Mechanics of non-Newtonian fluids and analysis of selected ...malek/new/images/Lecture2.pdf · Mechanics of non-Newtonian ﬂuids and analysis of selected problems Josef M´alek Mathematical

Mechanics of non-Newtonian fluids andanalysis of selected problems

Josef Malek

Mathematical institute of Charles University in Prague, Faculty of Mathematics and PhysicsSokolovska 83, 186 75 Prague 8

February 28, 2011

J. Malek (MFF UK) Mechanics of non-Newtonian fluids February 28, 2011 1 / 27

Contents

1 Introduction

2 Continuum mechanics, Fluids, Newtonian fluids

3 Non-newtonian fluids and phenomena

4 Applications


Part #1

Introduction


Non-Newtonian fluid mechanics - introduction

Goals:

Answer the following questions

Q1. What do I mean by mechanics?Q2. What is a fluid?Q3. What is a Newtonian fluid?Q4. What is a non-Newtonian fluid?Q5. What are materials that are modeled by non-Newtonian fluidmodels?

Recent advances in the constitutive theory

Importance of implicit constitutive theory


Part #2

Continuum mechanics, Fluids, Newtonian fluids


Continuum physics

Balance equations: mass, linear and angular momentum, balance of energy andthe second law of thermodynamics (general assumptions)

,t + div(v) = 0

(v),t + div(v ⊗ v) − div T = b

TT = T(

(e + |v|2/2))

,t+ div((e + |v|2/2)v) + div q = div (Tv)

. . . density

v . . . velocity

e . . . internal energy

b . . . external body forces

T . . . the Cauchy stress

q . . . heat flux


Continuum physics

Balance equations: mass, linear and angular momentum, balance of energy andthe second law of thermodynamics (general assumptions)

,t + div(v) = 0


TT = T(

(e + |v|2/2))

,t+ div((e + |v|2/2)v) + div q = div (Tv)

. . . density

v . . . velocity

e . . . internal energy

b . . . external body forces

T . . . the Cauchy stress

q . . . heat flux

Constitutive equations: involving T; q; assumptions defining idealized materials,representing certain aspects of behavior of natural materials

Continuum mechanics focuses on the mechanical issues - involving T.


What is a fluid?

Several definitions

Fluid is a body that takes the shape of container

Fluid is a body whose symmetry group is the unimodular group (group of allorthogonal transformations)

Fluid is a body that cannot support the shear stress


What is a fluid?

Several definitions




Drawbacks

do not make any difference between liquids and gases (they behavedifferently)

do not cover anisotropic fluids

Bingham or Herschel-Bulkley fluids can support the shear stress

do not take into account the time scale (how long should one wait)


What is a fluid?

Several definitions




Drawbacks

do not make any difference between liquids and gases (they behavedifferently)

do not cover anisotropic fluids

Bingham or Herschel-Bulkley fluids can support the shear stress

do not take into account the time scale (how long should one wait)

Maxwell: In the case of a viscous fluid it is time which is required, and if enough

time is given, the very smallest force will produce a sensible effect, such as would

require a very large force if suddenly applied.


Long-lasting physical experiment

In 1927 at University of Queensland: liquid asphalt put inside the closedvessel, after three years the vessel was open and the asphalt has started todrop slowly.

Year Event1930 Plug trimmed off1938 (Dec) 1st drop1947 (Feb) 2nd drop1954 (Apr) 3rd drop1962 (May) 4th drop1970 (Aug) 5th drop1979 (Apr) 6th drop1988 (Jul) 7th drop2000 (28 Nov) 8th drop


What do we mean by fluid-like behavior?

Most of the materials are mixture of constituents - no sharp interface betweensolid and fluid behavior

Fluid-like behavior - balance and constitutive equations expressed in terms ofthe velocity and its gradients

Solid-like behavior - balance and constitutive equations expressed in terms ofthe displacement and its gradients


Incompresibility

Definition

Volume of any chosen subset (at initial time t = 0) remains constantduring the motion.

for all t: |Vt | = |V0| ⇐⇒ detFχ = 1

Taking the derivative w.r.t. time and using the identity

d

dtdetFχ = div v detFχ

we conclude that

div v = 0


Incompresibility

Definition

Volume of any chosen subset (at initial time t = 0) remains constantduring the motion.

for all t: |Vt | = |V0| ⇐⇒ detFχ = 1

Taking the derivative w.r.t. time and using the identity

d

dtdetFχ = div v detFχ

we conclude that

div v = 0

compressible fluidincompressible fluid with variable densityincompressible fluid with constant density ∗


What is a Newtonian fluid?

T = −p()I + 2µ()D(v) + λ() div v I

T = −pI + 2µ()D(v)

T = −pI + 2µ∗D(v) with µ∗ > 0 D(v) = 12(∇v + (∇v)T )


What is a Newtonian fluid?

