WEBVTT
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hello in todays lecture we will be discussing
about the fluid in the cardiovascular fluid
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mechanics which is blood so the topic for
today is rheology of blood so let us look
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at the definition of this term rheology first
rheology comes from two terms rheo and logy
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so as you have a would have a studied in a
number of terms that logy or logus means
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study and rheo rheo means it is a old greek
term and it means flow or stream or current
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so rheology etymologically or literally means
study of flow or study of current ok the term
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was coined by two rheologists professor
eugene c bingham and professor markus reiner
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these two are famous rheologists if
you have done a course on fluid mechanics
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you might have had a sub topic on non newtonian
fluids where you would have studied about
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bingham plastic fluid in this lecture also
you will be studying what bingham plastic
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fluids are so that those bingham plastic fluids
are named after professor bingham and professor
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markus reiner he has also given a number of
rheological models for complex fluids so if
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you do a detailed course on rheology or if
you study in detail about rheo rheology
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of non newtonian fluids then you will come
across about the work of professor markus
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reiner ok
so the rheology literally means the flow of
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study of flow so to understand rheology
when we apply a a stress or a force on a fluid
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what happens the fluid starts continuously
deforming unlike a solid when we apply a force
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on a solid then what happens the solid deforms
and then and then it is stop after sometime
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now once the force has been removed from the
solid then the solid may retain its position
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completely or partially so if the solid retains
or comes back to its position completely
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then it is called perfectly elastic solid
or if its partially comes back to its original
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position then it is called partially elastic
solids and the energy of deformation is stored
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as elastic energy in the solid and then
that energy is used for the solid to come
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back to its original position so the solids
they deform and then after sometime they stop
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so the change or the deformation the solid
is non dimensionally is termed as strain whereas
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fluids they deform continuously so a strain
or a term strain which measures only the length
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or the or the dimension of deformation is
not appropriate for fluids because fluids
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deforms continuously so what is used there
is the measure of deformation in fluids is
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rate of a strain which is a tensor and it
measures the rate of deformation of the material
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so you might have seen this image in many
places when a force or a stress is applied
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then the fluid have a velocity and that
velocity profile is linear in case of newtonian
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fluid so we will look at further what rheology
is so the objective of rheology is that how
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does the fluid flow respond to the applied
stresses or applied forces as you know that
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stress is
force per unit area ok so the objective of
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rheology is that how does the fluid stress
is generally represented by the term tau ok
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so you can write this tau is equal to f over
a now the objective of rheology is to understand
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that how does a fluid respond to the applied
forces for simple fluids for example simple
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fluids can be water gases etcetera which does
not have large chain molecules or particles
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suspended in it they are generally simple
fluids in such case newtons law of viscosity
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is followed by the fluid so newtons law of
viscosity is a relationship between a stress
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and strain so tau is proportional to strain
and when then equality is brought into it
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tau is equal to mu this is sorry this is
not a strain this is rate of a strain so the
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shear stress is proportional to rate of a
strain in case of a newtonian fluid and a
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we plot it on a graph stress verses strain
you will get a linear relationship and the
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slope is called viscosity so the fluids which
follow this simple relationship between stress
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and strain are called newtonian fluids
now one must remember that this is not coming
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from fundamental principle but this is an
empirical formula which come which has come
