INTERMEDIATE FLUID MECHANICS
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1 INTERMEDIATE FLID MECHANICS Lecre 1: Inrodcion Benoi Cshman-Roisin Thaer School of Engineering Darmoh College Definiion of a Flid As opposed o a solid a flid is a sbsance ha canno resis a shear force iho moion. Hone is a flid becase an nconained pile slmps iho end. Perolem is a flid becase i can sand p as an immobile colmn. I spills! 1
2 Ke ariables sed in describing a flid flo Densi mass per olme a scalar denoed Greek leer rho meric nis: kg/m 3 Veloci Pressre disance raeled per ime a ecor denoed meric nis: m/s force per area a scalar denoed p meric nis: Pa = N/m 2 All hese ariables and occasionall more! depend on he folloing independen ariables: spaial coordinaes and ime nderling hese definiions is he assmpion of a coninm. Densi and hermal epansion coefficien of common liqids. Liqid Densi in kg/m 3 Densi in lb/f 3 a Temperar e Thermal epansion coefficien Waer / o C Seaaer o C/50 o F / o C Crde perolem aerage o C/60 o F / o C Aomobile gasoline o C/60 o F / o C SAE 30 oil / o C Mercr / o C Milk like aer Beer aries ih pe o C/50 o F like aer Vodka / o C Ehanol / o C Densi of aer is nearl 1 kg per lier hich is no b chance. The kilogram as iniiall defined as he mass of one lier of aer. The densi of liqids aries slighl ih emperare becase of hermal epansion: nder rising emperare he same mass of a liqid occpies a greaer olme. The noable ecepion is aer nder 4 o C hich epands as emperare falls oard he freeing poin of 0 o C. In qoing he densi of a liqid i is imporan herefore o sae a hich emperare i corresponds. For moderae ariaions in emperare he densi of a liqid ma be approimaed b a linear relaionship: 0 1 T T0 2
3 For gases one sall ses he ideal-gas la: in hich T is he absole emperare. p R T Absole emperare in degrees Kelin = emperare in o C Absole emperare in degrees Rankine = emperare in o F Gas R in J/kg oc R in BT/lb of Nirogen Ogen Air 78% nirogen - 21%ogen 1% oher Hdrogen Helim Waer apor Carbon monoide Carbon dioide Mehane Propane R-ales for common gases Pressre is a Scalar Inac a sone colmn holds p b if broken along an obliqe direcion i needs an encircling brace o sa prigh. Ths in a one-piece colmn here ms be an inernal obliqe force a all leels ha preens ha is aboe from sliding off from ha is belo. P in oher ords an inac solid colmn resiss an inernal shear force. In conras a colmn of flid is nable o resis in he same manner becase a flid b definiion is nable o resis a shear force. There are o possibiliies eiher he flid spills oer or here is anoher force ha negaes he shear force and keeps he flid in is posiion. 3
4 Pressre has he same ale in all direcions. Pressre has a niqe ale regardless of he direcion in hich i is ealaed. The obliqe pressre on op presses oer a broader area hs conribing o a larger force han he oher pressre forces b is projeced componens in he horional and erical direcions amon eacl o ha he righ and boom pressre forces are as he press on smaller projeced areas. Ths ih eqal pressre in all direcions he edge of flid is in eqilibrim. In he meric ssem pressre is measred in Pascals ih 1 Pascal Pa eqal o he pressre eered b a force of 1 Neon on a 1 m 2 area. Becase he Pascal is a relaiel small ni people ofen se he kilo-pascal 1 kpa = 1000 Pa or he megapascal 1 MPa = 1000 kpa = 10 6 Pa. In he Briish ssem he ni is he pond per sqare inch psi hich is eqal o 6895 Pa. The millibar mb is a qasi-meric ni sed in meeorolog orh 100 Pa. Is adanage is ha a drop of 1 millibar in he loer amosphere corresponds qie closel o a rise of 1 meer. Sandard amospheric pressre is Pa or 1013 mb = bar. An older ni sill is se for lo pressres is he millimeer of mercr 1 mmhg = 133 Pa. Tpical pressre in Pa Tpical pressre in psi Tpical pressre in mmhg Amosphere a sea leel Aomobile ire Biccle ire Base of a 120 m 394 f dam Propane ank Birhda helim balloon Blood pressre Tpical ales of common pressres. So-called gage pressre is pressre relaie o he sandard amospheric pressre of 1 amosphere Pa. A negaie gage pressre signifies a sb-amospheric pressre ha is noneheless in he direcion of compression. 4
5 5 Acceleraion The Maerial Deriaie Acceleraion is he change of eloci oer ime. In a floing flid here is he era complicaion de o he fac ha a a laer ime he flid has moed o a ne locaion. Ths is acceleraion ma parl be becase he flo has changed oer ime and parl becase i has moed o a place here he eloci is differen. ih D D D D ime no ne a Maerial deriaie Eample of acceleraion iho change in ime Going hrogh a nole he flid increases is speed and hs acceleraes. Ye he flo ma be nchanging oer ime. The acceleraion is enirel de o he spaial gradien of eloci. L L D D This clearl shos ha: 1. Acceleraion can occr iho change oer ime; 2. Acceleraion de o spaial ariaions can be qie large. For = 2 m/s and L = 1 m he acceleraion reaches 16 m/s 2 hich is larger ha he graiaional acceleraion g!
6 6 Conseraion of Mass: The Conini Eqaion Becase mass is consered he difference beeen he amon ha flos ino a domain mins he amon ha flos o ms be accmlaing inside he domain. Ths V d V d V d m d m d d draion mass a mass a Accmlaion Oflo ime olme olme mass ime mass Inflo Ping i all ogeher: 0 The eqaion is mos ofen epressed in he folloing a o aoid negaie signs: I is occasionall called he Conini Eqaion o indicae ha he flid occpies he enire olme once and onl once no holes and no doble se of he same space.
7 sefl approimaion for liqids Liqids are no er compressible and in mos applicaions one ma assme ha he behae as perfecl incompressible flids. Their densi ma hen be aken as consan. Wih = consan i has no deriaie and he conini eqaion redces o: 0 A consan mass/olme raio conseraion of mass is eqialen o conseraion of olme. The preceding eqaion epresses ha olme is consered b ensring ha diergence eloci gradien > 0 in a cerain direcion is compensaed b conergence eloci gradien < 0 in anoher direcion. 7
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