We were discussing about different flow visualization
lines in our last class and we will
discuss one more flow visualization line which
is called as timeline.
So, what is the time line? If you have a snapshot
at a particular time in the flow field
where you mark nearby particles, so nearby
fluid particles which are located in the flow
field at a given instant of time, if you somehow
mark those particles by somewhere then
if you now get the snapshot at different times,
it will give a picture of evolution of the
flow field as a function of time and that
is known as a timeline.
So, it is nothing but like snapshot of nearby
fluid particles at a given instant of time
that
is called as a timeline. So, let us look into
a small movie to see that what we mean by
a
timeline. So, if you see now this gives a
snapshot at different instance of time of
nearby
fluid particles and in a way, it gives a sense
of the velocity profiles at different instance
of time you can see that in this example the
flow passage is narrowing another flow
passage is narrowing the fluid is moving faster
to make sure that the mass flow rate is
conserved we will see later on that formally
this is described by the continuity equation
and in may be a differential form or an integral
form.
But at least this gives us a visual idea of
what the timeline is all about. Now with this
background on the flow visualization lines
we have now understood that how we can
visualize the fluid flow in terms of some
imaginary description like through the stream
lines, strip line, path line or maybe the
timeline. Next we will go into the description
of
acceleration of fluid flow. So, we have discussed
about the velocity. The next target is
the acceleration. Let us say that you have
fluid particle located at a position P at
specifically the location P 1 at time equal
to t and how the velocity is described here?
The velocity described is described here through
a velocity vector v which is a function
of r 1 that is the position vector of the
point P 1 and the time t.
This is nothing, but the Eulerian description,
if you write it in terms of components; you
can write an equivalent scalar component description
that you have u as a function of x,
y z and t; v as a function of x, y, z, t and
w as another function of x, y, z t. So, we
are
trying to describe it in terms of Cartesian
coordinates, it is not always necessary to
do
that, but it may be a simple way to demonstrate
one we use other coordinate systems as
well. So, if you are using a Cartesian coordinate
system 3 independent coordinates space
coordinates plus time coordinate that together
give the velocity at a particular point.
So, if the fluid particle is located at P
1, the velocity at that point is basically
the velocity
of a fluid particle located at that point
and that is given by these components. Now
let u
say that at a time of t plus delta t, this
thing get changed now at a time plus delta
t what
happens? This fluid particle is no more located
at this point the fluid particle is located
at
a different point. So, let us say that the
fluid particle is located at a point P 2.
So, at the
point P 2, now let us say that the velocity
is whatever some arbitrary velocity. So,
initially it may be velocity at the point
1 say v 1.
Now, it is v 2, which is again a function
of its local position and time. So, you have
this v
2, this one a function of what? So, let us
say that it is given by its components u plus
delta u, v plus delta v, w plus delta w, these
are functions of what these are functions
of
the new position vector, the new position
vector say is r 1 plus delta r 1. So, in terms
of
scalar components it may be x plus delta x,
y plus delta y, z plus delta z and the time
has
also now changed, it has become t plus delta
t. So, we are thinking about a small interval
of time delta t over which the fluid particle
has undergone some displacement which is a
change in position vector having components
delta x delta y and delta z that is what we
are trying to understand.
So, we can clearly see that there is an original
velocity in terms of its 3 components,
there is a change velocity in terms of its
3 components and if you want to find out the
acceleration see the basic definition of acceleration
is based on a Lagrangian reference
frame that is the rate of time rate of change
of velocity in a Lagrangian frame not in an
Eulerian frame the all the basic definitions
in newtonian mechanics that we have learnt
earlier are based on Lagrangian mechanics.
So, when you say that its rate of time rate
of
change of velocity then that has to deal with
the time rate of change of velocity of maybe
an identified fluid particle which earlier
was at P 1.
