WEBVTT
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in this lecture we will review some of the
basics of fluid mechanics specially the once
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that will be required throughout this course
so before looking into the fluid mechanics
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we need to know or we need to remind our
self what does a fluid mean or what constitutes
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if fluid or what what is the definition of
a fluid so if you go back to your early childhood
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you would have learned definitions of liquid
and gases which are both fluids so they
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are the materials you might have learned the
definition that the materials which takes
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the form of the vessel in which they are contain
or which takes the shape of the vessel in
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which they are contained or are called fluids
the another definition
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now or a more complete definition for the
fluids is that any material or any matter
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that when a shear stress is applied on the
fluid so under the application of a shear
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stress if a material deforms continuously
then it is known as the fluid so for example
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let us consider a fluid material that is confined
between two plates which are kept parallel
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and in this consider a fluid element or a
small fluid volume and let us name that as
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a b c and d now if a shear stress is applied
on this plate the shear stress let us remind
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ourselves that shear stress is a stress that
is directed tangentially to the material surface
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so a stress is force per unit area and shear
stress is the one that is directed tangentially
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to the surface
so a tangential force is applied on this surface
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and because of this the material will deform
to a location let us say these locations are
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c dash and d dash if you look at this after
sometime then the material will further
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will have located to another location c double
dash and d double dash so because of the application
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of a shear stress a shear rate or the the
material shear so the material deforms keep
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deforming continuously so that is why at two
different time instant so let us say that
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c dash is a time t and c double dash is t
plus this is t one and this is t two so
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the material deforms continuously under the
application of a shear stress now if we want
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to differentiate or we want to remind our
self that how is fluid different from a solid
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then
under the application of a shear the solid
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material will deform and
so there will be a deformation but not it
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will it will not deform continuously there
will be a a with the application of force
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there will be a certain deformation of the
fluid but it will not keep deforming continuously
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whereas in the fluid that deforms continuously
and the application of a shear stress so the
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fluid they have the properties of flow or
they flow under the application of a driving
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force whereas solids do not
now philosophically speaking or depending
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on the time scales everything flows so
there is a famous verse in the bible
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in the song of debora that says that the
mountain gushed before the lords so a that
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means that the mountains flowed before the
lords but not before the man so what does
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that mean that means that even the mountains
which are considered to be solid they also
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flow but not at the time scale of few hundred
or or at the time scale of hundred years which
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is the life of a man but the god who is considered
to live forever in his untimed scale the mountains
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flow they change shape so even so every material
does flow but depending on the time scale
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we may or may not perceive or experience that
this material is flowing anyway the materials
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that we are considering at this this course
because this course is concerned with a
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cardiovascular fluid mechanics where blood
or plasma is the fluid that we are dealing
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with so the plasma or blood is a if fluid
in any case ok so another fundamental
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thing that we would like to revise or we
would like to look up on is eulerian and
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lagrangian approach so there can be a different
approach to analyse the problems in engineering
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mechanics in fluid mechanics in general two
approaches are very popular the first one
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what we call is lagrangian approach
so in the lagrangian approach a particular
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mass of the fluid or particular fluid particles
they are tracked and the governing equation
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for the each particle can be solved by
say newton's second law of motion that f is
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equal to m d v by d t so this is what we call
lagrangian approach or another approach is
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eulerian approach so in the eulerian approach
what happens that one considers a control
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volume and look at the flow that is coming
in the control volume and going out the control
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volume so he is concerned or in the eulerian
approach we are concerned with the flow that
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is there in the control volume so the fluid
that comes in fluid goes out and the fluid
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that is there in the control volume so the
flow properties such as density viscosity
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and the velocity and pressure is studied in
this control volume and because the fluid
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flows so the fluid particles that are there
in the control volume at time t may not be
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there at time t plus delta t but irrespective
