MCG3143: Biofluid mechanics Lectures notes Summer 2015

Transcription

1 1 Faclt of Engineeing Depatment of Mechanical Engineeing MCG3143: Bioflid mechanics Lectes notes Smme 015 Maianne Fenech

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3 3 CHAPTER CARDIOVASCULAR PHYSIOLOGY... 5 SUGGESTED EXERCISES FROM THE TEXTBOOK:... 5 CHAPTER... 6 FUNDAMENTALS OF FLUID MECHANICS INTRINSIC PROPERTIES OF FLUID... 6 CONSERVATION LAWS Mathematical tools Mass Consevation Consevation of momentm fom of flid motions eqations Dimensional analsis Eneg consevations & Bioheat Eqation of Mammalian Tisse EXERCISES SOLUTIONS REFERENCES CHAPTER MATHEMATICAL SOLUTIONS FOR BIOFLUID PROBLEMS HOW TO SOLVE A PROBLEM? BOUNDARY CONDITIONS MATHEMATICAL SOLUTIONS FOR BIOFLUID PROBLEMS Shea stess on ateial endothelial cells NS in a pipe Validit of the Hagen Poisseille elationship in the cadiovascla sstem Plsatile flow Effect of plsatilit Womesle soltion EXERCISES SOLUTIONS ASSIGNEMENT REFERENCES CHAPTER COMPUTATIONAL FLUID DYNAMIC (CFD) AND MEASUREMENT TECHNIQUES IN BIOMEDICAL COMPUTATIONAL FLUID DYNAMICS FLOW MEASUREMENT IN THE CARDIOVASCULAR SYSTEM EXERCICES PROJECT : MEDICAL DEVICE DESIGN USING CFD CHAPTER FLOW OVER IMMERSED BODY (INCOMPRESSIBLE) GENERAL EXTERNAL FLOW CHARACTERISTICS LIFT AND DRAG CONCEPT Definitions Dag fo diffeent shapes Dag coefficient, fo a sphee in stokes flow ( Re 1 )... 7

4 4.4 Tanspot of mico paticles CHARACTERISTICS OF FLOW PAST AN OBJECT BOUNDARY LAYER CHARACTERISTICS Bonda Lae Stcte and Thickness on a Flat Plate Bonda lae thickness Momentm Integal Bonda Lae Eqation fo a Flat Plate Pandtl/Blasis Bonda Lae Soltion TURBULENT BOUNDARY LAYER PRESSURE GRADIENT EFFECT ON FLOW SEPARATION EXERCISES SOLUTIONS REFERENCES CHAPTER RHEOLOGY OF BLOOD RHEOLOGY OF BLOOD AND NON NEWTONIAN EQUATIONS EXERCISES SOLUTIONS CHAPTER INTRODUCTION TO FLUID MACHINERY INTRODUCTION TO FLUID MACHINERY EXERCISES SOLUTIONS FORMULA

5 5 Chapte 1 Cadiovascla Phsiolog Tetbook : K.B. Chandan et al. Bioflid Mechanics: The Hman Ciclation. Talo & Fancis d edition. :Chapte 3 p Sggested Eecises fom the tetbook: 3.1;3.;3.3;3.4;3.5;3.6

6 6 Chapte Fndamentals of flid mechanics 1. Intinsic popeties of flid Tetbook : K.B. Chandan et al. Bioflid mechanics: the hman ciclation. Talo&Fancis d edition.: Chapte 1 pp 4 8 Consevation laws.1. Mathematical tools.1.1. Del opeato Dell is a vecto opeato (o Nabla opeato). This opeato makes the eqation easie to ndestand and wite. Calclations ae jst like with vectos, ecept that the actall opeate on what follows (not jst mltiplies them). e e i) Gadient: Del applied to a scala Gadient of scala is a vecto e gad p p p p p Gadient is the vecto field to descibe a scala field.

7 7 Diection: diection of steepest ascent Magnitde: ate of ascent Scala field : Gadient : ii) Divegence Scala podct of Del opeato (o Nabla) and a vecto: Eamples of calcls: The phsical meaning is hade to ndestand than fo the gadient. Fo the moment, emembe that divegence of the velocit descibes a edction o an epansion of Volme. We will pove that late.

8 8 Eamples: iii) Cl Vecto podct of Del opeato and a vecto: cl Cl is sed to descibe a otation Eample The otation cold be done b the diffeence of adjacent vectos: U = U = o if the sccessive vectos do not have the same diection: U = U = 1/

9 9 iv) Laplacien of a scala S. S S of a vecto : S S S iv) Rles E.g. scala fnctions of position; A, B vecto fnctions of positions fg fggf Gadient AB ABBAA BB A fa f A Af Divegence AB BA AB fa f A Af Cl AB B A A B A B B A Second deivatives Laplacian opeato cl gad 0 A div cla 0.1. Mateial deivative This deivative epesents the time deivation of a scala o of a vecto. The eqation applies to a flid element which is a small blob of flid that contains the same mateial at all times ding flid movement. Flid elements ae defomed as the move bt the ae not boken p. Note that the mass of a flid element is constant. Mateial deivative consides a popet (e.g. tempeate, densit, velocit component) of the flid element. In geneal, this will depend on the time, t, and on the position (,, ) of the flid element at that time. So

10 10 γ = γ (,,, t) = γ (, t) Eample : Tail pipe: Follow the mass and mease how the popeties change t t t t dt d B definition, the velocit of the flid element is t t t U,, ),, ( Hence t dt d d/dt is the ate of change moving with the flid element. t / is the ate of change at a fied point in space. In flid dnamics, the time ate of change fo a flid element is sall denoted b D/Dt. Ths in Catesian epansion t Dt D In geneal fom. t Dt D Eample: fo the tempeate (scala):

11 11 Mateial deivative of a vecto: t t. t Dt D. t. t. Eample: Velocit when o ae walking on a montain: U t U Dt DU. ➀ ➁ ➀ How the things ae changing at a fied location ➁ How things change as the move with the flid.1.3 Einstein notation The Einstein notation is a notational. It was intodced b Albet Einstein in Accoding to this convention, when an inde vaiable appeas twice in a single tem, it implies that we ae smming ove all of its possible vales. In tpical applications, the inde vales ae 1,, 3 1,, 3 epesenting the thee dimensions of phsical Eclidean space (,,) Eamples A mati o aij o a a a a a a a a a a vecto o i o (1,,3) fo thee dimensions of the space

12 1 U a vecto o i o (1,,3) Einstein notation: aii = i i U div ) ( ii = aijj =.1.4. Konecke s delta The Konecke is a fnction that etns 1 if the inde is eqal and 0. Othewise: Eamples : j i j i F A j i j i ij

13 13. Mass Consevation..1 Mass Consevation : Integal fom Conside a fied volme. Flid moves into o ot of acoss the sface A. da is the element of sface, with its magnitde denoting the aea of the element and diection of the nomal pointing ot of U is the velocit vecto at the position of the element. Note: the component of U paallel to da denoting the tansfe flid ot of U.dA is the aveage of mass fl thogh the sface element leaving the Volme. Whee is the flid densit. U.dA So, is the ate of loss mass fom. Note that is negative if the mass is inceasing in. d Also is the total mass in the total consevation mass is : (ate of mass in) ate of change o + of mass = 0 Rate of mass ot in d U. da d 0 dt Since the Volme is fied we can wite (Leibni s fomla) U. da d 0 t (.1)

14 14 This is the consevation mass in integal fom. Remembe thee is balance between accmlating mass inside and loss of mass acoss the bonda of the Volme. We wold like to epess the flid eqation in diffeential fom i.e. in tems of the deivate of etc U,, fo eample etc t v t,, We want to find the deivation eqation at a fied point in space... Mass consevation: diffeential fom Small Volme in Catesian coodinates: We want to wite U.dA the fl fo each face (6 times!) Recall Talo seies:... a f a f f f a a a Right face: Position Taking and a... f f f Finall, since f f f... f f f

15 15 Let s wite U and ρ fo this face Right face : Position da () o da () 1 o Left face : Position da () 1 o da Fo the 6 faces: Sm : U.dA = w v We get the fist pat of the mass consevation eqation The second pat is ve eas: t d t Pt togethe:

16 16 v w 0 t U 0 and t We get o answe! How do we epess that with the mateial deivative? 1 Epand Sot 3 We get anothe fom of the mass consevation: D U Dt 0 (.4) Divegence: D We know that Dt gives the densit changes in a flid element. Bt what abot U? To have an idea, we can conside an ncompessible flid flid does not change. In this condition U =0 fo ncompessible flid (Divegence of the velocit) D 0 Dt becase the densit of We saw that divegence efes to the Volme change of an element of flid (with fied mass). We can ewite the eqation of the consevation of mass as: Dm Dt 0 It cold also be witten: Dm Dt D D D 0 Dt Dt Dt

17 17 Witing: Mass consevation fom 1 = ( mass consevation fom ) we get: D D D U 0 Dt Dt Dt 1 D U Dt (.5) It means that the nomalied ate of change of Volme occpied b the lmp of flid is connected with the velocit. divu eflects the Volme change!..3 Smma of diffeential statements (.1) Accmlation in a fied Volme + net fles ot acoss the sface contol = 0 (.3) Accmlation in a fied Volme + net fles acoss the sface = 0 (.4) Rate of densit change following a flid element + change of Volme of the flid element =0! All thee foms ae eqivalent. The give the same infomation fom diffeent points of view!

18 18.3. Consevation of momentm.3.1 Integal fom The pinciple of consevation of momentm was initiall fomlated fom Newton s second law of motion, which states that the sm of the foces (ΣF) acting on an object is eqal to its mass (m) times its acceleation (a) F ma Rewiting (a) as DU/dt and binging the mass (m) in the diffeential: dmu F dt dmu dt is now the time ate of change of the momentm (mu). Fo a contol Volme, the time ate of change of the momentm of the Volme is the sm of: Time ate of change in the Volme + Change thogh the sface t Ud UU. da (Demo, o book flid I chap 4) F is the sm of foces acting on the Volme (bod foces) and foces acting on the sface of the sface contol (sface foces) Bd tda Eample of bod foces: Foce of gavit g Acceleation de to the choice of the efeence fame Eample: Foces de to the otation (coiolis) Electomagnetic foce Sfaces foces: Pesse (p) Viscos foce (τ)

19 19 Finall we get the consevation of momentm in integal fom: Ud UU. da Bd tda t (.6).3. Diffeential fom We eamine the flid mechanical eqivalent of Newton s second law, ΣF=ma, called the momentm eqation. Since we ae following a flid element of fied mass m, m=ρ.volme, we can wite: Σf= ρ a Whee f ae the foces pe nit of Volme (f=f/volme). a As we follow a flid element D Dt (Mateial deivative) Recall in Catesian coodinate: t. D. Dt t t. t. t Now we have to check foces acting on the flid element. This incldes: Bod foce Hdostatic pesse Viscos foce Two tpes of foces ae consideed: sface foces acting on the sface of the nit element o Volme and bod foces which ae distibted thoghot the element o Volme. a Bod foce Bod foce cold be Foce de to the gavit: f= ρ g b Hdostatic pesse In the coodinate diections we have bod foces

20 0 Unit Volme of flid showing that pesse foces act nomal to the Volme Talo epansion gives s: Pesse ight face:... P P P Foce ight face : P P Pesse left face:... P P P Foce left face : P P Net foce in the diection: P P P P P So, net foce pe nit of Volme in diection is P Similal, we can demonstate that the net foce pe nit of Volme in and diections ae espectivel: P and P Vecto fom of the foce pe nit Volme is: P e P e P e P f δ δ δ P

21 c Viscos foces Foces acting on a nit aea, A, ae efeed to as stesses. Thee ae stesses nomal to the sface and stesses tangential to the sface. Viscos stesses oppose elative movements between neighboing flid paticles. The tangential, shea stess cold be visalied b consideing two paallel plates sepaated b a flid. Each plate has a diffeent velocit. Becase the flid does not slip on the sface plate, the flid element will be sbjected to a shea Stess. 1 Velocit = Velocit is popotional to The nomal shea stess de to viscosit is moe difficlt to visalie. It is acting when viscos flid (like hone) does not fall with gavitational acceleation, becase of the viscos inteaction. These nomal viscos stesses can be viewed as de to the stickness of the flid. 1 Hone In elation to o nit Volme in the Catesian coodinate sstem, the nomal stess to the diection is defined b τ= F / A

