Good morning, I welcome you all to the session
of fluid mechanics. Well, in the last class
we discussed the oilers equation of motion
or oilers equation that is the equation of
motion for an ideal flow, flow of an ideal
fluid with respect to cartesian and cylindrical
polar coordinate system.
So, we derived from the principle of conservation
of momentum applied to a fluid system, and
we considered the fluid to be an ideal fluid
that viscous force was not present in the
flow of fluid, and then derived the equation
for motion that is the conservation of momentum
with respect to a cartesian coordinate. Then
we converted it to a cylindrical polar coordinate
system. Then we recognized the vector form
of the equation of motion, and we also discussed
this in cylindrical polar coordinate system,
and we also discussed that this can be obtained
also in different coordinate systems. So,
these equations were known as oilers equation
that is the equation of motion for the flow
of an ideal fluid. Now, we will derive the
same equation of motion, that is the oilers
equation with respect to streamline coordinates
or along a streamline.
So, let us see how does it look? That means,
if we consider the flow along a streamline
like this, and we consider a fluid element
in this streamline, let us consider this is
the direction of the streamline. So, at any
point here, we consider a fluid element like
this whose length is for example, delta S
along the streamline, and let us consider
this area is DA or simply A, not DA cross
sectional area is A or del A cross sectional
area is del A small fluid element that is
the cross sectional area perpendicular to
this plane.
Now, this is moving let at a velocity at this
point. The velocity vector is V. I am not
using the vector notation because now V here
is identified as a scalar component because
this is always along the direction. When I
take the streamline in the direction things
become one dimensional. That means the velocity
at this point is along the direction of this
streamline. That means, along the tangent
to this. So, this is the situation.
Now, when we have assumed the flow to be ideal;
that means, the fluid is an ideal fluid, then
the only pressure forces acting on the fluid
is only, sorry surface forces acting on the
fluids are pressure P at the surface and the
downstream surface, it is P plus del P del
S into delta S. That means, the change over
delta S, that is the fluid apart from what
other external forces acting on the fluid
body is its weight that is mass times the
gravity. That means, because of the gravitational
body force field, that is rho into delta A
delta S that is the volume times g.
Let us consider the angle made by this vertical
line that is the line directing the weight
of the body acting the gravitational force,
the line of action of the gravitational force.
That is the vertical line with the tangent
of the direction of the streamline along which
the velocity exists. Let these two be theta.
That means, this angle, let this to be theta.
Then a simple force balance in this direction
of motion can be written as from the equation
of motion or the conservation of momentum,
whatever you tell that mass rho times delta
A delta S. This is the mass times the change
in velocity.
Let us write this as A DV Dt is equal to the
net force acting in this direction, and the
net force acting in this direction is contributed
by the pressure force and the component of
the gravity along this direction. Therefore,
at this point if we write the net force will
be P into delta A in this small elemental
volume minus P plus del P del S into delta
S into delta A, well minus the component of
the gravity is in this direction minus the
rho delta A delta S into g. That is acting
vertically downward component is cos theta.
That is if this be the theta, so this component
will be cos theta. Well, now from a simple
geometry, we can write that if we consider
this height of this element that is this dimension,
the vertical direction these del is delta
z. That means, this is the vertical displacement
from this point to this point or from this
point to this point delta z. Then we can write
cos theta is delta z by delta s. This is this
delta z. That means, this vertical displacement
divided by delta s, the length from a simple
geometry this triangle if we consider a triangle
like that. That means, this is delta s, this
is delta z. That means this vertical displacement
between these two points is delta z delta
s limit of these because theta is changing
from point to point.
If you define theta, the angle of implantation
at any point with the vertical and the direction
tangent to the streamline that will be delta
z delta s. Delta s tends to 0. Physically,
this is defined like that which gets a mathematical
shape as d z d s. That means with the change
in A’s along the streamline, how the z coordinate
is changing? Simple common sense. We can tell
that is the implantation cos theta that is
the angle theta between the vertical and the
direction tangent at any point to the line
is d z d s. Obviously, so if we substitute
this value of cos theta here and make a simplification
u c p del A p, del A cancels del s, del A
del s del A del s del A cancels from both
the sides. Then simply we get rho DV Dt is
equal to minus p del minus del p del s. Sorry,
p cancels out minus del p del s minus rho.
