FLUID STATICS. Types of Problems

Transcription

1 FLUID STTICS Tes of Poblems In fluids at est, thee is no elative motion between fluid aticles. Hence thee is no shea stess acting on fluid elements. Fluids, which ae at est, ae onl able to sustain nomal stesses. In fluids undegoing igid-bod motion, a fluid aticle etains its identit and thee is no elative motion between the aticles. Hence, in fluids undegoing igid-bod motion, onl stess comonent esent is the nomal stess as in the stationa fluids. Objectives In this chate, an exession fo the essue distibution in a stationa bod of fluid will be deived, and the essue foces acting on submeged sufaces will be studied.

2 THE BSIC EQUTION OF FLUID STTICS Ou ima objective is to obtain an equation that will enable us to detemine the essue field within the stationa fluid. Conside a diffeential element of mass dm, with sides dx, d, and d. The fluid element is stationa elative to stationa coodinate sstem. Fo a fluid aticle, Newton s second law of motion gives df dma da Two tes of foce ma be acting on the fluid element. - bod foce gavitational foce - suface foce essue foce The foce acting on fluid element is sum of the bod and suface foces, df df d B F s Bod foce can be exessed as, df B dmg dg dxdd g () ()

3 3 Suface Foce Let the essue at the cente O, of the element be P(x,,,t). To detemine the essue at each of the six faces of the element, we use Talo seies exansion about the oint O. The essue at the left face of the diffeential element is ) ( d d L L ) ( d Similal, at the ight face, Pessue foces on the othe faces of the element ae obtained in the same wa. Combining all such foces gives the net suface foce acting on the element dxdd k j i x dxdd k j i x dxdk d dxdj d ddi dx x F d S The tem in aentheses is called the essue gadient and can be witten as gadp o P. In ectangula coodinate sstem, k j i x P gad P Pdxdd gad P dxdd F d S (3) dxdd df P P gad S Foce acting in x-diection is obtained as df S = P P d dxd P + P d dxd = P d dxd

4 Phsicall, the gadient of essue is negative to the suface foce e unit volume due to the essue. We note that the level of essue is not imotant in evaluating the net essue foce. Instead, what mattes is the ate at which essue changes occu with distance, the essue gadient. Combining equations () and (3) in Eq. () df dfs dfb gad P gdxdd O on unit volume base Fo a fluid aticle, Newton s second law of motion gives df dma da Fo a static fluid, the acceleation df gad P gdxdd 0 gad P g 0 Comonents of this vecto equation ae a is eo. Thus, bod foce e unit volume at a oint essue foce e unit volume at a oint x-com.. 4

5 bove equations descibe the essue vaiation in each of the thee coodinate diections in a static fluid. To simlif futhe, it is logical to choose a coodinate sstem such that the gavit vecto is aligned with one of the axes. If the coodinate sstem is chosen such that -axis is diected veticall, then gx 0, g 0 and g g and equations become: x-com o d d d g d d d (4) Basic equation of fluid statics Note: The essue does not va in a hoiontal diection. The essue inceases if we go down and deceases if we go u in the liquid. Objective Pessue Vaiation in Stationa Fluids - Constant densit fluids - Vaiable densit fluids 5

6 PESSUE VITION IN CONSTNT-DENSITY FLUID If the densit of the fluid is constant, we can easil integate Eq. (4) to get an exession fo essue distibution. 0 fee suface g x n exession fo essue distibution can be obtained b solving the basic equation of fluid statics as follows: To be comleted in class Fo liquids, it is often convenient to take the oigin of the coodinate sstem at the fee suface, and measue the distance as ositive downwad fom the fee suface. With h measued ositive downwad, then 0 h 0 gh is called hdostatic essue whee o is the essue at the fee suface of the liquid. 6

7 BSOLUTE ND GGE PESSUES Pessue values must be stated with esect to a efeence level. If the efeence level is a vacuum, essues ae temed as absolute. Pessue levels measued with esect to atmosheic essue ae temed gage essue. 7

8 Examle: tank which is exosed to the atmoshee, contains m of wate coveed with m of oil. The densit of wate and oil ae 000 kg/m 3 and 830 kg/m 3, esectivel. Find the essue at the inteface and at the bottom of the tank. lso detemine the essue distibution at the tank wall. The atmosheic essue is 0.35 kpa. Find: h oil, o h o = m wate, w h w = m x Solution: 8

