Chapter 3 Newtonian Fluids

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CM4650 Chapter 3 Newtonian Fluid 1/23/2020 Mechanics

Chapter 3: Newtonian Fluids

CM4650 Polymer Rheology Michigan Tech

Navier-Stokes Equation

 v    vv   p  2 v   g  t 

1 © Faith A. Morrison, Michigan Tech U.

Chapter 3: Newtonian Fluid Mechanics

TWO GOALS

•Derive governing equations (mass and momentum balances •Solve governing equations for velocity and stress fields

QUICK START V W

x First, before we get deep into 2 v (x ) H derivation, let’s do a Navier-Stokes 1 2 x1 problem to get you started in the x3 mechanics of this type of problem solving.

2 © Faith A. Morrison, Michigan Tech U.

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EXAMPLE: Drag flow 𝑣 between infinite parallel plates 𝑣 𝑣 𝑣 •Newtonian •steady state •incompressible fluid •very wide, long V •uniform pressure W

v1(x2) H

x1 x3

EXAMPLE: Poiseuille flow between infinite parallel plates

•Newtonian •steady state •Incompressible fluid •infinitely wide, long W x2

2H x1 x3 v (x ) x1=0 1 2 x1=L p=Po p=PL

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Engineering Quantities of In more complex flows, we can use Interest general expressions that work in all cases. (any flow)

volumetric 𝑄 𝑛⋅𝑣 𝑑𝑆 flow rate

∬ 𝑛⋅𝑣 |𝑑𝑆 average 〈𝑣 〉 velocity ∬ 𝑑𝑆

Using the general formulas will Here, 𝑛 is the outwardly pointing unit normal help prevent errors. of 𝑑𝑆; it points in the direction “through” 𝑆

5 © Faith A. Morrison, Michigan Tech U.

The stress tensor was Total stress tensor, Π: invented to make the calculation of fluid stress easier. Π ≡𝑝𝐼𝜏

b (any flow, small surface) dS nˆ Force on the S 𝑛⋅ Π 𝑑𝑆 surface 𝑑𝑆 V (using the stress convention of Understanding Rheology)

Here, 𝑛 is the outwardly pointing unit normal of 𝑑𝑆; it points in the direction “through” 𝑆

6 © Faith A. Morrison, Michigan Tech U.

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To get the total force on the macroscopic surface 𝑆, we integrate over the entire surface of interest.

Fluid force on 𝐹 𝑛⋅ 𝑝𝐼 𝜏 𝑑𝑆 the surface S

𝐹 𝑝 𝜏 𝜏 𝜏 𝐹 𝑛 𝑛 𝑛 ⋅ 𝜏 𝑝 𝜏 𝜏 𝑑𝑆 𝐹 𝜏 𝜏 𝑝 𝜏

𝑛, 𝜏 and 𝑝 evaluated at the surface 𝑑𝑆

7 (using the stress convention of Understanding Rheology) © Faith A. Morrison, Michigan Tech U.

Engineering Using the general formulas will Quantities of help prevent errors (like Interest forgetting the pressure). (any flow)

force on the surface, S 𝐹 𝑛⋅ 𝑝𝐼̳ 𝜏 𝑑𝑆

z‐component of force on 𝐹 𝑒̂ ⋅ 𝑛⋅ 𝑝𝐼̳ 𝜏 𝑑𝑆 the surface, S

8 (using the stress convention of Understanding Rheology) © Faith A. Morrison, Michigan Tech U.

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Engineering Quantities of Interest (any flow)

Total Fluid Torque on a 𝑇 𝑅 𝑛⋅ Π 𝑑𝑆 surface, S

𝑅 is the vector from the axis of rotation to 𝑑𝑆

9 (using the stress convention of Understanding Rheology) © Faith A. Morrison, Michigan Tech U.

Common surface shapes:

rectangular: dS dxdy circular top: dS r drd surface of cylinder: dS Rd dz sphere:()sinsin dS Rd  r d R2  d d

Note: spherical coordinate system in use by fluid mechanics community For more areas, see uses 0 𝜃𝜋as the angle from the Exam 1 formula 𝑧=axis to the point. handout:

pages.mtu.edu/~fmorriso/cm4650/formu la_sheet_for_exam1_2018.pdf

10 © Faith A. Morrison, Michigan Tech U.