of that we look at the fluid particles that
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are inside the control volume and we do not
follow a the entire follow the same particles
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for the entire time of study for which the
analysis is being made for the entire time
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period for which the analysis is being made
so one example which we generally look
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at the considering the flow of a boat and
you have a consider that you and your friend
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so your friend is going going on the boat
and crossing the river so if he looks at the
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motion and that is that if you track the motion
of the friend in his reference frame so he
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is looking at that is if you follow the
motion of the friend or motion of a particular
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boat while crossing the river then it is
the lagrangian frame of reference or because
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you are tracking one particular boat on the
other hand if you consider a certain area
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in a river and look at what is coming in and
what is going out in that certain area or
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a certain volume if you consider the depth
also then that is eulerian analysis ok
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so coming to the cardiovascular fluid mechanics
we will not be looking at in most of the cases
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the eule[rian]- a the lagrangian approach
or rather we will take the eulerian approach
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so we will consider a control volume and
look at the blood coming and going out from
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this control volume so another important
approximation or another important assumption
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that is made that fluid is a continuum
we all know that each and every material is
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constituted of molecules and atoms which are
particles of very small size so there is always
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some space between those atoms now depending
on the length scale we can say that if
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lambda is very very small than d then flow
can be considered or the fluid can be considered
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as continuum so we can describe the fluid
in a as a continuous medium or as a continuous
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field on the properties of the fluid such
as pressure and velocity they can be considered
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as a field so what is lambda lambda is
the typical distance between the molecules
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or the fluid molecules so the mean free path
of the molecules for a fluid is lambda and
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d is the length scale of the problem that
you are considering
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so there is a non dimensional number associated
with it what is known as knudsen number so
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k n is equal to lambda by d now if knudsen
number is small then one then fluid can become
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zero as continuum and we can describe this
as a continuum so all the discussion that
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we will have in this course we can assume
the fluid to be continuum because the mean
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free path is order of few nanometres and the
smallest scale that we will be talking about
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is of the order of few microns so we can
safely consider the fluid to be a continuum
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or the blood to be a continuum
now we will continuously deal with the
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stresses in fluids as we just while
defining fluid we say that the under the application
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of a sheerest stress the the fluid which
deform continuously or or the material that
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deform continuously is known as known as fluid
so we need to also define stress stress
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is the measure of all forces that is acting
on a volume so if we define stress as sigma
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sigma is equal to f over a of force per unit
area now stresses will have a nine components
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it is a second order tensor so you are not
going to cover the details of vectors and
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tensors in this but it is strongly recommended
that you read a bit about the vectors and
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tensors or any student of fluid mechanics
should have a good idea about of vectors and
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tensors so they will have a will write down
this as a matrix sigma x x sigma x y sigma
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x z sigma y x sigma y y sigma y z sigma z
x sigma z y and sigma z z so if you look at
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a typical component of a stress there are
two subscript into it x and y so x is x denotes
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the surface on which the stress acts and y
denotes the direction of the force ok
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now in this representation where
the on a cubic element different components
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of a stresses are shown so its in cartesian
coordinate system x y and g so let us take
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one component here on this surface you
can see that sigma x x so sigma has been represented
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sigma represents here normal stress and
tau is denoted as denoted as shear stress
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that is the normal convention that you will
see in a number of books so in this case on
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this surface which is a x surf[ace]- surface
having area vector a normal to it and the
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normal stress is sigma x x and it has on this
x surface the force acting in the y direction
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on the extra setting in the y direction is
tau x y similarly the force acting in the
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z direction is tau x z so there are two shear
stress components acting on this surface and
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one normal stress component similarly on the
other x component and you can see the same
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thing for the other components as well
now a for conservation of angular momentum
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it is necessary that tau y x is equal to tau
x y similarly tau y z is equal to tau z y
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and tau z x is equal to tau x z so that means
that there are only six ind[ependent]-
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out of a nine components of a stress there
are only six stresses that are independent