Rho is there, rho g d z d s or we can write
in another form that DV Dt is equal to minus
1 upon rho del p del s minus g d z d s. That
means this is the form of oilers equation
that is a one dimensional type of thing, where
V is in the direction of base along a streamline.
So, s is the coordinate along a streamline
and V is the velocity along the streamline
because velocity will be always along a streamline.
V is the resultant velocity of the fluid particle.
This is the form. This can also be splitted
as a del v del t here. You see that V is a
function of one dimension; V is a function
of base of the coordinate along the streamline
and the time. So, therefore, with simple split
up of these substantial derivatives in terms
of its temporal derivative and the convective
derivative is this. So, therefore, one can
write this is equal to right hand side as
a t del p del s minus g d z d s. So, these
two equations, either this or this at the
oilers equation along a streamline, all right.
Now, I will tell you the most important theorem
in this context which is very important in
solving different problems, physical problems
or applied problems of engineering applications.
This important theorem is known as Reynolds
transport theorem which is very important,
Reynolds transport.
Now, I will come to, probably you can tell
that this is the most important part of the
section which allows us to utilize all the
conservation laws for analyzing fluid flow
problems in practice. So, for the time being,
I just talk the discussion on conservation
of energy. We have discussed conservation
of momentum mass and conservation of momentum.
Before that we discussed the Reynolds transport
theorem. Now, what is Reynolds transport theorem?
Reynolds transport theorem. Now, Reynolds
transport theorem is a very important theorem.
The proof of which is not that important or
I will not do here. If you are interested,
you can see in any book, standard book, but
what is more important is to know the theorem
very correctly. Now, if you see all the physical
laws for example, the laws of conservation
of mass momentum energy, this type of all
physical laws, a thermodynamic law, thermodynamic
second law. All physical laws are basically
stated with respect to a system or you can
tell the birth of all physical laws have taken
place with their implication or explanation
with respect to a system. For example, one
is asked what is the definition of, what is
the statement of Newton’s second law of
motion. What is the conservation of momentum?
That is rate of change of momentum of a system
or a particle.
Particle is a differential concept of a system
or a system is an assemblage of particles.
So, it is the rate of change of momentum of
a system or a particle which is equal to the
force in the same direction. For example,
the conservation of mass is defined at mass
of a system remains unchanged. Nobody can
tell that mass of a control volume remains
unchanged. So, whenever we tell that is conservation
of mass principle, that mass remains unchanged.
It is neither created nor destroyed. Total
mass remains constant. So, the definition
of system is inherent. For example, the conservation
of energy, this energy of a system is equal
to the energy of a system within. It is equal
to the total energy interacted by the net
energy interacted by the system with the surrounding.
So, therefore, the basic laws are first stated
in terms of systems, but it has been found
while analyzing the practical problems or
even the physical problems. For the sake of
academics, it is much easier to go for an
analysis with respect to a control volume.
So, therefore, to apply the principle of conservation
or any physical laws to control volume, one
has to know the statement for that. For example,
if I tell what is the law of conservation
of momentum for a controlled volume, can you
tell the rate of change of momentum for a
controlled volume because control volume is
a fixed control volume memo that I will come
afterwards. Control volume may move even in,
it can move with a change in velocity or with
uniform velocity, but there is another provision
that control volume is fixed in most of the
engineering applications will take a fixed
control volume well.
So, when control volume is fixed, the question
of its movement does not come over, change
of momentum does not come. So, therefore,
the question comes that how would you explain
the Newton’s second law with respect to
a control volume. Similarly, the conservation
of mass for the definition of a control volume
mass of a control volume may not remain fixed.
So, what will be the definition of the conservation
of mass for a control volume? For the conservation
of mass without knowing any theorem, it becomes
very simple. From our simple that mass conservation
is like that the net mass, the mass coming
in minus mass going out is the mass which
is being accumulated within the control volume.
That comes from one simple intuition, but
it may not be so for other physical law. So,
Reynolds transport theorem is a theorem which
helps us to relate or simply we can tell which
relates the statement of any physical law
for a system to that for a control volume.
That means, it relates this again. I am telling
the statement of a physical law for a system
to the statement of the same physical law
for a control volume. How these two are related
is given by the theorem known as Reynolds
transport theorem. So, without giving a proof
here, you can see proof is not that important,
but more important is to know the theorem
very correctly.
