeed of sound, c

•For gases (isentropic process)

ρρvE

d

dpc ==

kRTkp

c ==ρ



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1.13Ideal Gas Law•The equation forideal or perfect gasesknown asequation of stateforan ideal gas is:

• This equation closely approximates the behavior of gases under normalconditions when the gases are not approaching liquefaction.

RT

p=ρ•p is the absolute pressure•ρ the density•T the absolute temperature•R the gas constant

(psi)in

lb7.14kPa33.101

2==

+=

atm

atmgageabs

p

ppp



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1.14Surface Tension•At the interface between a liquid and a gas, or between two immiscible liquids, forces develop in the liquid surface which cause the surface to behave as if it were a “membrane” stretched over the fluid mass.•These types of surface phenomena are due to the unbalanced cohesive forces acting on the liquid molecules at the fluid surface.• Molecules in the interior of the fluid massare surrounded by molecules that areattracted to each other equally. However,molecules along the surface are subjectedto a net force toward the interior.



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•As a result of this unbalanced force a hypothetical membrane is created at the interface.

•A tensile force may be considered to be acting in the plane of the surface along any line in the surface.

•The intensity of the molecular attraction per unit length along any line in the surface is called the surface tension(σ). σ = F/l.

•For a given liquid the surface tension depends on temperature and the other fluid it is in contact with at the interface.

•The dimensions of surface tension are FL−1. With units of lb/ft and N/m.

• The value of the surface tension decreases as the temperature increases.



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•If the spherical drop is cut in half (as shown), the force developed around the edge due to surface tension is 2πRσ. •This force must be balanced by the pressure difference, ∆p, between the internal pressure, pi, and the external pressure, pe, acting over the circular area, πR2.

Determinationof the Pressure inside a Drop

Rppp

RpR

ei

σπσπ

2

2 2

=−=∆

∆=



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•Another phenomena associated with surface tension is the rise (or fall) of a liquid in a capillary tube. •If a small open tube is inserted into water, the water level in the tube will rise above the water level outside the tube. In this situation we have a liquid–gas–solid interface. •In this case, there is an attraction (adhesion) between the wall of the tube and liquid molecules which is strong enough to overcome the mutual attraction (cohesion) of the molecules and pull them up the wall.• Hence, the liquid is said to wet the solid surface



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•The height, h, is a function of σσσσ, R, γ, and the angle of contact, θ, between the fluid and tube. •An equilibrium analysis yields the following relations

•The angle of contact is a function of both the liquid and the surface. •For water in contact with clean glass θ ≈ 0°. •h is inversely proportional to R.

Rh

hRR

γθσ

γπθσπcos2

cos2 2

=

=



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•If adhesion of molecules to the solid surface is weak compared to the cohesion between molecules, the liquid will not wet the surface and the level in a tube placed in a nonwetting liquid will actually be depressed.•Mercury is a good example of a nonwetting liquid when it is in contact with a glass tube. •For nonwetting liquids the angle of contact is greater than 90°, and for mercury in contact with clean glass θ ≈ 130°.



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Pe



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Read Sections:1.7 Compressibilityof Fluids1.7.1 BulkModulus1.7.2 CompressionandExpansionof Gases1.7.3 Speedof Sound1.9 Surface Tension

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