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Outline ofta3220- openinglectures1. laminar f l owofNewtonian and non-Newtonian fluids

A . Fluidproperties: Newtonian fluids1. thought experiment definingviscosity2. magnitudesof viscosity;trendswi thT, p3. Newton's law inFTI4. meaning ofshearstressintensorform5. how one canmeasureviscosity withoutinfiniteplates

B . shell-balanceproblems for Newtonian fiuids in laminar fiow1. procedure(cf.alsoSect.5.6 of FTI2. examplesinFTI

a. shear fiowbetweenparallelplates- nopressureor gravitydrivingforceb. flow in aslit- nopressuredrivingforcec. fiow in atube- no gravitydrivingforced. film condensation (fal l ing film)3. examplesf r o m iSZa. fal l ing filmb. fiow in atube(bothpressureand gravity)c. flow in anannulus(bothpressureand gravity)d. fiow in aslit(bothpressureand gravity) - only finalequation given

4. limitsto laminar fiow intermsof ReC. shell-balanceproblems for non-Newtonian fluids

1. properties o fnon-Newtonian fiuidsa. Bingham plastic (bewareo fhowi^77writes this equation )b. power-law fiuidc. effective viscosity for a non-Newtonian fiuid

2. flow derivationsa. flow of Bingham plastic intubeb. power-law fluid in atubec. definitionof effective viscosity fortube flowd. flow in aslit

i . Newtonian fluid reconsideredi i . Bingham plastici i i . power-law fluid

e. non-Newtonian fluids in anannulus3. cf derivationsin FTI

a. Bingham plastic inslit(only gravity, nopressuredrivingforce)b. power-law fluid inslit (nopressuredrivingforce)

4. conditions forsuspendingsolid particles in Bingham plastic

+

4

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L Laminarf lowof Newtonian and non-Newtonianfluids

A . Fluidproperties: Newtonianf luid1. Thought experiment

F/A=

O =T -Py,x * y.\

Units:SIunit of viscosity: Pa s

2. Magnitudes of viscosity

3. Newton's law in FTI (see Eq. 2.4)

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Trends ofviscositywitht,pforpure, single-componentliquids:

forgases:

forcmdeoilswithdissolvedgas

4. Meaningofshear stressintensorform

^ m yx

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is

F IN IT E P A R A L L E L P L A T E SAREI M P O S S I B L E , OF C O U R S E .T H E N HOW CAN ONE M E A S U R E V I S C O S I T Y ?U S IN G C O N C E N T R I C C Y L I N D E R S . IF GAP WIDTH ^0

G A P A P P R O X I M A T E S P L A N A R G E O M E T R Y .

IF GAP WIDTH -hO, N E E D C O R R E C T I O N F A C T O R S

. . .THE IDE A B E H I N D F A N NV I S C O M E T E R

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-5 . Q>, outlineof shellmomentumbalance approachB SLch.2)

P r o c e d u r e : ( c/ B SL section 2.1)1. SELECT COORDINATE SYSTEM; DEFINE CONTROL V O L U M E2. STATEBOUNDARY CONDITIONS *3. PERFORM M O M E N T U M B A L A N C E * *4. TfflCICNESS ^ 0 (-^ dif. eq. for x)a) (optional): solve dif. eq. for i apply b.c. - IF b.c.appliesto xalone5. RE LATE x TO dv/dx (apply constitutive equation)6. SOLVE DIF . EQ. FOR v; APPLY B.C. *

a) (optional) COMPUTE w(massrateoff l o w ) , etc.