9 Examle: Wate flows though ies and B. Oil, with secific gavit 0.8, is in the ue otion of the inveted U. Mecu (secific gavit 3.6) is in the bottom of the manomete bends. Detemine the essue diffeence, P -P B. 9

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11 Pessue Vaiation in a Vaiable-Densit Fluid If the densit is vaiable, we must elate it to the essue /o elevation befoe we can integate the equation, d d g common case might involve an ideal gas. In such gases, densit can be exessed as a function of essue and temeatue. Pessue and densit of liquids ae elated b the bulk comessibilit modulus o modulus of elasticit. E v d d / dp E v d If the bulk modulus is assumed to be a constant, then the densit is onl a function of the essue. Fom two above exession, exession fo essue distibution in liquids is obtained as follows: To be comleted in class

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13 Examle: The essue, temeatue and densit of standad atmoshee at the sea level ae 0.35 kpa, 5. C, and.5 kg/m 3, esectivel. Calculate the ecent eo intoduced into the elevation of 8 km, b assuming the atmoshee as, a) to be incomessible b) to be isothemal c) to be isentoic d) lineal deceasing temeatue with ate of K/m. The actual essue at an elevation of 8 km is known to be kpa. The gas constant of ai is 87 J/kgK. Solution: a) Incomessible ai, =constant To be comleted in class 3

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15 HYDOSTTIC FOCE ON SUBMEGED SUFCES When a suface is in contact with a fluid, fluid essue exets a foce on the suface. This foce is distibuted ove the suface; howeve, it s often helful in engineeing calculations to elace the distibuted foce b a single esultant. To comletel secif the esultant foce, we must detemine its magnitude, diection and oint of alication. Tes of oblems: We shall conside both lane and cuved submeged sufaces.. HYDOSTTIC FOCE ON PLNE SUBMEGED SUFCE Magnitude of esultant foce F? Point of alication x'?,? 5

16 Foce acting on suface df d d Minus sign indicates that foce acts against the suface The esultant foce acting on the whole suface is found b summing (integating) the contibution of the infinitesimal foces ove the entie aea. Thus, F d. () In ode to calculate the integal, both essue,, and the aea element d must be exessed in tems of the same vaiables. The basic essue-height elation fo a static fluid can be witten as 0 d dh g h d gdh h0 h is measued ositive downwad fom the liquid fee suface. 0 gh.. () 0 is the essue at liquid fee suface (h=0) This exession can be substituted into Eq. (). Then to efom integation, h and d should be exessed in tems of x and/o. (Ex: h = Sinq, q = constant). Integation of Eq. gives the esultant foce due to the distibuted essue foce. The oint of alication of the esultant foce must be such that the moment of the esultant foce about an axis is equal to the moment of the distibuted foce about the same axis. 6

17 Let be the osition vecto of the oint of alication of the esultant foce F and be the osition vecto of an oint on the suface. Moment of esultant foce = Moment of distibuted foce F df F Pd xı j xı j F Fk d kd ccoding to the coodinate sstem used, ( xı j ) ( F k ) ( xı j) Pdk Evaluating the coss oduct, we obtain, x F j F ı ( xpj Pı ) d Consideing the comonents of this vecto equation, we obtain F xf xd x F NOTE : F F d Diection of F d F is nomal d xd to the suface Magnitude of F 7

18 Examle: The inclined suface shown, hinged along, is 5 m wide. Detemine the esultant foce F of the wate on the inclined suface. Solution df w = 5 m To be comleted in class 8

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20 LTENTIVE PPOCH FO CLCULTION OF HYDOSTIC FOCE (LGEBIC EQUTIONS) Note: Oigin of the coodinate sstem is laced at the intesection of the lane of the gate and the fee suface. Now we will fomulate an aoach to detemine the esultant hdostatic foce and coodinates of its oint of alication. Conside the exessions develoed befoe, i. e. F d Consideing that the fee suface is oen to atmoshee, the magnitude of the esultant foce can be witten as F ghd g sinqd g sinq d g sinq gh c c NOTE: d c is the fist moment of the aea with esect to the x-axis. Whee c is the coodinate of the centoid of the aea measue fom the x axis, which asses though O, and c sinq=h c. h c is the vetical distance fom the fluid suface to the centoid of the aea. 0

