WEBVTT
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hello up until now we have been looking
at the fundamentals of rheology what rheology
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basically is the properties of different time
independent non newtonian fluid and different
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simple models which can model the non newtonian
behaviour of complex fluid including that
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of blood now all these models are need to
be validated the parameters in these models
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are need to be found out so this is done
by doing careful experiments to measure the
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viscosity or apparent viscosity of complex
fluids
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in this lecture we will be looking at the
a simple and frequently used viscometers and
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rheometers specially in the context of blood
so the measurement of viscosity or apparent
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viscosity we will discuss in this lecture
three different viscometers or rheometers
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so there are three viscometers that have
been listed here capillary viscometer concentric
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cylinder viscometer and cone and plate viscometer
you might wonder the difference between viscometer
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and rheometer so the viscometer in my opinion
is an instrument using which the viscosity
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of a newtonian fluid can be measured whereas
rheometer is a general term by which the rheological
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characteristics including non newtonian viscosity
or apparent viscosity which is generally defined
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as the ratio of shear stress and shear rate
is measured
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so the difference between viscometer and rheometer
essentially is that viscometer the term viscometer
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is used for measurement of the viscosity of
newtonian fluids because their viscosity is
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constant at any particular temperature whereas
rheometer is used for non newtonian fluids
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so the three viscometers that we have listed
here are capillary viscometer concentric cylinder
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viscometer and cone and plate viscometer
capillary viscometer is it basically works
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on hagen poiseuille principle that a known
amount of liquid is flown is it the liquid
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flows through a small capillary so that the
capillary is small or channel is small so
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that the reynolds number is small flow becomes
fully developed and the exit and entrance
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effect can be neglected and when the time
of the flow of a fluid is measured and this
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time of flow of a known volume of fluid is
compared with the time required for the flow
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of a known fluid or the flow of a fluid of
known viscosity at the same temperature and
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then by comparing the times for the flow of
that two fluids one can define or one can
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find out the relative viscosity of the fluid
so it is generally used for newtonian fluids
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the next viscometer that we will discuss in
this lecture is concentric cylinder viscometer
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in which they are two cylinders which are
concentric they have same axis and the gap
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between the cylinder is very small when it
is compared with the radius of the either
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cylinder one of the cylinder generally it
is the outer cylinder is rotated at a known
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angular speed by doing which one can impart
a known amount of shear rate and the torque
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on the inner cylinder which is fixed is measured
by doing this one can calculate what is
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the shear stress being exerted on the inner
cylinder and by comparison and by the
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ratio of the two shear stress and shear rate
one can measure the viscosity of the fluid
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another very popular and frequently used
viscometer or rheometer is cone and plot cone
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and plate viscometer in which a cone and plate
plate arrangement is there over a flat plate
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a cone of very high apex angle is rotated
so again the flow the cone rotates and
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by this rotatary motion and from the angular
velocity the shear rate is defined and
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the shear stress over the cone for this particular
shear rate is measured and from the ratio
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of two the viscosity can be found out so
we will look at this it a bit in detail
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now
so capillary viscometer first so this is
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a simple diagram of a capillary viscometer
in which a known amount of liquid it flows
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through a channel what is the driving force
here the driving force here is gravity
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driven one can also have another or an
external pressure difference to drive the
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flow when it is gravity driven then it is
very important that the orientation of the
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channel should be vertical if not vertical
then the angle that it makes from the vertical
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direction is required so that the if the
effective value of gravity can be used in
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the calculations ok however this problem
can be eliminated if the experiments are done
