ME 321: FLUID MECHANICS-I

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1 8/7/18 ME 31: FLUID MECHANICS-I Dr. A.B.M. Toiqe Hasan Proessor Dearmen o Mechanical Engineering Bangladesh Uniersi o Engineering & Technolog BUET, Dhaka Lecre-13 8/7/18 Dierenial Analsis o Flid Moion eacher.be.ac.bd/oiqehasan/ oiqehasan@me.be.ac.bd Dr. A.B.M. Toiqe Hasan BUET L-3 T-1, De. o ME ME 31: Flid Mechanics-I Jan Dierenial Analsis The basic eqaions o lid dnamics in inegral orm or a inie conrol olme C are: 1 Conini eqaion: olme lo rae, Q A consan or incomressible lo mass lo rae, m A consan or an lo : comressible/incomressible Bernolli Eqaion: consan Bernolli consanor incomressible lo g These inegral eqaions are sel hen e are ineresed in gross behaior o a lo ield and is eec on arios lo ssems. Hoeer, he inegral aroach does no enable s o obain he deailed oin-b-oin knoledge o he lo ield. For eamle, he inegral aroach cold roide inormaion on he li generaed b a ing; i cold no be sed o deermine he ressre and shear sress disribions ha rodce he li on he ing. To see ha is haening in a lo in deail, i is reqired o se he dierenial eqaions o lid dnamics hich inoles an ininiesimal conrol olme, in conras o inie conrol olme. This aroach is knon as dierenial analsis. Dr. A.B.M. Toiqe Hasan BUET L-3 T-1, De. o ME ME 31: Flid Mechanics-I Jan. 18 1

2 8/7/18 Conseraion o Mass Mass can neiher be creaed nor desroed. Consider a er small olme o sace ininiesimal conrol olme hrogh hich a lid is loing. For simlici, a D lo is considered and he conrol olme is bonded b he sraces and as shon in igre. According o he la, he ne olo o mass hrogh he sraces srronding he olme ms be eqal o he decrease o mass ihin he olme. The mass lo rae is eqal o he rodc o densi, eloci comonen normal o srace and he area o ha srace. In ecor orm; s m nˆ da inlo e ρ, ρ olo +e olo +e inlo e Dr. A.B.M. Toiqe Hasan BUET L-3 T-1, De. o ME ME 31: Flid Mechanics-I Jan Conseraion o Mass A irs-order Talor series is sed o ealae he lo roeries a he aces o he elemen, since he roeries are a ncion o osiion coninm aroach Lecre-1. The ne olo o mass er ni o ime er ni deh is Talor series h h h ! olo +e area olo +e area inlo e inlo e inlo e ρ, ρ olo +e olo +e inlo e Dr. A.B.M. Toiqe Hasan BUET L-3 T-1, De. o ME ME 31: Flid Mechanics-I Jan. 18 4

3 8/7/18 3 Conseraion o Mass hich ms be eqal o he rae a hich he mass conained ihin he elemen decreases: mass in de o decrease e ; 1 mass= densi olme Eq aing he abo e o e ressions and di iding b Eqaing he aboe o eressions and diiding b - I -dimension is considered, he dierenial orm o he aboe eression comes as Dr. A.B.M. Toiqe Hasan BUET 5 L-3 T-1, De. o ME ME 31: Flid Mechanics-I Jan. 18 hich is knon as dierenial conini eqaion in ecor orm.,, oeraor, del and,, here ; Conseraion o Mass In case o sead los, Then he sead lo conini eqaion in dierenial orm becomes as- di di Then he sead lo conini eqaion in dierenial orm becomes as- Incomressible los ρ= consan Comressible los ρ consan Dr. A.B.M. Toiqe Hasan BUET 6 L-3 T-1, De. o ME ME 31: Flid Mechanics-I Jan. 18

