Momentum Heat Mass Transfer

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Momentum Heat Mass Transfer
MHMT3
Kinematics and dynamics. Constitutive equations
Kinematics of deformation, stresses, invariants
and rheological constitutive equations. Fluids,
solids and viscoelastic materials.
Rudolf Žitný, Ústav procesní a zpracovatelské
techniky CVUT FS 2010
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Constitutive equations
MHMT3
Material (fluid, solid) reacts by inner forces
only if the body is deformed or in nonhomogeneous
flows.
deformation
Stresses in elastic solids depend only upon the
stretching of a short material fiber.
Stretching of fibers depends upon the initial
fiber orientation. Stresses are independent of
the rate of stretching and of the rigid body
motion (translation and rotation). Deformation is
reversible, after removal of external forces,
original configuration is restored.
l0
?l0
Material with memory
Stresses in viscous fluids depend only upon the
rate of changes of distance between neighboring
molecules. The distance changes are caused by
nonuniform velocity field (by nonzero velocity
gradient). Stresses are again independent of the
rigid body motion (translation and rotation).
Process is irreversible and molecules have no
memory of an initial configuration (stresses are
caused by relative motion of instantaneous
neighbors).
Material without memory
Many materials are something between
viscoelastic fluids have partial memory (fading
memory). Example polymers, food products
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Constitutive equations
MHMT3
Kinematic variables in solids and fluids
Reference and deformed configuration is
distinguished. Motion is described by
displacement of material particles.
reference configuration
displacement
l0
?l0
stretch
Solids
Only the current configuration is considered
(changes of configuration during infinitely short
time interval dt). Motion is described by
velocities of material particles.
velocity
time tdt
time t
Fluids
Velocity is the time derivative of displacement
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Constitutive equations FLUID
MHMT3
Motion of viscous fluid is a fully irreversible
process, mechanical energy is converted to
heat. Fluid has no memory to previous spatial
configuration of fluid particle, there is no
recoil after unloading external stress. Newtons
law
Macke
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Fluids Kinematics of flow
MHMT3
In case of fluids without a long time memory the
role of displacements (differences between the
current and the reference position of material
particles) is taken over by the fluid velocities
at near points. Viscous stresses are response to
changing distances, for example between the near
points A,B during the time dt.
B
Gradient of velocity
A
Arbitrary tensor can be decomposed to the sum of
symmetric and antisymmetric part
B
A
Spin (antisymmetric) Rate of deformation
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Fluids Kinematics of flow
MHMT3
Result can be expressed as Helmholtz kinematic
theorem, stating that any motion of fluid can be
decomposed to translation rotation
deformation
Vorticity tensor
Rate of deformation tensor
Viscous stresses are not affected by translation
nor by rotation (tensor of spin, vorticity),
because these modes of motion preserve distance
between the nearest fluid particles.
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Fluids stresses
MHMT3

• Tensor of total stresses can be decomposed to
  pressure and viscous stresses
• Hydrostatic pressure (isotropic, independent of
  relative motion of fluid particles). For pure
  fluids the pressure can be derived from
  thermodynamics relationships,
• e.g. where u is internal
  energy and s entropy.
• Viscous stresses are fully described by a
  symmetric tensor independent of the rigid body
  motion. Viscous stress is in fact the momentum
  flux due to molecular diffusion.

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Fluids Constitutive equation
MHMT3
Constitutive equations represent a relationship
between Kinematics (characterised
by the rate of deformation for fluids) Viscous
stress (dynamic response to deformation)
The simplest constitutive equation for purely
viscous (Newtonian) fluids is linear relationship
between the tensor of viscous stresses and the
tensor of rate of deformation (both tensors are
symmetric)
Second (volumetric) viscosity Pa.s
Dynamic viscosity Pa.s
Index notation
In terms of velocities
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Fluids Constitutive equation
MHMT3
The coefficient of second viscosity represents
resistance of fluid to volumetric expansion or
compression. According to Lambs hypothesis the
second (volumetric) viscosity can be expressed in
terms of dynamic viscosity ?
This follows from the requirement that the mean
normal stresses are zero (this mean value is
absorbed in the pressure term)
Constitutive equation for Newtonian fluids
(water, air, oils) is therefore characterized by
only one parameter, dynamic viscosity
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Fluids Constitutive equation
MHMT3
Viscosity ? is a scalar dependent on
temperature. Viscosity of gases can be
calculated by kinetic theory as ?(?lmv)/3 (as a
function of density, mean free path and the mean
velocity of random molecular motion, see the
previous lecture) and these parameters depend on
temperature. This analysis was performed 150
years ago by Maxwell, giving temperature
dependence of viscosity of gases Viscosity of
liquids is much more difficult. According to
Eyring the liquid molecules vibrate in a cage
of closely packed neighbors and move out only if
an energy barrier is surpassed. This energy level
depends upon temperature by the Arrhenius term
Viscosity of gases therefore increases and
viscosity of liquids decreases with temperature.
Typical viscosities at room temperature ?water0
.001 Pa.s ?air0.00005 Pa.s
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Fluids Constitutive equation
MHMT3