So, I write this way. The Reynolds transport
theorem, it tells like that in statement that
rate of change, I write this thing rate of
change of any property within a system or
for a system, better you write rate of change
of property for a system. This rate means
time rate of change. You can write time rate
of change of any property for a system is
equal to the rate of change. The ultimate
outcome of this theorem, rate of change of
the property, the same property rate of change
of the property. Here you write within a control
volume 
plus the net rate of a flux of the property
from the control volume. It is not a simple
intuition. It has to be proved for mass. It
comes from simple intuition, otherwise it
is not.
So, now, I see that what is the definition
that rate of change that is transport theorem
gives us the time rate of change of any property
for a system is equal to time rate of change.
Here also, rate means with respect to time
usually in physical system. Rate means with
respect to time is equal to time rate of change
of the same property within a control volume
plus the net rate of a flux of the property
from the control volume. So, therefore, this
equation relates the statement for a system
to that for a control volume and this is operative
for any property which is for any extensive
property.
So, now if you look, this can be written for
any extensive property. Well, you can see,
so this can be written in a mathematical form
like that if we consider N as the extensive
property and we define it as any property
per unit mass, then this time rate of change
of property for a system, we can write this
definition as like this. Now, DN Dt for in
a mathematical shape of this theorem is like
that system is equal to time rate of change
of the property within the control volume.
If theta is the property per unit mass, this
first term on the right hand side can be written
eta rho d v is small elemental volume. It
is over the entire control volume here. I
write specifically c v, not the v because
this implies the control volume, where this
implies the system plus the net rate of a
flux of the property from the control volume.
You know how it can be written in mathematical
symbol? If theta is the property per unit
mass, it will be what eta into rho v dot.
Now, here I am not bringing n as the unit
vector for a area because N sometimes is confusing.
So, simply I used dA as the vector. That means,
without writing N dA, sorry we are right without
writing N dA to show this as a vector that
dA is the scalar magnitude of this of a small
elemental area, and N is the unit vector along
the normal to the area taken positive directed
outwards. Without taking that I simply use
sometimes, it is used dA as a vector whose
magnitude is the area scalar magnitude, and
direction is along the normal positive directed
outwards. So, this is simply over the entire
control surface c s. Its control volume is
the net rate of a fluid. So, this for the
second term in the right hand side, this is
for the first term in the right hand side.
So, therefore, right hand side is the statement
with respect to control volume, whereas the
left hand side is the statement with respect
to system and to 
control volume.
So, therefore, we can now say mathematical
shape is like that. This is the rate of time,
rate of change of any property N in a system
is this time rate. Sorry, I am very sorry,
del del t. So, time rate of change of the
property within the control volume plus the
net rate of a flux of this property from the
control volume. So, now in steady state, obviously
this term will be 0. So, for a steady state,
DN Dt is equal to eta rho without D, where
eta is the property per unit mass and N is
that property. So, therefore, you see for
a steady state, automatically this will become
0. So, this is the mathematical form of the
equation. Now, let us apply this for different
cases. Now, first of all we want to apply
this for conservation of mass.
Let us consider the conservation of mass,
and let us think in this way that we have
learned the conservation of mass first with
respect to a system. As we learnt it first
at school level that what is that for a system
Dm Dt is equal to 0. We have learned it. Now,
you want to apply it in influx system. The
mass of a system remains unchanged. That means,
this is simply the equation of continuity
if we use it for a system. That means the
conservation of mass applied to a system gives
this equation as simply the equation of continuity.
Now, if we apply the conservation of mass
for a control volume, we take the help of
Reynolds transport theorem. How here n is
the property of the mass. So, therefore, the
value of eta is 1. So, with this concept because
eta is the property per unit mass, we write
this, we will say that rate of change of mass
for a system Dm Dt is equal to del del t.
What is this rho d v over the entire control
volume plus rho v dot. Sorry, dA over the
entire control centre and since by the law
of conservation of mass, this has to be 0.