T = −p()I + 2µ()D(v) + λ() div v I

T = −pI + 2µ()D(v)

T = −pI + 2µ∗D(v) with µ∗ > 0 D(v) = 12(∇v + (∇v)T )

Balance equations

,t + div(v) = 0


reduce due to incompressibility constraint to

div v = 0 , t + v · ∇ = 0


and if the density is constant one obtains

div v = 0

∗ (v,t + div(v ⊗ v)) − div T = ∗b

Newtonian fluid = Navier-Stokes fluidJ. Malek (MFF UK) Mechanics of non-Newtonian fluids February 28, 2011 11 / 27

Part #3

Non-newtonian fluids and phenomena


What is a non-Newtonian fluid?

Definition

Fluid is a non-Newtonian if it is not a Newtonian fluid

Departures from behavior of Newtonian fluids (non-Newtonian phenomena)

Dependence of the viscosity on the shear rate (shear thinning/thickening)

Dependence of the viscosity on the pressure (pressure thinning/thickening)

The presence of activation or deactivation criteria (such as yield stress)

The presence of the normal stress differences in simple shear flows

Stress Relaxation

(Nonlinear) Creep


Viscosity

Definition

Coefficient proportionality between the shear stress and the shear-rate

Simple shear flow: v(x , y , z) =

v(y)00

D = 12

0 v ′ 0v ′ 0 00 0 0

Newton (1687):

The resistance arising from the want of lubricity in parts of

the fluid, other things being equal, is proportional to the

velocity with which the parts are separated from one another.

Txy = µv ′(y)

Experimental data shows that the viscosity depends on the shear-rate,pressure, . . .


Dependence of the viscosity on the shear-rate

Generalized viscosity

µg (κ) :=Txy (κ)

κ, kde κ = v ′

Shear thinning/thickening Generalized viscosity

1 Viscosity increases with incresing shear-rate (shear thickening)

2 Viscosity decreases with increasing shear-rate (shear thinning)

3 Constant viscosity (Newtonian fluid - provided that the fluid does notexhibit other effects)


Classical power-law models

Simple models that are able to capture such fluid behavior

T = −pI + 2µ|D|r−2D

or

T = −pI + 2µ(

1 + |D|2)

r−22 D

r > 2 Viscosity increases with shear rate (shear thickening)

r = 2 Viscosity is constant

r < 2 Viscosity decreases with shear rate (shear thinning)


Dependence of the viscosity on the pressure

Incompressible fluid with the viscosity depending on the pressure

T = −pI + µ(p, |D|2)D

(Q. What do we mean by the pressure?)

T = −

(

−1

3tr T

)

I + µ

(

−1

3tr T, |D|2

)

D

Note that tr T is the first invariant of T and |D|2 = tr D2 is the secondinvariant of D


Normal stress differences in simple shear flow

v(x , y , z) =

v(y)00

For the model T = −pI + ν(p, |D|2)D

T11 − T22 = −p + p = 0

T22 − T33 = −p + p = 0

The presence of non-zero normal stress differencesin simple shear flows is associated with the effectssuch as

Die swell

Delayed die swell

Rod climbing


Yield stress and activation criteria

Bingham and Herschel-Bulkley fluids


Stress relaxation

Sudden jump discontinuous change of deformation

Response at stress relaxation test for linear spring and linear dashpot


Stress relaxation

Response at stress relaxation test for natural materials: solid-like response(left) and fluid-like response (right)


(Non-linear) creep

Sudden jump discontinuous change in the shear stress

Response at creep test for linear spring and linear dashpot


(Non-linear) creep

Response at creep test for natural materials: solid-like response (left) andfluid-like response (right)


What all shall we neglect in what follows?

Thermal effect

The consequences of the second law of thermodynamics

Compressibility of the fluid

Visco-elastic properties (normal stress differences, stress relaxationand creep)

chemical reactions, electric and magnetic effects

Models for non-newtonian fluids are non-linear of

differential type

rate type

integro-differential type

others

We shall focus on the first three phenomena modeled by the

differential type constitutive equations.


Part #4

Applications


Selected areas of application

Newtonian fluid is exception

1 Food materials such as milk, oil, tomato products, products ofgranular type (such as rice)

2 Chemical suspensions, gels, paints, ....

3 Biological materials such as blood and synovial fluid

4 Geophysical materials such as rocks, soil, sand, clay, lava, the earth’smantle, glacier

Common properties

Complex mixture of solid-like components in a (Newtonian) fluid

Microstructure is very complicated, frequently with not completeunderstanding - it suffices (remains) to model such a material as asingle continuum

Same comment concerns (sometimes) chemical reactions

Observed non-Newtonian phenomena


Recent new approaches in continuum thermodynamics

K.R. Rajagopal (since 1995)

1 Concept of natural configuration associated to the currentconfiguration of the body

2 Principle of maximization of the rate of entropy production

3 Implicit constitutive theory

4 Consequences on the mixture theory


Documents

Mechanics of non-Newtonian fluids and analysis of selected ...malek/new/images/Lecture2.pdf · Mechanics of non-Newtonian ﬂuids and analysis of selected problems Josef M´alek Mathematical