Good morning. I welcome you all to this session
of fluid mechanics. So, this class I will
start a new section conservation equations
in fluid flow, but before that as usual I
like to have a closure of the earlier section
kinematics of fluid. And before an even prior
to that I like to continue with two more problems.
Because last class, because of due to the
lack of time we solved only one problem, two
very simple problems we go through horribly.
And then with a quick closure of the earlier
section kinematics of fluid, we will start
the new chapter or new section that is conservation
equations in fluid flow.
So, let us go to a problem related to earlier
section that is a kinematics of fluid. So,
just a simple problem the straight for one
applications of our theory, the velocity components
in a two dimensional flow fluid for in compressible
fluid or given by u is this, v is this u is
e to the power x cos hyperbolic of y. And
the y component of velocity minus e to the
power of x sin hyperbolic of x, well determine
the equation of streamline for this flow.
So, in a very straight forward application
for the equation of streamline as you know
the equation of streamline is that dy dx if
you recall for a two dimensional flow field.
The equation of the line, which is the streamline
represents the velocity vector as the tangent
at every point to this line is given by v
by u. So, this is the equation of streamline.
So, fluid mechanics ends here, what remains
is a simple school level a mathematics that
is dy by dx or dy dx as you tell is v by u,
that is minus what is that? Sin hyperbolic
x by cos hyperbolic y. So, if you make it
like this d sin, sin hyperbolic x d x plus
cos hyperbolic y d y is 0.
So, we integrate this, this is the differential
equation if you integrate this you get the
form like this. What is the integration of
sin hyperbolic x dx? Cos hyperbolic x; cos
hyperbolic x, and this will be sin with the
same sin for the hyperbolic function and that
will be a constant integration of 0. So, this
is precisely the equation in the x y plane
for streamline; this is precisely the equation
in x y plane for this streamline. So, this
constant the values of these constants can
be found out. If we define these streamline
that this constant value for a giving value
of x and y, this you will come afterward I
newly introduce the concept of steam function.
So, constantly represent the parameter; that
means a series of cos representing this streamlines
with differ in values of this constants.
See for example, as the parameter is a very
simple straight forward application. Another
straight forward application for well in kinematics
of fluid regarding the fluid motions you see.
A three dimensional velocity field as you
know the I told you yesterday the three dimensional
flow means were the 3 components of velocity
exist and all the 3 components are functions
of x y z all the 3 space co ordinates in general.
So, this is known as a three dimensional flow
field which is given by the u component if
component of velocity of could s get is not
a function of j x y and u 0. This u 0 is a
constant this is define in the problem, y
component of velocity is given by this and
z component of velocity is given by this.
In this equations C w 0 and u 0 these are
the constants C w 0 u 0; these are the constants.
So, variables are x y z as the independent
variable and dependent variable at the velocity
components 0 v w which are function of x y
z. These are study velocity field as you see
because there is no dependence with time.
So, what we have to find out? Find the components
of strain rates and rotational velocities.
It is again a straight forward applications,
of the fluid kinematics that how do you define
the strain rates? Let x the rate x co ordinates
strain rate is a sin on x which is given by
del u del x epsilon on dot y; that means,
the rate of strain in y direction. That means,
these are the gradient of the velocity components
in that direction with respect to the space
co ordinates in that direction. So, it very
simple to remember this strain rate in z direction
is the differential of z direct components
velocity with j. So, it is simple now class
school level thing that del u del x is C.
So, it is C well, what is del v del y? Again
C. So, is a constant strain rate in y direction
del w del z is minus 2 C. So, these are the
values of epsilon on dot x straight forward
application epsilon on dot z.
Next is the, next is the share rate or angular
deformation, rate of angular deformation,
angular strain rates whatever you called.
We know gamma dot x y in the x y plain it
is del v del x plus del u del y similarly,
gamma dot x z x z plain what will be the value
del w del x plus del u del j. And well gamma
dot y z is equal to what gamma dot y z is
del w del y plus del v del z. Now, what is
del v? Del is now straight forward of substitution
it is 0 v is not a function of this. But del
u del y is not 0 it is twice w 0; that means,
it has got a fixed angular strain rates in
x y plane which is twice w 0.
Similarly, for x z plane del w del x del w
del x is 0 since w is not a function of x
del u del z it is also 0; that means, it does
not deal any angular strain, rate of deformation
rate or shear rate these are the terminologies
used in the x z plane. Similarly, if we inspect
these strain angular strain rates in y z plane
del w del y del w del y is 0 del v del z is
0; that means, it has got only strain rates
in x y plane. Now, again if we think of rotation
it is very simple rotation the same application.
