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# Week 3 Notes CIVILEN 3130

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This 26 page Class Notes was uploaded by Aaron Bowshier on Friday January 30, 2015. The Class Notes belongs to CIVILEN 3130 at Ohio State University taught by Colton Conroy in Spring2015. Since its upload, it has received 63 views.

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Date Created: 01/30/15

CIVIL EN 3130 SPRING 2015 LECTURE 6 Contents 1 Chapter summary 2 Reading Chapter 1 of the text book Homework NA CIVIL EN 3130 LECTURE 6 CHAPTER SUMMARY 2 1 Chapter summary We conclude this chapter by highlighting the major points we covered 9 Continuum J Considering the MOIGCUIar StrUCture of matter physical uid characteristics such as density are not well de ned at every point J Thus in order to formulate a mathematical basis for uid mechanics it is necessary to replace the actual molecular structure of uid with a hypothetical continuous medium called a J The continuum has NO h0es or VOidS and therefore has properties that can be described by continuous functions 2 De nition of a uid J A uid is de ned as a substance that deforms continuously Or FIOWS when subjected to a non zero shear stress T no matter how small that shear stress may be J The Rate of Deformation of the uid an be ob Dl 5 served to be directly proportional to the applied shear stress 739 ie 739 oc i J When the relationship between applied shear stress and rate of defor mation is Linear we call the uid aqgewtonian uid In non Newtonian uid there is aelationship between applied shear stress an rate of deformation Assume Newtonian Linear vel dudy CIVIL EN 3130 LECTURE 6 CHAPTER SUMMARY 3 Dimensions and units J In dealing with uid characteristics it is necessary to develop a System by which characteristics can be described both qualitatively and guantitatively S l Qualitativer in terms of primary quantities or basic dimensions LTM or LTF1 In turn these primary quantities are used to describe secondary quantities eg area stress velocity etc J These characteristics are described DSC J The Quantitative description consists of both a num ber and a unit by which various quantities can be compared Two com mon systems of units were discussed the International System SI and M the US Customary System USC Q1 Viscosity J From a simple plate experiment NGWtOH39S Law Of Viscosity t7 ududy lcan be observed where the the constant of proportionality u is termed the dynamic or absolute viscosity of the uid Kinematic J The viscosity of a uid is a measure of the uids resistance to the rate at which it ows Fluids with High viscosity ow at slower rates than those Wlth lOWGI39 VISCOSItIGS When subjected to the same shear stress J To arrive at Newton s law of viscosity it was shown that the an gular rate of deformation 7 of the uid is equal to the vertical Div Mass weight and concentration variables J For Simple Fluids Floats in of one molecular structure threejv te quantities can be de ned that are related to the density of the uid Sinks in water the W the speci c weight 7 2 pg and the speci c S 7 t 39t graVIyS ppggg a IVAHa J For Multicomponents mixtures we can de ne the quantities of mixture density mass fraction mass concentration and vol ume concentration m 2m l quoti CIVIL EN 3130 LECTURE 6 CHAPTER SUMMARY 4 Temperature and thermodynamic variable J Units of AbSOIUte temperature scales in the SI and USC systems are Kelvin gKg and degrees Rankine OR respectively J More common are the Relative temperature scales of de grees Celsius PC and degrees Fahrenheit OF Which are related through m the equationlon 9g5OC l 32l Pressure and a perfect gas uid is component the Two different mea J The pressure p in a related to Normal Stress sures of pressure are generally used absolute pressure pabs and gage