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Review:

Chapter 3: Newtonian Fluid Mechanics

TWO GOALS

•Derive governing equations (mass and momentum balances •Solve governing equations for velocity and stress fields We got a Quick Start with Newtonian problem QUICK START V W solving… x First, before we get deep into 2 v (x ) H derivation, let’s do a Navier-Stokes 1 2 x1 problem to get you started in the x3 mechanics of this type of problem solving.

2 © Faith A. Morrison, Michigan Tech U.

Now… Back to exploring the origin of the equations (so we can adapt to non‐Newtonian)

Chapter 3: Newtonian Fluid Mechanics

TWO GOALS

•Derive governing equations (mass and momentum balances •Solve governing equations for velocity and stress fields

Mass Balance Consider an arbitrary control volume V enclosed by a surface S

rate of increase net flux of       of mass in CV  mass into CV 

12 © Faith A. Morrison, Michigan Tech U.

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Mathematics Review Polymer Rheology b dS nˆ S

13 © Faith A. Morrison, Michigan Tech U.

Chapter 3: Newtonian Fluid Mechanics Polymer Rheology

Mass Balance (continued) Consider an arbitrary volume V enclosed by a surface S rate of increase d        dV of mass in V  dt  V  outwardly pointing unit net flux of  normal   mass into V     nˆ v dS    through surface S S

14 © Faith A. Morrison, Michigan Tech U.

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Chapter 3: Newtonian Fluid Mechanics Polymer Rheology

Mass Balance (continued)

d     dV    nˆ v dS Leibnitz dt   rule  V  S  dV   nˆ v dS Gauss   Divergence V t S Theorem   v dV V      v  dV  0    V t

15 © Faith A. Morrison, Michigan Tech U.

Chapter 3: Newtonian Fluid Mechanics Polymer Rheology

Mass Balance (continued)

     v  dV  0 Since V is   t  arbitrary, V

Continuity equation: microscopic mass balance   v  0 t

16 © Faith A. Morrison, Michigan Tech U.

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Chapter 3: Newtonian Fluid Mechanics Polymer Rheology

Mass Balance (continued)

Continuity equation (general fluids)      v  0 t      v  v    0 t D     v  0 Dt For 𝜌=constant (incompressible fluids):

  v  0

17 © Faith A. Morrison, Michigan Tech U.

Chapter 3: Newtonian Fluid Mechanics Polymer Rheology

Momentum Balance Consider an arbitrary control volume Momentum is conserved. V enclosed by a surface S

rate of increase  net flux of   sum of          of momentum in CV  momentum into CV   forces on CV 

resembles the resembles the Forces: rate term in the flux term in the body (gravity) mass balance mass balance molecular forces

18 © Faith A. Morrison, Michigan Tech U.

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Momentum Balance Polymer Rheology b dS nˆ S

19 © Faith A. Morrison, Michigan Tech U.

Momentum Balance (continued) Polymer Rheology

rate of increase  d       v dV of momentum in V dt   Leibnitz V rule    v dV V t

net flux of      nˆ vv dS Gauss momentum into V S Divergence Theorem   vv dV V

20 © Faith A. Morrison, Michigan Tech U.

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Momentum Balance (continued) Polymer Rheology

Forces on V

Body Forces (non-contact)

 force on V     g dV dueto g    V

21 © Faith A. Morrison, Michigan Tech U.

Chapter 3: Newtonian Fluid Mechanics Polymer Rheology

Molecular Forces (contact) – this is the tough one

choose a surface stress   through P f   at P  dS   on dS the force on P that surface

We need an expression for the state of stress at an arbitrary point P in a flow.

22 © Faith A. Morrison, Michigan Tech U.

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Molecular Forces (continued)

Think back to the molecular picture from chemistry:

The specifics of these forces, connections, and interactions must be captured by the molecular forces term that we seek. 23 © Faith A. Morrison, Michigan Tech U.