• Viscosity ? dependent on the rate of deformation
  and stress.
• There exist many liquids with viscosity dependent
  upon the intensity of deformation rate
  (apparent viscosity usually decreases with the
  increasing shear rate), these liquids are called
• generalized newtonian fluids (viscosity depends
  only upon the actual deformation rate, examples
  are food liquids, polymers)
• There are also materials which flow like liquids
  only as soon as the intensity of stress exceeds
  some threshold (and below this threshold the
  material behaves like solid, or an elastic solid)
• yield stress (viscoplastic) fluids (example is
  toothpaste, paints, foods like ketchup)
• And there exist also liquids with viscosity
  dependent upon the whole history of previous
  deformation, changing an inner structure of
  liquid in time
• thixotropic fluids (examples are thixotropic
  paints, plasters, yoghurt).

Viscosity ?(T,t, rate of deformation, stress) is
a scalar, so the intensity of deformation and the
characteristic stress should be also scalars.
However the rate of deformation and the stress
are tensors
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Fluids Invariants
MHMT3
How large is a tensor? Magnitude of a stress
tensor or intensity of the deformation rate are
important characteristics of stress and kinematic
state at a point x,y,z, information necessary for
constitutive equations but also for decision
whether a strength of material was exhausted (do
you remember HMH criterion used in the structural
analysis?) and many others. Easy answer to this
question is for vectors, it is simply the length
of an arrow. Magnitude of a tensor should be
independent of the coordinate system, it should
be INVARIANT. We will show, that there are just 3
invariants (3 characteristic numbers) in the case
of second order tensors, telling us whether the
material is compressed/expanded, what is the
average value of the rate of deformation, density
of deformation energy and so on (it depends upon
the nature of tensor).
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Fluids Invariants
MHMT3
Any tensor of the second order is defined by 9
numbers arranged in a matrix. However these
numbers depend upon rotation of the coordinate
system. For the symmetric tensors (like stress,
or deformation tensors) the rotation of
coordinate system can be selected in such a way
that the matrix representation will be a diagonal
matrix (?, see the first lecture)
In view of orthogonality of R we obtain by
multiplying the equation by RT
This is so called eigenvalue problem given the
matrix ?3x3 calculate three eigenvectors
(columns of the matrix RTn1,n2,n3)
and corresponding eigenvalues ?1, ?2, ?3, that
satisfy the previous equation.
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Fluids Invariants
MHMT3
The eigenvalue problem can be reformulated to a
system of linear algebraic equations for
components of the eigenvector
This system is homogeneous (trivial solution
n1n2n30) and non-trivial solution exists only
if the matrix of system is singular, therefore if
Expanding this determinant gives a cubic
algebraic equation for eigenvalues ?
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Fluids Invariants
MHMT3
The values I? , II? , III? are three principal
invariants of tensor ?. Eigenvalues ? are also
invariants and generally speaking any combination
of principal invariants forms an invariant too.
For example the second invariant of the
deformation rate is frequently expressed in the
following form and called intensity of the strain
rate (it has a right unit 1/s).
Example at simple shear flow u1(x2)?0, u2u30
holds The first and third invariants of rate
of deformation are no so important. For example
the first invariant of incompressible liquid is
identically zero and brings no information
neither on shear nor elongational flows.
shear rate
This is also the explanation why the constant 2
was introduced in the previous definition
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Fluids GNF (power law)
MHMT3
Generalised Newtonian Fluids are characterised by
viscosity function dependent upon the second
invariant of the deformation rate tensor
The most frequently used constitutive equation of
GNF is the power law model (Ostwald de Waele
fluid)
K-coefficient of consistency n-flow behavior index
Example for the simple shear flow and
incompressible liquid the power law reduces to
Graph representing relationship between the shear
rate and the shear stress is called rheogram
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Fluids Yield stress (Bingham)
MHMT3
Even for fluids exhibiting a yield stress it is
possible to preserve the linear relationship
between the stress and the rate of deformation
tensor (Bingham fluid).
Constitutive equation for Bingham liquid
(incompressible) can be expressed as
(this is von Mises criterion of plasticity)
?p-plastic viscosity ?yyield stress
Example for the simple shear flow and
incompressible liquid the Bingham model reduces
to
?y0 Newtonian
Bingham fluid
This is orginal 1D model suggested by Bingham.
The 3D tensorial extension was suggested by
Oldroyd
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Fluids Thixotropic (HZS)
MHMT3
Thixotropic fluids have inner structure,
characterised by a scalar structural parameter ?
(?1 fully restored gel-like structure, while ?0
completely destroyed fluid-like structure).
Evolution of the structural parameter (identified
with the flowing particle) can be described by a
kinetic equation, depending upon the history of
rate of deformation.
a-restoration parameter, b-decay parameter,
m-decay index