21 Point of lication of the esultant Foce Exessions fo the coodinates of the oint of alication of the esultant foce can be obtained b equating the moment of the esultant foce to the moment of the distibuted essue foce. F df g sinq d ' F g sinq d g sinq c g sinq d c d ' d I x I x c is the second moment of the aea (moment of inetia), with esect to an axis fomed b the intesection of the lane containing the suface and the fee suface (x axis). Thus, we can wite Using aallel axis theoem I x I xc c whee I xc is the second moment of the aea with esect to an axis assing though its centoid and aallel to the x axis. Thus, ' I c xc c The x coodinate, x, fo the oint of alication of the esultant foce can be detemined in a simila manne as follows: x' x I xc c x c whee I xc is the oduct of inetia with esect to an othogonal coodinate sstem assing though the centoid of the aea. The oint though which the esultant foce acts is called the cente of essue.

22 Geometic oeties of some common shaes

23 Examle: Solve the evious examle using the algebaic equations method. To be comleted in class 3

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25 PESSUE PISM METHOD The concet of the essue ism ovides anothe tool fo detemining the magnitude and oint of alication of the esultant foce on a submeged lane suface. h h h gh gh gh x d Consideing the gage essue at the fee suface is eo, the infinitesimal essue foce, df, acting on the submeged lane suface is, df Pdk ghdk d whee d and gh ae infinitesimal base aea and imagina height of the essue ism, esectivel. Thus, oduct of d and gh eesents the infinitesimal volume dv P of the essue ism. fte integation, the magnitude of the esultant foce ma be obtained as, F k dp Pk P is the volume of the ism. P Theefoe, the magnitude of the esultant foce acting on a submeged lane suface is equal to the volume of the essue ism. P k 5

26 Point of alication of the esultant foce, x and F F xpd P Pd P xghd P ghd P P xd X P d Y G G whee X G and Y G ae the coodinates of the centoid of the essue ism. 6

27 Examle: Solve the evious examle using the essue ism method. D+Lsinq D=m wate Dg g(d+lsin30) 30 L=4m Solution: To be comleted in class 7

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29 HYDOSTTIC FOCE ON CUVED SUBMEGED SUFCES Conside the infinitesimal cuved suface element shown in figue. The hdostatic foce on an infinitesimal element of a cuved suface, d, acts nomal to the suface. Howeve, the diffeential essue foce on each element of the suface acts in a diffeent diection because of the suface cuvatue Usuall, to sum a seies of foce vectos acting in diffeent diections, we sum the comonents of the vectos elative to a convenient sstem. The essue foce acting on aea element df d The esultant foce is F d F can be witten as F ı F x d jf is kf Whee F ae comonents of in x, and, F x and F F diections. 9

30 To evaluate the comonent of the foce in a given diection, we take the dot oduct of the foce with the unit vecto in the given diection. Fo examle, taking the dot oduct of each side of the above equation with unit vecto i gives F ı d ı F x d x In geneal, magnitude of the comonent of the esultant foce in the l diection is given b F d l l l whee d l is the ojection of the aea element on a lane eendicula to l-diection. The line of action of each comonent of the esultant foce is found b ecogniing that the moment of the esultant foce comonent about a given axis must be equal to the moment of the coesonding distibuted foce comonent about the same axis. Because we ae dealing with a cuved suface, the lines of action of the comonents of the esultant foce will not necessail coincide; the comlete esultant ma not be exessed as a single foce. 30

31 Examle: n oen tank which is shown in the figue is filled with an incomessible fluid of densit,. Detemine the magnitudes and lines of action of the vetical and hoiontal comonents of the esultant essue foce on the cuved at of the tank bottom. To be comleted in class 3

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33 LTENTIVE PPOCH FO CLCULTION OF ESULTNT FOCE CTING ON CUVED SUFCES The esultant fluid foce acting on a cuved submeged suface can be detemined b integation as in the above examle. This is geneall a athe tedious ocess, and no simle geneal fomulas can be develoed. s an altenative aoach we will conside the equilibium of the fluid volume enclosed b the cuved suface of inteest and the hoiontal and vetical ojections of this suface. Conside the section BC shown in the figue above. This section has a unit length eendicula to the lane of the ae. - We fist isolate a volume of fluid that is bounded b the suface of inteest, in this instance section BC, and the hoiontal lane suface B and the vetical lane suface C. - Daw the fee-bod diagam fo this volume as shown in Fig. c. - The magnitude and location of foces F and F can be detemined fom the elationshis fo lana sufaces. - The weight, W, is siml weight of the fluid in the enclosed volume. - Foces F H and F V eesent the comonents of the foce that the tank exets on the fluid. - Fom the foce balance, we can obtain F H and F V as follow: F H F FV F W The esultant foce of the fluid acting on the cuved suface BC is equal and oosite in diection to that obtained fom the fee-bod diagam. 33

