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# Fluid Mech. Week 6 Notes CIVILEN 3130

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This 22 page Class Notes was uploaded by Aaron Bowshier on Saturday February 21, 2015. The Class Notes belongs to CIVILEN 3130 at Ohio State University taught by Colton Conroy in Spring2015. Since its upload, it has received 76 views.

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Date Created: 02/21/15

CIVIL EN 3130 SPRING 2015 LECTURE 15 Contents 1 Relative equilibrium 2 11 Uniform linear acceleration OJ Reading Section 29 of the text book Homework Problems 2121 2123 2126 and 2128 from your text book CIVIL EN 3130 LECTURE 15 RELATIVE EQUILIBRIUM 2 1 Relative equilibrium 0 Up to this point we have been primarily concerned With uids at REST 0 However recall that in a previous lecture we developed the general EQUATION OF MOTION for a uid un dergoing acceleration under the assumption that there are no SHEAR SRESSES o In vector form this equation is 7 F l 1 EL 1 J Where a axi ayj azk is the acceleration vector recall i j and k are the unit vectors in the 3 y and 2 directions respectively and Vp is the PRESSURE GRADIENT de ned by o In scalar form the equations are Where we have used the fact that p 79 CIVIL EN 3130 LECTURE 15 11 Uniform linear acceleration 3 11 o A general class of problems involving uid motion when equations 1 2 and 3 are applicable that is when there is acceleration but no shearing stresses occurs when a mass of uid undergoes RIGIDBODY MOTION o For example if a container of uid ACCELERATION along a after some initial sloshing has died out with each particle having the same straight path the uid will move as a RIGIDBODY acceleration rigid body the uid is said to be in RELATIVE EQUILIBRIUM 0 When moving as a 0 Two cases are considered in your text book 1 2 UNIFORM LINEAR ACCELERATION UNIFORM ROTATION ABOUT A VERTICAL AXIS 0 We will only be concerned with the rst of these two Uniform linear acceleration 0 Suppose an open container of liquid is translating along a straight path with a constant acceleration a as illustrated below 0 The equations of motion 1 2 and 3 in this case are d 0quot YIaigt 911 oLy 3 lto d2 v d f DU 3 2 I MLFIXA UNLIKE STATIC CASE 39 em ATM CIVIL EN 3130 LECTURE 15 11 Uniform linear acceleration 4 0 Note that in this case the PRESSURE p is a function of BOTH the a and y directions 0 Thus the total differential dp of the pressure is given by d d AFraEAxdny J39 0 Substituting the expressions for 81383 and 819 g from the previous page CAL 9L lt13 Y 0 XmlH T 8 h J 39 V l o This expression can be integrated to obtain 07 PXIVj 5 x YKI P L Whoi0 f o where c is a constant of integration 0 The constant of integration can be evaluated provided we know the pres sure at ONE POINT say p0 and assign this point the coor dinates a y O 0 Thus our pressure equation is CLx O Pr SX VIT7FO 4 0 Let s verify that this equation would simplify to our usual p yh in the case of a static uid with a free surface CIVIL EN 3130 LECTURE 15 11 Uniform linear acceleration 5 EXERICISE Verify that in the case of a static uid with a free surface the general pressure equation 4 from the previous page simpli es to the equation pw SOLUTION CIVIL EN 3130 LECTURE 15 11 Uniform linear acceleration p0 L1nes of constant pressure p1 P2 y L 0 Another important piece of information we can obtain from the pressure relationship is the SLOPE of the free surface of an accelerat ing liquid 0 Setting p 0 in equation 4 and solving fo e obtain a a gtltXF Y 739 f173 V Y339 I o This is the equation of a line y 2 ma I refer to gure above With Qv q I l SLOPEm39 am 9 pM l39xbfl 3900 work v wyg 0 Furthermore this slope is the slope of all lines of constant pressure throughout the liquid 0 To see this rst note that along lines of constant pressure our previously derived expression for the total pressure differential is zero that is 1 AF 0 0 Solving this expression for dyda slope of lines of CONSTANT PRESURE 7 we have 1 ax