The structural parameter ? increases at rest
exponentially with the time constant 1/a. Rate
of structure decay increases with the rate of
deformation.
Value of the structural parameter ? determines
the actual viscosity or the consistency
coefficient of the power law model and the yield
stress (Herschel Bulkley)
HZS model (Houska, Zitny, Sestak), see Sestak J.,
Zitny R., Houska M. Dynamika tixoropnich
kapalin. Rozpravy CSAV Praha 1990
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Fluids Rheometry
MHMT3
Experimental identification of constitutive
models -Rotational rheometers use different
configurations of cylinders, plates, and cones.
Rheograms are evaluated from measured torque
(stress) and frequency of rotation (shear
rate). -Capillary rheometers evaluate
rheological equations from the experimentally
determined relationship between flowrate and
pressure drop. Theory of capillary viscometers,
Rabinowitch equation, Bagley correction.
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Fluids Summary
MHMT3
Constitutive equations for fluids assume that the
stress tensor depends only upon the state of
fluid (characterized by the rate of deformation
and by temperature) at a given place x,y,z. It is
also assumed that the same coefficient of
proportionality ? holds for all components of the
viscous stress tensor and the deformation rate
tensor, therefore
It does not mean that the constitutive equations
are always linear because the viscosity ? can
depend upon the deformation rate itself (and
there exist plenty of models for viscosity as a
function of the second invariants of deformation
rate and stresses power law, Bingham, Herschel
Bulkley, Carreau model, naming just a few).
Nevertheless, some features are common, for
example the absence of normal stresses (?rr,
?zz,) as soon as the corresponding component of
the deformation rate tensor is zero. The
exception are rheological models of the second
order, for example
(Rivlin), exhibiting features typical for
viscoelastic fluids, like normal stresses,
secondary flows in channels, nevertheless these
models are not so important for engineering
practice.
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Solids
MHMT3
Reversible accumulation of external loads to
internal deformation energy Time is of no
importance, unloaded material recoils immediately
to initial configuration. Hooks law
Macke
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Solids Kinematics-deformation
MHMT3
In case of solids the deformation means that an
infinitely short material fiber is stretched.
Kinematics of motion can be decomposed to
stretching followed by a rotation
(the same decomposition was done with fluids, but
in solids the decomposition cannot be expressed
in terms of velocities because time is not
considered)
b
Right stretch tensor Rotation tensor Deformation
gradient Displacement gradient
stretched material line
a
B
material line
A
Reference configuration (unloaded body)
Deformed (current) configuration (loaded body)
Reference ( ) and deformed ( )
configurations are distinguished.
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Solids Kinematics-deformation
MHMT3
The stretch tensor U transforms a material fiber
dXi to the vector UijdXj which is extended or
compressed (stretched). Fiber extension can be
expressed in terms of the Green-Lagrange strain
tensor Eij
In the case of small displacements the quadratic
term can be neglected and the Green Lagrange
tensor of large strains can be approximated by
the tensor of small deformations
Please notice the perfect analogy with fluids
instead of rate of deformation (gradient of
velocities ) there is a
deformation (gradient of displacement)
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Solids Deformation energy
MHMT3
The deformation tensors enable calculation of
deformation energy (reversibly accumulated in a
deformed body) according to different
constitutive models. Knowing the deformation
energy W as a function of stretches (or
deformations) it is possible to calculate the
stress tensors as partial derivatives of W,
e.g. stress ? displacement deformation
energy change, e.g.
W J/m3 is deformation energy related to unit
volume, ?ij are components of Cauchy stresses and
?ij are components of strains. Example Hooks
law
wherefrom the component of stresses can be
calculated by partial derivatives
E-Youngs modulus of elasticity Pa, ?-Poissons
constant
This is the Hooks law for perfectly elastic
linear material
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Example Hook
MHMT3
Homogeneous extension of an elastic rod (1,2,3
are principal directions)
X1
For stretches close to one (?1) you can write
?2-1(?-1)(?1)2(?-1)
Hooks law
Special case of one dimensional loading (?2?30)
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Solids Deformation energy
MHMT3
There is a large group of simple constitutive
equations based upon assumption that the
deformation energy depends only upon the first
invariant of the Cauchy Green tensor
Then the Cauchy stresses can be expressed as
Neo-Hook (linear model similar to Hook, but not
the same, has only 1 parameter). Suitable only
for description at small stretches (for rubber lt
50)
Yeoh model (third order polynomial)
Gent model characterised by a limited
extensibility of material fibers (Imax is the
first invariant corresponding to stretches giving
infinite deformation energy)
Arruda Boyce model having 7 parameters (?max is a
limiting stretch). This is the 8 springs model
based upon idea of 8 springs connected in a cube.
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Solids Experimental methods
MHMT3