22 The stess acting tangential to the nomal bt in the diection is τ= F / A The stess acting tangential to the nomal bt in the diection is τ= F / A The pesse and shea stesses fo all thee coodinate diections ae shown in the Fige: Unit Volme of flid showing pesse and shea stesses in the thee coodinate diections. ij =viscos stess in the j diection on the face nomal to i ais is: Note that nde conditions of eqilibim (τ= τ, τ=τ, τ=τ), we can se the sface foces acting on the nit element of flid. In developing the eqations of motion fo a flid, we se the diffeential Volme again and conside that stesses va fom point to point in the flid. We ths epess the stesses on the vaios faces of the diffeential Volme in tems of the stess acting on one face of the element and the coesponding change in the stess fo a given coodinate diection:

23 3-1 d δδ 1 d δδ 1 d δδ - 1 d δδ 1 d δδ - 1 d δδ Looking at the nit Volme of flid showing shea stesses (τ) in the coodinate diection onl, we have the sm of the foces given b F The sm of the foces fo the and diections follows similal F F The net viscos foce pe nit Volme is: f, viscos f, viscos f, viscos O we can wite: f viscos ij ij j

24 4 Whee, a epeated inde means that we add ove that inde. (Einstein notation) In geneal, the stess is second ode tenso. The ate of defomation of the flid is also a second ode tenso (also called ate of stain tenso). We epect to have a elationship between the applied stess and the ate defomation of a flid element. Case of Newtonian flid: The Newtonian appoimation assmes that the stess is lineal dependent on the ate of stain, whee popotional coefficients ae the chaacteistic of the sbstance. The elationship between viscos stess and the defomation ate of the flid element fo a Newtonian flid is given b: U j ij i j i ij Whee i, j stand fo,,. μ is the viscosit and λ is called second viscosit, o coefficient of blk viscosit o as the Lamé constant as in linea elastic theo. ij is an eta smbol, it is a sefl notation, it means: j i j i ij ij 0 1 Note that μ, the viscosit is easie to get epeimentall than λ becase λ appeas copled with U, so fo the impessible flid U =0! So the developed fom of U j ij i j i ij will be: v w v w v w Then we can wite the net viscos foce pe nit Volme with the Newton appoimation: viscos f,

25 MGC 3143, M. Fenech. And similal fo the othe components v w Fo a Newtonian incompessible flid ( U =0) and constant viscosit we get: f, viscos Note: Blood does not have a constant viscosit at low stess. Howeve, we can assme Newtonian fo high shea ate when the ed blood cells ae disaggegate in a lage ate withot diseases. Bt we have to be caefl with that becase shea stess is involved in some elevant biological phenomena as haemolsis, ateioscleosis fomation o changing shape of endothelial cells..4 fom of flid motions eqations.4.1 Geneal fom (Cach eqations) D p Dt Dv p Dt Dw p Dt O in vecto notation: D U Dt g f g f P g f b b b g (i) (ii) (iii) (iv) (v) (i) m/volme.a (ii) Pesse foce (iii) Net viscos foce pe nit Volme (iv) Net weight foce pe nit Volme (v) Othe bod foce pe nit Volme f b (.7) Note, fo this geneal case, thee ae too man nknowns fo the nmbe of eqations given!.4. Navie Stokes eqations Navie & Stokes eqations efe to the combination of mass consevation and momentm consevation fo viscos, Newtonian (constant viscosit), and ncompessible flow. Viscos flow with constant viscosit consevation of momentm is fond b: 5

26 MGC 3143, M. Fenech. D p g Dt Dv p v v v g Dt Dw p w w w g Dt O in vecto notation: D U Dt P g U (.8) Unknown:,v,w, ρ, and we have 4 eqations (3 fo momentm and 1 fo continit) we can solve given the initial and bonda condition! If the flow is compessible, then eneg consevation needs to be solved at the same time as continit and consevation eqations. This is becase change in velocit ma affect the tempeate, and vise vesa. These eqations wee named in hono of Fench mathematician, L.M.H. Navie ( ), and English mathematician, Si G.G. Stokes ( ), who wee esponsible fo thei fomlation. Becase these eqations ae second ode, nonlinea patial diffeential eqations, thee ae onl a few eact soltions available. The Navie Stokes eqations ae a set of second ode, nonlinea patial diffeential eqations that ae developed fom the pincipal of consevation of mass (continit eqation) and fom the consevation of linea momentm. The few eact soltions to these eqations that eist ae in ve close ageement with epeiments. The soltions of these eqations inclde tblence, tonadoes, waves, bonda lae and othe complicated flid flow phenomena..4.3 Ele eqation: inviscid So if the pesse foce onl is involved, consevation of momentm is done b the EULER EQUATION. Note that assme an inviscid flid: DU p Dt Eample: 6

27 MGC 3143, M. Fenech Stokes flow Stokes flow is a tpe of flid flow whee inetial foces ae small compaed with viscos foces. The Renolds nmbe is low, i.e. Re< 1. This is a tpical sitation in flows whee the flid velocities ae ve slow, thei viscosities tend to be ve lage. Fo this tpe of flow, the inetial foces ae assmed to be negligible and the Navie Stokes eqations simplif to give the Stokes eqations: Fo this tpe of flow, the inetial foces ae assmed to be negligible and consevation of momentm is simplified and is given b : 0 P ij g f b In the common case of an incompessible Newtonian flid, the Stokes eqations ae (momentm and continit): P g U U 0 (.9) Eamples: Lbification Dac eqation.5. Dimensional analsis Dimensional analsis is a tool sed to ndestand the popeties of phsical qantities independent of the nits sed to mease them. What do we gain b sing Dimensional Analsis? An consistent set of nits will wok We don t have to condct an epeiment on the eal sie O eslts will even wok fo diffeent flids O eslts ae nivesall applicable We can assess the elative impotance of tems in the model eqations It avoids ond off de to maniplations with lage/small nmbes 7

28 MGC 3143, M. Fenech. Note: CFD code ses dimensionless eqations..5.1 Renolds nmbe V The Renolds nmbe ma be descibed as the atio of inetial foces to viscos V foces D and, conseqentl, it qantifies the elative impotance of these two tpes of foces fo given flow conditions. Re VD Whee ρ is the flid densit, μ is the flid dnamic viscosit, D is the pipe diamete, and V is the flid velocit. Lamina flow occs in a flow envionment whee Re<000. Conseqentl, tblent flow is pesent if Re>4000; the tansition ange is between these citical vales. The Re nmbe is also sefl fo pedicting the entance length in pipe flow. The atio of entance length XE and the pipe diamete fo lamina stead flow is given b: X E 0.65 D fo Re<50 X E 0.06 Re D fo lamina flow Re>50 X E 0.693Re 1/ 4 D fo tblent flow Most of the blood flow in the hman ciclation is lamina, having a Re of 300 o less. An estimation of the time aveage Renolds nmbe Re in the hman aota is abot This vale is below the citical vale (000). Bt at the peak flow ate ding the sstole the Re can each 5000! Howeve, the aota is distensible, so the citical Re nmbe detemined in a igid staight pipe is not applicable in this sitation. In vivo epeiments don t show evidence of sstained tblence in the hman ciclation (in absence of disease!)..5. Womesle nmbe The Womesle nmbe, o alpha paamete, is anothe dimensionless paamete. It is a dimensionless epession of the plsatile flow feqenc in elation to viscos effects. i.e. it is the compaison between nstead inetial foce and viscos foces. The Womesle nmbe, sall denoted α, can be witten as: 8

29 MGC 3143, M. Fenech. is the vessel adis ω is the fndamental feqenc, tpicall the heat ate, the nit mst be ad/s ρ is the flid densit μ the dnamic viscosit of the flid When α is small (1 o less), it means that the feqenc of plsations is sfficientl low that a paabolic velocit pofile has time to develop ding each ccle. It also means that the flow will be ve neal in phase with the pesse gadient, and can be appoimated b Poiseille's law, sing the instantaneos pesse gadient. It is a qasi stead flow. When α is lage (10 o moe), it means the feqenc of plsations is sfficientl lage that the velocit pofile is not paabolic. Inetial foces become moe impotant and stat to be dominate. Some tpical vale of α: Hman aota α = 0 Canine aota α = 14 Feline aota α = 8 Rat aota α = Similait The velocit field of a flow cold be investigated sing a lage o smalle model fo convenience. Howeve, it mst conseve hdodnamic similait (i.e. in ode to obtain similait between a flow in nate and the simlation model flow, the atios of actating foces have to be the same: Re and α have to be conseved)..5.4 Dimensional eqations Based on the pinciples of dimensional analsis, the vaiables U, p,,,, and t cold be witten as a fnction of dimensionless vaiables: U*, p*, *,*, *, and t*. Fo eemple: *=/V v*=v/v w*=w/v *=/L *=/L *=/L t*= t V /L P*=P/ (ρv ) o P*=PL/ (μv ) Note that depending on the flow chaacteistic we want to emphasie, the dimensionless vale of P cold be diffeent. Fo eample: P*=P/ (ρv ) will be sed fo flow whee the inetial foce is dominated P*=PL/ (μv ) will sed in flow whee viscosit is dominated Eample: Rewiting NS: D stead 9

30 MGC 3143, M. Fenech. Incompessible Two dimensional Newtonian Flid Dimension vaiables will be sbstitted in the eqation to obtain an eqation with phsical qantities independent of the nits. =*.V v=v*.v w=w*.v =*.L =*.L =*.L t= t *.V /L P=P*. (ρv ) Using dimensionless vaiables: Lage Re When 1/Re is neglected, we find the Ele eqation. In Inviscid flow, we mst be caefl becase close to the wall in the bonda lae the second deivative is lage and we cannot neglect the effect of the viscosit. Low Re Stokes flow o ceeping flow viscosit dominate. As the viscos foce dominates, can we jst keep the viscos tem? To veif that it is logical to take keep the viscos foce when we scale pesse, let s take: P*=P/ (μ V /L) Then we can show that, 30

31 MGC 3143, M. Fenech. DU * Re P * U * Dt * Now the fist tem becomes negligible and we obtain the stoke eqation in dimensionless fom. The flow is evesible!!!! (video) Note on Womesle nmbe What abot the Womesle nmbe which is specific to the plsatile flow as flow in ate? To intodce Womesle nmbe in the patial deivation eqation will take a little longe. Yo will do that in o fist assignment (with gidance of cose!) To scale epeimentation with a plsatile flow o need to keep the same Renolds nmbe and the same Womesle nmbe as o did in the fist ttoial..6 Eneg consevations & Bioheat Eqation of Mammalian Tisse.6.1 Fist Law of themodnamics fo a closed sstem The fist law of themodnamics is sall witten as: m e tot Q W (I) (II) (III) The vaiation in eneg = Heat tansfe + Wok done Whee Q is heat; W is wok and the tem etot incldes nmeos tpes of eneg: (I) Total eneg Intenal eneg CvT It is the fom of stoed eneg which can be diectl inflenced b a heat tansfe. Kinetic eneg 1 U U De to the velocit of the flid paticle Potential eneg g Mins becase g is pointing in diection of deceasing potential eneg. is the position vecto 31

32 MGC 3143, M. Fenech. Yo can add othe tems (electomagnetic, chemical, ) to the epession of eneg. Howeve, we will sall limit oselves to the thee tems above. Note: The elative magnitde of the thee components of eneg is often qite diffeent. The statement of a poblem shold give a qick cle to which tpes of tems will pedominate. Modest velocities will eslt in negligible changes in kinetic eneg. Similal, small changes in elevation will eslt in negligible changes in potential eneg. To appl the fist pinciple, we assme that thee is themodnamic eqilibim between two states following a flid Volme. We can epess the fist law nde the instantaneos time ate fom and pe nit Volme as: De D 1 tot CvT U U g Dt Dt (.6.1) (II) Rate of heat inpt into the element Conside a cbic element; we want the net heat tansfe ate into it. The local heat fl vecto: f f f +d f d f +d (,, d d f +d f Net heat into the element: (f f+d)dd+(f f+d)dd+(f f+d)dd Using Talo seies: f+d= f + f/ + So, the Net heat into the element: f - ( f f )ddd - f ddd 3

33 MGC 3143, M. Fenech. Also, the local heat fl is elated to the local tempeate gadient b the Foie s law of heat condction: f k T whee k is the local condctivit. So, Heat into flid element b condction fom neighboing elements pe nit Volme T T T k T k k k (.6.) (III)Rate of wok done on the flow b sonding flid: Pesse and viscosit Net ate of wok done b pesse foces: (p p+d +d) dd +(pv p+d v+d) dd+(pw p+d w+d) dd Using talo seies: P+d= P + P/ + U+d= U + U/ + Net ate of wok done b pesse foces: p pv pw - ( )ddd Net ate of wok done b pesse foces pe nit of volme: W p - ( pu ) (.6.3) Poceeding as we did befoe, we get : Net ate of wok done b viscos foces pe nit of volme : W visc ( U. ) 33

