WEBVTT
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so until now we have looked at the time dependent
behaviour time independent behaviour of blood
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or the time independent rheological behaviour
of the blood and we learned that under certain
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conditions the blood behaves as a newtonian
fluid but for other conditions the blood behaves
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as a non newtonian fluid but all those measurements
all those analysis have been done under steady
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state conditions or when the shear flow was
at a steady state
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now if the shear is time dependent that means
if the flow is unsteady which is the case
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in the cardiovascular system as we know that
in the cardiovascular system the flow is pulsatile
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and the it is unsteady so it changes with
time so that means to understand the rheological
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behaviour of blood we need to consider the
time dependence of time dependent rheological
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behaviour of blood so as we have discussed
while understanding the morphology of blood
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that the red blood cells white blood cells
they have viscoelastic behaviour so we need
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to so as a result blood also might show
some viscoelastic behaviour so in this lecture
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we will look at fundamentals of viscoelastic
fluids what is viscoelasticity and then briefly
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touch upon the viscoelastic behaviour of the
fluid ok
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so let us now consider what is viscoelasticity
so let us look at the two terms that the
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term viscoelasticity is made up made up of
one term is viscosity and the other term is
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elasticity so elasticity is generally a property
of solids you might have studied about elasticity
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while undergoing a course on solid mechanics
might have heard the term elastic solid or
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perfectly elastic solid so if we throw a ball
on the ground and it bounces back and comes
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up to the same height then such a elastic
such a solid we call elastic an elastic solid
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it follows an elastic solid follows hookes
law so what does the hookes law say hookes
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law say that the stress which is say tau is
proportional to the strain note we have been
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talking about the fluids in this course until
now so we have been saying the relationship
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between stress and strain rate whereas for
a solid it is the relationship between a stress
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and a strain so for an elastic solid the stress
is proportional to strain that is what is
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called hookes law and then when it becomes
equality tau is equal to g gamma where g is
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called youngs modulus
so elastic solid what do they do for elastic
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solid as i said earlier that the material
it stores when a stress is applied on the
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material then it because of that stress
the energy that is there the material stores
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that energy and after the removal the of the
stress it regains it say so that is energy
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helps in regaining the material it say but
after a certain yield stress the creep
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occurs that is the solid material starts flowing
so that is what an elastic behaviour is which
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is shown by the solids and it is generally
a property of the solids
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now a viscous fluid all the fluids that we
see show little or more amount of viscous
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behaviour so a viscous fluid as we have studied
earlier for a viscous fluid they follow newtons
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law viscosity and the shear stress that is
applied on the fluid that is proportional
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to shear rate so material deforms continuously
and what we are concerned here is not the
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deformation but the rate of deformation of
the material so that is a difference that
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we experience between solids and fluid in
the solids we are concerned about the deformation
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of the solid whereas in the fluids we are
concerned about the rate of deformation of
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the fluid so viscoelastic material is a can
say transient from solid to fluid or fluid
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to solid and these materials show both solid
and fluid behaviour which is they show elastic
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as well as viscous behaviour
so because of viscous effects as you know
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that when there is flow of a viscous fluid
in a channel then there is some pressure loss
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and you might remember while discussing flow
in a channel we derive this relationship that
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tau is equal to minus d p by d z into r by
two so what i am trying to say here that we
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the d p by d z is the pressure loss because
of the viscous systems so because of visc[ous]-
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viscous effect or because of the viscosity
there is loss of energy or the dissipation
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of energy the energy in is dissipated in overcoming
the viscous effects whereas as i said just
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now that elastic effect that means for a solid
the energy is stored in the fluid and after
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the stress is removed this energy is released
so that is a storage of energy so viscous
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due to viscous effect there is loss of energy
whereas in during the elastic effect the energy
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is restored
so because viscoelastic material shows both
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viscous and elastic effects so they have the
ability like the elastic materials to store
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and recover the energy but not completely
