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1 Chapter 3: Newtonian Fluid Mechanics TWO GOAL Derive governing equations (mass and momentum balances olve governing equations or velocity and stress ields QUICK TART W First, beore we get deep into derivation, let s do a Navier-tokes problem to get you started in the mechanics o this type o problem solving. x 3 x 2 x 1 v 1 (x 2 ) H 73 EXAMPLE: Drag low between ininite parallel plates Newtonian steady state incompressible luid very wide, long uniorm pressure W x 2 v 1 (x 2 ) H x 3 x

2 Chapter 3: Newtonian Fluid Mechanics TWO GOAL Derive governing equations (mass and momentum balances olve governing equations or velocity and stress ields Mass Balance Consider an arbitrary control volume enclosed by a surace rate o increase net lux o o mass in C mass into C 75 Mathematics Review b 76 2

3 Chapter 3: Newtonian Fluid Mechanics Mass Balance rate o increase o mass in net lux o mass into through surace d dt d v outwardly pointing unit normal Consider an arbitrary volume enclosed by a surace 77 Chapter 3: Newtonian Fluid Mechanics Mass Balance Leibnitz rule d dt d d t t v v vd v d 0 Gauss Divergence Theorem 78 3

4 Chapter 3: Newtonian Fluid Mechanics Mass Balance ince is arbitrary, t v d 0 Continuity equation: microscopic mass balance t v 0 79 Chapter 3: Newtonian Fluid Mechanics Mass Balance Continuity equation (general luids) v t v t D Dt 0 v 0 For =constant (incompressible luids): v 0 v

5 Chapter 3: Newtonian Fluid Mechanics Momentum is conserved. Consider an arbitrary control volume enclosed by a surace rate o increase net lux o o momentum in C momentum into C sum o orces on C resembles the rate term in the mass balance resembles the lux term in the mass balance Forces: body (gravity) molecular orces 81 b 82 5

6 rate o increase o momentum in d dt t v d v d Leibnitz rule net lux o momentum into vv vvd Gauss Divergence Theorem 83 Forces on Body Forces (non-contact) orce on dueto g g d 84 6

7 Chapter 3: Newtonian Fluid Mechanics Molecular Forces (contact) this is the tough one stress at P on the orce on that surace choose a surace through P P We need an expression or the state o stress at an arbitrary point P in a low. 85 Molecular Forces Think back to the molecular picture rom chemistry: The speciics o these orces, connections, and interactions must be captured by the molecular orces term that we seek. 86 7

8 Molecular Forces We will concentrate on expressing the molecular orces mathematically; We leave to later the task o relating the resulting mathematical expression to experimental observations. First, choose a surace: arbitrary shape small stress at P on What is? 87 Consider the orces on three mutually perpendicular suraces through point P: x 2 P x 1 ê 1 x 3 ê 3 ê 2 b a c 88 8

9 Molecular Forces a b c is stress on a 1 surace at P a surace with unit normal ê1 is stress on a 2 surace at P is stress on a 3 surace at P We can write these vectors in a Cartesian coordinate system: stress on a 1 surace in the 1- direction a a1e ˆ1 a2 2 a Molecular Forces a ae 1ˆ1 a2 2 a3e ˆ b be 1ˆ1 b2e ˆ2 b3e ˆ c ce 1ˆ1 c2 2 c3e ˆ a b c is stress on a 1 surace at P is stress on a 2 surace at P is stress on a 3 surace at P o ar, this is nomenclature; next we relate these expressions to orce on an arbitrary surace. tress on a p surace in the k-direction pk 90 9

10 Molecular Forces How can we write (the orce on an arbitrary surace ) in terms o the pk? 1 is orce on in 1-direction e e e 1ˆ1 2ˆ2 3ˆ3 2 is orce on in 2-direction 3 is orce on in 3-direction There are three pk that relate to orces in the 1-direction: 11 21,, Molecular Forces How can we write (the orce on an arbitrary surace ) in terms o the quantities pk? e e e 1ˆ1 2ˆ2 3ˆ3 1, the orce on in 1-direction, can be broken into three parts associated with the three stress components: ,, 31 irst part: projection o da onto the surace 1 orce area area e ˆ

11 Molecular Forces 1, the orce on in 1-direction, is composed o THREE parts: stress on a 2 -surace in the 1- direction irst part: second part: third part: projection o da onto the 1 surace projection o da onto the 2 surace projection o 3 surace da onto the n e ˆ 1 ˆ 2 3 the sum o these three =