Now, is at P 2. So, if you want to find out
the chain. So, you can write of course, you
can
write it in terms of the 3 different components,
but just for simplicity let us just write
for
the x component similar things will be there
for y and z component. So, how can you
write u plus delta u as a function of u? So,
u plus delta u is now dependent on the local
position of the particle and the time that
has elapsed. So, it is a function of it depends
on
what it depends on the original u plus the
change. So, what was the original you that
was
u plus see it is a function of 4 variables.
So, you again it 
is a same mathematical problem
that there is a function of 4 variables it
is known at a given condition.
Now, you make a small change in each of these
variables and you want to find out the
new function again, you can express it through
a Taylor series expansion. Now it is a
function of multiple variables instead of
a single variable. So, we will use the Taylor
series expansion into keep in mind that now
you are having 4 variables. So, let us first
consider the time variable maybe because it
is bit different in characteristic then the
earlier one. So, this is with regard to the
time then with regard to the space plus higher
order terms this we have just written the
first order term in the Taylor series.
Since it is a function of 4 variables, you
have 4 first order derivative terms, similarly
you
will be getting second order derivative terms
and so on but we will neglect the higher
order terms by considering that these delta
x, delta y, delta z and delta t are very small.
So, we have to keep in mind that all these
are tending to 0 and because all they are
tending to 0, we are neglecting their higher
orders. So, you can first thing what you can
do you can cancel u from both sides and what
is the definition of acceleration along x
for
from a particle mechanics view point, well
Lagrangian view point. So, you have to find
out the change in velocity x component of
velocity because we are writing acceleration
along x divided by the time delta t in the
limit as delta t tends to 0.
Very simple straightforward Lagrangian description,
when you do that basically what we
are doing? We are dividing the left hand side
by delta t. So, right hand side is also
divided by delta t and the limit is taken
as delta t tends to 0. So, the first term
is straight
forward, let us look into the next terms.
So, first we will evaluate the limits, limit
as delta
t tends to 0, delta x divided by delta t that
multiplied by the derivative with respect
to x,
similarly the other terms let us just complete
it. So, what we are doing is we are trying
to
find out that because of the changes in velocity
component along different directions,
what is the net effect in acceleration and
these terms are basically representatives
of that.
We will formally see that how they represent
such a situation. So, now, let us concentrate
on this limiting terms say the first limiting
term what it is representing it is representing
the time rate of change of displacement along
x of the fluid particle over the period delta
t. Now we have to keep in mind that we are
thinking about a limit as delta t tends to
0.
This is a very important thing, what is the
significance of this limit as delta t tends
to 0
when delta t tends to 0, P 1 and P 2 are almost
coincident; that means, let us say that P
1
P 2 all those converts to some point P and
that point is a point at which say we are
focusing our attention to find out what is
the change of velocity that is taking place.
So, when in the limit delta t tends to 0 we
are considering the Eulerian and Lagrangian
descriptions merge, this is very very important.
So, we are trying to see, what is our
motivation? We know something and we are trying
to express something in terms of
what we know. What we know? We know the straight
forward Lagrangian description of
acceleration, we are trying to extrapolate
that with respect to an Eulerian frame, to
do
that we must have an Eulerian Lagrangian transformation
and essentially we are trying to
achieve the transformation in a very simple
way that as the delta t tends to 0, Eulerian
and Lagrangian descriptions should go inside.
And then what does it represent it represents
the instantaneous velocity x component of
the instantaneous velocity of the fluid particle
located at p; that means, it represents the
x
exponent of the fluid particle located at
P, since you are focusing on attention on
pay
itself and the velocity of the fluid particle
if it is neutrally buoyant is same as the
velocity
of flow, we can write that this is same as
what this is same as u at the point P see
writing,
this as u is these very straight forward understanding
its it conceptually is not that trivial
and straight forward is the Eulerian and Lagrangian
descriptions did not merge we could
not have been able to write this because this
is on the basis of a Lagrangian descriptions
and this is the Eulerian and velocity field.