Galr3 0 Thus lines of constant pressure are PARALLEL to the free surface quot L M 4 4 Tz fIYLGviE rm 6 Wank Luannw m 53m nests M W xx 05 m1 cm Titusquot139 Farm 1 AQELEIA r K M A Farmd 9F X a Y 3939 TK T I L m 65 or coH M r P smi C PAnALLEL 11 FREE SoILFALE U lFotw 3 Ac eLELATuN quotquot VMI Raw 939 MOTH i P15IE 9 AW Pun39r P 339 3 ufL 3 5 I P P 39a 9 FREE SnpAce 7 SOLVE m V Fuut w 5P gaming J andquot Y i x a i my 2quot Y39 W K 9 EE EMEW 0F LEE Artfulimu39 u xpt w I NE Am i mnw N yulmscnm CIVIL EN 3130 SPRING 2015 LECTURE 16 Contents 1 Introduction to uid dynamics 2 2 Flow concepts and kinematics 21 Analysis approaches 22 Flow classi cation 23 Pathlines streaklines and streamlines IOOCJO Reading Section 31 of the text book Homework None CIVIL EN 3130 LECTURE 16 INTRODUCTION TO FLUID DYNAMICS 2 1 Introduction to uid dynamics 0 In the rst two chapters we studied the basic properties of uids and considered various situations involving uids that are either 1 AT REST STATIC or are 2 moving in a rather simple way such as rigid body motion 0 Although we saw that there are a number of situations where one of these two conditions is met most engineering uid mechanics problems are concerned with FLUID FLOW o This fact should not come as a surprise given that the application of the slightest SHEAR STRESS to a uid will cause it to ow 0 Thus in this chapter we will begin our study of uid dynamics 0 In order to analyze the problems of uid dynamics we will need at our disposal a basic set of equations that can be used to predict the FLUID MOTION that results from the application of spe ci c forces 0 These equations can be derived from the following fundamental princi ples or laws of physics CD CONSERVATION OF MASS lt2 NEWTONS SECOND LAw FMA 3 FIRST LAW OF THERMODYNAMICS CONSERVATION OF ENERGY C4 SECOND LAW OF THERMODYNAMICS INCREASING ENTROPY INEQUALITY o The derivation and subsequent application of the basic equations ob tained from the consideration of these laws will be the primary focus of this chapter CIVIL EN 3130 LECTURE 16 FLOW CONCEPTS AND KINEMATICS 3 21 Flow concepts and kinematics Prior to considering the equations of uid dynamics we will discuss vari ous aspects of uid motion without being concerned with the actual forces necessary to produce such motion The study of the motion of particles or bodies collections of particles again without consideration of what causes the motion is known as KINEMATICS KINEMATIC GREEK The variables considered in kinematics are the POSITION velocity and acceleration of particles As we shall see below in the next subsection there are two general ap proaches that can be used to describe these and other variables the LAGRANGIAN and EULERIAN descriptions J Using these kinematic variables and descriptions uid ow can be clas si ed in a number of ways This will be covered in subsection 22 Additionally various ow concepts can be developed to help visualize and analyze ow elds Speci cally in subsection 2 3 we will look at the concepts Of PATHLINE STREAKLINE and STREAM LIN E Analysis approaches 0 As we mentioned above there are two general approaches used in analyz ing uid mechanics problems or problems in any branch of engineering mechanics or physical sciences for that matter 0 We consider these on the next two pages beginning with the LAGRANGIAN DESCRIPTION XYZ CIVIL EN 3130 LECTURE 16 21 Analysis approaches 4 Lagrangian Description 1 3 j E O O O PARTICLE A K I 5 F10 Consider a mass of particles as shown above which may represent a SOLID or a FLUID This collection of particles is an example of a SYSTEM By de nition a system is a xed amount of matter that may may move distort in shape and interact with its surroundings but that will always THE SAME contain particles It may consist of a relatively LARGE amount of mass or it may be in nitesimally small ie a single particle In any case we follow the particles of the system as they move about PROPERTIES and determine how the associated with those particles velocity acceleration