So, therefore, this plus this is equal to
0 which is well known continuity equation
in integral, that is the time rate of change
of mass within the control volume plus the
net rate of mass. A flux from the control
volume is 0. Yes, a boy at school level can
tell what is use of all these theorems. Everybody
knows that come from the common sense that
if there is the control volume fixed region
is space where the mass is coming in and going
out, the balance mass will be accumulated
by the control volume, but this may not be
so from simple intuition for other laws, so
that we have to invoke or we have to recall
this theorem what is known as Reynolds transport
theorem that relates the change in property
of a system to that for a control volume,
all right. Now, let us see what happens for
the conservation of momentum which is not
so simple conservation of momentum.
Let us write the conservation of momentum.
Well, conservation of momentum, you know momentum
may be linear or angular. Let us consider
the conservation of linear momentum as we
have done for oilers equation. Now, let us
first see that we have learned this conservation
of momentum as we have really learned even
from our e school days with respect to a system
because all the statement explanation for
conservation of momentum or law of motion
is given for a system. What is that? That
is rate of change of momentum for a system
that mass into velocity is equal to the net
force, the total force acting in the same
direction. This is precise. The conservation
of momentum or equation of motion that is
being learnt at the first step because the
basic laws are always expressed in terms of
the system. That means rate of change of momentum
is equal to the force in the same direction
of what of a system or a particle.
So, this is always implied. Sometimes, a very
tough school teacher may deduct marks if you
do not write that. So, rate of change of momentum
of a system or a particle is equal to the
net force total sum of the forces acting in
that direction. This is precisely for a system.
Let us apply for a control volume. Now, then
routine work is like that what is N in this
case is the momentum. Let us denote the momentum
with a vector. So, v is a vector momentum.
So, what is eta? It will be simply the velocity
vector v because by definition, eta is the
property per unit mass. In this equation you
have to know all the nomenclature. So, therefore,
we can simply write this Dt of m D or we can
write this way Dm b Dt for a system. This
is for a system is equal to Dm b Dt for a
system, rather this looks very odd D Dt of
m v this way you write for a system.
All right, it is equal to what del del t.
Now, it is routine mathematical proceed here
control volume eta is v rho. So, we can write
rho g bar d v plus, please any problem you
ask me, control surface eta is v. So, rho
v v dot, this should be little like that this
is a vector and this is also a vector. These
becomes a scalar v dot dA. Simply I am writing
these equations with the value of eta v. So,
this is precisely the Reynolds transport theorem.
That means, precisely the form of the Reynolds
transport theorem for conservation of momentum.
That means, we are using the conservation
of momentum statement for system and control
volume with a equality sign with the help
of the Reynolds transport theorem, which states
physically that the rate of change of momentum
of a system is equal to rate of change of
momentum within the control volume plus flux
within the control volume net rate of momentum.
So, well this is the precise definition of
the Reynolds transport theorem for the equation
of motion.
Now, what is this is nothing, but the sigma
F. So, therefore, sigma F becomes equal to
I am writing again this thing because this
is so important. Control volume rho v d v
plus control surface v b dot dA. So, this
is what is known as momentum theorem. That
means, this is what is very important known
as, please rho is missing. Is rho correct?
This is what is known as momentum theorem.
This is what is known as momentum theorem.
That means, this is the equation of motion
or conservation of momentum applied to a control
volume. So, now onwards, we will recall this
formula that the rate of change of momentum
within the control volume plus the net rate
of momentum a flux across the control surface,
that is from the control volume across the
control surface is equal to the net force
acting in the same direction for which momentum
you will take is equal to the net force acting
on the control volume in that direction.
Similarly, we can derive this for conservation
of momentum angular. Similarly, conservation
of momentum we can do it for angular momentum,
simple. In case of angular momentum, if we
exploit this equation what is our N is m times
mass times, you know the angular momentum
is defined in this fashion that this is the
angular momentum of any particle or any fluid
element about any point or perpendicular axis
at a point is the cross product of the radius
vector. That means, if you join these two
points, then we get a radius vector and the
velocity vector. That means, physically it
is the distance of that particular point from
the point about which the angular momentum
is considered times the velocity component
perpendicular to this radius vector which
is denoted as r cross v. It may be v cross
r. V sign changed. That depends upon the sign
convention r cos v. This you know how the
angular momentum is specified. So, eta will
be r cross v.
So, this is the only thing. Then again we
write this thing in terms of the D Dt of r
cross v for a system is equal to del del t
eta is 1. That means, first I write rho r
cross v, this d v well plus vertices over
the control volume. So, what the control surface
rho r cross v into v dot dA. So, therefore,
we see that the rate of change of angular
momentum of a system is equal to the b is
missing. Where is b missing? Del t of Del
del t D Dt of m is missing.