partially only so they have the ability to
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store and recover the energy partially and
some part of energy is lost so when the stress
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is applied on it some part of energy is lost
to overcome the viscous effect and some part
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of the energy is stored by the fluid so depending
on where the energy is going more the fluid
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will show that kind of behaviour if the more
energy is stored then it will be more like
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an elastic solid or the visco elastic material
will have more viscoelastic or more elastic
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behaviour if more energy is lost then it will
more like a viscoelastic or the viscous fluid
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behaviour another interesting behaviour
that is seen in viscoelastic material is that
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due to the shearing motion due to the shearing
of the fluid in general for a fluid what we
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see is that there are shea[r]- because of
the shearing motion or because of the shear
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rate when the fluid experiences the shear
rate shear stresses are generated
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but in a viscoelastic material in addition
to this shear rate the material also experiences
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some normal stresses or what we call normal
stress differences and these normal stress
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difference is because if the material is
isotropic that means if the if if the normal
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stresses are isotropic that means if they
are same in all the directions then there
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will be no deformation of the material so
for the deformation for the deformation of
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the material to happen it is important to
have the difference between the normal stresses
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so what is important for deformation is the
difference of the normal stresses not the
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normal stresses itself so that is why we consider
the normal stress differences and there can
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be two independent normal stresses which are
called first normal stress difference and
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second normal stress difference
so under shear effect the fluid is characterised
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by three properties under simple shear flow
a viscoelastic material require three properties
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this is called let us say phi one this is
phi two so simple shear flow
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and there are three properties for the viscoelastic
material that characterises viscosity first
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normal stress difference and second normal
stress difference ok so let us look at some
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of the effects or some of the physical phenomena
that as that are peculiar to viscoelastic
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fluids so one such effect is called rod climbing
effect or a or weissenberg effect so if we
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take a beaker and a newtonian fluid into it
and then put a rod in the beaker and then
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rotate the rod about it as about its axis
then what will happen due to the centrifugal
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effect the fluid will move away from the rod
and reach near the wall so after it reaches
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it steady state we will get a we will see
a fluid profile which will look something
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like this for a newtonian fluid the top meniscus
will take a shape like this where the level
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of the fluid is lower near the rod and higher
near the walls because of the centrifugal
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effect however if the fluid is viscoelastic
what is seen is this that when the rod is
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rotating and because of that rod motion the
fluid rises up on the rod and this effect
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is known as rod climb[ing]- climbing effect
or after weissenberg is known as weissenberg
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effect
another related and similar effect is called
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die swell effect so when a newtonian fluid
comes out of a die or comes out of a say
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vessel from a small hole then the fluid stream
the diameter of the fluid stream either remains
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constant or it starts decreasing when the
velocity of the fluid increases whereas for
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a viscoelastic material when specially or
this phenomena is observed in polymers which
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are viscoelastic which show viscoelastic behaviour
the diameter of the die increases just after
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the after it exists from the die
this is because there are particles soft particles
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suspended or large molecular chains in
the fluid and once they come out in the air
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where the pressure is low so the molecules
polymer molecules large polymer chains they
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relax and the volume increases so the diameter
of this fluid z that is coming out it increase
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so this effect is known as die swell effect
another important behaviour is what we
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call memory effect so for a simple newtonian
fluid the stresses internal stresses are
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proportional to the instantaneous deformation
or a a instantaneous instantaneous rate
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of deformation it does not remember what
had happened in the past but for a viscoelastic
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fluid what is important that it the fluid
behaviour or the internal stresses in the
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fluid are proportional to the deformation
history of the fluid and that is understandable
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because viscoelastic fluids show the viscous
behaviour which is like a newtonian fluid
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but also elastic behaviour so elastic behaviour
that means it stores energy inside it and
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when the viscoelastic fluid stores energy
inside it and the energy is released so that
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means the entire storage history need to be
