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 time line. 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 some way.
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 time line, so it is nothing
but like snapshot of nearby fluid particles
at a given instant of time that is called
a time line. So, let us look into a small
movie to see that what we mean by a time line.
So, if you see now this gives snapshot at
different instants of time of nearby fluid
particles and in a way, it gives a sense of
the velocity profiles at different instants
of time you can see that in this example,
the flow passage is narrowing and as the 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 maybe a differential form or an integral
form.
But at least, this gives us a visual idea
of what the time line 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 streamline,
streak line path line or maybe the time line.
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 a fluid particle located
at a position P at specifically the location
P1 at time = t and how the velocity is described
here; the velocity is described here through
a velocity vector v, which is a function of
r1 that is the position vector of the point
P1 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 may 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 P1, 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
us say that at a time of t + delta t, these
things get changed.
Now, at a time t + 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 P2. So,
at the point P2, now let us say that the velocity
is whatever at some arbitrary velocity, so
initially it may be velocity at the point
1, say v1, now it is v2, which is again a
function of its local position and time.
So, you have this v2, this one a function
of what; so, let us say that it is given by
its components u + delta u, v + delta v, w
+ delta w, these are functions of what? These
are functions of the new position vector,
the new position vector say is r1 + delta
r1, so in terms of scalar components, it may
be x + delta x, y + delta y, z + delta z and
the time has also now changed, it has become
t + 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 which 3 components.
And if we 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,
all the basic definitions in Newtonian mechanics
that we have learnt earlier are based on Lagrangian
mechanics. So, when you say that it is a 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 P1, now
is at P2. So, if we want to find out the change,
so you can write of course, you can write
it in terms of the 3 different components
but just for simplicity, let us write for
the x component, similar things will be there
for y and z component. So, how can you write
u + delta u as a function of u.
So, u + 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 u? That
was u +; 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,
you have to keep in mind that now you are
having 4 variables. So, let us first consider
the time variable may be because it is bit
different in characteristic than the earlier
one, so this is with regard to the time, then
with regard to that space okay, 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, 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 think what you can do, you can cancel
u from both sides and what is the definition
of acceleration along x from a particle mechanics
viewpoint or a Lagrangian viewpoint?
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.
So, 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 a limit is taken as delta t
tends to 0.
So, the first term is straightforward, let
us look into the next terms, so first we will
evaluate the limit; 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 these 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, you 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, P1 and P2 are almost coincident, right
that means, let us say that P1, P2 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 straightforward 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 that transformation in a very simple
way that as the delta t tends to 0, Eulerian
and Lagrangian description should coincide.
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 component of the fluid particle located
at P, since you are focusing our attention
on P 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 very straightforward
understanding it conceptually is not that
trivial and straightforward, if 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
description and this is Eulerian 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 our attention
at that particular point, this replaces the
velocity of the fluid particle, if the fluid
particle is neutrally buoyant with the flow,
then it is like an inert tracer 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 is neutrally
buoyant then it becomes same as u of the flow
field. If it is an inert tracer particle 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
we have to keep in mind.
So, if you complete this description of this
term what we will get; you will get ax is
=; that is the straightforward 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 use 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, another gives the spatial
derivative.
You will see that this expression 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 these 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 flow field, so when
you go there, there are 2 ways by which your
velocity gets changed, how? One is maybe 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 a solely time dependent
phenomenon. How can you understand what is
the component of the time dependent phenomenon?
If you did not move to 2 but say you confine
yourself to 1, say you are not moving with
the flow field, you are confining yourself
with 1, then you are freezing your position
but still at the point 1, there may be a change
in velocity because of change in time, if
it is an unsteady flow field. So, because
of that it might be having an 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 it is 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 a change in position, it might
be having an acceleration, so that acceleration
is not directly because of the time rate of
change of velocity at a given point but because
of the spatial change, since the particle;
the fluid particle by the time has travels
to a different location, where it finds a
different flow field.
And since, we are considering that it is an
inner tracer particle, it has to have the
same velocity locally as that is there in
a new position, so because; so the next combination
of terms, it represents the change in velocity
solely due to change in position, so the total
the net change is because of 2 things; one
is if you keep position fixed and you just
change time because of unsteadiness in the
flow field, there may be an acceleration.
The other part is even if the flow field is
steady but you go to a different point because
of non uniformity because of a 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 may be because
of 2 reasons; 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 on.