pressure pgage J A perfect or Ideal Gas is a gas that satis es the perfect gas law relatin ssure p and density pzl p pRT iNher the speci c gas constant a d is the absolute temperature Bulk modulus of elasticity J The bulk modulus of elasticity K or simply bulk modulus is used as a measure of the compreSSibility of uids A lar e value of bulk modulus indicates that a uid is relatively J The bulk modulus is de ned as IE dp dV V kwhere dp is the differ Pressure ential change in needed to create a differential b change in volume dV of a volume V J Most common liquids have very Large bulk moduli and ca n lerefore be considered to bampincompressible gt Fab PATquot P6A61 VA c UUM CIVIL EN 3130 LECTURE 6 CHAPTER SUMMARY 5 Vapor pressure J Vapor pressure is de ned as the pressure exerted by a Vapor on its liquid counterpart when the two are in equilibrium and is a function of temperature only J Bomng of a liquid occurs when the vapor pressure of that liquid equals the surrounding atmospheric pressure acting on the uid J CaVitation is the formation and subsequent collapse of vapor bubbles in owing uid In severe cases it can cause structural damage to hydraulic systems Pump System F Surface Tension J Surface tension is a property of a liquid that causes the surface to act as a membrane that can support a ten sile force Speci cally it is a force per unit lengih 0 l J Phenomena in which surface tension is an important factor include such things as liquid drop formation capillary Rise and the ow of thin lms Bubble formation Homework Problem 163 CAVITATION 3 TABLE all 97 025 gQOL WOG CIVIL EN 3130 SPRING 2015 LECTURE 7 Contents Reading Section 21 of the text book Homework NA CIVIL EN 3130 LECTURE 7 INTRODUCTION TO FLUID STATICS 2 1 Introduction to uid statics 0 Just as in your study of solid mechanics we will begin our study of uid mechanics by considering uid Stat iCS 0 That is considering Newton s second law ZFMQ9 lt1 a we will speci cally consider the case where acceleration advdt0 in this chapter 0 This occurs when the uid is at Res39t or the entire uid body is moving with a constant velocity 0 In both cases there will be no Shea Iquot stresses in the uid and only forces due to normal stresses or pressure will be acting on the uid Maes 0 Thus our primary objective in this chapter is to formulatzrelationships for Pressure and its variation throughout a uid in the absence of shear stresses o This will allow us to calculate for example pressure forces exerted on Surfaces submerged such as dams and tank walls o In the nal section of this chapter we will consider the special case of a uid accelerating as a Rigid bOdy or in relative equilib rium where there are no shear stresses present 0 Although Are not strictly speaking rigid body motion problems uid statics problems they are covered in this chapter because the pressure relationships and the analysis of such problems are similar to those for uids At FESt CIVIL EN 3130 LECTURE 7 FORCE STRESS AND PRESSURE AT A POINT 2 Force stress and pressure at a point Illd Consider the pressure p at some arbitrary point located within a uid mass at rest The static pressure at a point is de ned as SEA AA A question that immediately arises is the following 3 How does the pressure p at a point vary with the orientation CT the pLane paSSihg through that A J pUiIIL QUESTION ANSWER To answer this question we consider the F Fee BOdY Diag ram on the following page 7 CIVIL EN 3130 LECTURE 7 IMQ 331 Fquot Fssin50 gtFA WW 52 womanma o 2F 0 F2 F5lt B W O PEWCCX Sy Cosbt Y39KX SQL Zgt 3EI GW J W S gs ms l PYgXJZFSSXSZ O FORCE STRESS AND PRESSURE AT A POINT 4 W 52 g 5w ngicyypgz gyz Gnul Terra ya 19yquot P 0 I 50W Slag 1 2quot 105 X 1 0 P2 9 RIP P5Py J CIVIL EN 3130 LECTURE 7 FORCE STRESS AND PRESSURE AT A POINT 5 CIVIL EN 3130 LECTURE 7 FORCE STRESS AND PRESSURE AT