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ok so a that is just recapitulate that
stresses are some of the or they are the measure
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of the forces that act on the surface now
pressure is a normal stress that act normal
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to a surface and in the navier stokes equation
or in the momentum conservation equation it
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is taking out taken out or sometimes in computational
fluid dynamic applications when we are looking
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at the stresses the pressure is clubbed with
the normal stresses so one need to take this
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into account or one need to keep this into
keep this in mind ok so then the forces
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they can be on they can be recomposed into
normal and the shear forces so the normal
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forces are represented here by sigma and shear
forces are represented by tau ok
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now before we discuss the two fundamental
conservation principles based on which we
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can calculate the unknown or the the unknown
quantities in fluid mechanics the pressure
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and velocities let us look at the general
conservation principle which is known as reynolds
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transport theorem we are not going to derive
this in this short course but it is strongly
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recommended that till you follow any standard
fluid mechanics text book and look at the
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derivation of reynolds transport theorem so
what does reynolds transport theorem states
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let us say w w is any property of the system
any arbitrary property of the system and it
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says that the time rate of change of any such
system property so the rate of change of time
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so how this property changed with time that
is equal to the time rate of change of a property
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within the volume of the interest so how
does this property now you have two w here
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one is capital w and one is small w
so capital w is the arbitrary system property
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or the any extensive property of the system
and small w is the intensive property of the
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system so small w is capital w per unit mass
or specific properties so that is del over
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del t so that is time rate of change of a
property within this volume of interest so
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you can consider any control volume let
us say the volume v and the boundary of this
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is represented by this area so this is a and
a is area is a vector you must remember that
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area is a vector and a its direction is outward
normal to the surface or now of which area
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you are considering so the rate of change
of any arbitrary system property is equal
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to time rate of change of a property within
the volume of interest plus the flux of the
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property so the flux that is coming in flux
of the property out of the surface of interest
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or into the surface of interest so the total
integral of the flux that is coming in and
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going or going out would that integral combine
so that says the rate of change of any arbitrary
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property in a control volume will be equal
to that rate can be effected because of two
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factors one is that the rate of the the the
property changes within the volume itself
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or the property is brought in the system or
it goes out of the system
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so that makes quite sense say
in general a but it is also important to describe
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this mathematically and understand this mathematically
so if we consider that this property w is
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mass of the system then the small w the
intensive property become one and if we consider
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the property as momentum of the system then
w becomes the velocity so based on this one
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can derive the mass conservation using a reynolds
transport theorem one can derive the mass
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conservation equation and momentum conservation
equation and a we might have one of this
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assignment problem but we are not going to
derive it here so the mass conservation
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principle states that del rho over del t plus
del dot rho v is equal to zero for an incompressible
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fluid so a if you look at this del rho over
del t is rate of change of density so for
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an impressible incompressible fluid the density
is constant that means del rho over del t
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is equal to zero so we are we have what we
have is del dot rho v is equal to zero because
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rho is constant so we can write that del dot
v is equal to zero for an incompressible fluid
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now when blood is an incompressible fluid
generally at the atmospheric temperature all
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the liquids can be treated as incompressible
fluid so most of the time in this course we
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will using only this as a mass conservation
fluid
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now when it comes that the flow is not steady
even then the density does not change with
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time so del dot v is equal to zero is good
for steady as well as unsteady flow because
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rho is anyway constant with time now if you
want to expand this then del is you can write
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del v x over del x plus del v y over del y
plus del v z over del z is equal to zero for
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a one dimensional equation you will have the
equation as del v x over del x is equal to
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zero and so on so forth ok so that is mass
conservation equation now before we
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go to describe a momentum conservation equation
i would like to bring your attention to this
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term what is called material derivative or
it is also known as substantial derivative
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so that has two terms into here into it that
d v over d t capital d over d t is known as