taken into account so that is why this memory
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effect is shown by the viscoelastic materials
so to characterise this memory effect a relaxation
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time or a spectrum or a distribution of relaxation
time is required to characterise the rheological
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behaviour of a viscoelastic material so on
the two extremes the viscous fluid which does
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not have any memory the relaxation time is
zero whereas the elastic fluid which has a
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perfect memory it will have a finite relaxation
time ok so let us come to a interesting number
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which is called deborah number and this name
is not proposed by or this number is not
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proposed by deborah whereas this has been
proposed by professor markus reiner you might
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remember that at the start of the rheology
chapter we discussed that professor reiner
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was one of the professors who gave the term
rheology ok
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so he also defined he also given he has
also given this number or the nomenclature
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deborah number so the name is inspired by
a verse in the bible which says the mountains
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flowed before the lord and this verse is
by profiteers deborah so let us look at
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the text from the article by professor reiner
in which he introduced the deborah number
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so what he says that deborah knew two things
first that the mountains flow as everything
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flows but secondly that they flowed before
the lord and not before man for the simple
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reason that man in his short life time cannot
see them flowing while the time of observation
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of god is infinite we may therefore well define
a non dimensional number the deborah number
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which is a ratio of time of relaxation and
time of observation
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so if we try to understand this what he says
that everything flows it is the time of observation
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depending on which one can see that the material
under observation is flowing or not flowing
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so he takes an example of mountain or the
deborahs description of mountain flows so
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he says because the observation time of a
man is about hundred years in which there
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is a non appreciable deformation of mountains
whereas observation time of god is very large
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or and in that the deformation of mountains
can be observed so it depends on the observation
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time of a particular person of or of a particular
phenomena or depending on which one can see
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up the material is flowing or not so the
deborah number is defined as time of relaxation
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and time of observation so if the relaxation
time of a material is negligible or small
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with respect to the time of observation then
one can observe the fluid to be if the
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relaxation time is very small then one can
see the flow happening and it will be fluid
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like behaviour whereas if the relaxation time
is very large as compared to the time of observation
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as in the case for a human observing the motion
of mountains so it will be solid like behaviour
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so the deborah number is time of relaxation
and time of observation
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now there is an another non dimensional number
which has a similar definition it can also
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be defined we the number is weissenberg
number and it is defined as the ratio of elastic
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forces and viscous forces so it compares the
two effects elastic effect and viscous effect
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and so if we say in terms of a strain the
strain that is received and the strain that
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is not recovered the elastic strain that will
be that can be recovered as the viscous cannot
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be recovered so it represent the recoverable
strain in the fluid in terms of the relaxation
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time again it is ratio of the characteristic
relaxation time to a characteristic time measure
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of shear rate so notice the difference that
in the deborah number the ratio of relaxation
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time was with the time of observation whereas
for weissenberg number it is time measure
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of shear rate so if the time of observation
its same as time measure of shear rate then
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the two numbers will be equal but it is not
always necessary that the two times will be
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equal so it is not necessary that the weissenberg
number will be same as deborah number ok
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so until now we have established that the
viscoelastic fluids they have two properties
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viscous and elastic behaviour so if we want
to develop a simple model for a viscoelastic
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fluid for relationship between stress and
strain now because it has a stress strain
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and a strain rate so we might take into account
the all the things all three things stress
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strain and rate of a strain so to model
develop a simple model we can consider
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a fluid which has two components one component
which shows visco viscous behaviour and another
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component which shows elastic behaviour so
that is where these spring dashpot models
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come into picture spring is a elastic component
which shows the elastic behaviour and dashpot
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is a viscous component which has or which
is a viscous damper so which shows viscous
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behaviour
so it is a combination of spring and dashpot