34 MGC 3143, M. Fenech. W visc v w v w v w (.6.4) (.6.1) becomes: Consevation of eneg pe nit of volme: k T - ( pu ) (. ) D 1 CvT U U g U Dt (A) (B) (C) (D) (.6.5) (A) ate of change of the total eneg following a flid element (B) ate of heat tansfe into the following element (C) wok done b the neighboing flid via pesse (D) wok done b the neighboing flid via viscos foce We can obtain an altenate fom b changing the wok tem of pesse: ( pu ) p U U p U Fom mass consevation: p D ( pu ) U p Dt 1 D Dt (.6.6) p D Dp D p Dt Dt Dt Also : (.6.7) D p 1 Dp D(1/ ) 1 Dp D(1/ ) D 1 Dp 1 D ( ) p p p ( Dt Dt Dt Dt D Dt Dt Dt ) ( pu ) The (.6.6) eqation becomes DP P U. P With Dt t (.6.7) becoming p D p ( p U ) t Dt Dp Dt D p Up Dt (.6.9) Sbstitting the eslt in (.6.5) we get : D 1 p D p CvT U U g k T ( U. ) Dt t Dt (.6.10) (.6.8) 34

35 MGC 3143, M. Fenech. 35 This eslt fo stead, inviscid and adiabatic flow is the Benoilli s eqation fo compessible flow. 0 1 g U U p CvT Dt D. 1 Const g U U p CvT. 1 Const g U U CpT with. p CvT CpT the enthalp Along a steamline ( is the diection of the flow) (.6.11) NOTE: p is becase the flow woks:.6. Altenative fom of eneg eqation To have a pe themodnamic fomlation, we will combine the momentm and the eneg eqation: (D) fom eqation (.6.5) can be witten as: j i ij U U U ). (. The last tem is called viscos dissipation Φ, fo Newtonian flid it is: U w w v v w v U j i ij So the eneg consevation becomes: U p U Dt D p T k g U U CvT Dt D 1 Now egading

36 MGC 3143, M. Fenech. U momentm eqation gives D U D U. U U. U. p U. ij U. g Dt Dt D Dt g U. g The eneg consevation becomes : D p D CvT k T Dt Dt Anothe sefl statement of the eneg eqation: (.6.1).6.3 Second law of the themodnamics. Consideing the themodnamic Gibb s eqation abot the entop s : p Tds Cv pd CvdT d p CvdT Tds d D Ds p D CvT T Then Dt Dt Dt In tems of entop, s, (.6.1) becomes: T Ds Dt Cv DT Dt What does it mean? p D Dt k T (.6.13) The eneg dissipation fom heat and viscos ae esponsible fo the entop changes of the flid element. (.6.13) fo incompessible gives s: Eneg eqation fo incompessible flow: DT Cv k T Dt T Cv t T T T T T T k k k (.6.14) Note: In the case of incompessible flow, if we know U, we can solve eneg eqation with heat tansfe. 36

37 MGC 3143, M. Fenech. Isentopic flow B definition, isentopic flow means Tds 0 DcvT p D 0 So, Dt Dt p D 1 Dp Dp / Witing Dt Dt Dt We get: DCvT Dt D p / Dp 1 Dt Dt (.6.10) becomes: Dp D 1 p U U g ( U. ) Dt Dt t With k T =0 becase no heat is echanged when isentopic p 1 1 p Up U U g U U U g ( U. ) t t t So fo stead, inviscid flow: 1 Up U U U g 0 dp 1 d U U g 0 The Benoilli eqation fo isentopic, compessible, stead, inviscid flow is : dp 1 U U g const Along a steamline We ecognie: Benoilli eqation fo isentopic, incompessible, stead, inviscid flow is: 1 p U U g const Along a steamline 37

38 MGC 3143, M. Fenech..6.4 bioheat eqation of mammalian tisse (pennes, 1948) Fom Bioflid Dnamic, C Kleinstee, Talo and Fancis The blood tempeate in the heat s venticles and the majo ateies emains essentiall constant, i.e., when bod pats ae sddenl being oveheated o sbcooled, tisse tempeate eqilibation occs as the blood passes thogh the smalle ateies. Both local blood and tisse tempeates ae the same ntil blood mies at vaios conflences as well as in the vena cava and the heat s ight atim. A mathematical desciption of the themal echange in tisse is complicated b two sets of blood vessels in the millimete to micomete ange, shapl vaing mateial popet vales, geometic ieglaities, metabolic activit, etc. Nevetheless, eneg eqation 3.13 can be witten as: k T St T T T T Cv v w t Whee St hee is the heat soce (hee we gain heat, compaed to viscos dissipation Φ whee we lose heat) Pennes, 1948 wite this eqation in the following fom in a one dimensional diection: T Cv bcvbb ( TA T ) k T St t (I) (II) (III) (IV) (I)accmlation (II) convection (III)condction (IV)heat soce Whee is the volmetic flow ate of blood pe nit Volme of tisse, TA the ateial blood tempeate, and T the tisse tempeate. This eqation is known as the bioheat eqation of mammalian tisse. Its ndeling assmptions inclde constant mateial popeties, nifom distibtion of blood capillaies in the tisse Volme, constant metabolic heat geneation, and constant ateial blood tempeate. Assming a idealied tisse Volme, bioheat eqation of mammalian tisse has been sed to pedict the tisse tempeate in space and time de to ecessive bod sface cooling (e.g. cosge o fost bites), sface heating (e.g. skin bning o hpethemia), and whole bod feeing. Hint: To pass fom 3.13 to the bioheat eqation of mammalian tisse: Assming T changes jst in the diection (pependicla to the vessel) : T T T T dt Q dt v w dt ( T TA ) d dd d 38

39 MGC 3143, M. Fenech. 3 Eecises Eecise.1* Alveola sface tension. Use the Laplace s law to calclate the pesse diffeence pi po (ai pesse liqid film pesse) in a single alveola with the nmeical vales: R = 150 μm and σ = 7 dnes cm 1 Eecise.* Benolli s eqation. Sometimes, in hemodnamic field, Benolli s eqation is edced to p1 p = 4(v v1) when p is epessed in mm Hg and v is epessed in m/s. Using the coect fom of the Benolli eqation, p1 p = ρ(v/ v1/), which applies when pesse, densit ρ, and velocit ae epessed in an consistent set of nits, compte the eact vale of the coefficient in the fist eqation. The densit of blood is 1.05 g/cm3. (Fom Moton H. Fiedman coses notes) Eecise.3* Stenosis. Conside a case whee thee is a focal stenosis of 6mm diamete femoal ate in which the coss section diamete is edced to one thid of nomal. How is the velocit V at the stenosis compaed to the psteam velocit V1? Detemine V if V1 is eqal to 50 cm/min. Detemine the pesse at the stenosis if the pesse at the psteam was 100 mmhg. Eecise.4 Renolds nmbe. Estimate the Renolds nmbe fo the blood flow fo each tpe of vessel descibed in the following table: Vessel Diamete (cm) Velocit (cm) Aota.5 48 Lage ateiole Ateiole (etinal micociclation) Capilla Do o think that some of these flows ae tblent? Eecise.5* Womesle nmbe. Recall the meaning of the Womesle nmbe. The heat ate of a 400 kg hose is appoimatel 36 beats pe minte (bpm); the viscosit of hose blood is Ns/m. We assme the same blood viscosit acoss the mammal species. The heat ate of a 3 kg abbit is appoimatel 10 beat pe minte (bpm). The abbit blood viscosit is Ns/m. Allometic stdies of mammals show that that the aotic diamete gows in a cetain elationship with the sie of the animal. Fom Li, 1996: D=0.48 W 0.34 Whee D is the aotic diamete, given in centimete of the aota, and W the weight given in kilogams. Compae the Womesle nmbe in the hose aota to that in a abbit aota. What does this mean? 39

40 MGC 3143, M. Fenech. Eecise.6* Similait. An aspiing gadate stdent wants to std the flow of blood in the coona ateies, bt becase a tpical vessel is onl 3 mm in diamete and too small to make measements in, she makes a eplica that is fo times the sie of eal coona ateies. Based on pblished data fo coona ate dimensions and flow, and the known popeties of blood, she calclates that the time aveage Renolds nmbe of the in vivo flow is 90 and the Womesle nmbe is 3.1. The flow sstem she is sing has a fied peiod of 3 sec. (a). What shold be the kinematic viscosit of he woking flid? (b). What shold the aveage velocit be at the inlet to the model to obtain similait to the in vivo case? Note: The convention in plsatile flow calclations is to se the tbe diamete (D) as the chaacteistic length in calclating the Renolds nmbe (Re = ρdu/μ = DU/υ) and the tbe adis as the chaacteistic length in calclating the Womesle nmbe [α = (D/)(ω/υ) 0.5 ]. (Fom Moton H. Fiedman coses notes) Eecise.7 A saline soltion (densit 1050 kg/m 3 ) is ejected fom a lage singe, thogh a small needle, at stead velocit of 0.5 m/s. Estimate the pesse developed in the singe. Neglect viscos effect. Assme that the velocit of the flid in the lage singe is appoimatel eo, when compaed with the velocit in the needle. Eecise.8 Hagen Poisseille. A 100 cm long cathete, with an inside diamete of 0.4 mm is connected to a singe. In a tpical infsion pmp, the plnge is diven at a constant velocit. The diamete of the singe is 5 mm. Fo a velocit of 50 mm/min, what volme ate of flow will dischage thogh the Cathete? Assme the flid has a viscosit of 0.00 Ns/m and densit of 1000 kg/m 3. Estimate the pesse developed in the singe. Neglect viscos effect. Assme that the velocit of the flid in the lage singe is appoimatel eo, when compaed with the velocit in the needle. Eecise.9 Hagen Poisseille. What pesse will be eqied to foce 1cc/s of blood sem thogh an intavenos tbe of adis 0.5 mm and length 3 cm into an ate with a mean pesse of 100 mmhg? (Assme: blood sem viscosit, 7 cp) Eecise.10 Qi W010 Saline soltion, with the same densit as wate and five times the viscosit of wate, is to be administeed continosl into a vein thogh a needle, in a pefsion. The inside diamete of the needle is 0.4 mm and it is 50 40

41 MGC 3143, M. Fenech. mm in length. To geneate flow, the bag of flid is hng highe than the patient. Given that 1 cmho 100 Pa, detemine how high shold the bag be hng to geneate a flow ate of 1ml pe minte. Note: o can neglect venos blood pesse (4 cmh O). State cleal o assmptions. Eecise.11* Qi W010 Similait. Hman Spematooa. Some Biomed stdents ae pepaing a demonstation fo UOttawa Da. The want to show the paticla wa that spematooa swim; with a nidiectional otation of the tail. The plan to cop the bbe mechanism shown in thei bioflid class, bt since a spem cell is so small, the have to make a bigge model. The fond in the liteate that a spem cell is 50 µm long, and its velocit is aond 00 µm/s in wate. (a) Compte the Renolds nmbe. (b) Which concentation of glceine shold the se to get the pope viscosit if the make a model 1000 times bigge, swimming with a velocit of 1 mm pe second? (Assme ρglceine= ρwate) (c) Is it ealistic? Eecise.1 Hangen Poiseille Qi S010A patient has atheoscleosis, which podces a stenosis of his aota of 16% diamete edction. a What is the edction in flow ate (assme the heat delives the same pesse) b Assming lamina stead flow, how mch pesse incease is necessa to compensate fo this edction? Eecise.13 Midtem W010. Similait.Yo want to std the flow of blood in the femoal ateies. Howeve, becase a tpical vessel is too small to make measements, o have to make a eplica that is fo times the sie of eal femoal ateies. Based on pblished data o know that femoal ateies have a diamete Da, a time aveage velocit Va, and that the heat feqenc is fh and the popeties of blood υb. a) Epess the time aveage Renolds nmbe Rea and the Womesle nmbe α a in the eal femoal ate as a fnction of Da, Va, fh and υb. b) The available epeimental flow sstem has a fied feqenc of fe. What shold be the kinematic viscosit υe of o woking flid to obtain a simila flow to the in vivo case? c) What shold the inpt aveage velocit Ve be in the epeimental model to obtain a simila flow to the in vivo case? d) Nmeical application. Given: Da =. mm, Va = 1 mm/s, μa = 4 cp, ρa = 1.05 g/cm 3, fh =1 s 1, and fe = 0.1 s 1; compte υe and Ve 41