• Biaxial testers
• Sample in form of a plate, clamped at 4 sides to
  actuators and stretched
• Static test
• Creep test
• Relaxation test

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Solids Experimental methods
MHMT3

• Inflation tests
• Tubular samples inflated by inner overpressure.
• Internal pressure load
• Axial load
• Torsion

CCD cameras of correlation system Q-450
Pressure transducer
Pressurized sample (latex tube)
Laser scanner
Axial loading (weight)
Confocal probe
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Elastic Solids (summary)
MHMT3

1. Relationships between coordinates of material
   points at reference (X) and loaded (x)
   configuration must be defined. For example in the
   finite element method the reference body is a
   cube and the loaded body is a deformed hexagonal
   element with sides defined by an isoparametric
   transformation.
2. Function x(X) enables to calculate components of
   the deformation gradient F and the Cauchy Green
   deformation tensor (by multiplication CFTF) at
   arbitrary point x,y,z.
3. In terms of the Cauchy Green deformation or
   strain tensor the density of deformation energy
   W(C) can be expressed.
4. Components of stress tensors are evaluated as
   partial derivatives of deformation energy with
   respect to corresponding components of strain
   tensor.

Anyway, what is typical for solid mechanics, the
constitutive equations are expressed in terms of
deformation energy W. This is possible because
the deformation of elastic solids is a reversible
process, time has absolutely no effect upon the
stress reactions, and it has a sense to speak
about potential of stresses. Nothing like this
can be said about fluids, where viscous stresses
depend upon the rate of deformation, and viscous
friction makes the process irreversible.
Therefore the constitutive equations of purely
viscous fluids cannot be based upon the
deformation energy.
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Viscoelastic FLUIDs
MHMT3
Exhibit features of fluids and solids
simultaneously. Deformation energy is partly
dissipated to heat (irreversibly) and partly
stored (reversibly). Both the deformation and the
rate of deformation should be considered.
Macke
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Viscoelastic FLUIDs
MHMT3
The simplest idea of viscoelastic fluids is based
upon the springdashpot models
Maxwell fluids are represented by a serial
connection of spring and dashpot
These materials are more like fluids, because at
constant force F no finite (equilibrium)
deformation is achieved. Only during the time
changes a part of mechanical work is converted to
deformation energy, later on all mechanical work
is irreversibly degraded to heat.
Voight elastoviscous materials are represented by
a parallel connection of spring and dashpot
These materials are more like elastic solids. At
constant force F finite (equilibrium)
deformation is achieved but not immediately. A
sudden change of deformation results to infinite
force response.
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Viscoelastic FLUIDs Maxwell
MHMT3
Viscoelastic fluids of the Maxwell body (serially
connected spring and dashpot) can be described by
ordinary differential equations (k-stiffness of
spring, z attenuation)
giving after elimination of x1
F
This 1D mechanical model is the basis of the
Maxwell model
where ? is a relaxation time, the time required
for the stress to relax to (1/e) value of the
initial stress jump (due to sudden extension).
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Viscoelastic FLUIDs Maxwell
MHMT3
There exist many objections about the previous
generalisation of the Maxwell model, first of all
against the way how the time derivative of the
stress tensor was evaluated.
Time derivative does not fulfill the principle of
material objectivity, which means that the time
derivative depends upon to motion of observer
(material objectivity requires, that even the
time derivatives should satisfy the tensor
transformation rules AR.A.RT ).
The material objectivity is satisfied either by
