CM4650 Chapter 3 Newtonian Fluid 1/23/2020 Mechanics

Chapter 3: Newtonian Fluids

CM4650 Polymer Rheology Michigan Tech

Navier-Stokes Equation

v vv 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.

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

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

x2

v1(x2) H

x1 x3

3

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

4

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

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.

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

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

Π

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.

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

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.

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

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)

© 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

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.

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

Mathematics Review Polymer Rheology b dS nˆ S

V

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.

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

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.

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

Momentum Balance Polymer Rheology b dS nˆ S

V

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

Molecular Forces (continued)

Assembling the force vector:

f dS nˆ 11eˆ1eˆ1 21eˆ2eˆ1 31eˆ3eˆ1

12eˆ1eˆ2 22eˆ2eˆ2 32eˆ3eˆ2

13eˆ1eˆ3 23eˆ2eˆ3 33eˆ3eˆ3 3 3 dS nˆ pmeˆpeˆm pm1 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.

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

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.

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

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

39 © Faith A. Morrison, Michigan Tech U.

Momentum Balance (continued) Polymer Rheology

Microscopic v momentum vv g 0 balance t

After some rearrangement:

v Equation of vv g Motion t Dv g Dt

Now, what to do with ?

40 © Faith A. Morrison, Michigan Tech U.

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

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?

41 © Faith A. Morrison, Michigan Tech U.

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

Now, what to do with ? This becomes the central question of rheological study 42 © Faith A. Morrison, Michigan Tech U.

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

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)

43 © Faith A. Morrison, Michigan Tech U.

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

44 © Faith A. Morrison, Michigan Tech U.

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

Momentum Balance (continued) Polymer Rheology

Microscopic momentum v Equation of vv g Motion balance t

In terms of the extra stress tensor: v vv 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

45 © Faith A. Morrison, Michigan Tech U.

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

(IFM choice: Note: 𝜏̃ 𝜇 𝛻𝑣 𝛻𝑣 (opposite) 46 © Faith A. Morrison, Michigan Tech U.

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

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

47 © Faith A. Morrison, Michigan Tech U.

Momentum Balance (continued) Polymer Rheology

Back to the momentum balance . . .

v Equation of vv 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

48 © Faith A. Morrison, Michigan Tech U.

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Momentum Balance (continued) Polymer Rheology

Navier-Stokes Equation

v vv 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.

49 © Faith A. Morrison, Michigan Tech U.

Momentum Balance (continued) Polymer Rheology

Navier-Stokes Equation

v vv p 2 v g t Newtonian Problem Solving

50 © Faith A. Morrison, Michigan Tech U.

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

x2

v1(x2) H

x1 x3

51

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

52

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

L

vz(r)

fluid R

53

EXAMPLE: Torsional flow between parallel

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

z

H r R

54

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

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

55 © Faith A. Morrison, Michigan Tech U.

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

56 © Faith A. Morrison, Michigan Tech U.

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