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from observation and measurement of a stress
and rate of a strain in a number of fluids
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so the relationship between shear stress
and shear rate is linear in a newtonian fluid
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and the law is name[d]- is known as newtons
law viscosity
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now what happens when the fluid is not simple
so all non simple fluids can be put together
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under a term which is called complex fluids
so what can be complex fluids complex fluid
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blood is an example of a complex fluid because
blood has number of particles suspended in
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it and blood is further complex because the
size of the particles size of the molecules
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that are suspended in blood varies it has
long chain molecules such as proteins anti
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coglo coagulating agents or the antibodies
not antibodies and number of things
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which are suspended into it in the plasma
plus particles red blood cells white blood
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cells and platelets which are of the size
of few microns so a few nanometres to few
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microns the size of the molecule suspended
in the blood and they effect the rheology
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or the rheological behaviour of the blood
significantly so blood is a complex fluid
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that is why we are studying this topic
rheology ok another example of complex fluids
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can be any suspension or any colloidal suspension
from chemical engineering perspective all
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the polymers behave as complex fluids long
chain polymer molecules are suspended in
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in in the liquid so they are all
non newtonian or complex fluids so the fluids
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which do not behave or which do not follow
newtons law viscosity are called non newtonian
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fluids and most of these fluids are complex
fluids
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so the response of the fluid in such case
is not simply linear that is it is not necessary
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that tau is proportional to rate of a strain
or the shear stress is not directly proportional
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to the strain rate so in such cases it is
important to understand the rheological behaviour
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or the a develop a constitutive equation between
a stress and rate of a strain and that is
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the objective of rheology and such equation
which is obtained from rheology which called
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constitutive equation and this can be done
using experiments which has been done
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for few last few centuries now and now
it is more and more also being done by
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a molecular dynamic simulations where people
are modelling the behaviour of molecules
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and then finding a constitutive relationship
between stress shear stress and rate of strain
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so the objective of rheology is to finally
give a constitutive equation between applied
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stress and rate of deformation for complex
fluids because we know that for simple fluid
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what is the law
so based on the behaviour of these complex
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or non newtonian fluids they can be divided
into three different categories time independent
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behaviour time dependent behaviour and viscoelastic
behaviour now this category or categorization
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is only for the understanding purpose so we
can demarcate different behaviours it is
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really possible that a fluid a complex fluid
shows only one type of behaviour it may be
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a the fluid is showing all three kind of behaviour
or it can show two different kind of behaviour
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so time dependent behaviour means that the
response of the fluid or rheological behaviour
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of the fluid is independent of time so it
does not change with time some examples
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are shear thinning fluid ok so the time dependent
behaviour can be shear thinning or shear thickening
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or another example as i was telling bingham
plastic fluids so they show a non linear kind
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of or non newtonian behaviour but that does
not depend on time the fluid does not have
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a memory whereas time dependent fluid the
behaviour is time dependent it depends on
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the time and the examples are thixotropic
fluid and rheopectic fluids so it may happen
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that the shear thinning behaviour a fluid
showing a shear thinning behaviour but it
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is also time dependent
in addition to that the fluids also show viscoelastic
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behaviour complex fluid also show viscoelastic
behaviour and blood is one example so in this
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case the fluid have viscous behaviour plus
elastic behaviour so as we discussed some