Molecular Forces (continued)

•We will concentrate on expressing the molecular forces mathematically; •We leave to later the task of relating the resulting mathematical expression to experimental observations.

First, choose a surface: nˆ •arbitrary shape •small f dS  stress   at P dS  f What is f ?   on dS 

24 © Faith A. Morrison, Michigan Tech U.

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Consider the forces on three mutually perpendicular surfaces through point P: x2

P x1 eˆ1 x3

eˆ 2 a eˆ3 b c

25 © Faith A. Morrison, Michigan Tech U.

Molecular Forces (continued)

a is stress on a “1” surface at P

a surface with unit normal eˆ1 b is stress on a “2” surface at P

c is stress on a “3” surface at P

We can write these vectors in a Cartesian coordinate system: a  a1eˆ1  a2eˆ2  a3eˆ3  11eˆ1  12eˆ2  13eˆ3 stress on a “1” surface in the 1- direction 26 © Faith A. Morrison, Michigan Tech U.

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Molecular Forces (continued)

a  a1eˆ1  a2eˆ2  a3eˆ3 a is stress on a “1” surface at P  11eˆ1  12eˆ2  13eˆ3 b is stress on a “2” surface at P b  b1eˆ1  b2eˆ2  b3eˆ3 c is stress on a “3” surface at P  21eˆ1  22eˆ2  23eˆ3

c  c1eˆ1  c2eˆ2  c3eˆ3

 31eˆ1  32eˆ2  33eˆ3

So far, this is  pk nomenclature; next we relate these Stress on a “p” expressions to force surface in the k-direction on an arbitrary surface.

27 © Faith A. Morrison, Michigan Tech U.

Molecular Forces (continued)

How can we write f (the force on 𝑛 an arbitrary surface dS) in terms of the P ? pk 𝑓 𝑑𝑆

f  f1eˆ1  f2eˆ2  f3eˆ3 f3 is force on dS in f1 is force on dS in 3-direction 1-direction f2 is force on dS in 2-direction

There are three Ppk that relate to forces in the 1-direction: 11, 21, 31 28 © Faith A. Morrison, Michigan Tech U.

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nˆ Molecular Forces (continued) f dS How can we write f (the force on an arbitrary surface dS) in terms of the f  f1eˆ1  f2eˆ2  f3eˆ3

quantities Ppk?

f1 , the force on dS in 1-direction, can be broken into three parts associated with the three stress components: .

11, 21, 31

nˆ eˆ1 dS

 projection of    first part: 11  dA onto the   11nˆ eˆ1 dS    1 surface   force    area  area  29 © Faith A. Morrison, Michigan Tech U.

Molecular Forces (continued)

f1 , the force on dS in 1-direction, is composed of THREE parts:

 projection of  first part:   11  dA onto the   11nˆ eˆ1 dS    1 surface   projection of  second part:   21  dA onto the   21nˆ eˆ2 dS    2  surface   projection of  stress on a   2 -surface third part: 31  dA onto the   31nˆ eˆ3 dS in the 1-   direction  3 surface 

the sum of these three = f1

30 © Faith A. Morrison, Michigan Tech U.

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31 © Faith A. Morrison, Michigan Tech U.

Molecular Forces (continued)

f1 , the force in the 1-direction on an arbitrary surface dS is composed of THREE parts.

f1  11nˆ eˆ1 dS  21nˆ eˆ2 dS  31nˆ eˆ3 dS

stress appropriate area

Using the distributive law:

f1  nˆ 11eˆ1  21eˆ2  31eˆ3 dS

Force in the 1-direction on an arbitrary surface dS

32 © Faith A. Morrison, Michigan Tech U.