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 different velocity and the resultant
acceleration is the combination of these 2.
So, let us; let us take a very simple example
to understand it. Say, you are traveling 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 2,
2 and 1/2 hours you reach Bombay and you find
that it is a very sunny weather. So, the question
is now if you want to ask yourself a question,
does it mean that when you departed from Calcutta
it was raining in Bombay or when you departed
from Calcutta, it was sunny at Bombay or when
you have reached Bombay, is it still raining
at Calcutta or is it still sunny at Calcutta.
It is not possible to give an answer to any
one of these because the net effect that you
have seen is a combination of 2 things, you
have traversed with respect to time, so you
have elapsed 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 migrated
to a different location and because of the
change in location maybe it was before 2 years
raining at Bombay, now it is sunny or it might
so happen that it was sunny 2 hours back in
Bombay, still it is sunny, so you can see
that individual effects, you can maybe 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
a change because of position and because of
time and that is why, this ax or maybe ay
or az, this is called as the total derivative
of velocity. So, it is given a special symbol
capital D Dt.
So, capital D Dt has a special meaning, it
is called as total derivative it is to emphasize
that it is a resultant 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, so these are certain names, which
are given.
Again by the name, local it is 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 a 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 a 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 we ask a question, is it possible that
there is an acceleration of flow in an in
a steady flow field that the flow field is
steady but there is an acceleration, it is
very much possible because if it is steady
only the first term will be 0 but if the velocity
components change with position, then the
remaining terms may not be 0. So, this is
of 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 - 10 meter
per second square so the sum of that 2 is
0.
But individually, each are not 0 that means,
it is possible to have a time dependent velocity
field but zero acceleration and it is possible
to have a non zero acceleration even if you
have a time independent velocity field, so
these are certain contrasting observations
from the straightforward Lagrangian description.
So, you can write the x component of acceleration
in this way and I believe it will be possible
for you to write the y and z components, which
are very straightforward.
And you have to keep in mind that when you
write y component this D Dt operator will
act on v and when you write the z component,
it will act on w, so you can write the individual
components of acceleration vector and the
vector sum will give the resultant acceleration.
Now, you can write these terms in a somewhat
compact form, so this you can also write as
v dot del, where del is the operator given
by; okay.
And v is the velocity vector you know that
is ui + vj + wk, so if you clearly make a
dot product of these 2, you will see that
this expression will fall. So, it is a compact
vector calculus notation of writing the convective
component of the derivative okay. So, we have
got a picture of what is the acceleration
of flow, how we describe acceleration of flow
in terms of expressions through simple Cartesian
notations and maybe also through vector notations.
Next, what we will do; we will start analysing
the deformation of fluid elements. Why this
is very important? Because we have seen that
fluids are characterized by deformation, they
undergo continuous deformation on the under
the action of even a very small shear force
and the relationship between the shear force
and or the shear stress and the rate of deformation
is something, which is unique to the constitutive
behaviour of different fluids.
So, we must first understand that how to characterize
deformation of fluid elements in terms of
the velocity components. Once, we understand
that it will be possible for us to mathematically
express different types of deformations in
terms of the velocity components the uv and
w. When we do that we have to keep in mind
that we will be essentially bothering about
2 types of deformations; one is the linear
another is the angular deformation.
When we talk about the linear deformation,
it may eventually give rise to a changing
volume of the fluid element also because if
you have a length element and the length element
gets changed, a volume element is comprising
of several such length elements, so if length
linear dimension gets change, the volume is
also likely to get changed. Initially, we
will think of how we can say, estimate the
linear deformation.
So, we will start with the linear deformation,
to understand or to get a visual feel, we
will consider a fluid element like this, maybe
we may consider even a 3 dimensional fluid
element, if you want but that will not make
the thing more complicated because at the
end, we will be dealing with linear deformations
in individual directions. See, why we use
a coordinate system for analysing a problem?
The reason is like say, when you think of
x, y, z the Cartesian coordinate systems,
these are independent coordinates, a combination
of which describe the total effect in the
system. So, when you are thinking of a linear
deformation along x, you may be decoupled
from what is the linear deformation along
y and z and these individual effects, you
can superimpose because you are dealing with
linearly independent components.