A POINT 6 CIVIL EN 3130 LECTURE 7 FORCE STRESS AND PRESSURE AT A POINT 7 This important result is known as PascaL I 5 LaW named in honor of Blaise Pasca1 1 We record this result below Pascal s law The pressure p at a point in a uid at rest is I ndependent of direction or equivalently we have The pressure p at a point in a uid at rest is the Same all directions Notes 0 This result also holds for uids in MOtion if there are no v shear stresses consider the exercise on the following page o If there are shear stresses present in the uid the normal stresses at a point are not necessarily The same in all directions x o In such cases we de ne the pressure as the Average of any three mutually perpendicular Normal stresses at that point see equation 213 of the text book p I Fz fPYTPle Shuxf Sireg 1Blaise Pascal June 19 1623 August 19 1662 was a French mathematician physicist and religious philosopher who made several important contributions to the eld of uid mechanics The SI unit of pressure is named in his honor Pascal s most in uential theological work the Pens es Thoughts which was not completed before his death at the young age of 39 is widely considered to be a masterpiece and includes the famous argument known as Pascal s Wager CIVIL EN 3130 LECTURE 7 FORCE STRESS AND PRESSURE AT A POINT 8 EXERCISE Extend the results of the previous analysis which demonstrated that the pressure at a point is independent of direction for a uid at rest to the case of a uid in motion that hasQEShear stresses ANALYSIS IF THERE ARE No SHEAR STRESSES THEN THE ONLY CHANGE To THE ANALYSIS IS THE ADDITION OF A NON ZERO RIGHT HAND SIDE IE 57 ZF may IT ZFZT maxLL WHERE 1y ANDQ Z ARE THE ACCELERATIONS IN THE y AND 2 DIRECTIONS RESPECTIVELY NEXT THE MASS CAN BE WRITTEN IN TERMS OF THE DENSITY 0 AND THE VOLUME V OF THE TRIANGULAR WEDGE OF FLUID LE 1 n CIVIL EN 3130 LECTURE 7 FORCE STRESS AND PRESSURE AT A POINT 9 FOLLOWING THE SAME LINES AS BEFORE AFTER COMMON TERMS ARE CANCELED WE HAVE RPI graced Pquot AZTSEEQQY p lt Fz Ps Eg zy z ii w AGAIN AS WE SHRINK TO A POINT 62 gt 0 WE HAVE Py GJTD P2 OLQ A 09 pAAf in Q QIUAd In MO Hoq j inCcho lbLm4quot CIVIL EN 3130 SPRING 2015 LECTURE 8 Contents Reading Section 22 of the text book Homework Problems 28 215 and 217 QUV I O SHEAtzVIsw31T7 C 2 3 KM SurCCLLL TLA SIOq lt7e CIVIL EN 3130 LECTURE 8 BASIC EQUATIONS OF FLUID STATICS 2 1 Basic equations of uid statics o In the previous lecture we addressed the question of how pressure at a point varies with D i FECt ion 0 Speci cally we showed that the P res S U re at a point is in dependent of direction for a uid with no Shea Iquot St res S o This result is known as Pascals LaW 0 However we now want to ask an important follow up question QUESTION How does the pressure vary from point to point ANSWER To answer this question we will again consider a small element of uid taken from some arbitrary position within the uid mass The F0 rce 5 acting on the element consist of two types 0 Body Forces A force that acts over the V01ume of a body or a force acting throughout the a mass of a body The most common body force and the only one we will consider here is the G ravity Force or weight of the ele ment W 0 SU FfaCe FOFCES A force that acts across an internal or external Su rface element in a body The only surface force considered here is due to Pressure re member we are looking at the speci c case where there if no shear stress CIVIL EN 3130 LECTURE 8 BASIC EQUATIONS OF FLUID STATICS 3 F053 5W V L 81962 EEW5 i I 62 F A 81963 ii A iygV ay366 8y2 7032 I 1 613 Vzriy39aa 2 m o The pressures on each face are expressed as Taylo Iquot Se ries expansions about the center point of the differential element 0 Recall that the Taylor series expansion of a function f about a point ais given by Oquot PNO P03 l l W M f ltagtltc a f 0 o The