42 MGC 3143, M. Fenech. Eecise.14 Demonstate some of the following les: f,g scala fnctions of position; A, B vecto fnctions of positions fg fggf Gadient AB ABBAA BB A fa f A Af Divegence ABB A A B fa f A Af Cl AB B A A B A B B A Second deivatives Laplacian opeato cl gad 0 A div cla 0 Eecice.15 Epess the mateial deivative fo one dimensional flow and give an intepetation of each tem. Eecice.16 Compte the mateial deivative of t U 1 t e 3te e Eecice.17 Continit law (i) Epess the continit law fo a one dimensional flow and give an intepetation of each tem. (ii) Eplain wh U = 0 efes to an incompessible flid. What is this tem fo a one dimensional flow? Eplain what it epesents phsicall. Eecice.18 Eqation of continit. Wite the special cases of the eqation of continit fo a) stead compessible flow in the plane. b) nstead incompessible flow in the plane. 4

43 MGC 3143, M. Fenech. c) nstead compessible flow in the diection onl. d) stead compessible flow in, θ coodinates. Eecice.19 Radial flow Incompessible: Fo a adial flow in the θ plane, V=f(), Vθ= 0. Find f() fo incompessible flow. Eecice.0 Navie Stokes eqation: What is the phsical meaning of each tem in the Navie Stokes eqation? (. ). f v p t What wee the assmptions involved in the deivation of the last tem? Simplif the eqation fo a stead flow in the plane in the absence of an bod foce. Eecice.1 Stock s eqation. Show that the momentm eqation fo Newtonian incompessible flow, withot bod foce, is edced to the following fom when viscosit dominates: 0 P * U * Whee: P*=P/ (μ Uinf/L), U*=U/Uinf. Give sitations in the biomedical field whee this eqation cold be sed. Eecice. The Renolds nmbe What is the phsical meaning of the Renolds nmbe? Compte the Renolds nmbes fo vaios flid dnamic eamples in eveda life (flow aond o ca, flow aond o ca s antenna, a bid fling, a fish swimming, a spoon stiing o coffee, insect fling, blood flow, flow aond o bod while o ae walking to school, etc ) and conclde which eamples fall in the High Renolds nmbe catego, and which ones fall in the Low Renolds nmbe catego. Eecises fom the tetbook: 1.1;1.;1.3; 4. Soltions Soltion.1 7.mmHg 43

44 MGC 3143, M. Fenech. Soltion. P1 P=3.93(V1 V) Soltion.3 V=450 cm/min P1 P=0.0mmHg Soltion.5 α hose =3; α abit =17 Soltion.6 ν= m /sec V=0.059m/s Soltion.11 a)re=0.01 b)5000cp c)no 5. Refeences M. Radlesc, Univesit of Ottawa, lecte notes 008 K.B. Chandan et al. Bioflid mechanics: the hman ciclation. Talo and Fancis. d edition. Bioflid Dnamic, C Kleinstee, Talo and Fancis THE NABLA OPERATOR online esoce 44

45 MGC 3143, M. Fenech. Chapte 3 Mathematical soltions fo bioflid poblems Navie and Stokes eqations ae called the One million dollas eqations. The ae called One million dollas becase one million is waiting to be won b anone who can solve one of the gand mathematical challenges of the 1st cent: Navie Stokes eqations. NS ae eceptionall sefl becase the descibe the phsics of man things of academic and economic inteest. The ma be sed to model the weathe, ocean cents, wate flow in a pipe, the ai's flow aond a wing, and motion of stas inside a gala. The Navie Stokes eqations in thei fll and simplified foms help with the design of aicafts and cas, the std of blood flow, the design of powe stations, the analsis of polltion, and man othe things. Howeve, onl few poblems can be solved mathematicall becase of the compleit of the eqations. Let s solve these eqations fo poblems elated to bioflid stff. 45

46 MGC 3143, M. Fenech. 1 How to solve a poblem? Bonda conditions Tpicall, if an solid bonda eists (i.e. wall, an object in the flow ) the velocit at the sface is eqal to eo. This acconts fo the viscos stesses (fiction) Velocit nomal to sface =0 (ecept fo poos sface) Velocit tangential to sface = velocit of the sface = 0 if the sface does not move. It is the NO SLIP condition. video 3 Mathematical soltions fo bioflid poblems 3.1 Shea stess on ateial endothelial cells Epeimental sitations aise whee it is necessa to se the Navie Stokes eqations fo pedicting shea stess impated to the bonda. One sch epeiment is to detemine the effect of flid shea stess on hman ateial endothelial cells that ae clted on flat plates []. An epeimental flow ig is constcted simila to the eample poblem above and a flid with a known viscosit is pmped at a constant velocit acoss the cells. Shea stess is calclated b mltipling the slope of the velocit pofile at the wall times the viscosit of the flid. The eslts of sch a std ae shown below 46

47 MGC 3143, M. Fenech. Hman ateial endothelial cells ae clted in a paallel plate flow ig to detemine the effect of flid shea stess on cell stcte. At phsiologic shea stess, the cells elongate in the diection of flow. These kinds of flid mechanics stdies eveal that hman endothelial cells will change thei shape if the do not epeience a cetain magnitde of flid shea stess. This has been shown to affect thei fnction in a negative wa. Stdies like this point to a gowing bod of evidence that flid mechanics plas a ole in the fomation of atheoscleosis, the nmbe one kille of adlts in the westen wold. It has been fond that the disease foms lesions onl in specific locations in the bod. Epeimental and comptational flow stdies in these egions point to low magnitde and oscillating wall shea stess as a common flid mechanical chaacteistic in egions whee the disease foms [3]. An eample of an eact soltion to the Navie Stokes eqations follows. This is also a ve pactical one. We will conside the eample of viscos flow between two fied paallel plates: h h g Viscos flow between two fied paallel plates showing a paabolic velocit pofile. Let s wite NS in Catesian coodinate fo: D stead Incompessible Two dimensional 47

48 MGC 3143, M. Fenech. Newtonian Flid In this poblem, thee is stead lamina flow between the plates, flid is moving in the diection onl (v=0, w=0). We will also ignoe gavit since it will have ve little inflence on the flow scenaio. Given that we know the viscosit of the flid, we want to know the velocit pofile acoss the plates. So, =() and these conditions ae sed in the Navie Stokes eqations so that the simplif to p 0 p 0 g p 0 We ve set g = g = 0 and g = g. These conditions make the eqations simplif to something manageable. The second ode eqation fom above can be ewitten as d d 1 p and integated to give d d 1 p c 1 and integated et again to give a soltion fo velocit in the diection. 1 p c c 1 The bonda conditions detemine c1 and c. If the two plates ae fied, then =0 at =+h de to the fact that a viscos flid has eo velocit at the wall. This condition is satisfied when c1=0 and 48

49 MGC 3143, M. Fenech. c 1 p h Theefoe, the velocit distibtion becomes p ( 1 h ) The obseved epeimental velocit pofile between paallel plates ve closel matches that pedicted b the Navie Stokes eqations. Also the maimm velocit is: 1 p ma h Then ma (1 ) h Also mean velocit: mean h h d h 3 ma And the shea stess is given b (Newtonian flid): v p p 1 ( h ) Note: The othes tems ae nll Close the bottom wall : 3. NS in a pipe p h R Now we come to deive the most popla application of the intenal flows, commonl known as Hagen Poiseille Flow o, simpl pipe flows. Since pipes have clindical geomet, we se the clindical fom of the momentm eqations. Let s assme an incompessible, stead flow thogh a cicla pipe withot an appeciable bod foces. Assming a paallel flow in the 49

50 MGC 3143, M. Fenech. diection, V 0, bt V V 0. Continit eqation 1 V V V V 0 As in the case of Plane Poiseille flow, witing ot the momentm eqations in and p p 0 diection will simpl eslt in. Theefoe, let s focs on diection. p 1 = 1 We can fthe assme V 0 becase of the clindical smmet. p 1 = o, integating twice ove, we get V dp ( ) C C1 ln 4 d (C1, C = Constants) Since the pipe adis is R, the bonda conditions ma be witten as dv ( 0) 0 and d. ( V R) 0 The second bonda condition is de to flow smmet at = 0, wheeas the fist one is de to no slip condition. Solving the constants C1 and C we get V ( ) R dp 1 4 d R As in the case of Plane Poiseille flows, dp 0 d fo this flow to eist (i.e., Q > 0). 50

51 MGC 3143, M. Fenech. Some additional eslts ae: 4 R dp R 4 p Q Q 8 d Hagen Poisseille elationship 8L R dp V 8 d, V Z V ma, dv dp d d [Note: Yo mst se an annla aea element da d ê to deive V and Q eslts.] d 3. 1 Validit of the Hagen Poisseille elationship in the cadiovascla sstem The simplest model fo blood flow thogh a vessel wold be: Lamina flow Stead Newtonian flid Staight tbe with a constant cicla coss section In this condition, Hagen Poisseille gives the elationship: R 4 p Q 8L Whee Q is the flow ate, Δp is the dop of pesse, L is the length of the tbe, μ is the viscosit, and R is the tbe adis. We shold citicall eamine the validit of these assmptions in models descibing blood flow in ateies. Newtonian flid As we peviosl discssed, the blood viscosit depends on the shea ate especiall at low SR. Bt fo high shea stess highe than 100s 1, the viscosit coefficient appoaches a constant vale. Ths, fo flow in lage blood vessels, whee low SR can be epected ding sstole, a Newtonian desciption appeas to be easonable. Lamina flow As we saw peviosl, the assmption of lamina flow in the model also appeas easonable. 51

52 MGC 3143, M. Fenech. Stead flow Flow in the hman ateies is cleal plsatile, consisting of sstolic and diastolic phases; theefoe, the assmption of stead flows is NOT valid in the majo pat of the ciclato sstem. Rigid wall The ateial walls ae visco elastic and distend with the plse pesse. The inteaction of the wall and the flid is an impotant facto in hemodnamics. This assmption is NOT valid. Howeve, in cetain cases of stead flow models, the distensabilit don t affect the soltion. Constant cicla coss section This is a good appoimation fo most of the ateies in the sstemic ciclation. Bt the vein and the plmona ateies ae moe elliptical in shape. 3.3 Plsatile flow Effect of plsatilit This is adapted fom Bioflid mechanics, the hman ciclation, K Chandan, p191 The pevios section focsed on stead flow. Howeve, we know that the blood flow in heat and ateies is plsatile. When the heat contacts ding sstole, a pesse is geneated b the left venticle, and the wave tavels de to the elasticit of the sstem. The plsatile nate of the flow affects: Pesse distibtion Velocit pofiles Accoding to the instantaneos Re ding the pick of sstole, we ae epecting a tblent flow (5000). Howeve, it has been obseved that aotic blood flow emains lamina and well steamlined nde these conditions. The eason is patiall de to the stabiliing effect that sstolic acceleation has on the flow and also becase is thee is not sfficient time available fo flow to become tblent. In a health ate, despite a lage Re, the flow is still lamina. Fo a not health case, the flow will become tblent becase the Re is not appopiate with a plsatile flow! The Womele nmbe α is also se to chaacteie the peiodic nate of blood flow. As α inceases, the inetial foces become moe impotant and stat to dominate, initiating at the cente of the tbe. As eslt, a dela can be obseved in the blk flow, and the velocit pofile becomes moe flat in the cental egion of the tbe. 5

53 MGC 3143, M. Fenech Womesle soltion Womesle soltion is one wa to estimate nstead velocit in a staight pipe. Othe soltions that ae moe comple inclde the elasticit of the tbe. These ae also poposed in the liteate. Hee we will focs on the Womesle soltion onl, afte solving togethe the eqation, o will have to se matlab to plot the velocit pofile in a tbe fo an nstead flow. Changing α, o will have the oppotnit to see the impotance of the inetial foces. Let s wite continit and Navie Stokes eqations withot bod foce in a clindical coodinate fo an nstead flow of an incompessible Newtonian flid, in diection. Vessels ae assmed non elastic (as staight pipe) Fo this plsatile flow case, we assme a igid wall. Then thee will also be no adial motion of the wall. Theefoe, we can assme that the adial velocit component will be eo. Also the flow is aismmetic. Jstifing each simplification, show that: V 1 p V 1 V t Eq. W1 Since p is a fnction of and t, dp/d will be a fnction of t onl. It is possible to wite p with a Foie seies: 53

54 MGC 3143, M. Fenech. p e a e int n0 n Eq. W Then we can wite each component of the pesse gadient as a comple eponential as following: p n a n e int Eq. W3 Whee an is a constant epesenting the amplitde of the hamonic n of the pesse gadient, ω is the fndamental feqenc, and i is the nit comple nmbe. In this contet of Foie seies, the fom velocit V is: V e V n n n0 f ( ) e V int n Eq. W4 Show that Eq.1 can be witten fo a hamonic n, independentl of the time as follows: d f d n ( ) 1 d f n ( ) n f n ( ) B Eq.W5 d whee n and B ae constants that o have to detemine. t n n n 54