And these vector components actually give
you linearly independent basis vectors like
components along x, y or z. So, similar concept
whenever, we are considering a change along
x, maybe we are bothered only with respect
to; like what is the change in the linear
dimension along x, disregarding what happens
along y and z. So, let us keep that target,
let us say that delta x is the length of the
fluid element, which is originally there.
And now, what is happening; now, we are having
a change in time and because of a change in
time now, you see that let us, consider the
front phase of this cuboid, so this left phase
over a time interval of delta t will traverse
a displacement; will undergo a displacement.
What is the displacement? If u is the velocity
at this location simply, u * delta t, we are
considering the time interval delta t to be
very small, so it is like just a product of
the velocity into delta t.
The right phase will also undergo some displacement,
what is that? So, if this is the x direction,
the new u here is not the same as the u at
the left phase but this is because of change
in u due to change in x, so this is the new
u times delta t. So, if you consider only
the front phase and only subjected to this
motion, say we freeze all other events just
for a clear picture, so maybe now, it is having
a new configuration shown by this dotted line.
So, what is the change in its length along
x; that is the final length minus the original
length, so what is the final length, so what
is the net change? See, the right hand phase
has got displaced by this amount, the left
hand phase has got displaced by this amount,
so the net displacement is the difference
between these 2, so what is that change in
length?
So, change in length along x, this is the
change in length along x. What is therefore
the strain along x? The change in length per
unit length, so the strain along x; this is
the elemental strain because we have considered
only a small part of the fluid, which is having
an extent of delta x, so this is elemental
strain along x. As we have discussed earlier,
we are not just interested about the strain
for a fluid.
Because if you allow it to grow in time, the
strain will be more, see if this delta t is
larger and larger and you integrate it over
a large interval of time, this will be trivially
more and more, so measuring strain in a fluid
is nothing that is important, it is just a
function of the time that is elapsed. What
is more important is the rate of deformation
or the rate of strain. So, the rate of strain
along x, what is that; it is basically this
divided by delta t as delta t tends to 0.
So, when we say rate that means, the time
rate, we always implicitly mean that so that
will be simply the partial derivative of u
with respect to x. We may give it a shorthand
notation say, epsilon dot x, similarly just
from your common sense, you can say what will
be epsilon dot y and what will be epsilon
dot z. So, what is epsilon dot y and epsilon
dot z is this, so we have been successful
in finding out a very simple thing, what is
the rate of linear deformation along x y and
z in terms of the velocity component.
So, if you are given u as a function of position,
v as a function of position and w as a function
of position by simple partial differentiation,
it will be possible to find out the rates
of change. Now, we are interested not only
just in terms of the rate of change in the
linear dimension but maybe rate of change
in the volume. So, to understand that what
is the rate of change in the volume; let us
say, that we are having this fluid element,
which has dimensions along x, y, z as delta
x, delta y and delta z.
So, we set up coordinate axis as this is x,
this is y and this is z, so delta x, delta
y and delta z. Now, what is the new length?
So, we are interested to get the new volume,
so what is the new volume? The new volume
is new length along x * new length along y
* new length along z.
So, what is the new length along x? That is
the old length plus the change. So, the old
length is delta x + the change is this one,
so we can take delta x common and write this
one. Similarly, it is possible to write what
is new length along y and new length along
z, so let us complete those expressions. So,
the new volume is a product of these 3, so
what is the new volume; minus the old volume,
yes.
So, what you have to do; you have to find
out the product of this then subtract the
old volume that is delta x * delta y * delta
z, you will see that the first the delta x
* delta y * delta z that term will go away,
will get cancelled, then out of the remaining
terms you have to neglect the terms of maybe
higher order in delta, so like if you have
products like delta t square or delta x * delta
t square that type of term you tend to neglect.
Because those are higher order terms, so retain
only the leading order terms because you have
to keep in mind that you are dealing with
a situation again as delta x, delta y, delta
z, delta t, all tending to 0 and then what
will be the term that is remaining here, yes,
so obviously a product of these 3 is there
then then, what is the remaining term; these
plus higher order terms will be there that
into delta t, other terms will be of order
higher than delta t.
So, what is the volumetric strain; the rate
of volumetric strain? So, the rate of volumetric
strain is the change in volume per unit volume
per unit time just like what we found out
for the linear strain. So, when you say find
out per unit volume, you are basically dividing
it by this delta x, delta y * delta z, so
this is like the original volume, let us give
it a symbol V with a strike through, so that
is the original volume.