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for the standard fluid which is used for the
calibration and for the fluid of which
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viscosity is to be measured if they are done
at the same orientation of the viscometer
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then the orientation effect can be ruled out
so the capillary viscometer is based on poiseuille
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law you might remember in the previous lecture
we looked at Newtonian fully developed and
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laminar flow in a circular channel the
flow was steady so we derived the following
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relationship for the velocity profile in the
channel one can then calculate the flow rate
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by just integrating this zero to r v z two
pi r d r and if we do that we will end up
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with r square over four mu minus d p by d
z integral zero to r r minus r cube by r square
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d r we can take two pi outside so this will
give us two pi r square over four mu minus
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d p by d z and if we integrate this that
will be and put the limits we will get r square
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by two minus r is to the power four divided
by four r square which will be r square by
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four effectively this will be four so that
is equal to sorry so this is r square
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so when we change this this will become r
square by four and we will get four
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eight pi r raise to the power four divided
by sorry pi r to the power four divided
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by eight mu minus d p by d z which is volumetric
flow rate to the capillary so you might notice
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that r is a property of the capillary and
this will remain fixed for a given capillary
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tube d p by d z in our case is equal to rho
g
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so one can see that q which can be written
as v over t where v is equal to volume of
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fluid generally in a viscometer the volume
is fixed between these two points so the volume
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is fixed q is equal to v over t pi r to the
power four by eight mu minus d p by d z and
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so one can see that t is directionally proportional
to mu and if the measurements have been done
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for a fluid already then one can just simply
use formula t one by t two is equal to mu
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one by mu two to calculate the viscosity of
a fluid ok so this equation can be rearranged
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to obtain the viscosity now as we have
seen that we can use capillary viscometer
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to measure absolute viscosity coefficient
as well as relative viscosity generally it
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is the relative viscosity that is measured
using capillary viscometer the pressure gradient
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it might be applied externally or gravity
can act as a pressure gradient to drive the
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flow
in simple viscometer which we use in our undergraduate
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labs or in our day to day simple experiments
it is capillary viscometer which in which
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the flow is driven by gravity however this
instrument has some limitations of the
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pressure gradient some of the energy of the
fluid because the the gravity the gravitational
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energy in our case some part of it will be
used to impart the kinetic energy to the fluid
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then as you might notice that the flow happens
from a bulb into a capillary so in at the
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entrance there will be entrance effect similarly
on the other side there will be divergence
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effect or the stream lines are going that
way so there will be divergence effects so
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these entrance and exit effects are neglected
we have not discussed in the previous class
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but one useful relation that you might have
or you probably would have studied in your
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undergraduate fluid mechanics course that
for developing flow in a channel the development
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length which is required for the flow to become
fully developed is equal to zero point zero
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five r e into diameter of the channel so this
tells us that we should keep reynolds number
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sufficiently small so that the flow quickly
becomes fully developed so it is important
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to take into account all these errors while
measuring the viscosity into capillary
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viscometer ok another important point here
is that as we said earlier that in a capillary
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the velocity profile is parabolic as you might
remember v z is equal to v z max into one
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minus r square by capital r square where capital
r is radius of the channel and r is any arbitrary
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radial coordinate
so we can quickly see that tau r z or the
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shear stress which is equal to minus mu del
v z over del r that will be not a constant
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but tau r z is proportional to it is proportional
to r so the shear rate is not constant so
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if the viscosity of the fluid is shear rate
dependent it varies the shear stress varies
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with the radius so if the viscosity of the
fluid is dependent on the shear rate then