55 MGC 3143, M. Fenech. V 1 p V 1 V t becomes: 4 Solve the Eq.W5 fo n=0. 55

56 MGC 3143, M. Fenech. 5 Detemined a paticla soltion of the Eq.W5. fo n 0 f ( ) paticla Const. Hint: t the eas one: n 6. The homogenos diffeential eqation of Eq.5 fo n 0 withot the second membe is the well known Bessel s eqation of the fist kind and eo ode: d f n ( ) 1 d f n ( ) f ( ) 0 n n Eq.W6 d d In this case the soltion of Eq.6 fo n > 0 is : f ( ) C1 Jo( ) n Homogenos n Whee Jo is the Bessel fnction of the fist kind (o do not need to know the eact epession of Jo fo the following). With the appopiate bonda condition pove that a Jo( ) n n f n ( ) 1 i n Jo( nr ) Eq.W7 56

57 MGC 3143, M. Fenech. 7 Dedce the epession of V as a fnction of and t sing the 3 fist hamonics i.e. n=0,1, (be caefl the soltion fo n=0 comes fom point 4). 57

58 MGC 3143, M. Fenech. 8 Integate nmeicall the velocit V(,t) ove the entie coss section of the vessel. The otpt is the flow ate, Q(t). 9 Show that the nomaliation of Eq.5 sing * = /R and α = R ωρ/μ is : d f n 1 d f n * f B * n n Eq.8 d * * d * whee n * and B* ae constants that o have to detemine. 58

59 MGC 3143, M. Fenech. 10 What is the phsical meaning of the paamete α? How will the velocit pofile look like if α<<1? 4. Eecises Becase the mateial is difficlt, poblem solving is the onl wa o will be able to etain it. The soltions will not be given fo all poblems and o ae esponsible fo knowing how to solve the poblems. Simila poblems will be also given on the midtem and final, so o shold know how to do them efficientl. Poblem Solving Pocede: The following pocede shold be sed in fomlating all witten poblem soltions in eams and homewok. 1. Cleal fomlate the assmptions o make.. Cleal fomlate the analtic soltion as fa as possible sing smbolic foms befoe sbstitting nmeical vales. plg and chg will be penalied. 3. Onl once the entie poblem is solved, o can sbstitte appopiate nmeical vales. 4. Conclde b discssing the validit of o soltion in view of the assmptions o have made. Eecise 3.1 Midtem Winte 010 Plana Coette flow. Conside blood as an incompessible Newtonian isothemal flid in a lamina, stead, fll developed flow between two paallel 59

60 MGC 3143, M. Fenech. plates. One plate is fied, the othe moves with a constant velocit U. Gavit is neglected. End effects ma be neglected. a) Give the simplified continit and momentm eqations that model this flow. State o assmptions concisel and fll jstif o simplifications. b) Wite the bonda conditions. c) Show that the velocit distibtion in the flid is 1 p U ( b) b p Whee and U ae constants d) Daw the shapes of the velocit pofiles fo the following cases: p 0 (i) ; U>0 p 0 (ii) ; U=0 p 0 (iii) Dedce the pofile shape fo the following case ; U>0 e) Compte the shea stess τ in the gap. p 0 f) Assme fo this qestion that and U 0. The ed blood cells in a saline soltion ae damaged (haemolsis) when the epeience a shea stess above a citical vale c. Given the blood viscosit μa=3.5 cp, b= 1mm, and c = 1500 dnes/cm, popose a condition on the velocit U to avoid haemolsis. Eecise 3. Navie Stokes eqations in clindical coodinate. The blood flow in an etacopoeal line is assmed lamina, fll developed, and stead; the blood viscosit is assmed constant and the vessel coss section cicla. Conside the blood in a hoiontal tbe. End effects ma be neglected becase the tbe length L is elativel lage compaed to the tbe adis R. The flid flows nde the inflence of both a pesse diffeence Δp and gavit. 60

61 MGC 3143, M. Fenech. a) Give the continit and Navie Stokes eqations simplified to model the flow of the flid. b) Detemine the stead state velocit distibtion in the flid. c) Detemine the elationship between the flow ate and the pesse dop. d) Detemine the maimm velocit Uma. e) Detemine the velocit distibtion U (diection of the tbe) in tem of Uma, and R. Eecise 3.3 Inclined plate sface. (Middtem S010). A liqid flows steadil down an inclined plane foming a lamina film of thickness h. The inclination angle of the plate is small sch that o can safel assme fll developed flow ( / =0) and negligible acceleation in the diection. The flid can be assmed incompessible with a constant viscosit μ. Since the pesse on the sface of the film is constant, we can assme ρ/ =0 fo this thin feesface flow. a) Simplif the continit and Navie Stokes eqations fo this flow field. State o assmptions concisel and fll jstif o simplifications. b) Give a phsical eplanation fo wh o can assme that the shea stess on the sface of the film can be assmed negligible? c) Show that the velocit pofile is given b d) Show that the shea stess distibtion in the flid is given b e) Show that the aveage shea stess is given b 61

62 MGC 3143, M. Fenech. g sinh f) A stdent has to std the sie of ed blood cell aggegate fnction of the aveage shea ate. She designed an epeimental set p with which it is possible to watch the blood flowing ove a sloping slide nde a micoscope. Changing the inclination angle, will change the aveage shea stess and then she will be able to watch the sie of aggegates thogh the micoscope. Give the fnction that will allow he to compte the mean shea stess in the blood film as a fnction of the angle of inclination of the slide. Blood is assmed Newtonian. The sstem is fed, sing a singe pmp, with a constant flow ate Q. The blood is flowing in a channel with a width L. Accoding to Nssel the thickness of a film neglecting the sface tension foce is: width. 3N h g sin 1/ 3 Q L whee N / is the feed ate pe nit of Given: Q = 1 ml/h L = 1mm Blood densit 1050 kg/m3 g = 10 m/s Blood viscosit = 4 cpoise Eecise 3.4* Stock flow: lbication. Flow between two concentic otating sphees: This cold be the flid lbication in hman posthesis hip joints. 6

63 MGC 3143, M. Fenech. Conside an incompessible Newtonian, isothemal flid in lamina flow between two concentic sphees, whose inne and ote wetted sfaces have adii of kr and R, espectivel. The inne and ote sphees ae otating at constant angla velocities i and o, espectivel. The sphees otate slowl enogh that the ceeping flow assmption is valid. a) Give the continit and Navie Stokes eqations simplified to model this flow filed. b) Detemine the stead state velocit distibtion in the flid (fo small vales of i and o, we can assme DU/Dt=0). c) Detemine the Shea stess with o=0. d) Regading the Shea stess, discss wh the snovial flid is a shea thinning flid Eecise 3.5* Radial Flow between disks. Stead, lamina flow occs in the space between two fied paallel, cicla disks sepaated b a small gap b. The flid flows adiall otwad de to a pesse diffeence (P1 P) between the inne and ote adii 1 and, espectivel. Neglect end effects and conside the egion 1 onl. Sch a flow occs when a lbicant flows in cetain lbication sstems. Fige. Radial flow between two paallel disks. a) Simplif the eqation of continit to show that v = f, whee f is a fnction of onl. b) Simplif the eqation of motion fo incompessible flow of a Newtonian flid of viscosit μ and densit ρ. c) Obtain the velocit pofile assming ceeping flow. d) Sketch the velocit pofile v (, ) and the pesse pofile P(). e) Detemine an epession fo the mass flow ate b integating the velocit pofile. f) Deive the mass flow ate epession in e) sing an altenative shot ct method b adapting the plane naow slit soltion. Eecise 3.6* Disk viscosmete. A paallel disk viscomete consists of two cicla disks of adis R sepaated b a small gap B (with R >> B). A flid of constant densit ρ, whose viscosit μ is to be meased, is placed in the gap between the disks. The lowe disk at = 0 is fied. The toqe T, necessa to otate the ppe disk (at = B) with a constant angla velocit Ω, is meased. The task hee is to dedce a woking eqation fo the viscosit when the angla velocit Ω is small (ceeping flow). 63

64 MGC 3143, M. Fenech. Fige. Paallel disk viscomete. a) Simplif the eqations of continit and motion to descibe the flow in the paallel disk viscomete. b) Obtain the tangential velocit pofile afte witing down appopiate bonda conditions. c) Deive the fomla fo detemining the viscosit μ of a Newtonian flid fom measements of the toqe T and angla velocit Ω in a paallel disk viscomete. Neglect the pesse tem. Eecise 3.7 Application of the biohead eqation (fom Bioflid dnamics, C. Kleinstee, Talo & Fancis) Conside blood pefsion of a tisse lae of thickness h whee at the fat tisse inteface T = T(=0) = T1 and at the tisse coe inteface T= T (=h) =T. The blood (ρ, Cp) entes the tisse with a constant flow ate and tempeate Ta < T1 < T Sketch: Assmptions: Stead 1 D flow nifom flow Negligible metabolic ate Constant popeties Appoach: Redced biohead eqation Diect integation a) Based on the stated assmptions edce the biohead eqation of mammalian tisse: k T St T Cv Cp( Ta T ) t Whee is the flow ate and St is the heat soce b) Show that the following poposition is the soltion of the eneg eqation in the pesent case and that it espects the bonda condition: 64

65 T Ta T1 Ta T Ta sinh( m) cosh( m) coth( mh)sinh( m) T1 Ta sinh( mh) MCG3143, M. Fenech Cp m Whee k c) Plot fo Ta = 3 C, T1 = 34.5 C, T = 37 C, Cp = J/(kg.K), k = 0.37 W/(m.K) and = 400 ml/min. Plot again fo = 400 ml/min Eecises fom the tet book Soltions Soltion Soltion Soltion Assignement The Womesle eqations: blood plsatile flow in femoal ate. Each stdent shold do thei own assignment, althogh o ma wok togethe. Yo ma NOT shae electonic copies of the soltion. Woking togethe means that o ma look at each othe s wok, ask qestions, discss soltions, bt please not cop paste! Objectives To appl the Navie Stokes eqations to nstead flow To gain pactice sing a Foie coefficients epesentation to geneate a pesse wavefom in Maple. To geneate a flow wavefom fom a plsatile pesse wavefom 65

66 MCG3143, M. Fenech e θ e e V PART 1: Resoltion of Navie Stokes eqations in class PART : Visalisation: Matlab The diamete of femoal ate is.5 mm. We assme a constant blood viscosit μ = Ns/m and ρ = 1060 kg/m 3. Qestion 1 The Foie coefficients shown below descibe the pesse gadient vesstime cve in femoal ate. The feqenc is 1 H (Recall ω=π/peiod). Plot the pesse gadient vess time cve. n 0 1 a n coefficient (Pa/m) i 531 Matlab hint: Yo can se i o j fo the nit comple nmbe, bt be caefl to not se the same lette fo anothe vaiable. Yo can take the eal pat of a nmbe sing eal( ). Qestion. Using the Womesle s Navie Stokes soltion, on 4 diffeent figes, plot the 3 tems of the velocit (n=0, n=1, n=) and the total velocit as a fnction of the adis at t=0.5s Matlab hint: the Bessel fnction command fo a eo ode Bessel fnction is BesselJ(0, agment). Qestion 3 On 4 diffeent figes, plot the 3 tems of the velocit (n=0, n=1, n=) and the total velocit pofile as a fnction of adis fo t=0 to t=1 (0 plots pe gaph ). Matlab hint se the commands hold on hold off. Qestion 4 Integate nmeicall the velocit V(,t) ove the entie coss section of the vessel. The otpt is flowate, Q(t). Please show the flowate in lite/minte. Qestion 5 In some diseases, the blood viscosit cold each 8 cp. Plot the velocit as a fnction of the adis as in qestion 3 sing this hpeviscosit. 66

67 MCG3143, M. Fenech Qestion 6 Discss the eslts obtained Qestion 3 and 5 highlighting α vale in each case. Qestion 7 To conclde, discss the assmptions made in this model and popose how to assess the model. Yo have to sbmit a had cop of o epot inclding an intodction and a conclsion. Fo each qestion mathematical fom of eqation, etact of matlab code, figes and desciption of eslts ae epected. 7. Refeences Bioflid mechanics, the hman ciclation, K Chandan and al., Talo &Fancis d edition. p41 43 Fid mechanics, Mnson Yong Okiishi. Ebook. 67