M is missing del m is missing, naught v is
missing, m is missing, yes. So, correct. Very
good. Now, try to understand. So, therefore,
this is equal to simply the rate of change
of angular momentum within the control volume
plus net rate of angular momentum from the
control flux control surface, all right. Now,
again in the similar fashion we have learned
the angle conservation of angular momentum
for a system first and what is this quantity.
Torque. Very good. The rate of change of angular
momentum to a system is torque. So, therefore,
first plus second. So, this is the angular
momentum theorem. That means, the torque applied
to a control volume must be equal to the time
rate of change of angular momentum within
the control volume plus the net rate of a
flux of the angular momentum from the control
volume. Obviously, for the steady case, this
part will be 0. Similarly, for the steady
case for angular conservation of linear momentum,
this part will be 0. The first part will always
be 0 for the steady state.
So, these are very important theorem, angular
momentum theorem and the linear momentum theorem
which will be discussed in analyzing the problem.
Now, I will come to the conservation of energy.
Now, conservation of energy again if you recall,
the conservation of energy if we first start
with the a system with the definition with
respect to a system, we know the conservation
of energy for a system is first given by the
first law of thermodynamics. As we have already
read at school at first year class that conservation
of energy and the first law of thermodynamic
is synonymous. What is that for A system?
Now, conservation of energy tells that the
total energy in universe or in an isolated
system which does not interact with the surrounding
means anything external to the system with
the surrounding. If we consider universe as
an isolated system, then what universe is
used in the physical science in a lose sense
of an isolated system that it does not interact
with anything outside. That means in universe
or in an isolated system, total energy remains
constant. So, there are many systems within
the universe or an isolated system, they are
interacting with the surroundings.
So, therefore, if we define its conservation
of energy for an interactive system, system
interacting with the surrounding means everything
external to the system. There may be number
of systems which are external to a particular
system on which we are concentrating our attention.
So, therefore, we can tell that for a system
interacting with surrounding, the conservation
of energy is obvious and it comes again from
the physical intuition from our school days
that the total energy remains constant. That
means, the net energy interaction by the system
with the surrounding must balance with the
energy contained in the system with the energy
contained in the system, and as you know or
you will read afterwards in more detail that
thermodynamic, first law gives the two distinct
status of energy. One is the energy transit
which always occurs in transit. For example,
heat and work and another is, the energy which
is contained or stored in a system. It is
known as internal energy.
So, therefore, the total energy interaction
by a system in the form of heat and work with
the surrounding must be balanced by a corresponding
amount of internal energy stored in the system.
That means, in a process if system exchanges
energy and it gives energy to the surrounding,
so its internal energy will be decreased or
it takes energy from the surrounding in a
net energy interaction process. Its internal
energy will be increased.
So, therefore, the first law of thermodynamics
for a system which is the law of conservation
of energy applied to a system which you learnt
at the basic level can be written like that.
Please see that if there is a system and if
we consider in conventional positive sense
that amount of heat delta Q is added to a
system and the system torque and work delta
W and by doing. So, if the system goes for
a change in energy delta E, internal energy
E, then from the first law of thermodynamics
or the law of conservation of energy whatever
you call delta Q minus delta W, this is the
net energy that system has gain mass b equal
to the increasing delta E.
So, this is the conservation of energy in
thermodynamics. You will see that all kinds
of energy interactions is divided into heat
and work. This work takes care of all kind
of energy interaction apart from heat. That
means, this will be mechanical work, this
will be magnetic work, this will be electrical
work, all kind of work. So, heat and work,
this difference is equal to the energy change
in internal energy. That means, heat added
to a system minus the work done by a system
is equal to delta for a finite process. You
can express this as differential expression
as well this is dE, rather we call it as dE,
because differential is dE. This is the finite
delta E finite change it call it dE Q minus
W is dE.
Now, on a rate basis if you write, we are
more interested on a rate that is the rate
of change of Q for a change in. That means,
if delta Q is added for an interval of delta
t and delta W for an interval of delta t,
so it is the rate of heat addition and this
is the rate of work done. As you know that
heat and work are the path function. They
are not the point function. We cannot write
in a strict differential sense. So, del Q
del t minus delta w, the rate of heat added
and the rate of work added work done. That
means, delta Q for a time delta t delta W
for a time. So, this is equal to d E d t which
simply can be written in a differential manner
with the real spirit of differential from
mathematics. That means the rate of change
of internal energy because internal energy
is a point function.