temperature etc change with time eg velocity of particle A 21A FREE BODY DIAGRAMS When we draw in statics or dy namics the body that we consider is our system an identi ed portion of matter that we follow during its interactions with its surroundings TRACK the movement of the particles and identify how the properties associ Such a description of a body in which we ated with that particle change as a function of time is called a LAG RANG IAN description The Lagrangian description is named for Joseph Louis Lagrange January 25 1736 April 10 1813 an Italian mathematician and astronomer who made a number of outstanding contributions to mathematics and classical mechanics CIVIL EN 3130 LECTURE 16 21 Analysis approaches Eulerian Descript ionb x39 g f l 9 AvT l FLOW X l CONTROL l VOLUME I n 39 l o In uid dynamics it is often quite dif cult to identify and keep track of a speci c quantity of matter or system as is done in a LAGRANGIAN DESCRIPTION o This is due to the fact that the FLUID PARTICLES move about quite freely unlike a SOLID that may deform but usu ally remains relatively easy to identify 0 For example it is much easier to follow a branch oating in a river than it is to identify and follow a speci c portion of water in the river consisting of the SAME PARTICLES 0 Thus in uid dynamics it is often more convenient to adapt an alterna tive approach where we focus on a FIXED REGION in space as opposed to a xed set of particles 0 Such a xed region in space as shown above is an example of a CONTROL VOLUME the size and shape of which are arbitrary and may even be in nitesimally small ie a single point 0 Now instead of tracking individual particles as in the Lagrangian approach we obtain information in terms of what happens at FIXED POINTS in space within our control volume as par ticles ow past those points 0 A description is thus given of the properties of the uid at each SPATIAL POINT 3 y and 2 within our control volume as a function of time eg velocity v va y z t o This type of description is known as a EULERIAN descrip tion bThe Eulerian description is named for Leonhard Euler April 15 1707 September 18 1783 a Swiss mathematician and physicist renowned for his work in mechanics optics and astronomy FOR THIS COURSE CIVIL EN 3130 LECTURE 16 21 Analysis approaches 6 In summary 0 With the LAGRANGIAN with a as it MOVES ABOUT description which is associated SYSTEM we follow the uid and observe its behavior 0 With the EULERIAN a CONTROL VOLUME the uid s behavior at a description which is associated with we remain stationary and observe FIXED LOCATION 0 These two descriptionsc are illustrated below Mam CAccording to one of the classical text books on uid mechanics Lamb Hydrodynamics 6th ed Cam bridge University Press 1937 both the Lagrangian and Eulerian descriptions are in reality due to Euler it is interesting to note that Lagrange was a doctoral student of Euler s Any attempt to correct this inaccuracy however would most likely cause more confusion than bene t CIVIL EN 3130 LECTURE 16 22 Flow classi cation 7 22 Flow classi cation Flow can be classi ed depending on how the ow eld changes in both SPACE and TIME quotTIMEquot Temporal classi cations M L til STEADY FLOW G 0111 STEADY FLOW the velocity eld and all other uid properties at all points in space do not vary in time EG AVIATO o For example if the velocity at a particular point in space is 3 m s in the 3 direction it Will remain that amount and in that direction for all time if the ow is steady UNSTEADY FLOW o In contrast to this UNSTEADY FLOW is when the velocity or any other uid property at any point in space varies With time EG AVIATO o In reality almost all ows are unsteady in some sense 0 As might be expected unsteady ows are usually more dif cult to analyze than STEADY FLOWS AV BE LO ERAGE FLUID HAVIOR OVER A NG TIME 0 However in the analysis of uid ow one can often make the ASSUMPTION of steady ow Without compromising the usefulness of the results CIVIL EN 3130 LECTURE 16 22 Flow classi cation 8 