How these 2 can be same, they can be same
only when we are considering a particular
case when in the limit as delta t tends to
0. So, wherever we are focusing on attention
at
that particular point this represents the
velocity of the fluid particle if the fluid
particle is
neutrally buoyant with the flow then it is
like can inner stress or particle moving with
the
flow and then it would have the same velocity
as that of the flow at that point at that
point at that instant; however, if the fluid
particle has a different density than that
of the
flow then this would be u of the fluid particle.
So, fundamentally this is u of the fluid
particle.
Not u of the flow field if it neutrally buoyant
then it becomes same as u of the flow field
if it is an inner stress are article in the
flow which is the definition of the fluid
particle
then it is definitely same as u at that point,
but if it is a fluid if it is a particle of
a
different characteristic different density
characteristic than that of the flow, it may
be
different from that of the velocity field
at that point so that you have to keep in
mind. So,
if you complete this description of this term
what you will get you will get a x is equal
to
that is the straight forward follow up of
this expression because the other limits you
can
express in terms of v and w again with the
same understanding as we expressed as we
used for expressing the first term.
Now, if you clearly look into this acceleration
expression there are 2 different types of
terms, one is this type of term which gives
the time derivative and other gives the special
derivative, you will see that this expression;
we will give you a first demarcating look
of
how the expression is different in terms of
what we express in a Lagrangian mechanics
in
a Lagrangian mechanics it is just the time
derivative that comes into the picture here
you
also have a positional derivative and what
do these terms represent we will give a formal
name to this terms, but before that first
let us understand that what these 2 terms
represent say you are located at a point 1,
now you go to a point 2 in the field.
So, when you go there; there are 2 ways by
which your velocity get changed how one is
may be from 1 to 2 when you go you have a
change in time and because of a change and
you also have a change in position you have
a change in velocity and that is. So, only
time dependent phenomena how can you understand
what is the component of the time
dependent phenomena if you did not move to
2, but say you confined yourself to one say
you are not moving with the flow field you
are confining yourself with one then you are
freezing your position, but still at the point
one may be a change in velocity because of
change in time it if it is an unsteady flow
field.
So, because of that it might be having acceleration.
So, the acceleration that acceleration
component is because of what the time rate
of change of velocity at a given point at
a
given location. So, that is reflected by this
one, but by the time when you are making the
analysis the fluid particle might have gone
to a different point even if its local velocity
that is velocity at a point is not changing
with time it has gone to a different point,
there
it encounters a different velocity field.
So, here it was encountering a particular
velocity
field because of its change in position. So,
what it has done? It has got advected with
the
flow it has moved with the flow and it has
come to a new location where it is
encountering a different u, v, w.
So, because of the change in u, v, w, with
the change in position it might be having
acceleration. So, acceleration is not directly
because of the time rate of change of
velocity at a given point, but because of
the special changed since the particle the
fluid
particle by the time has travels to a different
location where it finds a different flow field
and be and since we are considering that it
is an inner stress of particle it has to have
the
same velocity locally as that is there in
the new position. So, because the next
combination of terms it represents the change
in velocity, only due to change in position.
So, the total the net change is because of
two things, one is if you keep position fixed
and
you just change time because of unsteadiness
in the flow field and maybe an acceleration
the other parties even in the flow field is
steady, but you go to a different point because
of non uniformity because of change in velocity
due to change in position the fluid
particle might have a change in velocity.
So, the change in velocity in the fluid particle
maybe because of tourisms one is because of
the change in velocity due to change in
time even if it were located at the same position
as that of the original one and the other
one is not because of change in time, but
because of change in position as it has gone
to a
different position because of non uniformity
in the flow field it could encounter a define
velocity.
And the result and acceleration is a combination
of this two. So, let us let us take a very
simple example to understand it say you are
you are travelling by flight from Calcutta
to
Bombay. So, when you are taking the flight
before taking the flight you see that it is
raining very very heavily and then say you
take two to and half hours to reach Bombay
and you find that it is a very sunny weather.