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material date derivative or substantial derivative
so the first term del v over del t is known
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as local acceleration and del dot v is actually
it should have been a v dot del so that is
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called so far the in partial differential
equation if you have any property that depends
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on time as well as cartesian coordinate then
its a derivative the total derivative will
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be a with respect to del v over del t with
respect to time as well as with respect to
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the cartesian coordinates so considering the
local acceleration and convective acceleration
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one get the material derivative
so the momentum conservation can be written
28:59.220 --> 29:12.400
as rho d v by d t is equal to minus del
p minus del dot tau plus rho z rho g now this
29:12.400 --> 29:21.070
is we can expand so this when we write rho
out of it then this means that flow is incompressible
29:21.070 --> 29:48.440
we can write that rho del v over del t plus
del dot v v in the vector form is equal
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to minus del p minus del dot tau plus rho
g if the gravity is considered in the system
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so you can see that all the terms are vector
in this and it is quite clear from here that
30:18.510 --> 30:26.550
this v is vector g is a vector the gradient
of pressure is a scalar but the gradient of
30:26.550 --> 30:33.970
pressure will be a vector but del dot tau
tau is a second order tensor and its depth
30:33.970 --> 30:42.180
dot with del will result in first order tensor
that means it is also a vector similar v v
30:42.180 --> 30:50.890
it is dyadic product and it will
be a second order tensor but there product
30:50.890 --> 31:02.310
will also be a first order tensor so it is
a vector now this in this so we have looked
31:02.310 --> 31:10.560
at the two conservation equation mass conservation
and momentum conservation equation and in
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general what we will be concerned with that
we need to know the pressure field or the
31:15.900 --> 31:20.670
pressure distribution and the velocity distribution
and actually if we want to know only velocity
31:20.670 --> 31:25.950
distribution that to know velocity distribution
we need to know pressure distribution or other
31:25.950 --> 31:31.950
way around so this is the momentum conservation
equation
31:31.950 --> 31:43.309
now in this what we need is a we have two
unknown pressure and velocity so if we look
31:43.309 --> 31:52.420
at pressure one unknown and velocity as a
vector so two unknowns and we have two equations
31:52.420 --> 31:59.559
one is a mass conservation equation and another
is momentum conservation equation so in principle
31:59.559 --> 32:05.340
using these conservation equations and the
appropriate boundary condition one can solve
32:05.340 --> 32:14.900
for the fluids but we have a unknown here
or we have a new term here which is tau shear
32:14.900 --> 32:22.260
stress so before we solve these equations
we need to close tau or we need to find a
32:22.260 --> 32:28.890
relationship for tau in terms of velocities
which is called constitutive equation and
32:28.890 --> 32:37.330
that comes from rheology or and one of the
most common example for many fluids is what
32:37.330 --> 32:47.100
is known as newton's law of viscosity so newton's
law of viscosity relates tau and shear rate
32:47.100 --> 32:55.190
which is sometimes represented as gamma dot
or one can say that for one dimensional
32:55.190 --> 33:14.770
flow del v over del y so if over a surface
there is a flow then the shear stress tau
33:14.770 --> 33:26.730
can be written as minus mu del v this is x
direction and this is y direction del v x
33:26.730 --> 33:34.590
over del y now if we want to generalise this
so the generalised newton's law of viscosity
33:34.590 --> 33:43.240
is given as tau or the stress tensor is equal
to minus mu del v plus the transpose of velocity
33:43.240 --> 33:48.750
gradient plus two by three mu minus kappa
del dot v del
33:48.750 --> 34:07.710
now mu is the dynamic viscosity of the
fluid and kappa is called dilatational viscosity
34:07.710 --> 34:24.000
which is often zero for monatomic gases
and use we have just seen that del dot v is
34:24.000 --> 34:31.940
equal to zero for liquids in general which
because the liquid is incompressible fluid
34:31.940 --> 34:39.970
so this term is often zero in the fluid mechanics
and at at least the problems that we are
34:39.970 --> 34:45.179
going to consider for the flow of liquids
because the blood is a liquid so this term
34:45.179 --> 34:52.389
is going to be zero so looking at the say
some components of stresses tau x x or
34:52.389 --> 34:57.180
if you want to see say this in the normal
stress term then it can be sigma x x is equal
34:57.180 --> 35:05.510
to minus two mu del v x over del x and tau
x y is equal to minus mu del v x over del
35:05.510 --> 35:11.040
v y over del x plus del v x over del y what
i would like to bring your attention here
35:11.040 --> 35:19.070
that the viscous stresses can be normal stress
also it is a different matter because the
35:19.070 --> 35:29.140
del v x over del x term is often smaller than
the del v y over del x term so the gradient
35:29.140 --> 35:36.020
in the same direction are smaller than in
the oth[er]- transfers direction so
35:36.020 --> 35:42.600
that is why tau x x or del a the normal
stresses components are often neglected in
35:42.600 --> 35:47.619
our analysis as we will see in the later classes
ok
35:47.619 --> 35:53.440
so if we substitute this newton's law of viscosity
and then we will end up with rho d v by d
35:53.440 --> 36:01.820
t minus del p minus mu del two v plus rho
g here so this is navier stokes equation and
36:01.820 --> 36:12.880
the first term is known as the acceleration
term and it has combination as we have just
36:12.880 --> 36:18.020
for the definition of material derivative
it is is the combination of local acceleration
36:18.020 --> 36:30.810
which is with respect to time and the convective
acceleration and del p is the a pressure gradient
36:30.810 --> 36:45.740
or you can say it is a pressure term this
is called viscous term and as we know that
36:45.740 --> 36:55.600
viscosity is also diffusivity or it is
momentum diffusivity because of the viscous
36:55.600 --> 37:00.440
properties of the fluid the momentum is diffused
in the fluids or between two fluids layer
37:00.440 --> 37:09.750
so it is also in general called it can be
known as diffusion term and any other forces
37:09.750 --> 37:17.170
whereas the gravity is the general body
force gravity is the usual body force but
37:17.170 --> 37:24.450
one can also have other body forces here so
this is the body force ha
37:24.450 --> 37:30.930
so in summary what we have looked at today
is a some basics of fluid mechanics which
37:30.930 --> 37:36.870
are prerequisites or which are require to
understand the the problems that we solve
37:36.870 --> 37:43.540
later in the course for the fluid mechanics
course we will also look at some basics of
37:43.540 --> 37:51.180
solid mechanics in a small lecture so what
we have looked at today is that the what is
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a fluid and fluid can be treated as continuum
then we have looked at very briefly eulerian
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and
lagrangian
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description of fluid then we have also looked
at the reynolds transport theorem and mass
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and momentum conservation equations ok
so
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thank you