expressions for pressure in the gure above retain the rst TWO terms of the Taylor series expansion 0 We proceed in two steps First we Will look at the Taylor series expansions for the pressure Second making use of these expressions we Will write down the sum 01 forces in the 3 y and 2 directions The concept of a Taylor series was discovered by the Scottish mathematician James Gregory Go gure CIVIL EN 3130 LECTURE 8 BASIC EQUATIONS OF FLUID STATICS Q 1 7 k 1L L v imam TE 0L MP W 91 lt17 A J P F ffy39a V w i 2 EUM Mora3 S IZSFYSMQY lt LiL d7 2 l7 ydd l7 x 5may W W Lw q 52 9H r v j gt 3ng 1 gm Q7 D3 V 5v 3 D amp 9 731 XSYJE qux X CIVIL EN 3130 LECTURE 8 BASIC EQUATIONS OF FLUID STATICS 5 239 if 5 J2 quot nggyg t figMO lit 7 W J DEFINET SF JJFx HIFyg rIFz 32 QQx ay3QzQ Equot SMR V A1 dP A1 0 A W ozx a J T1 k 539 C X 1 CXJyJE DYMOL CIVIL EN 3130 LECTURE 8 BASIC EQUATIONS OF FLUID STATICS Cancel like terms PlEQEf 1 1amp7 A 0 F7 d Gradient of p Jx 139 3 T 3 f 0 an EVfX zf CIVIL EN 3130 LECTURE 8 BASIC EQUATIONS OF FLUID STATICS 7 CIVIL EN 3130 LECTURE 8 11 Pressure variation in a static uid 8 In summary the general equation of motion for a uid in which there are NO shear stresses is Vp Ykl9 1 A In the next subsection we look at how this equation simpli es for uids at rest 11 Pressure variation in a static uid 0 For a uid at RESt we have a 0 Thus Eq simpli es to Vp 7k0 quotVF YR J l quot Or in component form 8190 8190 3p 8 3 5 These equations show us that the P FESSU re dWm and y and therefore the pressure does not change from point to point as we move in the Ho r izontal It only changes as we move in the z direc tion in the ve rt lcal plane Sincel p depends on 2 only we can write the component equation for 2 above as an ordinary differential equation No acceleration d9 E 39Y 7 2 Equation T is the fundamental equation for uids at RESt CIVIL EN 3130 LECTURE 8 11 Pressure variation in a static uid 9 We note the following with respect to equation 111 o It can be used to determine how pressure changes with Elevation 0 The sign in front of 7 which is always positive indicates the Pressure gradient yl ggg galgl g bg 0 That is pressure decreases as we move Up in a uid at rest 0 The equation is valid for both con 5t ant values of speci c weight 7 subsection 111 of these notes as well as those that may have a z dependence subsection 112 of these notes r Pressure variation in a static incompressible uid 0 For an Incompressible uid there will be no change in volume and thus density when the uid mass is subjected to a compres sion force 0 Thus neglecting the small variations of g with elevation the speci c weight 7 pg of the uid is constant 0 That means equation T from the previous subsection can be di rectly integrated to obtain an expression for p as a function of Depth 0 We will consider this along with the case of a compressible uid CIVIL EN 3130 LECTURE 8 11 Pressure variation in a static uid 10 112 Pressure variation in a static compressible uid 0 When the compressible uid is a Pe rfect gas recall that we have the following relation the ideal gas law between pressure p and density p PIfRTl quotgt KT 0 Solving this equation for p and substituting the result into equation met 9i d1 RT 0 We consider the integration of this expression along with the case of an incompressible uid on the next two pages CIVIL EN 3130 LECTURE 8 11 11 Pressure variation in a static uid CASE 1 Incompressible Fluid w 2 Not a function CIVIL EN 3130 LECTURE 8 11 Pressure variation in a static uid 12 For pressure p at some depth h below the free surface pp ATM k Incompressible CASE 2 Compressible 3 SP d1 RT X f separate variables integrate assume gR T are constant PL 1339 5 1 amp l P 1739 79I P EL 7 39ZTZ31 2 Solve for p2 Compressible fluids 31 1 n deal gas Pa r ex LT

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