So, what is the rate of volumetric strain?
This change in volume divided by volume divided
by delta t and take the limit as delta t tends
to 0, when the all other higher order terms
in the limit will be 0, so it is not that
we are neglecting. The one delta t here will
remain even after division by delta t that
will be tending to 0 as in the limit delta
t tends to 0. So, then what will be the final
expression of this?
Some of these 3; so we may write it in terms
of the total derivative, see the volumetric
strain it may be due to many things; change
in time, change in position and a combination,
so we are not bothered about that what is
the individual effect, we are bothered about
the total effect, what is the net change in
the fluid element volume because of this,
so this should be expressible in terms of
the total derivative.
So, it is capital D Dt of the volume with
per unit volume, so this is the rate of volumetric
strain and in terms of the vector calculus
notation, you can also write it as the divergence
of the velocity vector. This leads to a very
important definition; the definition is with
regard to incompressible flow. So, when we
say that a flow is incompressible, so incompressible
flow by the name it is clear that we are looking
for a case, when the fluid element does not
change in volume.
So, incompressible flow 
it will have what signature; one and only
important signature, zero rate of volumetric
strain because the fluid element may not be
changing its volume that is the meaning of;
that is even the literal meaning of incompressible
that you cannot really compress it. So, zero
rate of volumetric strain 
and that boils down to the divergence of the
velocity vector is = 0.
So, if you are given a velocity field and
you are asked to check whether it is compressible
or incompressible flow, then it is possible
to check by looking into the fact, whether
it is satisfying this equation or not. If
it satisfies this equation, we say that it
is an incompressible flow, keep in mind the
distinction between this definition and incompressible
fluid definition. So, earlier we also introduced
the concept of incompressible fluid.
And we said that a fluid is incompressible,
if its density does not significantly change
with change in pressure, so that is incompressible
fluid. Now, we are talking about incompressible
flow and these two are again related but different
concepts that we have to keep in mind. So,
when you are having an incompressible flow,
it is possible to characterize the particular
flow in terms of its mechanism by which it
satisfies the overall conservation of mass.
To understand, how it does let us try to write
an expression for conservation of mass of
the fluid element. So, we will now write conservation
of mass for fluid for a fluid element, let
us say that m is the mass of a fluid element,
you can express it in terms of the density
and the volume, let us say that rho is the
density and v is the volume. Since the mass
is conserved of a fluid element, so there
will be zero rate of change of mass.
So, the; since we know already the expression
for the volumetric strain and in that volumetric
strain 1/v appears, it may be useful to utilize
that expression by taking log of both sides
and then differentiating because then 1/v
will automatically come out. So, let us take
the log of both sides and then differentiate
with respect to time, when we say we want
to differentiate with respect to time, it
has to be a total derivative.
So, because it is a fluid element now, it
may have change with respect to change in
position, time whatever, we are bothered about
now the total effect because the conservation
of mass is not for individual effects, it
is a combination of total effects that gives
rise to a mass of a fluid element is conserved.
So, when we write say, when we differentiate
it with respect to time by keeping that in
mind, we have the left hand side like this,
which again becomes 0.
Because the mass of the fluid element is conserved,
the right hand side 1/rho, so you have 0 = 1/rho
d rho Dt + 1/ v dv Dt is what; is the divergence
of the velocity vector or if you write in
terms of the Cartesian coordinates, when it;
till you write it in a vector form it is coordinate
system independent but when you write its
corresponding say, components then the components
depend on how you take your reference.
So, in a Cartesian reference it is this, now
let us write this one, what will be this;
use the definition of the total derivative,
okay, now you can multiply both sides by rho
because density of the fluid is not 0, so
you can multiply both sides by rho and then
if you multiply both sides by rho, what you
get?
You get okay; now, you can combine these types
of terms and write them in a compact form
by using the product rule of differentiation,
so this equal to 0, will imply; this = 0,
okay, just by using the product rule of differentiation.
It is again possible to write it in a compact
vector notation, this becomes divergence of
rho * velocity vector that is = 0, this equation
in a general understanding is supposed to
be the most fundamental differential equation
in fluid mechanics.