68 MCG3143, M. Fenech Chapte 4 Comptational Flid Dnamic (CFD) and measement techniqes in biomedical 1. Comptational flid dnamics Tetbook : Bioflid mechanics, the hman ciclation, K Chandan and al., Talo &Fancis d edition. Chapites 11.. Flow measement in the cadiovascla sstem Tetbook : Bioflid mechanics, the hman ciclation, K Chandan and al., Talo &Fancis d edition. Chapites Eecices Eecise 3.1 A patient s cadiac otpt is 5500 ml/min while his ateial ogen concentation is 0.ml/ml and he venos ogen concentation 0.15 ml/ml. Find this peson s spiomete ogen consmption Eecise 3. Using a continos wave Dopple with a caie feqenc of 7Mh, α =45, the speed of sond =1500m/s, and Dopple shifted feqenc 5000H find the blood velocit. 68

69 Eecise 3.3 MCG3143, M. Fenech Using the themodiltion method fo measing cadiac otpt, 10 ml of injectate is injected ove.5 seconds ove a peiod of 5 seconds. The cadiac otpt is 4.0 l/min. Use the following data to estimate the vale of T b dt t1 0 Volme of injectate = 10 ml, dt of injectate = 30 K Densit of injectate = 1005 kg/m3 Heat capacit of injectate = 4170 J/(kg K) Densit of blood = 1060 kg/m3 Heat capacit of blood = 3640 J/(kg K) Sggested Eecises fom the tetbook: 10.4;10.5;10.6; CFD simlation assignment Tet book page p413 69

70 MCG3143, M. Fenech Chapte 5 Flow ove immesed bod (incompessible) Adapted fom Fndamentals of Fid Mechanics, Mnson Yong Okiishi (FFM) And fom Mltimedia Flid mechanics, Cambidge Univesit pess (MFM) Goal: Std how the foces de to the flid flow act on a bod Eamples in: Bioflid mechanics Shea stesses affect endothelial cell Shea stesses affect ed blood cell (hemolsis) and othe cells Fish swim / Bid fl Ai in the espiato sstem / Paticle tanspot Eamples in: Classical flid mechanics Aifoil Cas Planes Ect 1 Geneal Etenal Flow Chaacteistics A bod immesed in a moving flid epeiences a esltant foce de to the inteaction between the bod and the flid sonding it. Cases: The flid is stationa and the bod moves thogh the flid with velocit U. The bod is stationa and the flid flows past the bod with velocit U. In an case, we can fi the coodinate sstem in the bod and teat the sitation as flid flowing past a stationa bod Steamlined o blnt Steamlined bodies : aifoils, acing cas, little effect on the sonding flid, 70

71 Blnt bodies : paachtes, bildings Stong effect on the sonding flid MCG3143, M. Fenech It is sometimes desiable to make an object as steamlined as possible. In othe sitations a blnt object is desied. Tpicall, steamlined objects have less dag than blnt objects. A kaak is a steamlined object that moves thogh the wate with minimal esistance and distbance to the flid. It eqies a elativel small poplsive foce. The ppose of the paddle is to impat the poplsive foce to the kaak. To do so it mst geneate a elativel lage esistance to motion thogh the wate. A paddle is a blnt object. The Renolds nmbes fo the paddle and the kaak ae in the ode of 100,000 to 1,000,000. Lift and Dag Concept.1 Definitions Foces at the flid bod inteface de to the inteaction between the bod and the flid occs: Wall shea stesses, de to viscos effects w Nomal stesses de to the pesse, p Total sface foce: Fp Fw Sface PdA da w Sface D FD dag : esltant foce in the diection of the psteam velocit L lift : esltant foce nomal to the psteam velocit Dag Coefficient = Dag foce/chaacteistic inetia foce often CD is an epeimentall epessed fnction of Renolds nmbe CD=f(Re) (see table 1) Lift Coefficient = Lift foce/chaacteistic inetia foce 71

72 MCG3143, M. Fenech.3 Dag fo diffeent shapes Table 5.1: Low Renolds Nmbe Dag coeficient [FFM] Fige 5.1: Eemple of dag [FFM]. Dag coefficient, fo a sphee in stokes flow ( Re 1) Fom Dag coefficient definition : Fom the table : C D 4 UD D sphee C D U D 4 7

73 So, D sphee 4 U UD D 4 MCG3143, M. Fenech Usefl eslt : D D sphee 6 U Note: U the elative velocit of the sphee in the flid (U flid U sphee ).4 Tanspot of mico paticles Biomedical applications: ed blood cell sedimentation, paticles in the ppe ai wa, aeosol Assming spheical dilte mico paticles in sspension with a negligible otation, we can wite the Newton law: dv p m p FD Fg FB FB Fo dt Mass. acceleation = dag + gavit +Bownian +Boant +othe (electostatic, magnetic, ) Dag foce: Fo a sphee in stokes flow f : flid velocit p : paticle velocit R : paticle adis ρ f : flid densit ρ p : paticle densit F F D D Cd f ( f p ) R( ) 6 f p Ap Gavit foce : Boant foce: F G F G R g p 3 R g f Bownian foce has to be consideed fo a sb micometic paticle. Paticle sedimentation If we assme spheical a micometic paticle, neglecting Bownian foce, qasi static (sedimentation) we get: R (0 p ) 4 R p g 4 R f g 3 3 R ( p f 9 p ) Dag foce dominates (eve shape) g 73

74 MCG3143, M. Fenech d p Cd m p f ( f p ) Ap dt We get a diffeential eqation. This eqation is sed in simlation of paticle deposition / aeosol in the ppe aiwa. Howeve, tblence has to be consideed. Theefoe, the tem of the flid velocit is actall moe comple, composed of an aveage velocit and a flctating velocit to model the tblence. 3 Chaacteistics of Flow Past an object Etenal flow past objects encompasses an etemel wide vaiet of flid mechanics phenomena. Cleal, the chaacteistic of the flow field is a fnction of the shape of the bod. Thee can be a wide vaiet in the sie of a bonda lae and the stcte of the flow within it. Pat of this vaiation is de to the shape of the object on which the bonda lae foms. Flows past elativel simple geometic shapes (i.e. a sphee o cicla clinde) ae epected to have less comple flow fields than flows that pass a comple shape sch as an aiplane o a tee. Howeve, even the simplest shaped objects podce athe comple flows. In this section we conside the simplest sitation, one in which the bonda lae is fomed on an infinitel long flat plate along which flows a viscos, incompessible flid and anothe in which the bonda lae is fomed aond a cicla clinde o an aifoil. Fige 5. :The nate of the flow past a bod depends stongl on whethe Re>>1 [FFM] Re<<1 o 74

75 MCG3143, M. Fenech Fige 5.3 Flow sepaation ma occ behind blnt objects. [FFM] Fige 5.4 : Effect on the dag coefficient on a cicla clnde [FFM] 75

76 MCG3143, M. Fenech 4 Bonda lae chaacteistics 4.1 Bonda Lae Stcte and Thickness on a Flat Plate Fige 5.5 : 1 Bonda Lae Stcte and Thickness on a Flat Plate [FFM] 4. Bonda lae thickness Fige 5.6 : 1 Bonda Lae Thickness [FFM] Fist definition: Thickness : Bonda lae displacement thickness 76

77 MCG3143, M. Fenech Momentm Thickness, θ Detail of calclation: 4.3 Momentm Integal Bonda Lae Eqation fo a Flat Plate Fige 5.10 : Contol volme sed in the deivation of the momentm integal eqation fo bonda lae flow. [FFM] One of the impotant aspects of the bonda lae theo is the detemination of the dag cased b shea foces on a bod. We conside the nifom flow past a flat plate and the fied contol volme. In ageement with advanced theo and epeiments, we assme that the pesse is constant thoghot 77

78 MCG3143, M. Fenech the flow field. The flow enteing the contol volme at the leading edge of the plate [section 1] is nifom, while the velocit of the flow eiting the contol volme vaies fom the psteam velocit at the edge of the bonda lae to eo velocit on the plate. The flid adjacent to the plate makes p the lowe potion of the contol sface. The ppe sface coincides with the steamline jst otside the edge of the bonda lae at section. It need not (in fact, does not) coincide with the edge of the bonda lae ecept at section. B definition the dag foce is D is the dag that the plate eets on the flid. Note that the net foce cased b the nifom pesse distibtion does not contibte to this flow. If we appl the momentm eqation in diection fo stead flow of flid within this contol volme we obtain Since the plate is solid and the ppe sface of the contol volme is a steamline, thee is no flow thogh these aeas. Ths, Whee fo a plate of width b Flow ate thogh section 1 mst eqal that thogh section : Mltipl b ρub: So we get fo the dag: Recall the momentm thickness:, we can wite: On an epeimental point of view, if o can mease the, o can compte θ and then dedce the dag. Note that this eqation is valid fo lamina o tblent flows. 78

79 MCG3143, M. Fenech As inceases, inceases δ, and the dag inceases. The thickening of the bonda lae is necessa to ovecome the dag of the viscos shea stess on the plate. This is conta to hoiontal fll developed pipe flow in which the momentm of the flid emains constant and the shea foce is ovecome b the pesse gadient along the pipe. The shea stess distibtion can be obtained fom this eqation b diffeentiating both sides with espect to to obtain The incease in dag pe length of the plate occs at the epense of an incease of the momentm bonda lae thickness, which epesents a decease in the momentm of the flid. Since (fom dag definition) it follows that And finall On an epeimental point of view, o need to mease the field of velocit to be able to compte θ at an point; this cold be done sing PIV fo eample. Appoimation of the BL thickness: The seflness of this elationship lies in the abilit to obtain appoimate bonda lae eslts easil b sing athe cde assmptions. Fo eample, if we knew the detailed velocit pofile in the bonda lae, we cold evalate the dag fom and the shea stess fom. Eample: assming linea velocit pofile in the BL =U/δ, estimate the bonda lae thickness fnction of the position 79

80 MCG3143, M. Fenech 80

81 MCG3143, M. Fenech Pandtl/Blasis Bonda Lae Soltion Fom mass consevation in an incompessible flid, the steamwise velocit vaiation mst be the same sie as the coss steam velocit vaiation. This eqalit gives s an estimate fo the coss steam velocit, V. Since we have assmed that the bonda lae thickness, δ, is small compaed to the bod length, L, we can see that is mch smalle than U Fige 5.7 : 1 Bonda Lae Thickness [MFM] Mass consevation Eact eqation: 0 v Appoimate eqation: V L U Means that U L V Fo the steamwise NS eqation, within the BL, the change in U in the steamwise diection, can be neglected compae to the change in the coss steam diection. Using Benoilli s eqation, we can eplace the pesse gadient b the velocit gadient in the ote flow. This means we can simplif the steamwise momentm eqation. Note, that fo stead flow since the velocit is constant, and fom Benoilli we know the pesse is constant, the vaiation of pesse cold cancelled. Momentm eqation fo stead flow diection Eact eqation : 1 p v Dimensional analsis : 0 U L U U U L L U Simplif BL eqation: v Momentm eqation fo stead flow diection Eact eqation : 1 v v p v v v Dimensional analsis : L U L U L U L U 3?

