chapter i. introductionhomepages.wmich.edu/~s9montef/me3560chapteri.pdf1836), gaspar coriolis...

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 from 283 to 133 BC the Hellenistic city ofPergamon (Turkey) built a series of pressurized led and clay pipelines,up to 45 km long that operated at a pressure exceeding 1.7 Mpa (180 mof head, 246.6 psi).

• ► 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 andexplored fluid 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 momentum equations.

•Bernoulli published in 1738 his treatise Hydrodynamica. 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 uniform motion.



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• ► Fluid mechanics theory up through the end of the 18th century hadlittle impact on engineering since fluid properties and parameters werepoorly quantified.

•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 werecoming from different fronts:•Jean Poiseuille (1799–1869) had accurately measured flow in capillarytubesfor multiple fluids.•Gotthilf Hagen (1797–1884) had differentiated between laminar andturbulent flow in 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 William Thompson, Lord Kelvin(1824–1907), William Strutt, 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 canbe derived into a layer near the walls, the boundary layer, where thefriction effects are significant and an outer layer where such effects arenegligible and the simplified Euler and Bernoulli equations areapplicable.



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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 bythe development of the digital computer. The ability to solve largecomplex problems, such as global climate modeling or to optimize thedesign of a turbine 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 flow when 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 how the presence of solid surfaces affects fluid flow.

•.Consider the flow of 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 the non–slip condition.• The fluid property responsible for thenon–slip condition is the viscosity.



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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 the surface 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 flow problems and it is convenient toclassify them based 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 them and the slower layer tries to slow down thefaster layer.

•This internal resistance to flow is quantified by the viscosity. 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 called viscousflows.



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

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

• The development of viscous and inviscidregions of flow as a result of inserting a flatplate parallel to a fluid stream of 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 flow region (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 flow in 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 is external flow.

•The flow in a pipe or a duct is internal 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 downstream of bodies.



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

•Incompressibility is an approximation and a flow is 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 when Ma < 0.3



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• The speed of sound in air (room temperature, sea level) is c = 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 called laminar.

• 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 flow over 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 terms steady and uniform are used frequently in engineering.

• The term steady implies no change at a point with time. The oppositeof steady is unsteady.

•The term uniform implies no change with location over a specifiedregion.



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• The terms unsteady and transient are often used interchangeably,however, in fluid mechanics unsteady applies to any flow that is notsteady, and transient applies to developing flows.

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

• In the most general case, a fluid flow is 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 flow can be 1–D or 2–D.



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

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

•A system may 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 it alwayscontains 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 put on 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 flow within, through, or around thatvolume.



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• In general, the control volume can 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 system approach.

•For example, “the mass of a system remains constant,” or “the time rateof change of momentum of a system is equal to the sum of 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 system fordescribing the fluid characteristics qualitatively and quantitatively.

•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 NFsma

smV



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•All theoretically derived equations are dimensionally 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 UnitLength Meter (m)Time Second (s)Mass Kilogram (kg)

Temperature Kelvin (K)

15.273 CK o



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Quantity UnitForce Newton (N)Work Joule (J)Power Watt (W)

2

2

m/s81.9m/s1N1J/s1W

m1N1J1m/s1kg1N

mg;gW



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-British Gravitational (BG) System.

Quantity UnitLength Foot (ft)Time Second (s)Mass Slug (slug)

Temperature Rankine (oR)Force Pound (lb)Work lbftPower lbft/s

2

2

oo

32.2ft/sft/s1slug1lb

459.67FR

mg;gW



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

Quantity UnitLength Foot (ft)Time Second (s)Mass Pound mass (lbm)

Temperature Rankine (oR)Force Pound (lb)Work lbftPower lbft/s

2

2

oo

32.2ft/s32.2lbm1slug

ft/s2.321lbm1lb459.67FR

mg;gW



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

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

•The experimental approach is expensive, time consuming and oftenimpractical. Additionally the system to 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 solution ofdifferential 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, a continuum.

• The continuum idealization 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 system considered 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 Fluid Mass and Weight1.9.1 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 ,

ftslug

mkg

Vm

lbmft

slugft

kgm

mVv

333

,,1

RTp



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1.9.2 Specific Weight•Specific weight () is weight per unit volume. g

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): =1.94 slugs/ft3=1000 kg/m3.

COHCOH oo

SG4@4@ 22



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1.10 Viscosity•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 bottom fixed plate has a zero velocity.•The fluid between the two plates moves with velocity u=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 the no-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:

yu

bU

AP

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

yu

•Absolute (dynamic) viscosity

LTM

LTL

LTLM

1/

1

GradientVelocityStressShear:Dimensions

22

sPasm

kgm

sN:Units 2

•1poise = 0.1 Ns/m2

Dynamic viscosity is property that relates shearing stress and fluid motion



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•Fluids for which theshearing stress is linearlyrelated to the rate ofshearing strain (also referredto as rate of angulardeformation) are designatedas Newtonian fluids.• = (T)•For gases increases as Tdoes.•For liquids decreases as Tdoes.



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

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

• kinematic viscosity•The dimensions of 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 from liquid to gas (boiling)occurs.•Under certain circumstances in flowing fluids low pressures can begenerated such that cavitation may occur.

1.11 Vapor Pressure (pv)

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



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1.12 Compressibility of Fluids1.12.1 Bulk Modulus•A property that is commonly used to characterize compressibility is thebulk modulus, Eν, defined as

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

// ddp

VdVdpEv



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1.12.2 Compression and Expansion of 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)

pEpv cons;

vpvp

vk

ccRcck

kpEp

;/

cons;



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1.12.3 Speed of Sound•The velocity at which small disturbances propagate in a fluid is calledthe acoustic velocity or the speed of sound, c

•For gases (isentropic process)

vE

ddpc

kRTkpc



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1.13 Ideal Gas Law•The equation for ideal or perfect gases known as equation of state foran ideal gas is:

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

RTp

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

(psi)inlb7.14kPa33.101 2

atm

atmgageabs

p

ppp



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1.14 Surface 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, Δp, between the internal pressure, pi, and the external pressure, pe, acting over the circular area, πR2.

Determination of the Pressure inside a Drop

Rppp

RpR

ei

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 , R, , 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

cos2cos2 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 Compressibility of Fluids1.7.1 Bulk Modulus1.7.2 Compression and Expansion of Gases1.7.3 Speed of Sound1.9 Surface Tension

chapter i. introductionhomepages.wmich.edu/~s9montef/me3560chapteri.pdf1836), gaspar coriolis...

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