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in such a case it is important that the viscosity
whatever viscosity that we measure using capillary
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viscometer that will not give us the correct
value of viscosity because viscosity is a
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function shear rate and shear rate is a function
of radius in a capillary viscometer so the
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viscosity will be varying across the cross
section in in the viscometer so generally
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it is useful to have capillary viscometer
to measure the viscosity of a non newtonian
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fluid of or a bingham plastic fluid then another
viscometer that we are going to discuss is
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coaxial cylindrical viscometer so in the capillary
viscometer the problem is that the shear rate
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is proportional to the radius whereas by the
arguments that we have made just now we
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would prefer to have a flow in which shear
rate is not constant so let us go back to
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our undergraduate fluid mechanics knowledge
and try to think that which is the flow in
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which shear rate is constant so if you remember
the flow between two parallel plates so two
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plates infinitely long plates which are parallel
to each other there is no pressure gradient
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in the flow but the upper plate is moved with
a velocity let us say capital v subscript
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z
if the upper plate is moved with a velocity
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v z there is no pressure gradient to drive
the flow then this flow is known as couette
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flow after couette in such case one can
show that the velocity if the distance between
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the plates is h and this cordinate is z and
this coordinate is y let us say then in such
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case the velocity profile v z will be equal
to capital v z which is the velocity of the
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upper plate divided by h into y so that means
v z is proportional to y which in terms mean
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that tau is a constant so in this case in
couette flow the shear rate is constant so
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the sorry tau is equal to so in this case
v z is proportional to y that means del v
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z over del y is a constant independent of
y so tau is also a constant the linear relationship
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between shear stress and shear rate is not
valid it is that shear rate is constant so
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tau is proportional to mu the tau is equal
to mu gamma dot sorry newtonian fluid so how
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can we realise this in practice because
it is not easy to have two infinite plates
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in parallel or the a simpler arrangement
is when two infinite cylinders they are
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moving relative to each other so this arrangement
is often used for designing a viscometer
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and this is also called bop cup viscometers
so in this case there are two concentric cylinders
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the outer cylinder is known as cup and the
inner cylinder is known as bob and the outer
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cylinder with the help of a shaft is rotated
with an angular velocity omega and because
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of this fluid motion the torque that is imparted
on the inner fluid is measured by this torsion
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wire so from this one can measure the
viscosity
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so let us look at how this is done there are
two concentric cylinders the flow between
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two concentric cylinder is known as taylor
couette flow and in this case the gap between
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the two cylinders are two minus r as we see
from this figure is very very small from either
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of course r two or r ok so we assume in this
case we make some assumptions to derive the
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relationship between the torque and the
angular velocity and find out the viscosity
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from there the flow is steady which is
when we do the measurements when the flow
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has become steady the flow is tangential
so there is only v theta component of velocity
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is there v that means v r is equal to zero
and v z is equal to zero and the flow is laminar
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ok so let us look at the governing equations
conservation of mass momentum in r theta z
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which is cylindrical coordinates and try to
look at different terms so because the flow
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is steady so all time dependent terms will
become zero there is no r so all terms containing
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v r or v z will be zero so this term goes
to zero this is zero this is zero this is
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zero this is zero because this is del
v theta over del theta is equal to zero and
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from here we can see that del v theta over
del theta is equal to zero that means that
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the[ta]- v theta is not a function of theta
v z is here so this is zero v r is here so
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this is zero v r is here so this is also zero
v z is here so this term becomes zero v z
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is here so this term becomes zero
now the gravity is acting in the z direction
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so these two terms become zero tau x y is
proportional to del v y over del x plus del