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time back that the elastic behaviour that
after the stress has been removed the material
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regains its original position is elastic behaviour
so partially the fluid regains its original
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position now this happens generally if the
fluid is suspended with flexible particles
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for example blood has red blood cells which
are flexible disc shape particles and those
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particles when the shear stress is removed
from them they regain their shape so because
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of that the fluid also show viscoelastic behaviour
some kind of elastic behaviour ok so we will
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look at time independent behaviour in
this lecture one example is for first fluid
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that we will be looking at bingham plastic
fluid so bingham plastic fluid they still
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have the linear relationship between a stress
and rate of a strain so we can say that tau
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is equal to tau y plus mu gamma dot so if
we plot this on a stress verses rate of a
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strain plot then we will get some this kind
of behaviour this intercept is known as tau
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y or yield stress so what does this mean physically
when we apply a stress on a fluid one of the
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examples of this is toothpaste so when we
push the toothpaste for sometime the paste
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does not come out only after a certain or
a critical amount of force the paste starts
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coming out and that is because the when tau
is less than tau y there is no deformation
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and when tau is greater than tau y so as you
can see on this graph that gamma dot is equal
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to zero for tau is less than tau y when stress
shear stress is less than yield stress then
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the deformation is zero and when is greater
than yield stress then it follows a behaviour
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like a newtonian fluid so here we plot a newtonian
fluid with the same viscosity it will be a
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line parallel but passing through the origin
whereas in case of a bingham fluid or bingham
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plastic fluid it will have certain amount
of yield stress and this is again a property
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that is shown by complex fluid which has particles
suspended in it so imagine a fluid which has
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number of particles which are which
have taken a certain kind of shape in the
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fluid they do not move until a certain
amount of force is applied on them and when
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the force increases that critical force or
when the stress increases beyond that critical
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stress they reorient themselves and the fluid
behaves as a bingham plastic fluid
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so the bingham plastic fluid the property
is that they show yield stress blood also
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shows and yield stress behaviour but not exactly
bingham plastic behaviour they have a some
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non linear relationship with it which we will
show so again let me emphasise here
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that the relationship between a stress and
a strain again is a empirical relationship
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or a model and in this case there are
two parameters in the model tau y which is
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yield stress and viscosity of the fluid which
is mu ok for shear another behaviour is
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shear thinning fluid so as you see here they
show non linear relationship so that means
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tau is not proportional to directly proportional
to gamma dot but it is it follows a power
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law where and is less than one so we plot
it on a stress verses rate of a strain curve
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it will still go through origin and this straight
line represents a newtonian fluid the power
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law behaviour will be it is also known as
pseudo plastic so the slope of the curve in
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this case is less than what for newtonian
fluid so the slope of curve will be you
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can also say this is apparent viscosity which
will be defined as tau over gamma dot we introduce
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a equality constant here then tau will be
equal to some m gamma dot to the power n where
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m is called consistency index
so blood also behaves as shear thinning fluid
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under certain conditions and they are a number
of complex fluid which show shear thinning
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behaviour they have a power law kind of behaviour
and n is less than one and is viscosity will
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be m gamma dot to the power n minus one and
which is what this slope represent ok sorry
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the image shown here is for tomato
ketchup ok which also behaves as a shear thinning
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fluid ok for shear thickening fluid as
the name suggest then when the shear is applied
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or when the shear rate increases the viscosity