So, this is the time rate basis the first
law applied to a system. Now, if I use the
Reynolds transport theorem again, if we recall
the Reynolds transport theorem, what is this
Reynolds transport theorem that D and Dt again
if we write now Reynolds transport theorem.
So, DE Dt, sorry big D. Usually D followed
DE Dt for a system. What is this del del t
of control? What will be eta? Let e where
e is the at internal energy per unit mass
without knowing express it form of the energy
in terms of other parameters. You cannot write
anything either for e or e. So, we define
small e as the counter part of big E per unit
mass. It is a total energy, total internal
energy per unit mass. So, therefore, this
simply becomes equal to d v plus what is this
please. Tell me any problem. Eta is e rho,
absolutely simple. So, this is that means
the right hand side represents that the rate
of change of internal energy within a control
volume plus net rate of a flux of energy from
the control volume across its control surface
is equal to d E d t for a system. So, what
is DE Dt for a system? Open it because what
we have learnt first, so we see my way of
approach first. We see what we have learnt
for any conservation law with respect to a
system then using the Reynolds transport theorem.
So, this is replaced by del Q del t minus
del w del t is equal to del del t of c v e
rho d v plus c s rho vita dA.
So, this is precisely then the conservation
of energy applied to a control volume which
tells that the heat added to a control volume
at the rate basis, the rate of heat added
to a control volume minus the rate of work
done from a control volume is equal to the
rate of change of energy within the control
volume plus the net rate of a flux of energy
from the control volume across the control
surface. This is known as a general energy
equation for a control volume general energy
equation for a control volume.
Now, next task is that this energy equation
is well known for a control volume. Next tas
task is to recognize what is the internal
energy E here because here also I have forgotten
to write E E has been. So, what is this E?
That means, what is the internal energy which
is associated with a flowing particle because
this is going out when it is going across
the control surface. So, our main task is
now to recognize the internal energy. So,
after this mathematics, I think we should
now physically try to recognize one of the
different forms that a fluid element or a
fluid particle possess when it is in flowing
codes of flow because if you consider a control
volume, so definitely when the fluid particle
crosses, its control surface comes into control
volume. It is some energy within the control
volume.
Similarly, when it is going out of a fluid
particle or lamp of fluid system going out
for the control volume, it takes away energy
from the control volume along with its flow.
So, therefore, now as it is essential to recognize
what are the different forms of energy that
a fluid particle possess or a fluid element
possess by virtue of its flow or in codes
of its flow. What are those energy’s? Can
you tell what are those energies? One is the
first for any system under any conditions
the energy possessed by it is the intermolecular
energy that is the energy by virtue of the
kinetic and potential energies of molecules
comprising that sustain. So, one is the intermolecular
energy, another energy is the kinetic energy
this is because of motion. So, when any particle
is in motion whether it is solid particle
or fluid particle by virtue of its motion,
it has energy which is the kinetic energy
whose magnitude is given by mass time, the
square of velocity by 2 m b square by 2, that
is the potential energy and another one is
the, sorry which is kinetic energy and another
one is the potential energy.
What is the potential energy? Please define
what is potential energy. So much of potential
energy you have. Read at school level. What
is potential energy? Yes, very correct. It
is the energy which a particle or a system
or any element possesses by virtue of its
position in a force. What type of force by
virtue of a conservative force field? Yes,
by virtue of its position in a conservative
force because work is done to place the body
in at a particular position in a conservative
body force speed if it is the only gravity
as the conservative body force speed will...
Now, the value on the potential energy is
m g h. If we consider h is the height from
any arbitrary datum level, where we take the
energy to be 0. That means, from that level
to bring the particle at a level height h,
the work in g h is being done. So, this is
the potential energy. Apart from that, there
is another energy known as flow work or pressure
energy which I will discuss in the next class.
Time is up. So, we just finish it here for
today. So, next we will be discussing another
form of energy along with these three forms
which appears for a flowing particle of fluid
or a flowing element of fluid, that is pressure
energy or flow one. This I will discuss in
the next class.
Thank you.
