Ch1: Introduction and Review

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//6 Ch: Inroducion and Review. Soli and flui; Coninuum hypohesis; Transpor phenomena (i) Solid vs. Fluid No exernal force : An elemen of solid has a preferred shape; fluid does no.

(yield; plasic) Flui deform coninuously, however small he shear sress is.

1 //6 Ch: Inroducion and Review. Soli and flui; Coninuum hypohesis; Transpor phenomena (i) Solid vs. Fluid No exernal force : An elemen of solid has a preferred shape; fluid does no. Under he acion of a shear force: Soli deform o cerain exen and hen sop. Soli deform coninuously if he shear sress excee a cerain limiing value. (yield; plasic) Flui deform coninuously, however small he shear sress is. (ii) Coninuum hypohesis Discree a microscopic scales Molecules are so dense ha gross behavior can be described a each poin in space (average manifesaion) e.g. Pressure he saisical average of force due o he consan bombardmen of he molecules. The size of he flow sysem (L) >> mean free pah (λ) Opposie : rarefied gases. (iii) Transpor phenomena (Phenomenological Law high concenraion diffuse low concenraion Fick s law : q m km C [ kg m s ] / Fourier s law : q k T [ J m s] Newon s law : τ [ kg m s m s] du μ mass / hea momenum μ Dynamic viscosiy [ kg m s] ν μρ Kinemaic viscosiy [ m s] Kinemaics ~ displacemen, velociy, acceleraion, deformaion, roaion c.f. namics ~ force () Eulerian and Lagrangian framework () Maerial (subsanial, paricle) derivaive () Flow lines (4) Fluid moion (5) Voriciy and circulaion (6) Sream funcion and poenial funcion 4

2 //6 Ch. Inroducion and Review. Eulerian and Lagrangian framework; Maerial derivaive (i) Eulerian and Lagrangian framework Eulerian framework: A conrol volume fixed in he space independen variable ( x, y, ) ( x, ) 5 Lagrangian framework A fixed paricular mass of fluid x, z x a ime y, Pass hrough ( ) Pass hrough ( x y, z) x x x( x,, ) ( ), a ime Lagrangian coordinaes x, idenify which fluid elemen is being considered. Time idenified is insananeous locaion. α ( x, ) α( x( x,, ), ) per uni volume Mass per uni volume αρ Momenum per uni volume αρu Kineic energy per uni volume α ρu u 6 (ii) Eulerian vs Maerial derivaive maerial derivaive : δα α ( x ( x, + δ), + δ) α( x( x,, ), ), Eulerian derivaive change rae measured a a fixed poin in space δ + δ x+ δ y+ δz x y z α( x, + Δ) α( x, ) lim Δ Δ Maerial derivaive change rae measured by following a fluid elemen x y z δ + δ + δ + δ x y z δα + u+ v+ w δ x y z Dα + u α D δα α ( x ( x, + δ), + δ) α( x( x,, ), ), Dα + D + u ( u ) α + u j x S j

//6 Ch.Inroducion and Review. Flow lines (i) Sreamlines Lines whose angens are everywhere parallel o he velociy vecor.

x( c e ( + )s z y + z z( y( s s ce + ce s s ce ce x( e y( z( ( ) s xs () ce + d z + z + ( + )s s s e + e s s e e d z z z( ) c + c y

3 //6 Ch.Inroducion and Review. Flow lines (i) Sreamlines Lines whose angens are everywhere parallel o he velociy vecor. d s (,, ) a infiniesimal segmen of a sreamline d s // u : : u : v : w u v w s : parameer for he parameric form of he sreamline. <example> <sol.> ( + ) u x v z w y + z Find he sreamline a ime ha passes (,, ) x( + ) z y + z is fixed ; x, y, z are funcions of s x( + ) ( + ) x x( c e ( + )s z y + z z( y( s s ce + ce s s ce ce x( e y( z( ( ) s xs () ce + d z + z + ( + )s s s e + e s s e e d z z z( ) c + c y ) c c x ( ) c ( ~parameric form (ii) pahline he ravelling orbi of a given fluid paricle. specify a fluid paricle he one passing ( x, y, z ) a ime Find x( ; x, y, z, ),y( ; x, y, z, ), z( ; x, y, z, ) d u ( x, y, ) ( x, y, ) v d w x, y, d ( ) I.C. x ( ) x, y ( ) y, z ( ) z

//6 <example> ( + ) u x v z w y + z Find he pahline of he fluid paricle ha passed (,,) a ime.

passed a specified poin (x,y,z )... A sreakline a collecion of,, u d find x ( x ;, τ) for fixed x, I. C.

4 //6 <example> ( + ) u x v z w y + z Find he pahline of he fluid paricle ha passed (,,) a ime. z () ce + ce y () ce ce + x() ce z() c + c I. C. y() c c x( ) c <sol> x u x( ) d + ( + ) d ln x + + C () exp ( ) x c x() e z () e e y () e e ~ parameer v z w y+ z d d d z + z + d d d d (iii) Sreakline a collecion of locaions of fluid paricles a a given ime ha ever passed a specified poin (x,y,z )... A sreakline a collecion of,, u d find x ( x ;, τ) for fixed x, I. C. x τ x ( ) <example> u x( + ) v z <sol> w y + z x( + ) d + x () ce z y( ) ce ce d z () ce + ce y z + d x (,, ) ( ) x τ ce I. C. y( τ ) c e c e z() τ c e + c e τ+τ τ τ τ τ ;, ;, ;, parameer τ 4

5 //6 The sreamline a ime ha passes (,,) is :, The pahline of he fluid paricle ha passed (,,) a ime is :, The sreakline a of hree paricles ha even passed (,,) a he same ime is :,.4 Fluid moion ~ displacemen, deformaion, roaion (i) Linear srain rae ~ he lengh change rae per uni lengh of a line elemen. * volumeric(bulk) srain rae ~he volume change rae per uni volume of a fluid elemen * volumeric(bulk) srain rae ~he volume change rae per uni volume of a fluid elemen V δ δ δ V δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ DV δ δv D D ρ u ρ D Dρ ρ ρ D D ρ + u ρ D ~ mass conservaion ~ Dρ ρ D Dρ ρ + ρ u + u ρ + ρ u D ( ) 5

parallel shear flow(d) The rae of shearing he rae a which CD and CB are

6 //6 (ii) Roaion and Rae of shear an an an,,,, an, an as Similarly, an The rae of clockwise roaion of he fluid elemen abou is cenroid is e.g. parallel shear flow(d) The rae of shearing he rae a which CD and CB are approaching each oher. D : Ω roaional ensor srain ensor voriciy vecor 為正 6

7 //6 Principal direcions : ~ elongae in he firs dir. conrac in he nd dir. ~ ABCD undergoes shear bu no normal srain. ~ PQRS undergoes normal bu no shear srain. ~ Boh area do no change in ime. ~ A circular elemen deforms o an ellipic elemen whose axes coincide wih he principal axes of he local srain rae ensor. 7

2.1: What is physics? Ch02: Motion along a straight line. 2.2: Motion. 2.3: Position, Displacement, Distance

2.1: What is physics? Ch02: Motion along a straight line. 2.2: Motion. 2.3: Position, Displacement, Distance Ch: Moion along a sraigh line Moion Posiion and Displacemen Average Velociy and Average Speed Insananeous Velociy and Speed Acceleraion Consan Acceleraion: A Special Case Anoher Look a Consan Acceleraion