of the fluid when it decreases then it is
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called shear thinning fluid when the rate
of shear increases and the viscosity increases
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then it is called shear thickening fluid so
when the fluid thickens because viscosity
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in general term or in general perception viscosity
represents the that how difficult is it for
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a fluid to flow
so if the it is difficult for the fluid to
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flow then we will say that this fluid is thin
if it is difficult to for the fluid to flow
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then it is thick if it is easy to for the
fluid to flow then it is called thin fluid
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so from that nomenclature we can understand
that shear thinning fluid if the shear rate
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when the shear rate is increased and a the
viscosity of the fluid is decreased then the
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that means the fluid thins then it is called
shear thinning fluid or pseudo plastic fluid
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whereas when the shear is increased or
rate of shear is increased and the viscosity
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of the fluid increases then it is called shear
thickening fluid they are also known as dilatant
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fluid they again follow a power law kind of
relationship tau is equal to m gamma dot to
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the power n but n is greater than one ok so
we plot it on a stress verses rate of a strain
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curve then now this is for newtonian fluid
and this is for the shear thickening or dilatant
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fluid
one example is corn starch corn starch
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and water solution so if you dissolve or
if you make a solution of corn starch in water
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and you apply the shear is a a low shear or
if you rotate it slowly then you will see
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that it flows but if you increase the rate
of shear if you rotate it fast then the fluid
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does not move at all so that is an example
of shear thickening fluid another example
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of shear thickening fluids is quicksand ok
so combining two we can give the power
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law model and in the power law model we
can say again tau is equal to m gamma dot
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to the power n where m is the consistency
index and n is greater than one then and n
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is less than one
so n is greater than one we called shear thickening
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fluids shear thickening behaviour shown by
the fluid and is less than one and it is called
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shear thinning fluid however this in this
model what happens that if shear stress is
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very low then the viscosity or apparent viscosity
of the fluid which is tau over gamma dot that
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will be approaching infinity that will be
very high viscosity similarly if shear rate
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is very high then viscosity will be approaching
zero both of which are not correct representation
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of the behaviour often so some limiting
behaviour is to be applied because this model
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power law which is given by ostwald de waele
which is also known as ostwald de waele model
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it cannot describe the viscosity at very low
and very high shear rates
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so number of a corrections to this behaviour
has been proposed number of models has been
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proposed to correct this anomaly in the
behaviour of the power law model one of the
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a examples is carreau yasuda model which is
given by this formula so you you see there
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that eta is non dimensionalized here so eta
minus eta infinity where eta infinity viscosity
26:36.140 --> 26:52.110
where eta is in this particular example
eta is viscosity and eta infinity eta at high
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shear rate viscosity at high shear rate and
eta naught is equal to eta at low shear rate
27:11.100 --> 27:20.620
if you see the left hand side is non dimensional
viscosity on the numerator you see the
27:20.620 --> 27:26.640
difference between the viscosity and the viscosity
at high shear and in the denominator you see
27:26.640 --> 27:33.410
difference in the viscosity at low and high
shear rates and this is equal to one plus
27:33.410 --> 27:40.100
lambda shear rate lambda into shear rate to
the power a total to the power n minus one
27:40.100 --> 27:48.270
over a so another new term here which is lambda
so you see here lambda you can easily identify
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or you can easily see from dimensional
analysis that because shear rate has unit
27:57.120 --> 28:07.800
of
time to the power minus one right because
28:07.800 --> 28:13.610
shear rate is what non dimensional deformation
per unit time so non dimensional deformation
28:13.610 --> 28:22.310
no unit and time second so per unit time is
second to the power minus one and lambda is
28:22.310 --> 28:30.460
a time constant because when we multiply
this to be non dimensionless sorry dimensionless
28:30.460 --> 28:40.220
then lambda is a lambda will have unit
of time and it is one over lambda is the
28:40.220 --> 28:48.970
critical shear rate at which viscosity starts
to decrease what is a a this power it determines
28:48.970 --> 28:54.760
the transition between low shear rate and
