We will now consider what are commonly called
as equations of fluid motion. You can call
it that the governing equations of fluid motion,
eventually the governing laws are nothing
but the statement of the physical conservation
laws, one of these physical conservation law
we have already considered and discussed,
which talk about conservation of mass and
we have derived the mass conservation equation
or equation of continuity.
Just to remind you once again, mass conservation
that is the first equation and if you remember
that we consider two three approaches to derive
this equation, in which in one case we considered
a control volume and then found how much mass
is entering that control volume and leaving
the control volume, and the difference of
that is a rate of change of mass within that
control volume. For that we consider say a
cubic control volume and you consider each
phase how much mass is entering, that is the
way we did it one case, we also used a straight
forward vectorial approach by which we found
out the total flux of mass out of that of
control volume, which again we equated with
the rate of change of volume rate of change
of mass within the control volume.
And the third alternative that we considered,
that we consider a material volume that is
a volume which is not fixed in space, but
a volume consisting of or comprising of certain
amount of fluid, then you said that the volume
of that material element can change because
of the movement of the boundary, how much
the boundary is moving, and that is the way
we found out the rate of expansion or dilatation
of course, the equation is same in all the
approaches and the equation to remind you
once again is d rho d t plus divergence of
rho u equal to 0, or can be written as like
this d rho d t plus rho divergence of u equal
to 0.
In index notation this also we wrote that
d rho d t plus rho d u i d x i, so this is
a different form of conservation of mass,
popularly known as equation of continuity
or continuity equation, and from here we said
that for an incompressible flow in which the
change in temperature is also not large, change
in temperature is also small so that density
remain constant, the equation of continuity
simply becomes divergence of u equal to 0,
or divergence of u is the dilatation or rate
of expansion so rate of expansion is 0, is
the form of continuity equation for incompressible
fluid.
So, for incompressible case write… in particular
I will ask you to recall that
we also defined that rate of change of the
material volume, rate of change of material
volume tau, tau is the material volume that
is volume comprising of certain amount of
metals, certain amount of fluid, not just
a volume in space, is I am sorry I am sorry
not divergence, or in terms of divergence
sorry write both, in terms of the movement
of the boundary where this s is the boundary
of this same material volume. So, this is
the amount by which the boundary is moving,
which can be written as… And you defined
that rate of dilatation, rate of change of
material volume per unit volume is divergence
of u, rate of change of volume per unit volume
is the rate of expansion so divergence of
u is the expansion or dilatation which of
course, you have done earlier, we need this
for our next equation so that is why I just
wanted to recall this.
The second conservation or which now we said
to derive is the conservation of momentum,
which is of course, known to you in the form
of Newton’s second law, see eventually that
is what gives the so called equation of motion
or governing equation conservation of momentum,
what it says that it relates the rate of change
of momentum with the force acting.
Now, when we discuss about rigid body mechanics
these are more or less obvious the rate of
change of momentum. Rate of change of momentum
is what? Obviously of the particle or the
body, in rigid but mechanics there is no ambiguity,
but when you come to fluid mechanics, because
of our nature of description of the fluid
motion, we have little ambiguity, rate of
change of momentum of what, so you have to
be very clear that this is the rate of change
of momentum of the material, rate of change
of momentum so in case we are using the eulerian
description, where everything we fixed in
space a volume a fixed volume in space, the
material is not fixed the volume is fixed.
So obviously the since the material is not
fixed the rate of change of momentum of that
volume is really not applicable in this context,
what we need is rate of change of momentum
of certain material fluid and similarly, the
force acting on it, so let us consider a small
or infinitesimal material volume, infinitesimal
material volume then what will be the momentum
associated with it. Let us consider a material
volume 
material volume element we are calling it
delta tau, so how much is the momentum associated
with this material. Rho u delta tau for all
the mass of this material element is rho delta
tau, and so momentum is into u of course,
integrated over this entire volume oh sorry
rho u d t integrated over that volume element
delta tau.
Now, the rate of change of this 
however we already mentioned that see this
description of fluid motion based on material
is not very convenient, is not very suitable,
the description based on the control volume
approach or point approach field approach
is more convenient and is more widely used.
So, this will now shift to that control volume
or volume based approach, and we see how that
be obtained, this we can write this u the
way we have defined is that eulerian description,
the velocity is described by eulerian description,
that velocity associated at a point, so this
derivative is basically the material derivative,
here we have derived it on the basis of the
material, so to look for the derivative of
the velocity this is basically the material
derivative, and we will use that same notation,
that this is now become this, is that all,
no you see even this the material volume that
is also changing, the material volume that
is also changing.
So, we have even this integration, and now
replace this by using that relation here,
using this by the rate of definition of rate
function, this we 
can further write sorry… And now see the
second term is 0, this term is 0 by equation
of continuity. The second term is 0 from the
equation of continuity, so we have only…
please look to this particularly carefully,
because this is a question of perhaps again
we will hear from you many times that in this
the density is out of the derivative term,
it is perhaps again you will say that this
is valid when the density is constant, or
say for incompressible flow, what you see
it is not so, we have started here where the
density is taken within the derivative, only
it has become like this, so here even though
this density is outside this derivative sign
we have never considered density to be constant,
it has become 0, because of this continuity
equation, and this continuity equation you
can see is valid when density is variable,
because in our final equation this is what
will remain, where you will not see that the
density is being differentiated throughout
the equation.
So, most of the time you try to believe or
you think to or you think that this is an
equation in which density is taken as constant,
but no see the density is been taken as a
variable, only because of this continuity
equation this has taken this form where density
has come out of the derivative. This rate
of change of momentum as you know is balanced
by the forces that are acting on the fluid
element, we have already discussed a great
deal about the forces that acts on a fluid
element, and what are these in general in
a specific case we will have specific forces,
but in general case the general forces, the
body forces and surface forces.