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two components which shows one component show
elastic behaviour another component show viscous
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behaviour so a combination of different components
different combinations can be used to model
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the fluid behaviour flow viscous visc[ous]-
the rheological relationship between the viscous
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and elastic fluids ok so in simple model is
called maxwell model in maxwell model the
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spring and dashpot are in series so if they
are in series that means they will have different
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deformations so let us say the deformation
in the two are gamma one and gamma two and
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they will experience the same stress which
is tau so we can write for spring tau is equal
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to g gamma one whereas g is the youngs modulus
for the spring and for dashpot we can write
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tau is equal to mu gamma two dot that is the
rate of a strain or the shear rate
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so as we said that the total strain will be
a sum of gamma one plus gamma two and if we
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differentiate this with respect to time that
means the total strain rate will be gamma
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one dot plus gamma two dot now if we substitute
gamma one dot and gamma two dot from previous
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equations let us call this equation one and
equation two and if we substitute then we
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can write from one and two we get gamma dot
is equal to this equation so you might notice
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that because we want to write gamma one dot
plus gamma two dot so to obtain gamma one
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dot we need to differentiate this so we can
write del d tau over d t is equal to g gamma
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one dot and we substitute that here we will
get a relationship between stress the rate
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of change of a stress and strain rate ok so
this model is known as maxwell model another
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combination which is frequently discussed
or which is simple model which parallel combination
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of spring and dashpot and in such case the
deformation will be same in both the cases
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whereas they will experience different stresses
let us say tau one and tau two so we can write
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tau one is equal to g gamma one and tau two
is equal to mu gamma two dot and if we add
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those tau is equal to tau one plus tau two
so we will have after substitution tau is
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equal to g gamma plus mu gamma dot so that
is another model that can be used to model
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the behaviour of a viscoelastic fluid so these
are the two simple models however people can
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develop a complex model which have these
spring and dashpot a number of them in
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series and parallel or in series and parallel
combination
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so for measuring viscosity what people
can do is they they have studied simple
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shear flow of the viscoelastic fluids and
the another behaviour that is or another
30:25.539 --> 30:33.550
flow that is often used to measure the behaviour
of the viscoelastic fluid is oscillatory test
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so the fluid under goes a oscillatory shearing
so a oscillatory shear rate and the stress
30:48.799 --> 30:58.080
is measured or the fluid under goes a
under a oscillatory shear stress and the shear
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rate is measured so the sample say for example
in a cone and pla[te]- cone and plate viscometer
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or coaxial cylinder viscometer the cylinder
outer cylinder which is rotating or the cone
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which is rotating is imparted a say oscillatory
motion for example sinusoidal motion in place
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of a unidirectional rotation so in such case
the oscillatory behaviour of the fluid can
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be studied generally this is done in a sinusoidal
manner and the plate can be oscillated say
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if this flow between two parallel plates it
can be oscillated at in two different directions
31:42.909 --> 31:57.809
or about a steady position so if the oscillation
is sinusoidal then it is often convenient
31:57.809 --> 32:08.309
to represent the sinusoidal behaviour in terms
of a complex number because it is easier to
32:08.309 --> 32:18.159
deal with while multiplying dividing or
differentiating one can easily deal with complex
32:18.159 --> 32:25.110
numbers so let us consider a complex shear
rate which is given as shear rate is equal
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to a gamma zero dot e to the power i i is
to represent the complex number omega t and
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omega is the oscillation frequency and from
that if the corresponding shear stress is
32:43.429 --> 32:50.870
tau is equal to tau naught e to the power
minus i delta e to the power i omega t and
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this minus i delta has been introduced to
take into account of the fact that the shear
32:59.350 --> 33:08.669
rate and the shear stress are not necessarily
will not generally be in phase or will be
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only under certain cases that they will be
in phase
33:13.480 --> 33:26.860
so that means in that case delta will be zero
so if we divide this complex shear stress
33:26.860 --> 33:35.590
by the complex shear rate we obtain a complex
viscosity and this viscosity will have two
33:35.590 --> 33:43.940
components so we can write this as tau naught
over gamma dot naught into e to the power
33:43.940 --> 33:54.529
minus i delta and if we write this in terms
of the two components real and imaginary component
33:54.529 --> 34:01.720
then the real component shows the viscous
behaviour and the imaginary component shows
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the elastic behaviour the viscous part is
e to the power minus i delta what we know
34:10.990 --> 34:27.990
is this is equal to e to the power i theta
is cos theta plus i sin theta so e to the
34:27.990 --> 34:34.750
minus i delta is cos delta minus i sin delta
this is viscous part and the elastic part
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will be sin delta and this delta is the phase
shift between shear rate and shear stress
34:43.339 --> 34:50.510
and this is what measures the viscoelasticity
so this the the phase shift between the