when the power law region starts
28:54.760 --> 29:05.740
so if you look at if shear rate is low that
means let us say if lambda gamma dot is very
29:05.740 --> 29:16.380
very less than one then what will happen eta
minus eta infinity divided by eta naught minus
29:16.380 --> 29:26.350
eta infinity is equal to one because this
term will be negligible with respect to one
29:26.350 --> 29:34.080
so that means that eta is equal to eta naught
and it is newtonian behaviour with the viscosity
29:34.080 --> 29:47.490
of eta naught where as at high shear rate
if lambda gamma dot is very very larger than
29:47.490 --> 29:53.820
one then you can see that this term will go
away or one will away or you can neglect one
29:53.820 --> 30:21.600
with respect to this term and this will give
you eta minus eta infinity is equal to ok
30:21.600 --> 30:29.030
so this model shows at low shear rate newtonian
behaviour as we saw here and at high shear
30:29.030 --> 30:36.570
rate and so power law behaviour so what
we have done in power law model we have
30:36.570 --> 30:45.340
combined the behaviour power law model
can represent or can show the behaviour of
30:45.340 --> 30:51.290
a power law fluid it may be shear thinning
or shear thickening in the time independent
30:51.290 --> 30:59.020
fluids we have shown may three kind of
behaviour one is yield stress behaviour another
30:59.020 --> 31:04.450
one is shear thickening behaviour and the
third one is shear thickening behaviour
31:04.450 --> 31:11.060
now shear thickening and shear thinning behaviour
can can be combined in power law model and
31:11.060 --> 31:18.500
depending on that the power n is more than
one then it is shear thinning behaviour and
31:18.500 --> 31:23.900
sorry if the power n is greater than one then
it is shear thickening behaviour if the power
31:23.900 --> 31:31.960
is less than one then it is shear thinning
behaviour now we can have a model which can
31:31.960 --> 31:40.720
combine the yield stress and the non linear
behaviour together then we can model
31:40.720 --> 31:47.280
a fluid which shows power law behaviour as
well as yield stress behaviour and that can
31:47.280 --> 31:54.980
be done using herschel berkley law or herschel
berkley model of for fluids this is
31:54.980 --> 32:09.320
given as tau is equal to tau y plus m shear
rate to the power n so here we have combined
32:09.320 --> 32:22.520
the yield stress as well as power law behaviour
of the fluid ok so that can be given as when
32:22.520 --> 32:41.250
n is equal to one when n is when n is greater
than one and when n is less than one ok so
32:41.250 --> 32:47.990
this model is known as herschel berkley fluid
so if a fluid is showing bingham plastic behaviour
32:47.990 --> 32:56.890
or n power law behaviour then its behaviour
can be given by herschel berkley model then
32:56.890 --> 33:05.230
another model which is very similar to
herschel berkleys model that means it combines
33:05.230 --> 33:14.740
the yield stress as well as power law behaviour
is known as casson fluid model we look
33:14.740 --> 33:22.740
at it in general what it can be shown or
it can be said that m is equal tau one to
33:22.740 --> 33:33.860
the power one over m is equal to or you can
give this another power let us say tau to
33:33.860 --> 33:46.700
the power one over p is equal to tau y to
the power one over p plus m gamma dot to the
33:46.700 --> 33:54.730
power one over p
now if we say that p is equal to one that
33:54.730 --> 34:13.460
means we will get tau is equal to tau y plus
m gamma dot which is for bingham plastic fluid
34:13.460 --> 34:25.770
whereas if p is equal to two then we will
get root tau is equal to root tau y plus root
34:25.770 --> 34:40.629
of m gamma dot and this fluid is known as
casson fluid or this model as known as casson
34:40.629 --> 34:59.220
model and this often represents for the
rheological behaviour of blood so in this
34:59.220 --> 35:07.039
for blood is p is equal to two and the
fluid model is known as casson fluid ok as
35:07.039 --> 35:16.990
we said earlier at the start of this lecture
that blood is a suspension of different particles
35:16.990 --> 35:26.349
in fluid so it is important or it is useful
to look at some of the a literature that describes
35:26.349 --> 35:34.190
the rheology of suspensions or rheology of
different particles suspended in a fluid ok
35:34.190 --> 35:42.789
so blood is a suspension of different kind
of particles in plasma and if the volume fraction
35:42.789 --> 35:50.849
of these particles are is small as well as
these particles are very small with respect
35:50.849 --> 35:57.559
to the length scale of the flow that means
the say if the flow is happening in a channel
35:57.559 --> 36:02.570
in a artery in a tube and if the particles
are very small with respect to the channel
36:02.570 --> 36:06.039
dimension then we can assume it to be homogeneous
fluid
36:06.039 --> 36:12.071
so homogeneous flow means the viscosity is
same in all the dimension and the fluids fluid
36:12.071 --> 36:20.440
as well as the particles move with the same
velocity ok and the viscosity of such suspensions
36:20.440 --> 36:27.210
will depend on a number of factors which include
the concentration or volume fraction of the
36:27.210 --> 36:33.089
particle let us say the volume fraction is
given by phi shape of the particle particles
36:33.089 --> 36:38.760
may be spherical may be ellipsoidal or it
may be disc shape which is the shape of r
36:38.760 --> 36:45.589
b cs flexibility of the particles and the
mechanical properties of the particles so
36:45.589 --> 36:54.569
if we take a simple case of spheres which
are rigid and the concentration of these spheres
36:54.569 --> 37:01.990
is very low it is so low that we do not need
to consider the interaction between the spheres
37:01.990 --> 37:09.041
and for that einstein has given a relationship
for the viscosity of the susp[ension]- sphere
37:09.041 --> 37:16.560
suspended viscosity of the fluid which
has low concentration of the sphere suspended
37:16.560 --> 37:22.319
in the fluid and this viscosity is given as
eta s which is equal to eta naught eta naught
37:22.319 --> 37:44.130
is the viscosity of the fluid
and phi is volume fraction of particle and
37:44.130 --> 37:49.779
this is valid for phi is less than zero point
zero one so if the concentration of particles
37:49.779 --> 37:56.700
is very small then the viscosity of the
particles can be given by this relationship
37:56.700 --> 38:04.510
and as you can see from here that the viscosity