Because no matter how complex or how simple
the flow field is, it should satisfy the law
of conservation of mass, so this is a differential
equation expressing the law of conservation
of mass for a fluid element and this is known
as continuity equation. So, if you are given
a velocity field, you must first check whether
it is satisfying the continuity equation,
if it does not satisfy the continuity equation,
it is an absurd velocity field.
It may be mathematically something but it
does not physically make any sense because
it has to satisfy the mass conservation. Now,
briefly let us look into certain special cases
of these, so what are the special cases?
The first special case, we consider as steady
flow. So, when you consider a steady flow,
then how this equation gets simplified to;
so steady flow means the first term at a given;
remember what is the definition of steady,
at a given position, any fluid property will
not change with time, so at a given position
that is why the partial derivative with respect
to time that means, keeping the position frozen
you are trying to find out the change in density
with respect to time.
So, that is 0, if it is a steady flow, so
steady flow will have that term = 0, so that
will boil down to; but that does not ensure
that rho is a constant; rho might not be a
function of time but might be a function of
position, so still rho remains inside the
derivative, it does not get disturbed. Let
us consider a second case, incompressible
flow. We have to keep in mind that there is
a very big misconcept that we should try to
avoid.
What is that? Many times, we loosely saying
incompressible means density is a constant,
it is a special case of incompressible flow
but it is not a general case of incompressible
flow because general case of incompressible
flow is what; the divergence of velocity vector
= 0 that is the definition. Now, where does
it ensure that rho is a constant that basic
definition never ensures that rho is a constant?
At the same time, it can be shown that if
rho is a constant then this will be satisfied,
so the converse is true that means, rho is
a constant is a special case of incompressible
flow but it is not a general case. What is
the general case, let us look into that. So,
when you are looking about the general case,
you have to see the continuity equation. So,
when you look at the continuity equation,
look into this primitive form that is not
the compact form but this form before that.
So, here when you have an incompressible flow;
which terms will go away; these terms will
go away because divergence of the velocity
vector is 0, so then what you are left with;
the total derivative with respect to; total
derivative of density with respect with time
is 0, so that means incompressible flow means
D Rho Dt; capital D Rho Dt = 0. See, a very
interesting thing, it does not mean that rho
is a constant.
Because rho might be a function of position
and time in such a way that this collection
of terms eventually gives rise to 0, if rho
is a constant, this collection will definitely
give it to be 0 but that is a trivial solution;
that is a trivial solution to the case that
is rho is a constant therefore, any derivative
of rho with respect to time or position is
0 but even if any derivative of rho with respect
to position and time is not 0, still the net
effect may be 0.
And then even though rho is a variable, we
will say that the flow is incompressible,
so a variable density flow may also be an
incompressible flow, this is a very important
concept. So, incompressible flow need not
always be a constant density flow. So, a typical
example is; let us say that you have a domain
like this, within this there, is a fluid.
Now, this fluid changes its phase, say it
was in a particular phase, it was in liquid
phase, now it becomes a vapour phase.
So, when it becomes vapour phase, it becomes
lighter, so the same mass now cannot occupy
this volume, so there is; so it wants to occupy
an extra volume but given a particular volume
what it will do; some extra mass we leave
because you are constraining the volume and
if you are having a change in density, you
must have a flow to accommodate a change in
density, so that whatever fluid is there now
is accommodated within the volume that was
given to you.
So, you can see that you might have a change
in density at a fixed position with time because
maybe with time the phase change has triggered,
so with time the density has changed, so this
has to be now adjusted with some u, v, w,
so that the net effect may still be 0, so
it might so happen that now here, the net
effect it may be 0, it may not be 0, so let
us take an example, where the net effect is
0.
What is that example? Maybe there was a fluid,
now it is getting frozen and because of freezing
its volume gets change, so it is possible,
so its density has got changed but we do not
call say liquids or solids as compressible
fluids, so what has happened because if with
freezing, there is a shrinkage then there
will be a deficit in volume here, maybe to
satisfy the deficit in volume there might
be a material supply from all sides.
So, it is possible that to make a balance
of what is happening locally and what is happening
over the volume element you might have to
adjust these things with velocity across the
different phases of the element. So, in summary
we can say that incompressible flow definition
is the total derivative of rho with respect
to t is 0 but not just rho is a constant,
okay. We will stop here today, we have just
seen one way of deriving the continuity equation
but we will learn more by having different
ways of deriving the continuity equation and
that we will do in the next class. Thank you.