34:50.510 --> 34:59.490
two is the viscoelastic behaviour so if delta
is zero then this term will be zero if delta
34:59.490 --> 35:10.369
is equal to zero and so that means it will
be if delta is zero then it is viscous fluid
35:10.369 --> 35:21.570
this term will be zero if the delta is ninety
degree and in between the fluid will behave
35:21.570 --> 35:31.570
as a viscoelastic fluid this analysis can
also be done in a different manner if one
35:31.570 --> 35:51.650
consider gamma is equal to or gamma t is equal
to gamma naught e to the power i omega t then
35:51.650 --> 36:00.900
one can define a complex modulus which is
the ratio of stress verses strain and then
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it will also have components
so one can represent either in terms
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of the modulus or module and one can also
represent the behaviour in terms of complex
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viscosity and one can even find out the relationship
between the complex modulus and the complex
36:31.380 --> 36:38.589
viscosity of for two different components
so if the modulus is complex modulus then
36:38.589 --> 36:47.720
in that case the elastic component is generally
known as storage loss and the viscous component
36:47.720 --> 36:54.930
the elastic component is known as the storage
modulus and the viscous component is known
36:54.930 --> 37:05.280
as the loss modulus ok so let us briefly
look at the viscous or viscoelastic behaviour
37:05.280 --> 37:11.670
of blood as we have said that the all the
measurements that we have looked into the
37:11.670 --> 37:20.530
about the rheology of blood has been steady
and the flow in the cardiovascular system
37:20.530 --> 37:27.910
is pulsatile it changed with time and is unsteady
so it is important to consider the time dependent
37:27.910 --> 37:37.210
rheological behaviour of the blood so it
has been shown it has been observed first
37:37.210 --> 37:46.500
time by professor g b thurston that the
blood behaves as a viscoelastic materiel because
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the red blood cells and white blood cells
they are viscoelastic component of the fluid
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and on the application of the stresses the
r b cs they undergo deformation and once the
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stresses are removed they come back to their
shape
38:05.180 --> 38:13.010
so the blood as a whole also so the viscoelastic
behaviour recently there is some literature
38:13.010 --> 38:20.510
which says that under some conditions plasma
may also behave as viscoelastic fluid and
38:20.510 --> 38:29.160
that can be attributed to the proteins that
are suspended in the plasma so these proteins
38:29.160 --> 38:37.810
so the because of the presence of these proteins
a plasma may behave as a viscoelastic fluid
38:37.810 --> 38:44.720
and this behaviour under oscillatory flow
conditions it will depend on the shear rate
38:44.720 --> 38:54.040
and the hematocrit value which is if you remember
that hematocrit is the percentage of red blood
38:54.040 --> 39:00.720
cells so the fraction of a red blood cells
which is primarily responsible for the viscoelastic
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behaviour of the blood their volume fraction
that determines the viscoelastic behaviour
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of the fluid and the shear rate as we have
seen that the blood viscosity is the shear
39:15.250 --> 39:26.170
dependent viscosity so what this graph
shows is the viscosity and these can be different
39:26.170 --> 39:42.860
viscosities as a function of shear rate so
the shear rate is the r m s value of the oscillatory
39:42.860 --> 39:52.260
shear and under this oscillatory shear what
has been plotted is the viscous that is real
39:52.260 --> 40:02.270
and imaginary that is elastic component of
viscosity so this is eta we can say eta
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v and this is eta elastic and the third graph
shows the viscosity under steady shear conditions
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so that is why on the x axis you also have
the velocity gradient or the shear rate
40:22.460 --> 40:28.450
for a steady state conditions
these measurements were done at a frequency
40:28.450 --> 40:36.950
of ten hertz and for fifty percent hematocrit
value but it has been suggested that similar
40:36.950 --> 40:43.230
trends are observed for hematocrit value or
the r b c fraction more than twenty percent
40:43.230 --> 40:50.740
so what you can see from this graphs that
at low values of shear about two per second
40:50.740 --> 40:57.470
the complex viscosity both the components
the elastic elastic component and the viscous
40:57.470 --> 41:04.350
component they are almost independent of shear
at very low shear rate about two per second
41:04.350 --> 41:10.410
or about one per second at that magnitude
when the shear rate is increased then the
41:10.410 --> 41:20.720
elastic component decreases continuously you
might notice that the scale here on the x
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and y axis is log scale so this is a log log
plot the elastic component which is imaginary
41:33.730 --> 41:41.420
component of a complex viscosity it decreases
continuously with the increase in shear rate
41:41.420 --> 41:47.250
whereas the viscous component of viscosity
it decreases slightly there is just slight
41:47.250 --> 41:54.470
decrease in the viscosity and then it becomes
constant and it remains constant we have already
41:54.470 --> 42:03.990
seen this behaviour for the steady state
viscosity which decreases continuously up
42:03.990 --> 42:11.730
to hundred when the shear rate is up to
hundred per second and above that the velocity
42:11.730 --> 42:17.339
is constant which we call shear thinning behaviour
so what is interesting to know that at high
42:17.339 --> 42:26.910
shear rates above hundred per second the complex
the real component of the complex viscosity
42:26.910 --> 42:38.060
or the viscosity under oscillatory shearing
is same as the steady state viscosity at high
42:38.060 --> 42:45.349
shear rates ok so these are some of the observations
for the viscoelastic properties of the blood
42:45.349 --> 42:52.440
by professor thurston who looked at this phenomena
in seventies
42:52.440 --> 43:03.790
so in summary what we can say that viscoelastic
fluids in simple shear flow apart from
43:03.790 --> 43:09.860
the viscosity we also need two normal stress
differences first normal stress difference
43:09.860 --> 43:17.410
and second normal stress difference to characterise
the rheological behaviour of a viscoelastic
43:17.410 --> 43:24.970
fluid the viscoelastic fluid of course they
so viscous as well as elastic behaviour
43:24.970 --> 43:34.730
the another important characteristic of viscoelastic
fluid is relaxation time and these relaxation
43:34.730 --> 43:44.600
time in the non dimensional form is defined
in terms of deborah number and weissenberg
43:44.600 --> 43:56.020
number the simple behaviour the rheological
behaviour of a viscoelastic fluid the simplest
43:56.020 --> 44:06.420
possible model are spring dashpot model in
which a elastic compound is spring and
44:06.420 --> 44:13.950
a dashpot component or a viscous component
a a viscous temper or dashpot is considered
44:13.950 --> 44:22.060
and this is called spring dashpot model
and we have also looked at the oscillatory
44:22.060 --> 44:53.100
shearing motion and the complex viscosity
of the blood briefly ok