^ BOUNDARY CONDITIONS (c/BSL section2.1)1. SPECIFYVA T S O L I DSURFACE

l a ) F L U I DV=S O L I D V E L O C I TYA T S O L I D W A L L2. SPECIFYXA T F L U IDSURFACE

2a) I N L IQ U I D,x - 0 AT GAS LF.2b) X,VCONTINUOUSACROSS L I Q U I D /L I Q U I D L F .3. T V NOT TT JFTNTTEANYAVHEREW I T H I NCONTROLV O L U M E

''ALL BOUNDARY CONDITLONS ARISE FROM NATURE''(i.e., f r o mproblem statement)

ELEMENTS OF M O M E N T U M B A L A N C EM O M E N T U MF L U X(a area);called (j) tensorinBSL(sect. 1.7)

1. CONVECTIONOFM O M E N T U MT H R USURFACE( p v v )2. SHEAR STI^SS xON SURFACE("molecular transport of

momentum")3. PRESSURE PRESSINGI NW A R DON SURFACE-pM O M E N T U M"GENERATION"or"SOURCE" (a volume)

4. B O D YFORCES W I T H I N V O L U M EM O M E N T U M A C C U M U L A T IO N(a volume) (not atsteady state )

5. ACCELERATIONOFSYSTEMMASS-d pv) /a t

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L B . 1. continued

Keyelements inshellbalances:

2. Notes on examples inFTI

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A nalternatederivationforVz(r):ThederivationofVz(r)forNewtonianf lowinan annulus inBSLsection 2.4 malcesuseofacleverchangeof variable in equation 2.4-3. Moststudents(and probablymost professors)wouldnotthinlcofmakingthis change. It is notnecessaryforsolvingthisproblem. Y o ushould be able to solvetliisproblemby the straight-aheadmethodillustratedhere. The equation numbers here, after 2.4-2,haveno particularrelationto the equation numbers ui section 2.4.

Startwithequation 2.4-2:r 0n 2L r +J } (2.4-2)

Equate v fromNewton's lawo fviscositywit l i fromthe momentum balance:r r 2L

\ CIr

dv.~dr (2.4-3a)

drr P - P

2juL r /Jr (2.4-4a)

VV 2juL

\ 2rJ 2

9l Inr+ C, (2.4-5a)

B .C . : v. =Oatr= R (2.4-6a)0 PQ-PL

\ RJ 2

9i Ini? + C (2.4-7a)

P,-PL2jLlL

"R' CJ 2

+ I Ini?insertingthisinto Eq.2.4-^agives

P -P \ R2 1\

R + ^ l n f ^

(2.4-8a)

(2.4-9a)

B .C . : v, =Oatr= K R (2.4-1Oa)0 I 2/zI

\ R2 1 R\2 (2.4-1la)

0V 2juL 2

K (2.4-12a)

c /In 1 V 2/zZ. 2

K (2.4-13 a)

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Plugthis expressionforC]into Eq.2.4-9a:P - P

2/JL\ R

2r1

\

V VR J

\/

/ 1 NIn 1 2/JL J 21 In

JGroup terms together withcommon factor:

P - PA/JL

\ R 1 r V l TP.-P.R\ r

J0 L R'hi Rr 1In 1K

4/JL\

J1 R

\2 KJ In /1 >1 In R\r J

ThisisEq.2.4-14 ofBSL.

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fflL FaAsssBBipDODDSBeDsDidBeirwatnoffiisDBDB I LCl fi= C- "slov//'rectilinearflow(exceptfiowaroundsphere) incompressible (p constant Newtonian(p.constant) floidsteadystate(v(x)indegeodesiof t) ignoreentranceandexiteffects no slip atwalls(v=wal lvelocityatwall; "BCtype 1")o fluid =continuumAssumptions breakdownwhenflowtoo fast: "turbulence"Quantifyconditionsforbreakdownin"Reynolds number:"

Re (iength)(velocity)(density) inertialeffects pV2

viscosity viscous f i V / L )

Typeo fFlowDef.ofRe

Trans, toturbulence Reference

Horizontal infiniteplates(BSLFig.1.1-1)

FallingFilm

YVp

4Sp

3000

20

Schlicliting,Boundary LayerTheory,1979, p. 590

SL,p. 46

4 5 p 2300 Wliite,Fluid Mechanics1979, p. 433

CircularTube 2100 BSL,p. 52

Annulus 2(R-icR)V p 2000 BSL ,p.56

Flow Ai'oundSphere 0.1 1* BSL,p.61

^ computed = V iswithinabout10%oftrue valueforRe