So, the question is now if you if you want
to
ask yourself a question does it mean that
when you when you departed from Calcutta it
was raining in Bombay or when you departed
from Calcutta it was funny at Bombay or
when you have reached Bombay is it still raining
at Calcutta or is it still is it sunny at
Calcutta.
It is not possible to give an answer to anyone
of this because the net effect that you have
seen is a combination of 2 things you have
traversed with respect to time. So, you have
ill of certain time by which maybe it was
raining at Calcutta, but right now it is not
raining at Calcutta, maybe it was sunny at
Calcutta and right now it has started raining.
So, it is like at a particular location the
weather has changed because of change in time,
but the other effect is that you have migrate
to a different location and because of the
change in location maybe it was before two
years raining at Bombay.
Now, it is sunny or it might. So, happened
that, it was sunny 2 hours back in Bombay,
still it is sunny. So, you can see that individual
effects you can; may be try to isolate, but
what is the net combination of changing with
respect to position in time that is the net
effect of this and 
it might not be possible to isolate these
effects. So, when you think
about the total acceleration. So, it is just
like a total change. So, when you have the
total
change it is the change because of position
and because of time and that is why this a
x
or may be a y or a z this is called as the
total derivative of velocity.
So, it is given a special symbol capital D
D t. So, capital D D t has the special meaning
it
is called as total derivative it is to emphasis
that it is result in change because of change
in position and change in time. So, with respect
to change in time if you have a change
then it is called as a temporal component
of acceleration. Temporal stands for time
temporal or transient or local these a certain
names which are given again by the name
localities clear local means confined to a
particular position only with respect to change
in time and this is known as the convective
component.
So, convective component is because of the
change in position from one point to the
other and this therefore, is the total or
sometimes known as substantial. So, the total
derivative is the very important concept mathematically
here we are trying to understand
this concept physically, but it is not just
restricted to the concept of acceleration
of fluid
flow it is applicable in any context in any
context where you are having an Eulerian type
of description.
And it is therefore, possible to write the
general form of the total derivative as this
way
where it has local component and a convective
component. So, we can try to answer
some interesting and simple questions and
see and get a feel of the difference of these
with again the Lagrangian mechanics.
So, if you ask the question, is it possible
that there is an acceleration of flow in a
steady
flow field that the flow field is steady,
but there is acceleration. It is very much
possible
because if it is steady only the first term
will be 0, but there, but if the velocity
components change with position then the remaining
terms may not be 0. So, this is the
like; these are certain contradictions that
you will first face when you compare it with
Lagrangian mechanics. In Lagrangian mechanics
if there is something which does not
change with time its time derivative is obviously,
0.
But here even if it does not change with time
the total derivative is it may not be 0. On
the other hand, it may be possible that it
is changing with time at a given location,
but
acceleration is 0 because I mean in a very
hypothetical case it may so happen that the
local component of acceleration say it is
10 meter per second square convective
component is minus 10 meter per second square.
So, the some of that to be 0, but
individually each or not 0; that means, it
is possible to have a time dependent velocity
field, but 0 acceleration.
And it is possible to have a non 0 acceleration
even if you have time independent
velocity field. So, these are certain contrasting
observations from the straight forward
Lagrangian description. So, you can write
the x component of acceleration in these way
and I believe it will be possible for you
to write the y and z components which are
very
straight forward and you have to keep in mind
that when you write y component this D D
t operator will act on v and when you write
the z component it will have act on w. So,
you can write the individual components of
acceleration vector and the vector sum will
give the result and acceleration.
Now, you can write these terms in a somewhere
in a somewhat compact form. So, this
you can also write as v dot del where del
is the operator given by and v is the velocity
vector you know that is u i plus v j plus
w k. So, if you clearly make a dot product
of
these 2, we will see that this expression
will follow. So, it is the compact vector
calculus
notation of writing the convective component
of the derivative. So, we have got a picture
of what is the acceleration of flow how we
describe acceleration of flow in terms of
a
expressions through simple Cartesian notations
and maybe also through vector notations.
Thank you very much.