But here is sin convention comes that means,
if you take this as x y z probably you recall
I discussed in the class that the omega x
omega dot x; that means the rotation about
the x axis. That means, in the y z plane will
be half the positive sin is this; that means,
del w del w del del w del y minus del v del
z minus del v del z. Then omega dot y will
be similarly, half del u del z minus del w
del x. Well similarly, omega dot x will be
half this was derive in the last class del
v del x minus del v del y. That means, about
any axis it will contain the velocities in
the components not in that then x above rotation
about x axis will contain the z and y components
velocities. And they are cross differential
z component with y component with z with a
minus sign.
So, it can come from the determinant concept
as you know i j k that is del del x again
I am repeating this things del del j u v w.
That means, this is the curl of the velocity
vector this is the definition of half of course,,
they are half of course,. So, from an analytically
you want can find out now the list. But is
to substitute the value accordingly and you
can find out the values that this left to
you a simply substitution of the value as
I did for this strain rate. So, this is the
formulae and you can find out. So, this is
all for the examples which highlight to show
you in this class. Now, I like to give you
close up, close up of the chapter 3 that what
we actually observed that read in chapter
3.
First is the, it is the fluid kinematics first
there are two approaches to describe the fluid
flow. One is the Lagrangian approach which
considers each and every fluid element of
given identity and to trace they are path
in the fluid flow. The identity is fixed by
fixing the position vector of a fluid element
at a given interval of time. So, therefore,
at a give in at particular interval at, they
particular use tend of time particular use
tend of time whereas, Eulerian approach solves
the fluid flow problem by concentrating at
the particular point, and describing the flow
velocities, at that point as a function of
time. So, they are putting an entire flow
field the velocity field is described the
acceleration field is describes that the function
of space co ordinates and time.
So, Eulerian approach describes the velocity,
acceleration an all hydrodynamic parameters
is a continuous function of x y z that t;
that means, the space co ordinates and time.
And this is the most convenient defining of
fluid flow then we recognize what is the study
flow and what is the non study flow. When
all hydrodynamic parameter become independent
of time in a flow fluid the flow is said to
be study, and if he does not do. So, the flow
is a non study flow similarly, a flow is said
to be uniform move in the velocity and accelerations
in a flow fluid are independent of the space
co ordinates they are same at each and every
point in the flow fluid. So, this is known
as the uniform flow the flowing general may
be on study and non uniform.
But any combinations of these four can occur
next we appreciated very important thing that
is the acceleration. The basic thing is like
that when any parameter changes because of
the time and also it due to convection, let
me is a parameter is changing because of the
convection the parameter is convective. So,
the time and the convection both are coupled
to make the change of the parameter with time
because the parameter moves with the flow
fluid. So, therefore, with time it has got
a movement. So, they are put the change composed
of change with rustic to time at a given point.
If you rustic the movement plus the change
along with the convection a simple example
is that if a fluid particle. If you trace
its change in velocity with time then as we
fluid particle moves with time its change
of velocity will depend.
Because, of this movement from 1 position
to other position along with the change in
the velocity field even at the particular
position with time. So, in general therefore,
this is the rule that total derivative with
respect to time contents a temporal derivative
which is the change with respect to time at
a fix point plus the convective derivative.
So, this gives the concept of temporal and
convective acceleration. So, total acceleration
or substantial acceleration consists of two
parts; one is the temporal acceleration which
is the change of velocity at a point with
respect to time.
Because of an stead in, in the flow and another
is the convection that is the change of velocity
even for a study flow for a fluid particle
moving from 1 point to other. Because of the
non uniform me to of the flow. So, they are
fort temporal acceleration plus convective
acceleration is total are substantial acceleration.
Then we recognize stream lines path lines
and straight lines stream lines and imaginary
lines down in a fluid flow. So that the tangent
at any point on this line represent the velocity
direction of the velocity vector at that point.
Now, the series of stream line changes from
to time to time a non study the path line
are the locus of the, differ end fluid particles
with differ and identity and straight lines.
We are defined as the locus of the feet of
the end points of several points several fluid
particle that across a particular points fix
points. So, straight line specified by that
fix point through which a number of particles
have crossed and it give in stent what are
the end points of those particles along they
are locus this is a straight line. In study
flow stream line, path line, straight lines
are coincident. Since the Lagrangian version
Eulerian version become identical then the
most important part and that is the last part
of that section which we a cognize that we
fluid particle in general had 3 distinct features.