So, you can write what is a body forces acting
on the fluid, that can straight away be written
body forces acting on the fluid, once again
if we follow the same notation that f is the
body force per unit mass, f is the body force
per unit mass then the total body force 
then the total body 
force is… so the most general form of the
equation of motion, however you see that the
equation in this form is perhaps not that
useful unless we know little more about the
stress tensors, particularly how the stress
tensor is related to the motion, how the stress
tensor is related to the motion, unless we
can specify that this equation cannot be used
this equation cannot be used.
As far as the body force is concerned you
see the most common cases the most common
case is the body force is the gravitational
force where you can write that f i is the
g, gravitational acceleration, in some problem
if there is different type of body forces
usually you will be able to write what is
that body forces, if it is electromagnetic
body forces it will come with a cross product
with the velocity and magnetic field, electric
field multiplied by the charge.
So, that sort of force stamps will come if
there is any other type of body forces like
electromagnetic or if you have some centrifugal
type of fictitious body forces that also you
can write in a in a specific case, but we
must know something about this stress tensor,
how it is related with the motion in general,
until we can do it there is no question of
solving this or doing anything with this equation.
However, before we go to look for something
about this stress tensor, we will try to write
this equation in another form, we try to write
this equation in another form known as an
integral form of course, we can say that we
have you have already derived in integral
form, but we will now do it whether we will
write this equation in another integral form.
See the first term here this d u i d t as
you know that this material derivative can
be written in this form think about a jet
coming out of a nozzle, let us say a nozzle
from this a jet of fluid is coming out, and
this jet of fluid is heating a plate, as you
can understand that this jet of fluid will
try to displace this plate it will act exact
certain force on it.
So, if you want to keep it fixed you have
to apply the opposite type of force, how much
is that force? consider say the diameter is
10 centimeter and the velocity here is just
8 meter per second, the velocity it has only
1 component of velocity the x component we
will call it say u 1 8 meter per second, you
can consider a control volume like this, and
apply this momentum and apply this momentum
equation in integral form, and you will get
the force or the solution straight away, in
this problem of course, we can neglect all
these viscous stresses, sorry we will we should
not call viscous stage, at this stage that
stress there is no stress acting on it, what
happen when this jet heats the plate? you
know whether jet speeds into up it follows
the plate, so there will be flow in this here
after coming here, when it heats the plate
it can no longer go in the straight direction
so it will go up and down following the plate.
Cannot field particles rebound field particles
can rebound also .
No see in this case rebound means with the
they will be again carried this flow is continuous
know, so they cannot come back this way, this
flow is going continuously so a particle cannot
come back wait, it will be again taken away
by other particles, so in this problem see
this i and j cannot take 1, 2 3, we will treat
this problem as a 2 dimensional problem, we
will treat this problem as a 2 dimensional
problem. So, i and j both can take 1 and 2,
and eventually that rho into u 1 that gives
the in to area of course, gives the mass flow
rate, the mass flow rate rho e 1 Into this
cross sectional area that will give you the
mass flow rate.
Yes see the force acting on the fluid, the
force acting on the fluid can be obtained
from this just by writing the left hand side
term only, the right hand side whatever is
there that is the force acting on the fluid.
So, that can be obtain by writing only, the
left hand side terms only so how much is that
rho u 1 a, now this u 2 that is the velocity
in this direction there is of course, what
in that there are both terms upward as well
as downward.
However, both are same, yes both are same
yes you take this area of course, here is
an assumption that the fluid is split in two
half exactly, but of course, there is no reason
why it should not be, anything else, yes also
in this direction is there rho u 1 a u 1,
what will be the sign of this term, if we
consider force acting on the fluid then it
should be negative, if we 
consider 
force 
acting on 
the plate then it 
is positive. So, actually this is 
what is 
the term ultimately, this is what is giving
us everything, so force on the plate. u 1
and u 2.
.
Sorry.
.
This is 1 is in the 
upward direction the other is in the downward
direction.
Sir why cannot we take a .
I perhaps not following u.
Sir we took all the combinations u 1 u 2.
U 1 u 2 again in the u 1 u 1.
Why did not we take u 2 u 2.
U 2 u 2.
Yeah.
No see this part if you write this equation
this i in the direction of x we are writing
only in the force in the x direction, force
writing in x direction, this i actually gives
the direction, as you can see this j is summed,
j is repeating so j mean this it will be sum
over j so actually this i will remain in the
final equation, in each terms you see this
i will remain so this is an equation in the
i direction, you are writing here the equation
in the x direction or the first 1 direction.
I is the potential function .
psi
.
No this is not velocity potential in this
case this is the.
.
This is the we said that the force is a potential
force so it is the potential force.
.
So, in this case this psi do not treat it
as a streamline, please do not give much weight
age to the symbol itself, look what that symbol
stands for, that whenever there is a psi;
the psi is streamline no, in this case the
psi is not streamline it is that force potential,
so there it is a potential energy per unit
mass, the numerical figures are not important
that can be obtained.
So, see this is the utility of this integral
form, in any situation wherever you will be
able to apply integral form, you see that
the answer is obtained almost without any
effect, but as already mentioned that you
will not get the details of the flow picture,
you will get the important information without
getting the details, this is of course, one
simple example but there are many applications,
you can consult any standard text book on
fluid dynamics, and they will give you the
application and momentum principle for various
numerical problems, perhaps we will also give
you a few more problems as tutorial problems
which you may try. Then I think we will stop
and discuss about the stress tensor in the
next class.