of the suspension increases then the viscosity
38:04.510 --> 38:12.920
of the fluid
however if the concentration of the particles
38:12.920 --> 38:19.350
is more than zero point zero one and in such
case particles interact then the viscosity
38:19.350 --> 38:24.749
of the particles is further increase which
is given by eta is equal to eta naught plus
38:24.749 --> 38:31.960
one plus two point five phi plus two point
five phi square you might notice that six
38:31.960 --> 38:41.789
point two five is this is just square of two
point five phi and this relationship is valid
38:41.789 --> 38:52.819
for volume fraction up to point three at
higher volume fractions the fluid starts behaving
38:52.819 --> 39:00.180
as a non newtonian fluid until volume fraction
less than point three phi is less than point
39:00.180 --> 39:06.580
three the fluid behaves as a newtonian fluid
when the particles the susp[ended]- particles
39:06.580 --> 39:15.289
suspended sufficiently small than the channel
dimension now if the particles are sphere
39:15.289 --> 39:25.299
but deformable spheres for examples droplets
because droplets will have spherical shape
39:25.299 --> 39:31.309
small droplets will have spherical shape
because of the surface tension and if their
39:31.309 --> 39:39.089
concentration is low so they show their behaviour
or their viscosity can be given by this formula
39:39.089 --> 39:54.499
you might notice where eta d is the viscosity
of the droplet ok when eta d tends to infinity
39:54.499 --> 40:12.940
that means the particles are rigid then they
will take in case of eta d is infinity then
40:12.940 --> 40:24.251
eta s will be is equal to eta naught into
one plus two point five phi so they will come
40:24.251 --> 40:32.559
back to the relationship which we saw in the
previous slides for rigid spheres ok
40:32.559 --> 40:41.519
so in such case when the spheres are deformable
the viscosity is less than for the rigid particles
40:41.519 --> 40:51.170
because the particles can respond to the stresses
and they can reorient themselves now if the
40:51.170 --> 40:57.609
particles are asymmetric that means they are
not symmetric in all the directions which
40:57.609 --> 41:05.020
is say non spherical particles and for low
concentrations they are viscosity is given
41:05.020 --> 41:14.830
by eta naught one plus k phi so this k is
equal to two point five if it is symmetric
41:14.830 --> 41:22.579
particles right if it is sphere then k is
equal to two point five and this k will depend
41:22.579 --> 41:29.259
on different fluids the for different kind
of particles it will have different number
41:29.259 --> 41:34.869
in this case what will happen the particle
will also rotate and they may occupy larger
41:34.869 --> 41:39.569
volume
so the viscosity of the particle will increase
41:39.569 --> 41:48.740
but if the shear rate is high then particles
will tend to orient along the direction of
41:48.740 --> 41:55.109
the fluid and the viscosity will decrease
so they will show a shear thinning behaviour
41:55.109 --> 42:00.809
ok so at low concentrations you will see this
kind of behaviour but at the same time this
42:00.809 --> 42:07.289
value of k will change when the shear rate
is increased so you can say that k is shear
42:07.289 --> 42:26.190
dependent and the particle show shear thinning
behaviour for rigid asymmetric particles if
42:26.190 --> 42:32.160
the particles are deformable and asymmetric
and the particles will rotate they can change
42:32.160 --> 42:38.989
their shape so that the friction is minimised
and viscosity for such case will be lower
42:38.989 --> 42:45.720
than for rigid asymmetric particles and because
the particles can respond to the fluid and
42:45.720 --> 42:50.690
if the shear rate is or if the shearing
is removed then they can regain their shapes
42:50.690 --> 42:58.119
so they will also show viscoelastic behaviour
ok
42:58.119 --> 43:05.400
so for rigid all deformable asymmetric particles
but when the concentration is high particles
43:05.400 --> 43:13.619
may form a continuous structure they will
show a yield stress behaviour so that when
43:13.619 --> 43:20.839
this structure is broken down and this yield
stress is a function of volume fraction so
43:20.839 --> 43:28.779
blood has number of red blood cells or
the the volume fraction of red blood cell
43:28.779 --> 43:34.759
there is about forty to forty five percent
so they show this kind of behaviour which
43:34.759 --> 43:42.470
have deformable asymmetric particles r b cs
are asymmetric particles and they form a continuous
43:42.470 --> 43:48.930
structure and yield stress behaviour is shown
by them and this yield stress will be a function
43:48.930 --> 43:56.069
of the volume fraction of r b cs ok so in
summary in this lecture what we have looked
43:56.069 --> 44:06.319
at it is the what is rheology what does
the term rheology mean and what can be the
44:06.319 --> 44:15.480
different kind of behaviours shown by fluids
we looked at time dependent or time independent
44:15.480 --> 44:19.569
behaviour and viscoelastic we categorized
into three categories at the same time we
44:19.569 --> 44:25.300
said that any fluid can show only one or two
or three kind of behaviour
44:25.300 --> 44:34.849
then we looked at some of the time independent
behaviours bingham plastic that means the
44:34.849 --> 44:40.920
fluid which show yield stress shear thinning
where with the increase of the shear rate
44:40.920 --> 44:49.660
the viscosity of the fluid decreases shear
thickening where with the introduction
44:49.660 --> 44:57.140
of high shear rate introduction of the
shear rate the viscosity of the fluid increases
44:57.140 --> 45:04.630
and we also looked at briefly the rheology
of suspensions we looked at different rheological
45:04.630 --> 45:13.010
models or different models for non newtonian
fluids and we said that for blood it shows
45:13.010 --> 45:30.239
yield stress it also shows shear thinning
behaviour
45:30.239 --> 45:36.430
and because it is a suspension of particles
and these particles are viscoelastic so it
45:36.430 --> 45:47.869
they also shows viscoelastic behaviour so
it turns out that with all the components
45:47.869 --> 45:54.309
of the blood the behaviour of the blood changes
so it is important to understand that what
45:54.309 --> 46:00.989
does the blood constitute and the properties
of those different particles so in the next
46:00.989 --> 46:13.510
section we will look at the morphology
of the blood briefly ok
46:13.510 --> 46:29.609
thank you