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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
J. Malek (MFF UK) Mechanics of non-Newtonian fluids February 28, 2011 2 / 27
Part #1
Introduction
J. Malek (MFF UK) Mechanics of non-Newtonian fluids February 28, 2011 3 / 27
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
J. Malek (MFF UK) Mechanics of non-Newtonian fluids February 28, 2011 4 / 27
Part #2
Continuum mechanics, Fluids, Newtonian fluids
J. Malek (MFF UK) Mechanics of non-Newtonian fluids February 28, 2011 5 / 27
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
J. Malek (MFF UK) Mechanics of non-Newtonian fluids February 28, 2011 6 / 27
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
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.
J. Malek (MFF UK) Mechanics of non-Newtonian fluids February 28, 2011 6 / 27
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
J. Malek (MFF UK) Mechanics of non-Newtonian fluids February 28, 2011 7 / 27
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
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)
J. Malek (MFF UK) Mechanics of non-Newtonian fluids February 28, 2011 7 / 27
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
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.
J. Malek (MFF UK) Mechanics of non-Newtonian fluids February 28, 2011 7 / 27
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
J. Malek (MFF UK) Mechanics of non-Newtonian fluids February 28, 2011 8 / 27
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
J. Malek (MFF UK) Mechanics of non-Newtonian fluids February 28, 2011 9 / 27
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
J. Malek (MFF UK) Mechanics of non-Newtonian fluids February 28, 2011 10 / 27
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 ∗
J. Malek (MFF UK) Mechanics of non-Newtonian fluids February 28, 2011 10 / 27
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 )
J. Malek (MFF UK) Mechanics of non-Newtonian fluids February 28, 2011 11 / 27
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
(v),t + div(v ⊗ v) − div T = b
reduce due to incompressibility constraint to
div v = 0 , t + v · ∇ = 0
(v),t + div(v ⊗ v) − div T = b
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
J. Malek (MFF UK) Mechanics of non-Newtonian fluids February 28, 2011 12 / 27
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
J. Malek (MFF UK) Mechanics of non-Newtonian fluids February 28, 2011 13 / 27
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, . . .
J. Malek (MFF UK) Mechanics of non-Newtonian fluids February 28, 2011 14 / 27
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)
J. Malek (MFF UK) Mechanics of non-Newtonian fluids February 28, 2011 15 / 27
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)
J. Malek (MFF UK) Mechanics of non-Newtonian fluids February 28, 2011 16 / 27
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
J. Malek (MFF UK) Mechanics of non-Newtonian fluids February 28, 2011 17 / 27
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
J. Malek (MFF UK) Mechanics of non-Newtonian fluids February 28, 2011 18 / 27
Yield stress and activation criteria
Bingham and Herschel-Bulkley fluids
J. Malek (MFF UK) Mechanics of non-Newtonian fluids February 28, 2011 19 / 27
Stress relaxation
Sudden jump discontinuous change of deformation
Response at stress relaxation test for linear spring and linear dashpot
J. Malek (MFF UK) Mechanics of non-Newtonian fluids February 28, 2011 20 / 27
Stress relaxation
Response at stress relaxation test for natural materials: solid-like response(left) and fluid-like response (right)
J. Malek (MFF UK) Mechanics of non-Newtonian fluids February 28, 2011 21 / 27
(Non-linear) creep
Sudden jump discontinuous change in the shear stress
Response at creep test for linear spring and linear dashpot
J. Malek (MFF UK) Mechanics of non-Newtonian fluids February 28, 2011 22 / 27
(Non-linear) creep
Response at creep test for natural materials: solid-like response (left) andfluid-like response (right)
J. Malek (MFF UK) Mechanics of non-Newtonian fluids February 28, 2011 23 / 27
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.
J. Malek (MFF UK) Mechanics of non-Newtonian fluids February 28, 2011 24 / 27
Part #4
Applications
J. Malek (MFF UK) Mechanics of non-Newtonian fluids February 28, 2011 25 / 27
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
J. Malek (MFF UK) Mechanics of non-Newtonian fluids February 28, 2011 26 / 27
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
J. Malek (MFF UK) Mechanics of non-Newtonian fluids February 28, 2011 27 / 27