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Molecular Forces (continued)

The same logic applies in the 2-direction and the 3-direction

f1  nˆ 11eˆ1  21eˆ2  31eˆ3 dS

f2  nˆ 12eˆ1  22eˆ2  32eˆ3 dS

f3  nˆ 13eˆ1  23eˆ2  33eˆ3 dS

Assembling the force vector:

f  f1eˆ1  f2eˆ2  f3eˆ3

 dS nˆ 11eˆ1  21eˆ2  31eˆ3 eˆ1

 dS nˆ 12eˆ1  22eˆ2  32eˆ3 eˆ2

 dS nˆ 13eˆ1  23eˆ2  33eˆ3 eˆ3

33 © Faith A. Morrison, Michigan Tech U.

Molecular Forces (continued)

Assembling the force vector:

f  f1eˆ1  f2eˆ2  f3eˆ3

 dS nˆ 11eˆ1  21eˆ2  31eˆ3 eˆ1

 dS nˆ 12eˆ1  22eˆ2  32eˆ3 eˆ2

 dS nˆ 13eˆ1  23eˆ2  33eˆ3 eˆ3

 dS nˆ 11eˆ1eˆ1  21eˆ2eˆ1  31eˆ3eˆ1

 12eˆ1eˆ2  22eˆ2eˆ2  32eˆ3eˆ2

 13eˆ1eˆ3  23eˆ2eˆ3  33eˆ3eˆ3

linear combination of dyadic products = tensor 34 © Faith A. Morrison, Michigan Tech U.

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Molecular Forces (continued)

Assembling the force vector:

f  dS nˆ 11eˆ1eˆ1  21eˆ2eˆ1  31eˆ3eˆ1

 12eˆ1eˆ2  22eˆ2eˆ2  32eˆ3eˆ2

 13eˆ1eˆ3  23eˆ2eˆ3  33eˆ3eˆ3 3 3  dS nˆ   pmeˆpeˆm pm1 1

 dS nˆ  pmeˆpeˆm f  dS nˆ 

Total stress tensor (molecular stresses)

35 © Faith A. Morrison, Michigan Tech U.

Momentum Balance (continued) Polymer Rheology

rate of increase  net flux of   sum of          of momentum in V momentum into V  forces on V  molecular  v dV   vv dV   g dV  V t V V forces molecular  We use a stress sign molecular   convention that   forces on  requires a negative forces  sign here. S    dS   nˆ   dS Gauss  Divergence S Theorem      dV V 36 © Faith A. Morrison, Michigan Tech U.

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Momentum Balance (continued) Polymer Rheology

rate of increase  net flux of   sum of          of momentum in V momentum into V  forces on V  molecular  v dV   vv dV   g dV  V t V V forces UR/Bird choice: molecular  molecular   positive   forces on  compression forces  (pressure is S    dS  positive)  nˆ   dS Gauss  Divergence S Theorem      dV V 37 © Faith A. Morrison, Michigan Tech U.

Momentum Balance (continued) Polymer Rheology

~ F on  nˆ   dS  nˆ  dS surface   SS  ~ yx  yx

UR/Bird choice: (IFM/Mechanics fluid at lesser y choice: (opposite) exerts force on fluid at greater y

38 © Faith A. Morrison, Michigan Tech U.

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Momentum Balance (continued) Polymer Rheology

Final Assembly:

rate of increase  net flux of  sum of          of momentum in V  momentum into V   forces on V    v dV   vv dV    g dV    dV V t V V V v     vv   g   dV  0 V  t 

Because V is arbitrary, we may conclude: v Microscopic  vv   g    0 momentum t balance

Momentum Balance (continued) Polymer Rheology

Microscopic v momentum  vv   g    0 balance t

After some rearrangement:

 v  Equation of   vv       g Motion  t  Dv       g Dt

Now, what to do with  ?

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Momentum Balance (continued) Polymer Rheology

Now, what to do with  ? Pressure is part of it.

Pressure

definition: An isotropic force/area of molecular origin. Pressure is the same on any surface drawn through a point and acts normally to the chosen surface.

 p 0 0   pressure  p I  p eˆ1eˆ1  p eˆ2eˆ2  p eˆ3eˆ3  0 p 0 0 0 p  123

Test: what is the force on a surface with unit normal 𝑛?