82 Ths Finall, U L p 0 U MCG3143, M. Fenech Bonda lae eqations (Newtonian, incompessible, stead): v 0 Continit: v Momentm : BC: (,0)=0 (,inf)=u v(,0)=0 This dimensional analsis give s the magnitde of the BL thickness at the length : U In geneal, the soltions of nonlinea patial diffeential eqations sch as the bonda lae eqations, ae etemel difficlt to obtain. Howeve, b appling a cleve coodinate tansfomation and change of vaiables, Blasis edced the patial diffeential eqations to an odina diffeential eqation. Blasis coodinate tansfomation is the following: Desciption of this pocess can be fond in standad books dealing with bonda lae flow. Soltion of BL eqation is give in the following table : 8

83 MCG3143, M. Fenech Fige 5.9 : Bonda lae eqation, Blasis soltion [FFM] The Blasis soltion cold be smmaied as: With the velocit pofile known, it is an eas matte to detemine the wall shea stess, whee the velocit gadient is evalated at the plate. The vale of at =0 can be obtained fom the Blasis soltion to give: In dimensionless fom, we can have the local fiction coefficient cf: c f w 1 U Re O fo the all sface, the dag fiction coefficient, CDf 83

84 C Df D f w U A l b d U lb 1.38 Re MCG3143, M. Fenech 5 Tblent bonda lae Fige 5.1 :Tblent flow [FFM] The analtical eslts ae esticted to lamina bonda lae flows along a flat plate with a eo pesse gadient. The agee qite well with epeimental eslts p to the point whee the bonda lae flow becomes tblent, which will occ fo an fee steam velocit and an flid povided the plate is long enogh. The vale of the Renolds nmbe at the tansition location is a athe comple fnction of vaios paametes involved, inclding: the oghness the cvate distbances in the flow otside the bonda lae. Eample: On a flat plate with a shap leading edge in a tpical ai steam, the tansition takes place at a distance fom the leading edge of a Re aond Actall, the tansition fom lamina to tblent bonda lae flow ma occ ove a egion of the plate, not at a specific single location. This occs, in pat, becase of the spottiness of the tansition. Tpicall, the tansition begins at andom locations on the plate The comple pocess of tansition fom lamina to tblent flow involves the instabilit of the flow field. Tpical pofiles obtained in the neighbohood of the tansition location ae indicated in the net fige. The tblent pofiles ae flatte, have lage velocit gadients at the wall, and podce a lage bonda lae thickness than the lamina pofiles. 84

85 MCG3143, M. Fenech Fige 5.1:Bonda lae pofiles [FFM] 6. Pesse gadient effect on flow sepaation The bonda lae discssions in the pevios pats have dealt with flow along a flat plate in which the pesse is constant thoghot the flid. In geneal, when a flid flows past an object othe than a flat plate, the pesse field is not nifom. Fige 5.13 :Pesse gadiant [FFM] Phsicall, in the absence of viscos effects, a flid paticle taveling fom the font to the back of the clinde coasts down the pesse hill fom to A to C and then back p the hill to fom point C to F withot an loss of eneg. Thee is an echange between kinetic and pesse eneg, bt thee ae no eneg losses. The same pesse distibtion is imposed on the viscos flid within the bonda lae. Howeve, becase of the viscos effects involved, the paticle in the bonda lae epeiences a loss of eneg as it flows along. This loss means that the paticle does not have enogh eneg to coast all of the wa p the pesse hill fom C to F and to each point F at the ea of the clinde. 85

86 MCG3143, M. Fenech This kinetic eneg deficit is seen in the velocit pofile detail at point C. Becase of fiction, the bonda lae flid cannot tavel fom the font to the ea of the clinde. The sitation is simila to a bicclist coasting down a hill and p the othe side of the valle. If thee wee no fiction, stating with eo speed, the ide cold each the same height fom which he o she stated. Cleal, fiction cases a loss of eneg making it impossible fo the ide to each the height fom which he o she stated withot sppling additional eneg Fige 5.14 :Pesse gadiant : Biccle analog [MFM] The flid within the bonda lae does not have sch an eneg sppl. Ths, the flid flows against the inceasing pesse as fa as it can, at which point the bonda lae sepaates fom the sface. At the sepaation location, the velocit gadient at the wall and the wall shea stess ae eo. Beond that location, fom D to E, thee is evese flow in the bonda lae. Fige 5.15 :Bonda lae pofile, Sepaation point [FFM] Compaed with a lamina bonda lae, a tblent bonda lae flow has moe kinetic eneg and momentm associated with it. Ths, the tblent bonda lae can flow fathe aond the clinde befoe it sepaates than the lamina bonda lae can. Fige 5.16 :Sepaation point pictes [MFM] Avoiding flow sepaation A loss of pesse in the sepaated flow egion behind blnt bodies cases an imbalance between the psteam and downsteam pesse foces, contibting geatl to an inceased net dag foce. In intenal flow BL sepaation has dastic conseqences to pesse losses. 86

87 MCG3143, M. Fenech Shape design limits the advese pesse gadient Fige 5.17 :Sepaation point, effect of the shape [MFM] Enegie the flow b bpass Fige 5.18 :Sepaation point, bpass [MFM] Enegie the flow b tblence Fige 5.19 :Sepaation point, tblence [MFM] Enegie the Blowing Fige 5.0 :Sepaation point, blowing [MFM] 87

88 Sction Effect MCG3143, M. Fenech Fige 5.1 :Sepaation point, sction [MFM] 7. Eecises Eecise 5.1* (fom Fid Mechanics, Mnson Yong Okiishi) A 0.3 m diamete socce ball moves thogh the ai with a speed of 10 m/s. Wold the flow aond the ball be classified as low, modeate, o lage Renolds nmbe flow?eplain. Eecise 5.* (fom Fid Mechanics, Mnson Yong Okiishi)A small 15 mm long fish swims with a speed of 0 mm/s. Wold a bonda lae tpe flow be developed along the sides of the fish? Eplain. Eecise 5.3* (fom Fid Mechanics, Mnson Yong Okiishi) Ai flow ove a flat plate of length l = ft sch that the Renolds nmbe based on the plate length is Re = 105. Plot the bonda lae thickness δ, fo 0 < < l Eecise 5.4 * (fom Fid Mechanics, Mnson Yong Okiishi) A lamina bonda lae fomed on one side of a plate of length L podces a dag D. How mch mst the plate be shotened if the dag on the new plate is to be D/4? Assme the psteam velocit emains the same. Eplain o answe phsicall. Eecise 5.5* (fom Fid Mechanics, Mnson Yong Okiishi) How mch less powe is eqied to pedal a acing stle biccle at 0 mph with a 10 mph tail wind than at the same speed with a 10 mph head wind? 88

89 MCG3143, M. Fenech Eecise 5.6 A vetical wind tnnel (Qi winte 010) A vetical wind tnnel can be sed fo skdiving pactice. Estimate the vetical wind speed needed if a 150 lb peson is to be able to float motionless when the peson (a) cls p as in a coching position o (b) lies flat. g = 3. ft/s Ai densit = lb/ft3 Eecise 5.7 (Qi winte 010) In the photogaphs shown, indicate appoimatel the egion in which o epect viscos effects to be impotant and the egion whee the flow can be assmed to be inviscid. In which egion can o appl Benolli s eqation? 89

90 MCG3143, M. Fenech Eecise 5 8 Gossame Condo vess Albatos (Final W010) In 1977 the Gossame Condo won the Keme pie b being the fist hman poweed aicaft to complete a pescibed figeeight cose coveing a total of a mile (1.6 kilometes). Fige 4: Gossame Condo Fige 5: Albatos lift and dag coefficients The following data petains to this aicaft: Detemine the lift coefficient fo the Gossame Condo Detemine the lift to dag atio of the Gossame Condo Detemine the powe to ovecome the dag Dedce the powe, eqied b the pilot of the Gossame Condo Detemine the mean lift to dag atio of the Albatos fom the Fige 5 (ed dotted line) Compae eslts obtained in b) and e) with the lift to dag atio given in table and eplain wh a highe lift to dag atio is tpicall one of the majo goals in aicaft design. 90

91 Flight aticle MCG3143, M. Fenech Boeing 747 Cessna Concode Hose spaow L/D atio Table : Repesentative L/D atios Eecise 5.9 Flow ove a Hmme (Final S008). The flow aond a Hmme H atomobile is being stdied sing a 1/18 scale model in a wate tnnel at the Univesit of Ottawa. The epeiment is intended to simlate the ai flow aond a eal Hmme diving in a staight line at 30 km/ho on a still da (no wind). 0.6m 0.11m Fige 4: Hmme model s shape and dimensions a) In the photogaph shown Fige A 1, indicate appoimatel the egion in which o epect viscos effects to be impotant and the egion whee the flow can be assmed to be inviscid. In which egion can o appl Benolli s eqation? b) Detemine the Renolds nmbe coesponding to the flow aond the Hmme H. GIVEN: Densit of ai 1. kg/m3. Viscosit of ai kg/(m s). Densit of wate 998 kg/m3. Viscosit of wate kg/(m s). c) The epeiment is pefomed b monting the model Hmme H pside down on a stationa plate in the wate tnnel and b pmping wate b the model at a constant velocit. What wate velocit shold be sed fo the epeiment to be dnamicall simila to the te flow aond a eal Hmme H on the oad? 0.11m Fige 5: Hmme H epeiment setp in the wate tnnel. d) A dag foce of FD = 8 N is meased on the model. Compte the dag coefficient of the ca. What foce does this coespond to on the eal ca? e) Do o foesee an poblems associated with this setp? Discss an soces of eo that ma be pesent. f) Bons : Repot the dag coefficient on the Fige A. Wh we do not b this kind of ca? Think geen! 91

92 MCG3143, M. Fenech Fige A 1 : Flow ove a ca. Fige A : The histoical tend of steamlining atomobiles to edce thei aeodnamic dag and incease thei miles pe gallon. Fom fndamental of flid mechanics Mnson edt. Eecise 5.10 Red blood cell sedimentation. 1) How fast does a ed blood cell of 90μm3 (assme it is a sphee) fall thogh the plasma? The dag fo a sphee in stoke flow is given b 6πμRU. The plasma viscosit is 1.4cP and the ed blood cell densit 1.09g/cm3, plasma densit 1.03g/cm3 ) The ed blood inclines to clste (aggegation). Compte the how fast does a spheical clste with a diamete of 0 ed blood cell fall thogh the plasma? Note: If o do a blood test sometime the sedimentation velocit is meased to have infomation abot how o ed blood cells incline to aggegate. Eecise 5.11 Flow of a white blood cell in a mico channel. (Midtem fo ndegadate Fench stdents, UTC). Fo the fist appoimation of the micociclation, the capilla vessels cold be modelled as clindes with adis b and the cells as solid bodies centeed on the ais of the clinde. We conside a stetched white blood cell (WBC) as a clinde of length L and adis b. The WBC is moving in the capilla with a constant velocit U becase of a pesse dop p1 p, whee p1 is the pesse at the entance of the film, p pesse at the eit. 9

93 L U a b MCG3143, M. Fenech p1 film We assme : h = b a, L >> h We neglect : End effects Gavitational effect p Capilla Wall White blood cell 1) Show that the eqation of consevation in clindical coodinate fo this poblem is given b: p 1 d d p ( ) 0 et 0 d d With =(). ) Wite the bonda conditions fo ( = a) and ( = b). 3) Detemine the velocit distibtion in the film as a fnction of the pesse dop p/ and U. Yo can se the following notation: 1 G A U a b ln( a ) 4 G = constant= p/ and b 4) Show that the shea stess distibtion on the white blood cell at = a is given b: G A a a 5) Dedce the dag foce de to the shea stess applied on the white blood cell. 6) What ae the foces appling on the WBC? Dedce the elationship between the white blood cell velocit and the pesse dop G. 7) What happens to the velocit pofile in the film when h << b? 1 Hint: saw, in the film b 8) Discss this model fo the flow of a white blood cell in a capilla (as shown on the fige) Eecise 5 1 Bonda lae ove the eath s sface. An atmospheic bonda lae is fomed when the wind blows ove the eath s sface. Tpicall, sch velocit pofiles can be witten as a powe law: whee the constants a and n depend on the oghness of the teain. As is indicated in the Fig. Below. Tpical vales ae fo ban aeas, fo woodland o sbban aeas, and fo flat open cont. If the velocit is 0 ft_s at the bottom of the sail on o boat, what is the velocit at the top of the mast If the aveage velocit is 10 mph on the tenth floo of an ban bilding, what is the aveage velocit on the sitieth floo? 93

94 MCG3143, M. Fenech 8. Soltions Soltion 5.1 Soltion 5. 94

95 MCG3143, M. Fenech Soltion

96 MCG3143, M. Fenech Soltion 5.5 ANS : 0.375hp 9. Refeences Fndamentals of Fid Mechanics, Mnson Yong Okiishi (FFM) Mltimedia Flid mechanics, Cambidge nivesit pess (MFM) 96

97 MCG3143, M. Fenech Chapte 6 Rheolog of blood 1. Rheolog of blood and non Newtonian eqations Tetbook : Bioflid mechanics, the hman ciclation, K Chandan and al., Talo &Fancis d edition. Chapte 4. Eecises Eecise 6.1 Compte the appaent viscosit of blood flowing thogh a tbe with a 100m diamete sing the fee maginal cell lae theo. Fee maginal cell lae fo hman blood: assme vales of plasma viscosit of 1. CP and whole blood viscosit of 3.3 CP at 37oC. Assme a cell fee lae thickness of 3 m. Eecise 6. (Qi W010) n The blood as a Non Newtonian flid can be chaacteised b the powe law: K Using a ln ln gaph, detemined gaphicall K and n with the following data: ln( ) ln( ) ln( ) 97

98 MCG3143, M. Fenech ln() Eecise 6.3 (Final W010) Capilla Viscomete Fige 6.: Pincipal design of capilla viscomete Given clindical coodinates and pesse diven lamina flow, anale the capilla viscomete. The flow is pesse diven b a pesse dop of P thogh a capilla with a length L (Fige 3). Show that in clindical coodinates the onl non eo components of the consevation eqation is: (jstif each simplification o make) P 1 ( ) Integate this eqation b pats sing the appopiate bonda condition and show that: P ( ) L ( P 0 ) Assming a Newtonia,n flid show that: 1 P ( R ) 4 L Fo a Newtonian flid, compte the flow ate and dedce the elationship between viscosit and Q, P, R and L: 4 R 8 Q P L Assming a powe law flid model, 1/ n n1 n1 n P n n ( R ) n 1 KL K n show that: 98