4 8/7/18 Conseraion o Momenm Linear Momenm Eqaion: The ne orce acing on a lid aricle is eqal o he ime rae o change o he linear momenm o he lid aricle. As lid elemen moes in sace, is eloci, densi, shae and olme ma change, b is mass is consered. Conseraion o momenm can be rien as- F m D D D direcion : F m D D direcion : F m D D direcion : F m D ;,, and F F, F, F 1 The eloci o a lid aricle is, in general, an elici ncion o ime as ell as o is osiion,,. Frhermore, he osiion coordinaes,, o he lid aricle are hemseles a ncion o ime. The deriaie in he aboe eression is reqenl ermed as aricle, oal or sbsanial deriaie D/D o eloci Lecre-: aricle acceleraion. Dr. A.B.M. Toiqe Hasan BUET L-3 T-1, De. o ME ME 31: Flid Mechanics-I Jan Conseraion o Momenm Since,,,,,,,,, D D D D oal local ; conecie,, Similarl D D D D A > A Area=A 1 A < A 3 1 Sead lo eloci increases 1 o eloci decreases o 3 a Conecie acceleraion Dr. A.B.M. Toiqe Hasan BUET L-3 T-1, De. o ME ME 31: Flid Mechanics-I Jan

5 8/7/18 Conseraion o Momenm The rincial orces ih hich e are concerned are hose hich ac direcl on he mass o he lid elemen, he bod orce, and hose hich ac on is srace, he ressre orces and shear orces. The sress ssem acing on an elemen o he srace is shon in igre: There is a oal o 6 shear sresses and 3 normal sresses acing on a lid elemen. The roeries o mos lids hae no reerred direcion in sace; ha is, lids are isoroic. Asa resl- ace shear shear ace ace normal Dr. A.B.M. Toiqe Hasan BUET L-3 T-1, De. o ME ME 31: Flid Mechanics-I Jan Conseraion o Momenm In general, he arios sresses change rom oin o oin coninm aroach. Ths, he rodce ne orces on he lid aricle, hich case i o accelerae. To simli he illsraion o he orce balance on he lid aricle, consider a D lo, as indicaed in igre. The reslan orce in -direcion or a ni deh in he -direcion is here is he bod orce er ni mass in -direcion. Inclding lo in he -direcion, he reslan orce in he -direcion- F Dr. A.B.M. Toiqe Hasan BUET L-3 T-1, De. o ME ME 31: Flid Mechanics-I Jan

6 8/7/18 6 Conseraion o Momenm Use his eression in eqn. 1 or -direcion: D D D F Similarl, or - and -direcions D Dr. A.B.M. Toiqe Hasan BUET 11 L-3 T-1, De. o ME ME 31: Flid Mechanics-I Jan. 18 These are he basic orms o Naier-Sokes eqaions NS eqaions. Oher lid mechanical relaions are obios o sole sch eqaions. NS eqaions are he mos amos eqaions or adanced analsis in lid dnamics. Sress-deormaion relaion: For incomressible Neonian lids i is knon ha he sresses are linearl relaed o he raes o deormaion and can be eressed in Caresian coordinaes,, as: Naier Sokes Eqaions sresses : normal For here μ is he moleclar iscosi o lid Dr. A.B.M. Toiqe Hasan BUET 1 L-3 T-1, De. o ME ME 31: Flid Mechanics-I Jan. 18 here μ is he moleclar iscosi o lid. Pa.s 1 1. Pa.s aer air

7 8/7/18 7 Naier Sokes Eqaions sresses : shear For No, se hese sress-deormaion relaions in NS eqaion in -direcion: Dr. A.B.M. Toiqe Hasan BUET 13 L-3 T-1, De. o ME ME 31: Flid Mechanics-I Jan. 18 Naier Sokes Eqaions conini eqaion ; = Dr. A.B.M. Toiqe Hasan BUET 14 L-3 T-1, De. o ME ME 31: Flid Mechanics-I Jan. 18

8 8/7/18 8 Naier Sokes Eqaions : Finall, he comlee se o Naier-Sokes eqaions in Caresian Coordinaes,, are: : : nd order arial dierenial eqaions Non-linear arial dierenial eqaions Dr. A.B.M. Toiqe Hasan BUET 15 L-3 T-1, De. o ME ME 31: Flid Mechanics-I Jan. 18 q q Since he Naier-Sokes eqaions are nonlinear, second-order arial dierenial eqaions, hese are no manageable or eac mahemaical solions ece in a e simliied lid lo cases. Nmerical solion is a ms o sole mch comlicaed arial dierenial eqaions NS eqaions. This oens a broad horion o mechanical engineering.

ME 425: Aerodynamics

ME 425: Aerodynamics 3/4/18 ME 45: Aerodnamics Dr. A.B.M. Toiqe Hasan Proessor Deparmen o Mechanical Engineering Bangladesh Uniersi o Engineering & Technolog BUET Dhaka Lecre-6 3/4/18 Fndamenals so Aerodnamics eacher.be.ac.bd/oiqehasan/