34 Examle: Solve the evious examle using the second method. To be comleted in class 34

35 BUOYNCY When a bod is eithe full o atiall submeged in a fluid, a net foce called the buoant foce acts on the bod. This foce is caused b the diffeence between the essue on the ue and lowe suface of bod. Conside the object shown in the figue immesed in a static fluid. We want to calculate the net vetical foce that essue exets on the bod. df ( 0 gh ) d ( 0 gh ) d g( h h ) d gd Thus the net vetical foce on the bod is F df gd g whee is the volume of the object. Thus the net vetical essue foce, o buoanc foce, equals the foce of gavit on the liquid dislaced b the object. This elation was eotedl used b chimedes in 0 B.C., it is often called chimedes Pincile. The line of action of the buoanc foce ma be found using the methods that used in the evious section. X B F B xdf g xd Note: The line of action of the buoant foce asses though the centoid of the dislaced volume. This centoid is called the cente of buoanc. d 35

36 Stabilit of Submeged and Floating Bodies The location of the line of action of the buoanc foce and the line of action of the foce due to gavit detemines the stabilit. W W Bage CG C CG C F B F B Stable CG W CG W C: centoid of oiginal dislaced volume C : centoid of new dislaced volume Slende Bod F B C Unstable C F B ovetuning coule 36

37 FLUIDS IN IGID BODY MOTION fluid in igid bod motion moves without defomation as though it wee a solid bod. Since thee is no defomation, thee can be no shea stess. Consequentl, the onl suface foce on each fluid element is that due to essue. Hence, the foce acting on a fluid element in igid bod motion is the same as in the case of static fluid, i.e. d F ( gad g) d o foce on a fluid element of unit volume df gad g d Using Newton s second law, we can wite df adm gad g a The hsical significance of each tem in this equation is gad g a essue foce bod foce mass acceleation e unit volume e unit volume e unit volume of fluid at a oint at a oint aticle Fom the above vecto equation, following scala equations can be witten g x g x g a a a x 37

38 Examle: n oen tank is used to tansot liquid as shown in the figue. What should be the maximum height of the liquid in tank to be sue that it will not sill ove duing the ti? d=? Solution To be comleted in class 38

39 39 FLUID OTTING BOUT VETICL XIS clindical containe, atiall filled with liquid, is otated at a constant angula velocit, about its axis. fte a shot time, thee is no elative motion; the liquid otates with the clinde as if the sstem wee a igid bod. Detemine the shae of the fee suface. P =? Exession fo fee suface =? Witing Newton s second law, we get, a g gad Scala comonents in clindical coodinate sstem can be witten as, g a g g a g 0 a g a g q q q q q q a g a g Theefoe, P = P(,) NOTE: The same exessions can also be obtained b aling Newton s second law in each diection to a suitable diffeential fluid element.

40 P = P(,). Using chain ule, total change in essue at an oint can be witten as, d d Substituting exessions fo d/d and d/d, we get, d d d gd To obtain the essue diffeence between a efeence oint (, ), whee the essue is P, and abita oint (,), whee the essue is P, we must integate d d ( ) ( ) g( ) gd Taking the efeence oint on the clinde axis at the fee suface gives atm, h 0, Then, we get, Solving fo, we get h =? atm g( h ) atm g( h ) 40

41 4 Since the fee suface is a suface of constant essue (= atm ), the equation of the fee suface is obtained as ) (aabola with vetex on the axis at the feesuface. Equation of ) ( ) ( 0 h g h h g We can solve fo the height h in tems of the oiginal liquid height h o and tank adius. To do this, we use the fact that the volume of the fluid must emain constant, i.e. Volume of liquid with no otation = Volume of liquid with otation h 0 With no otation, d dd With otation, g h g h d g h Then equating these two exession, we get g h h g g h ) ( 4 ) ( 0 Finall, substituting into above equation, we obtain the equation of the fee suface as 0 ) ( g h Equation of the fee suface Solving fo h, h = h 0 ω 4g