99 MCG3143, M. Fenech Eplain how to obtain the following epession fo the flow ate (Give some ke points bt do not do the fll calclation): 3 n R P Q R 3n 1 KL 1/ n Eplain qalitativel the wa to get K and n, in ode to chaacteise the non Newtonian flid (no comptation is needed). Eecice 6.4 QUIZ smme 010 h We conside blood flow in a channel with w>>h. The elationship between and of the blood follow the powe law that is given b: h g K Fo an flid thogh two plates, the balance between the pesse gadient and the shea stess is given b: p p Whee does the eqation come fom? Compte the velocit pofile. State cleal the bonda condition sed. Eecice 6.5* Bingham flid flow in a plane naow slit. Conside a flid (of densit ρ) in incompessible, lamina flow in a plane naow slit of length L and width W fomed b two flat paallel walls that ae a distance B apat. End effects ma be neglected becase B << W << L. The flid flows nde the inflence of a pesse diffeence Δp, gavit o both. n 99

100 MCG 3143, M. Fenech, W011 Fige : Flid flow in plane naow slit. a) Detemine the stead state velocit distibtion fo a non Newtonian flid that is descibed b the Bingham model. b) Obtain the mass flow ate fo a Bingham flid in slit flow. Eecice 6.6* Bingham flid flow in a cicla tbe. Conside a flid (of densit ρ) in incompessible, lamina flow in a long cicla tbe of adis R and length L. End effects ma be neglected becase the tbe length L is elativel lage compaed to the tbe adis R. The flid flows nde the inflence of a pesse diffeence Δp, gavit o both. Fige. Flid flow in cicla tbe. a) Detemine the stead state velocit distibtion fo a non Newtonian flid that is descibed b the Bingham model. b) Obtain the mass ate of flow fo a Bingham flid in a cicla tbe. 100

101 MCG 3143, M. Fenech, W011 Eecise 6.7 Cell fee Maginal lae. Assme that whole blood with a hematocit of 45% flows thogh a small diamete tbe. The total flow ate is 8 µl/h, althogh in the coe egion it is 6 µl/h and in the peipheical egion it is 4 µl/h. The blood cells accmlate in the coe egion with a volme of 5 µl and thee ae no blood cells pesent in the cell fee peipheical egion which has a volme of 3 µl. a.) Daw a fige showing the diffeent aeas: cell fee peipheical and coe egion b.) Detemine the hematocit in the coe egion. c.) Then detemine the aveage hematocit in the whole tbe. d.) What is the effect of the cell fee lae on the appaent viscosit of the blood in the tbe? Sggested Eecises fom the tetbook: 4.3;4.4; ;4.7; 3. Soltions Soltion 6.5 Bingham flid flow in a plane naow slit Soltion: Soltion 6.6 Bingham flid flow in a cicla tbe Soltion: 101

102 MCG 3143, M. Fenech, W011 Chapte 7 Intodction to flid machine 1. Intodction to flid machine Chapte fom the book Fid mechanics (Mnson Yong Okiishi) PPT ae posted online. Eecises Eecise 7.1* Centifgal blood pmps (Final W010). A centifgal blood pmp is designed to podce a volme flow ate of 11 L/min at an impelle otational speed of 3610 pm. The geometical specifications of the centifgal blood pmp impelle given b the constcto ae smmaied in Table1. The viscosit of blood is 1050 kg/m3. Fige 1: In vivo animal test ig: (a) anatomical animal std; (b) view of the impelle afte in vivo test; and (c) view of the volte casing afte in vivo test. 10

103 MCG 3143, M. Fenech, W011 Fige : Centifgal pmp flow in and ot Inlet diamete (mm) Eit diamete (mm) Inlet width (mm).70 Eit width (mm).70 Nmbe of blades 5 Tip cleaance (mm) 0.30 Length in aial diection (mm) 5.70 Blade angle at inlet (o) Blade angle at dischage (o) 67.9 Table 1: Specifications of the centifgal blood pmp impelle. a) Assming nifom flow at the inlet and the otlet, and that the flow entes and leaves tangent to blade: daw the inlet and otlet velocit diagam. Q Vt U b) Show that b cot. c) Calclate the mechanical powe and the theoetical head. d) Accoding to the Fige A 1 in the appendi, what is the pmp efficienc fo the woking condition descibed above? e) Dedce the powe needed and the pmp head. Plot o eslts on the Figes A and A 3. Eplain an diffeences o obtained. f) Calclate the specific speed and the specific diamete sing given eqations. On Fige A 4, which pesents data of efficient indstial tbomachines, plot the location of o centifgal pmp sing a sta smbol. 103

104 MCG 3143, M. Fenech, W011 Fige A 1: Blood centifgal pmp chaacteistic obtained fom Oh et al. with blood: Pmp efficienc Fige A : Blood centifgal pmp chaacteistic obtained fom Oh et al. with blood: Pmp inpt powe. Fige A 3: Blood centifgal pmp chaacteistic obtained fom Oh et al. with blood: Static pesse ise. 104

105 MCG 3143, M. Fenech, W011 Fige A 4 : Data of efficient indstial tbomachines [4] Balje, O. E. Tbomachines: a gide to design, selection, and theo, 1981 (John Wile, New Yok). Refeence: H W Oh, E S Yoon, M R Pak, K Sn, and C M Hwang. Hdodnamic design and pefomance analsis of a centifgal blood pmp fo cadioplmona ciclation. Poc. IMechE Vol. 19 Pat A: J. Powe and Eneg. p Eecise 7. Heat pmp head. Ae o feet still pefsed when o do a handstand? Neglecting the fiction loss, estimate the elevation of the blood that a health heat can achieve with a diastolic/ sstolic pesse 80/10 mmhg. Eecise 7.3 Calclation of pmp chaacteistics fom test data (Fom : Intodction to flids mechanics Fo and all) 105

106 MCG 3143, M. Fenech, W011 The flow sstem sed to test a centifgal pmp at a nominal speed of 1750 pm is shown. The liqid is wate at 80oF, and the sction and dischage pipe diametes ae 6in. Data meased ding the test ae given in the table. The electic moto spplied 460V, 3 phase, and has a powe facto of and a constant efficienc of 90%. Calclate the net head deliveed and the pmp efficienc at a volme flow ate of 1000gpm. Plot the pmp head, powe inpt, and efficienc as fnction of volme flow ate. Eecise 7.4* Centifgal pmp Fom : Flid mechanics Mnson Yong Okiishi Shown in the Fig. ae font and side views of a centifgal pmp oto o impelle. If the pmp delives 00 lites/s of wate and the blade eit angle is fom the tangential diection, 106

107 MCG 3143, M. Fenech, W011 detemine the powe eqiement associated with flow leaving at the blade angle. The flow enteing the oto blade ow is essentiall adial as viewed fom a stationa fame. Eecise 7 5* Similait laws Fom :Flid mechanics Mnson Yong Okiishi When the shaft hosepowe spplied to a cetain centifgal pmp is 5 hp, the pmp dischages 700 gpm of wate while opeating at 1800 pm with a head ise of 90 ft. (a) If the pmp speed is edced to 100 pm, detemine the new head ise and shaft hosepowe. Assme the efficienc emains the same. (b) What is the specific speed, fo this pmp? Eecise 7.6* Velocit tiangles Fom : Flid mechanics Mnson Yong Okiishi An aial flow tbomachine oto involves the psteam 11 and downsteam 1 velocit tiangles shown in the Fig. Is this tbomachine a tbine o a fan? Sketch an appopiate blade section and detemine the eneg tansfeed pe nit mass of flid. 107

108 MCG 3143, M. Fenech, W Soltions Soltion = = B1 = B = w = U1 = U = Q = e 04 b1 = b = p = 1050 Vt1 = Vt = Ts = Wm = H =.0848 HmmHg = efficienc = Wp =.811 Hp = HpmmHg = headcoef = flowcoef = specificspead = specificdiamete =

109 MCG 3143, M. Fenech, W011 Soltion 7.4 Soltion 7 5 Similait laws ANS: 40 ft, 7.41 hp;

110 MCG 3143, M. Fenech, W011 Soltion 7.6 Velocit tiangles ANS: tbine; _36.9 ft_s 110

111 MCG 3143, M. Fenech, W011 Fomla UNITS Length [L] Mass [M] Time [T] Sface [L ] Volme [L 3 ] Velocit [LT -1 ] Acceleation [LT - ] Foce [LMT - ] Feqenc [T -1 ] Pesse, stess [L -1 MT - ] Eneg, wok, qantit of heat [L MT - ] kinematic viscosit [L T -1 ] Densit [L -3 M] dnamic viscosit [L -1 MT -1 ] Sface tension [MT - ] SI Units Mete/kg/second CGS Units Cm/gam/second US cstoma Units Inch/pond/second mete: m centimete: 1 cm = inch: 1 in = m 0.01m feet: 1 ft = m kilogam: kg gam:1g = kg pond:1lb m = kg second: s m 1 cm = m 1 in = m m 3 1 cm 3 = 1ml =10-6 m 3 1 in 3 = gallon = 1.64e-005 m 3 m.s -1 cm.s -1 1 mi.h -1 = m.s -1 = ft.s -1 m.s - cm.s - ft.s - newton: N = m. kg.s - dne : 1 dn = 1 cm.g.s - = 1e-005 N het: H = 1.s -1 pascal: Pa = N.m - = kg.m -1.s - 1 cmh O = 98 Pa 1 mmhg = Pa 1 dn.cm - = 1 g. cm -1. s - = 0.1 Pa = cmho = mmhg (called also bae) jole: J = N m = m eg :1 eg = 1 g cm /s.kg.s - = 1e-007 J Ponds foce : 1 lb f = N = 4448 dn Psi =1 lb.in - = 6895 Pa = 7.68 inho =.036 inhg BTU :1 bt = 1.055e 10 eg =778 ft.lb f m s -1-1 stokes: 1 St = 1 cm s = m s 1 ft.s -1 = 900 St -1 = 9e4 cst Kg.m -3 1 g.cm -3 = 1000 Kg.m -3 1 lb m.in -3 = 16.0 Kg.m -3 Pa s = N.m -.s= m -1. poise :1 P = 1 dn.s.cm - kg.s -1 = 100 cp = 0.1 Pa s N.m -1 = kg.s - lb m.s - 1 lb f.s -1. ft -1 = 10 P = 1000 cp 111

112 MCG 3143, M. Fenech, W011 FLUID S DENSITY AND VISCOSITY ai wate plasma blood µ 18.3 µpa.s 0.89 cp 1.3 cp 3.5 cp ρ (kg/m3) CONSERVATION LAWS: Geneal fom Continit eqation (1 eqation) UdA. d 0 t Momentm consevation (3 eqations) (+ othe foces) Eneg Eqation (1 eqation) t Ud UU da Bd tda Ine. foces Bod foces tial foces Pesse foces Viscos Newtonian Flid Themal eneg Themal epansion Viscos dissipation Heat condction Viscos stess components fo a Newtonian flid: Catesian clindical 11

113 MCG 3143, M. Fenech, W Non-Newtonian Flid (clindical) Powe law Bingham Casson Cach eqation, Catesian coodinate 0 ) ( w v w v t f g p w v t f g p v w v v v t v f g p w w w v w t w Cach eqation, Clindical coodinate t ) ( 1 t = p f g t = p f g t = p f g n d d K d d K 0 d d 0 d d d d K

114 MCG 3143, M. Fenech, W Navie Stokes eqations, Catesian coodinate 0 w v g p w v t v v v g p v w v v v t v w w w g p w w w v w t w Navie Stokes eqations, Clindical coodinate 0 1 ) ( 1 t = p g t = p g t = p g Stokes eqations, Catesian coodinate 0 w v g p v v v g p w w w g p Stokes eqations, Clindical coodinate 0 1 ) ( 1 p = g p 1 = g p = g

115 MCG 3143, M. Fenech, W011 LIFT AND DRAG Dag fo a sphee in stokes flow F R( ) D 6 f p BL Thickness BL displacement thickness Momentm Thickness, θ TURBOMACHINERY:PUMP Ele tbo-machine eqation Tangential ot velocit Efficienc Hdalic powe Pmp head Net positive sction head: Mechanical Powe NPSH Ps g Vs g Pv g Head Sstem eqation p V p1 V H p H fl g g g g H fl KQ 115