Momentum Balance (continued) Polymer Rheology

back to our question, Now, what to do with  ? Pressure is part of it. There are other, nonisotropic stresses

Extra Molecular Stresses

definition: The extra stresses are the molecular stresses that are not isotropic

    p I (other sign convention: 𝜏̃ 𝛱𝑝𝐼 ̳

Extra stress tensor, i.e. everything complicated in molecular deformation

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Momentum Balance (continued) Polymer Rheology

Stress sign UR/Bird choice: convention affects fluid at lesser y any expressions with exerts force on or ~ fluid at greater y ,  ,~    p I

~ ~ (IFM/Mechanics    p I choice: (opposite)

Momentum Balance (continued) Polymer Rheology

Constitutive equations for Stress

•are tensor equations   f (v, •relate the velocity field to the stresses material properties) generated by molecular forces •are based on observations (empirical) or are based on molecular models (theoretical) •are typically found by trial-and-error •are justified by how well they work for a system of interest •are observed to be symmetric Observation: the stress tensor is symmetric

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Momentum Balance (continued) Polymer Rheology

Microscopic momentum  v  Equation of   vv      g Motion balance  t 

In terms of the extra stress tensor:  v    vv   p     g Equation of  t  Motion Cauchy Momentum Equation

Components in three coordinate systems (our sign convention): http://www.chem.mtu.edu/~fmorriso/cm310/Navier2007.pdf

Momentum Balance (continued) Polymer Rheology

Newtonian Constitutive equation

  v  v T 

•for incompressible fluids (see text for compressible fluids) •is empirical •may be justified for some systems with molecular modeling calculations

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Momentum Balance (continued) Polymer Rheology

How is the Newtonian Constitutive equation related to Newton’s Law of Viscosity?

v1 T  21     v  v  x2

•incompressible fluids •incompressible fluids •rectilinear flow (straight lines)

•no variation in x3-direction

Momentum Balance (continued) Polymer Rheology

Back to the momentum balance . . .

 v  Equation of   vv   p     g Motion  t 

  v  v T 

We can incorporate the Newtonian constitutive equation into the momentum balance to obtain a momentum-balance equation that is specific to incompressible, Newtonian fluids

24 CM4650 Chapter 3 Newtonian Fluid 1/23/2020 Mechanics

Momentum Balance (continued) Polymer Rheology

Navier-Stokes Equation

 v    vv   p  2 v   g  t 

•incompressible fluids •Newtonian fluids

Note: The Navier-Stokes is unaffected by the stress sign convention because neither 𝜏 nor 𝜏̃ appear.

Momentum Balance (continued) Polymer Rheology

Navier-Stokes Equation

 v    vv   p  2 v   g  t  Newtonian Problem Solving

25 CM4650 Chapter 3 Newtonian Fluid 1/23/2020 Mechanics

from QUICK START EXAMPLE: Drag flow between infinite 𝑣 parallel plates 𝑣 𝑣 •Newtonian 𝑣 •steady state •incompressible fluid •very wide, long V •uniform pressure W

v1(x2) H

x1 x3

from QUICK START EXAMPLE: Poiseuille flow between infinite parallel plates

•Newtonian •steady state •Incompressible fluid •infinitely wide, long W x2

2H x1 x3 v (x ) x1=0 1 2 x1=L p=Po p=PL

26 CM4650 Chapter 3 Newtonian Fluid 1/23/2020 Mechanics

EXAMPLE: Poiseuille flow in a tube r •Newtonian z •Steady state A cross-section A: •incompressible fluid r •long tube z

vz(r)

fluid R

EXAMPLE: Torsional flow between parallel

plates cross-sectional view: •Newtonian  •Steady state •incompressible fluid •𝑣 𝑧𝑓𝑟

 z

H r R

27 CM4650 Chapter 3 Newtonian Fluid 1/23/2020 Mechanics

Done with Newtonian Fluids. Let’s move on to Standard Flows

Chapter 4: Standard Flows

CM4650 Polymer Rheology Michigan Tech Newtonian fluids: non-Newtonian fluids:    vs.   

How can we investigate non-Newtonian behavior? MOTOR TORQUE CONSTANT 

t  fluid