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v x over del y so using this principle we
can say that if v r zero then tau r r will
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be zero tau theta r will be zero because it
is del over del theta del over del z tau z
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r this is zero tau theta theta will be zero
tau theta theta will be zero tau z theta will
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be zero tau theta r will be equal to tau r
theta because their stress stands for symmetric
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so this term will be zero and tau r z is zero
because v r and v z r zero v theta and v z
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so this will be zero because v z is zero and
v theta over v z is zero similarly tau z z
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is equal to zero so finally we will end up
with these terms from the continuity equation
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del v theta over del theta is equal to zero
from r momentum equation we get minus rho
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v theta square over r is equal to minus del
p over del r so that shows that how does the
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pressure vary in the radial direction and
it is dependent on v theta v theta is not
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a function of theta you can see from here
now the theta v theta equation so there
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v get the del p over del theta is equal
to zero we have not done this in the previous
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slide so we can just note that del p over
del theta is equal to zero because there is
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no pressure gradient in the angular direction
and the flow is happening because of the outer
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rotating cylinder ok and in this we can replace
tau r theta with minus mu r del over del r
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v theta over r and the third equation from
the z momentum equation is minus del p y del
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z plus rho g is equal to zero so that is basically
for the hydrostatic pressure or the dependence
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of the pressure in the a z direction
so if we want to find out the velocity profile
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we can substitute this here let us look
at that we can substitute tau r theta so
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we will get r square tau r theta is equal
to a constant say c one dash and when we substitute
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tau r theta is equal to minus mu r and with
r it becomes r cube del over del r v theta
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over r is equal to a constant let us say this
so c one dash and so we get this is equal
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to minus mu r cube we can now remove this
from here so we get v theta over r is equal
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to minus c one over mu r raise to the power
minus two divided by minus two plus a constant
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c two so that gives us v theta is equal to
c one let us call this c one dash two mu into
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one over r plus c two dash let us say c two
dash r so if we look at this equation c one
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by two mu one over r plus c two r now we have
two constants c one and c two so to find out
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this we need two boundary conditions and these
two boundary conditions within will be
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the no slip boundary conditions on the two
walls so that means at r is equal to capital
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r and at r is equal to kappa r let us say
that the radius of the outer cylinder is r
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and the radius of the inner cylinder is kappa
times r so kappa will be less than one so
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at r is equal to capital r which is the outer
cylinder it is rotating with a velocity omega
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so the v theta will be equal to omega into
capital r whereas v theta is equal to zero
32:45.879 --> 32:50.929
at the inner cylinder which is fixed so you
might notice here that we have applied no
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slip boundary condition but the velocity at
the wall is not essentially zero in this case
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so if one substitute these values here then
one will get at r is equal to capital r
33:13.700 --> 33:24.790
v theta is equal to omega r is equal to c
one by two mu one over capital r plus c two
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capital r and at r is equal to kappa r the
velocity is zero c one y two mu one over kappa
33:40.330 --> 33:50.720
r plus c two kappa r
now this gives us a relationship between c
33:50.720 --> 34:09.980
one and c two so we can say that c two r is
equal to minus c one by two mu one over kappa
34:09.980 --> 34:22.480
square r and if we substitute this in this
equation then we will end up with omega r
34:22.480 --> 34:40.260
is equal to c one by two mu one over r this
is one minus c two r will be c two r is
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minus c one two mu one over r into one over
kappa square so this only one over kappa square
34:47.149 --> 35:03.180
is there so this gives us c one over two mu
is equal to omega r square divided by you
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can take kappa square into kappa square minus
one so c one by two mu is equal to omega r
35:16.170 --> 35:23.510
square kappa square divided by kappa square
minus one you might notice that kappa is less
35:23.510 --> 35:29.579
than one so this term is going to be negative
so one might want to write this in this form
35:29.579 --> 35:39.420
so let us look at after substitution one
will get the velocity in this form and from
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that we can get what is tau tau r theta is
equal to minus mu r del over del r v theta
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over r so we can calculate v theta over r
is equal to omega kappa capital r over one
36:14.299 --> 36:32.190
minus kappa square one over kappa r minus
kappa r over r square now if we want to differentiate
36:32.190 --> 36:47.170
it then del over del r over of v theta over
r is equal to omega kappa r over one minus
36:47.170 --> 37:05.010