the model
or by the upper convected Maxwell (UCM)
Remark Both lower and upper convective Maxwell
models are acceptable from the point of view of
mathematical requirements, but they are not the
same. The UCM model prevails in practice, at
least in papers on polymer melts rheology.
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Viscoelastic FLUIDs Maxwell
MHMT3
Example Fiber spinning (steady elongation flow
of a polymer extruded from a dye). Axial velocity
w is determined by the radius of fiber R
This is continuity equation relating radial u and
axial w velocity (cylindrical coordinate system)
R
and this is a simplified solution based upon
assumption of linear radial velocity profile
L
z
Normal stresses for the upper convected Maxwell
F
This is the system of two ODE for the two
unknown normal stresses ?zz and ?rr. Given axial
force F it is possible to calculate axial profile
of thickness R(z)
see also the paper Tembely M.et al, Journal of
Rheology vol.56 (2012),pp.159-183, fiber spinning
using Oldroyd-B and the structural FENE CR
rheological model (you will see close similarity
of equations presented here).
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Viscoelastic FLUID integral models
MHMT3
Let us assume a fluid particle moving along a
streamline
rate of strain (e.g. rate of elongation in 1D
case)
Viscous fluids stress depends only upon the rate
of deformation at the current time t Thixotropic
fluids stress depends also only upon the current
rate of deformation but viscosity is affected by
the deformation history Viscoelastic fluids
stress at time t is a weighted sum of stresses
corresponding to the history of deformation
rates. Stress tensors calculated at previous
positions of particle must be recalculated to the
present position.
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Viscoelastic FLUID Boltzmann
MHMT3
Integral viscoelastic models are based upon the
Boltzmann superposition principle. The idea can
be expressed for 1D case (a dashpot, taking into
account only scalar stress and rate of strain) in
form of integral
strain rate
memory function
The monotonically decreasing memory function
describes the rate of relaxation (rearrangement
of fibers, entanglements) and can be identified
from the unidirectional relaxation experiments.
The unidirectional model can be extended to 3D
for example as
?i-relaxation times, ai-relaxation moduli
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Viscoelastic EFFECTs
MHMT3
Polymer melt flows against the centrifugal forces
towards the rotation axis
so that you will have a better idea about what is
going on, imagine that instead of a homogeneous
fluid there are entangled spaghetti fibers
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Experiments solid and fluids
MHMT3
Characteristic features of elastic, viscoelastic
and viscous liquids are best seen using
oscillating rheometer (usually cone and plate
configuration)
Sinusoidaly applied stress ? and measured strain
? (not the rate of strain!)
Weissenberg rheogoniometer from Wikipedia
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EXAM
MHMT3
Constitutive equations
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What is important (at least for exam)
MHMT3
Fluids
You should know Helmholtz decomposition of the
velocity gradient tensor into spin and rate of
deformation tensors
Decomposition of stress tensor
Newtonian fluid
41
What is important (at least for exam)
MHMT3
Non Newtonian Fluids Invariants
Power law liquids
Bingham liquids
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What is important (at least for exam)
MHMT3
Solids
Deformation gradient
(what is it x and X?)
Decomposition of deformation to stretch and
rotation tensors
(what is it R and U?)
Cauchy Green deformation tensor
Hooks law in terms of Kirchhoff stresses and
Cauchy Green deformation tensor
(what is it E and ??)
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What is important (at least for exam)
MHMT3
Viscoelastic fluids
Maxwell model
(draw a combination of dashpots and springs
corresponding to this model)