12 Molecular Forces 1, the orce in the 1-direction on an arbitrary surace is composed o THREE parts stress appropriate area Using the distributive law: Force in the 1-direction on an arbitrary surace 95 Molecular Forces The same logic applies in the 2-direction and the 3-direction Assembling the orce vector: 1e ˆ

13 Molecular Forces Assembling the orce vector: ee ˆ1 ˆ ee ˆ1 ˆ ee ˆ ˆ linear combination o dyadic products = tensor 97 Molecular Forces Assembling the orce vector: 11ee ˆ1 ˆ ee ˆ1 ˆ ee ˆ ˆ 3 p1m1 pm p m pm p Total stress tensor (molecular stresses) m

14 rate o increase net lux o sum o o momentum in momentum into orces on t vd vv molecular orces d molecular orces on molecular g d orces d We use a stress sign convention that requires a negative sign here. Gauss Divergence Theorem 99 rate o increase net lux o sum o o momentum in momentum into orces on t vd vv molecular orces d molecular orces on molecular g d orces d UR/Bird choice: positive compression (pressure is positive) Gauss Divergence Theorem

15 F on surace ~ yx ~ yx UR/Bird choice: luid at lesser y exerts orce on luid at greater y (IFM/Mechanics choice: (opposite) 101 Final Assembly: rate o increase net lux o o momentum in momentum t vd vvd g d v t sum o into orces on vv g d 0 d Because is arbitrary, we may conclude: v t vv g 0 Microscopic momentum balance

16 Microscopic momentum v vv t g balance 0 Ater some rearrangement: v vv g t Dv g Dt Equation o Motion Now, what to do with? 103 Now, what to do with? Pressure is part o it. Pressure deinition: An isotropic orce/area o molecular origin. Pressure is the same on any surace drawn through a point and acts normally to the chosen surace. p pressure p I p ee ˆ1 ˆ1 p 2 2 p Test: what is the orce on a surace with unit normal? 0 p p

17 back to our question, Now, what to do with? Pressure is part o it. There are other, nonisotropic stresses Extra Molecular tresses deinition: The extra stresses are the molecular stresses that are not isotropic p I Extra stress tensor, i.e. everything complicated in molecular deormation Now, what to do with? This becomes the central question o rheological study 105 tress sign convention aects any expressions with, ~ or, ~ p I UR/Bird choice: luid at lesser y exerts orce on luid at greater y ~ ~ p I (IFM/Mechanics choice: (opposite)

18 Constitutive equations or tress are tensor equations relate the velocity ield to the stresses generated by molecular orces are based on observations (empirical) or are based on molecular models (theoretical) are typically ound by trial-and-error are justiied by how well they work or a system o interest are observed to be symmetric ( v, Observation: the stress tensor is symmetric material properties) 107 Microscopic momentum balance v vv g t Equation o Motion In terms o the extra stress tensor: v vv p g t Equation o Motion Cauchy Momentum Equation Components in three coordinate systems (our sign convention):

19 Newtonian Constitutive equation v v T or incompressible luids (see text or compressible luids) is empirical may be justiied or some systems with molecular modeling calculations Note: ~ v v T 109 How is the Newtonian Constitutive equation related to Newton s Law o iscosity? v 1 T 21 v v x2 incompressible luids incompressible luids rectilinear low (straight lines) no variation in x 3 -direction

20 Back to the momentum balance... v vv t p g Equation o Motion v v T We can incorporate the Newtonian constitutive equation into the momentum balance to obtain a momentum-balance equation that is speciic to incompressible, Newtonian luids 111 Navier-tokes Equation v vv t 2 p v g incompressible luids Newtonian luids Note: The Navier-tokes is unaected by the stress sign convention

21 Navier-tokes Equation v vv t 2 p Newtonian Problem olving v g 113 EXAMPLE: Drag low between ininite parallel plates Newtonian steady state incompressible luid very wide, long uniorm pressure W rom QUICK TART x 2 v 1 (x 2 ) H x 3 x

22 EXAMPLE: Poiseuille low between ininite parallel plates Newtonian steady state Incompressible luid ininitely wide, long x 2 W x 1 x 3 2H x 1 =0 v 1 (x 2 ) x 1 =L p=p o p=p L 115 EXAMPLE: Poiseuille low in a tube Newtonian teady state incompressible luid long tube A z r cross-section A: z r v z (r) L luid R

23 EXAMPLE: Torsional low between parallel plates Newtonian teady state incompressible luid cross-sectional view: z H R r

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