kappa square this is a constant so this becomes
zero minus kappa r r to the power minus three
37:05.010 --> 37:22.670
multiplied by minus two so we take all this
into account we can make this as plus this
37:22.670 --> 37:46.710
also can go and this two can come here ok
and tau r theta will be equal to minus mu
37:46.710 --> 38:04.599
r omega kappa square r square divided by one
minus kappa square into r cube
38:04.599 --> 38:20.690
now if you one want to find out the torque
where t is torque on the inner cylinder so
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t is equal to force into distance so this
force is tau r theta at the inner cylinder
38:36.819 --> 38:48.490
so tau r theta at r is equal to kappa r into
the radial or the circumferential not the
38:48.490 --> 39:01.050
circumferential but the lateral area of the
cylinder two pi kappa r l which is the area
39:01.050 --> 39:18.260
multiplied by the radius which is kappa r
so this becomes minus mu this is this can
39:18.260 --> 39:33.720
be reduce to r square so let us just get rid
of this so minus mu omega kappa square r square
39:33.720 --> 39:48.900
divided by one minus kappa square kappa square
r square into two pi kappa square r square
39:48.900 --> 39:57.200
l so kappa square kappa square cancels out
so the torque that we obtain is the two is
39:57.200 --> 40:10.640
missing here so we substitute this two so
two mu so this is four pi mu omega r square
40:10.640 --> 40:17.990
kappa square divided by one minus kappa square
into l so that is the torque on the inner
40:17.990 --> 40:30.039
cylinder now if you see kappa and r are from
the geometry of the cylindrical arrangement
40:30.039 --> 40:40.359
omega is the velocity which imparts the shear
rate which is proportional to the shear rate
40:40.359 --> 40:49.000
and l is the length of the cylinder so again
this is also a geometrical parameter ok so
40:49.000 --> 41:13.460
we can say that tau torque is a constant times
mu into omega so by measuring torque and omega
41:13.460 --> 41:22.870
one can find out the viscosity of the fluid
that is there in between the two cylinders
41:22.870 --> 41:26.740
ok
so again we have made some assumptions here
41:26.740 --> 41:34.180
because this relationship has been derived
for the flow of fluid between two infinite
41:34.180 --> 41:44.109
cylinder but in practice what will we have
is the outer cylinder in which there is an
41:44.109 --> 41:59.150
inner cylinder so there will be the effect
of top surface plus there will be end effect
41:59.150 --> 42:10.559
at the top because at the top because of the
ministers for a non viscoelastic fluid
42:10.559 --> 42:17.710
the fluid surface will look something like
this so the surface area covered by the outer
42:17.710 --> 42:23.780
and inner cylinders will be different so we
might need to consider the relationship
42:23.780 --> 42:31.319
might be valid somewhere in between ok so
such corrections need to be taken into account
42:31.319 --> 42:39.880
when using coaxial cylindrical viscometers
so the another and the third important
42:39.880 --> 42:50.750
viscometer is cone and plate viscometer in
which there is a flat plate and over which
42:50.750 --> 43:03.500
a cone which has very large angle which
is this is apex angle and this apex angle
43:03.500 --> 43:12.990
is very small this angle size sometimes as
low as one degree so the cone of very large
43:12.990 --> 43:19.000
apex angle and a flat surface this is the
arrangement and the cone is rotated with a
43:19.000 --> 43:31.329
constant angular velocity so which gives us
a constant shear rate and the torque require
43:31.329 --> 43:41.359
to turn the cone is measured so from the torque
one get tau principle it is very similar
43:41.359 --> 43:50.180
to coaxial flat plate arrangement for very
large apex angle one can find out the velocity
43:50.180 --> 44:02.069
distribution that the and from that the
torque and one get that torque is proportional
44:02.069 --> 44:12.839
to gamma and the viscosity is the proportionality
constant so one can measure the velocity here
44:12.839 --> 44:22.800
and from that one can again calculate the
torque so this is shear stress and the torque
44:22.800 --> 44:34.609
so let us do a simple question and example
let assume that it is a capillary viscometer
44:34.609 --> 44:41.320
and the yield stress of the blood is about
zero point zero seven dyn per centimetre square
44:41.320 --> 44:49.640
there is a capillary viscometer which has
a length of twenty centimetre and the radius
44:49.640 --> 44:56.369
is one m m and what we need is we need to
calculate the required pressure difference
44:56.369 --> 45:07.740
for the blood to start flowing so if we remember
the relationship between shear stress and
45:07.740 --> 45:20.880
pressure we can write this pressure gradient
in terms of pressure difference so we can
45:20.880 --> 45:33.540
write let us say this as delta p which is
the pressure difference and l r by two so
45:33.540 --> 45:41.589
if we take this shear stress on the wall because
that is when the fluid it is start flowing
45:41.589 --> 45:50.300
and in this case r will be equal to the capital
r of the channel so in this the shear stress
45:50.300 --> 46:01.369
is zero point zero seven dyn per centimetre
square and that is equal to delta p which
46:01.369 --> 46:10.069
is pressure difference l is twenty centimetre
r is the radius of channel so let us write
46:10.069 --> 46:18.660
this also in centimetre so thats we get everything
in c g s units and this gives us delta p is
46:18.660 --> 46:35.480
equal to zero point zero seven into forty
divided by point one zero dyn per centimetre
46:35.480 --> 46:54.480
square so that will be twenty eight dyn per
centimetre square or two point eight pascal
46:54.480 --> 47:02.329
one can then change it to rosy using rho
g h one can then change it to m m h g and
47:02.329 --> 47:07.730
the final answer will be zero point zero two
one m m of h g
47:07.730 --> 47:14.660
so in summary in this lecture we have looked
at the capillary viscometer the coaxial
47:14.660 --> 47:21.599
or couette flow viscometer and cone and plate
viscometer the capillary viscometer is very
47:21.599 --> 47:29.960
simple easy to use and it can be used very
accurately for measuring the viscosity of
47:29.960 --> 47:35.890
newtonian fluids but it is not useful for
measuring the viscosity of non newtonian fluids
47:35.890 --> 47:43.339
because of the region that the shear stress
varies with radius then couette viscometer
47:43.339 --> 47:51.760
they can be useful for measuring shear dependent
viscosity because the shear rate is constant
47:51.760 --> 47:59.960
in the couette viscometers finally the
cone and plate viscometer cone and plate viscometer
47:59.960 --> 48:07.440
are most popular or most accurate among
the three and they require small amount
48:07.440 --> 48:14.769
of samples so for measuring the viscosity
of the samples which are costly they are used