Spatial classi cations quotSPACEquot Uniform ow 0 In uid variables in time as UNIFORM FLOW the velocity eld and all other are the same at all points in space at a given instant AVAX AVAYAVAZ 0 0 Note that NO CHANGE that in any of the uid variables in space at this de nition simply states there is a particular I o It makes no comment about the properties in NSTANT IN TIME CHANGE in the uid TIME Nonuniform o oIIl NONUNIFORM FLOW W the velocity eld or other uid variables time vary from point to point in space at any instant in E G AVAX ilAvAvylAvAZ 710 Your text book also describes What are called TURBULENT LAMINAR and ows in this chapter however we Will postpone this discussion to a more appropriate time CIVIL EN 3130 LECTURE 16 23 Pathlines streaklines and streamlines 9 TO VISUALIZE THE FLOW 23 Pathlines streaklines and streamlines Although uid motion can be quite COMPLICATED there are several geometric concepts that have been developed to help visualize and an alyze ow elds To this end we brie y discuss PATHLINES STREAKLINES and STREAMLINES in this sub section PATHLINE o A PATHLINE is a line or the trajectory TRACED OUT by a given particle as it ows 0 A PATHLINE can be produced in the laboratory by marking a uid particle dying a small uid element and taking a time exposure photograph of its motion EXPERIMENTAL Q STREAKLINE O A STREAKLINE is a line JOINING all particles in a ow that have previously passed through a common xed point 0 A STREAKLINE can be obtained in the laboratory by taking instantaneous photographs of marked particles that all passed through a given location in the ow eld at some time STREAMLINE o A S T R EA M LIN E is a line that is everywhere TANGENT to the velocity eld 0 The s TR EAM U N E is used more in mLYTm work than experimental V CIVIL EN 3130 LECTURE 16 23 Pathlines streaklines and streamlines 10 Some notes on these concepts o For STEADY FLOWS pathlines streamlines and streak lines are all the SAME o For UNSTEADY FLOWS none of these three types of lines need be the SAM E 0 See page 117 of your book for more information PROBLEM SOLUTIONS Q39q Problem 299 j 570 r39wmww E KJ FLgryvl H Wig173 239 1 FL 9311 Hi FL 2 r 39 WOW 4 Ar A 1 a I Lampt 5 W M W mm PROBLEM SOLUTIONS m w SUM Ma aN72 Q d f w W 39 0 Jb vr i Flef iwayltggtto g a Fwy t Uc ryg quot3 kiwqiii u 4 I quot quot quot quot quot w quotcquot 3 SCamp A37 gj f H 7l JHzQ C ifquot W QQS 4 I Q U a gig D Em LIESmquot It 1 W 7 V I 15 quot lt1 U A quot r 9 1 3 ma Ma PROBLEM SOLUTIONS Problem 2121 0 Q I I a i A tuwHWWMWWWMM I 50pr 1 3 r1 xiW 3 4 0w 2 01 I 5 739 039 3 k l ngw W m M N mw fi em MMMMM W 5 WW mm F55 2 We Paarff quot397 Ont01A H 39 39 quot W gt Hm f H 1 VJ y1 x P w u x v V 3 a 0gaqb 1L POW 3 3 43 LiaPH Pox7 I3 2 9 z 63 lieH U gt 3 7 3 39P Fquot Mm C 3Y39L 3 19 gs f39 Di z 2 334133 21 quotw fig Rag pm m on H 3 2 3 aFgtgt 79 gt 03 n to y 2 H PROBLEM SOLUTIONS Problem 2123 9 L13 1 7quot Slept mg Frw 3vt fm M a w i A quoti 300V 1 a 7 L WA 6 quot 3Zerigq g A Hquot WP M MMquot quot 57 L n 1 gm 6 I 7 5 q ax 0 Q05 5 I V V r i h M m mm a O 39 344 I I 0739 P A F quot 5 quot 39 quotM1 EVA 2 LPN WANT x 39o3mltm5i gt a 39m 39 I VGJ 1 2 x 21gt 224 U rm 1 L J B c fgtf 0333331 1quot w ECLJT quotQ 1 Oilzgznlj X 63 5 36 Om2 m I 9 gt9 I 2 2 Qt ui xy 4 1 P y g Fly 3 n yy VA I 39 V f FOzVT A P qJonz qQ V2 09232 1 909 k 2quot 8 2 pfxg vowza s gem 13 my mmmz rquot l 93 kfy l 63 00 3 wawfs z w i O 10WZZ 1 403quot quot 7 a M PROBLEM SOLUTIONS Problem 1128 31 132 39 4353 1 i N39t NJTOTAL gi ox 9 EiHmO g 5130 4 1 Sma i 2 306 m3 39 I 1 g 57 V 35 X tquot m3 x 019M 99636 MA2gtIl l giggj a J m V Cg PJ 039679JN ii m H I 21 tm j wsww R 3 Wk ffq h 39 Mw 344gt n gm w633 95 Mn Q 62353 Ms392 quot39 quot Whvw you was IN93 AL I 39quot W 129339g 5 m 3 3917quotquot mm Q fwfi 9 i K 4 5 i133 6ng 331quot 30 3 i Watquot 4 I a a 39 Okay wmsrm 0 a 7quot 3 r 2041 THE SbofE or T39Hg Ffl SvQFAca 2 11 a f II K 93903 7 3 m muwmww yuwnwmww LI 3 1 WW v 39939 M 023e SOLVE

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