One is translation simple translation it translated
without changing its shape without changing
any dimensions linear dimensions of the body
or angular dimensions plus rotation and deformation.
Deformation is the most distinct which able
feature almost important feature that distinct
which is a fluid element from solid element
that fluid in continues motion continuously
defunds is getting defund is shape the shape
get defund continuously. And the dimensions
linear dimensions and angular dimension goes
on changing continuously.
So, fluid elements have got 3 distinct, part
1; translation rotation and deformation where
are the solid body has translation and rotation
without deformation. That means, for solid
body if it is translate if it is translated
without any change in the linear dimension
and the angular dimension of the body dimension
remain as it is. Similarly, when fluid body
get a rotation is dimension remains same;
that means, all the particles in the solid,
solid body all the particles are the solid
body moves really same angular velocity. The
solid body does not have the deformation which
is the distinguishable feature for a fluid
body in its motion. Now with this event the
close the lecture and kinematics of fluid
today we will be discussing the conservation
equations in fluid flow. Now, you know that
heavy physical system on that any process
transferring energy exchanging energy or any
process which is performed by any physical
system mass obey 3 laws of conservation. One
is the conservation of mass; that means, mass
is neither, created not destroyed mass of
a system remains unchanged.
Another is the conservation of moment that
you have already come to know from your school
level that conservation of momentum is giving
by the Newton. The Newton second law of momentum
should be concern with respect to the force
applied on a system according to the laws
motion. Another is conservation of energy;
energy is neither, created not destroyed.
So, there is any conservation of energy transformation
of energy total energy remain same of course,
one thing as to be told in precaution that
they are all physical process were mass energy
is been mutually converted with to each other..
So, if you take care of those process then
the there are 3; there are this 3 conservation
equations are not independent. For example,
conservation of mass and conservation of energy
then jointly make is conservation statement
that conservation on mass and energy; that
means, mass plus energy remains constant.
However, if we discuss that particular part
of the physics with which at present we are
not interested in our fluid flow problem where
the mass energy mutual exchange is there we
can till the 3 independent laws of conservation.
Conservation laws; that conservation of mass
conservation of momentum and conservation
of energy have to be follow the obey by flow
of fluid also. As it has to obeyed by all
physical system under any condition or executed
or executing any process.
Now, before examining this, all three conservation
equations our main objective of this chapter
will be to apply this conservation equation
to a fluid flow problem. And finally, derive
an equation in the thing is hydrodynamic parameters
like velocity pressure and all this things.
And you till that this is an equation will
the thing to velocity, pressure which comes
from the conservation principle either conservation
of mass. And conservation of momentum and
conservation of energy generally give a name
to that particular equation, but to derive
those equations following the conservation
risible, we first no certain terminologies,
which will be very helpful in studying solid
mechanics. Of course, of course,, you are
learned in solid mechanics, fluid mechanics,
thermodynamics any physical system to learn
we must at must have to know some terminology.
Let us concentrate on those terminologies,
now what are those things how do you define
is system, how do you define a system. Now,
a system is usually defined as some amount
of mass, some amount of mass at some amount
of mass of the working fluid or working system
some amount of mass within a giving boundary.
So, system definition includes 2 things some
amount of mass and some boundary, boundary
is very important for defining a system. So,
a system distinguishes a giving mass, with
a giving boundary. So, boundary is a very
important for this system. So, system has
2 important characteristic; one is the mass
and another is within the boundary and everything
external to the boundary is known as surroundings,
surroundings.
So, this is the definition between this is
a system and this is a surrounding. So, surrounding
everything external to this system we are
may be another system. For example, we can
think of system a with a boundary and system
b interacting between each other. In that
case if we tell system a, and system, system
b will be, be surrounding to the system a
similarly, system a will be surrounding to
system b. So, surrounding means it is external
to a particular system on which we are paying
our attention. But now in this characteristic
feature we can still there are two types of
system which is very important; one is control
mass system control mass system you will get
in my book..
But I tell you that really in any book it
is define like this control mass system or
sometimes we calling that close system or
in a more liberal way convention we call these
as the system. Then you can ask me sir other
type of system you are told to that is not
called system usually not. So, this I am coming
cosmologically it is basically control mass
system, another is control volume system.
One is control mass system another is control
volume system. So, control volume system we
sometimes define as open system or we sometimes
define as control volume we do not refer to
system. So, when system and control volume
the 2 what that define system refers to control
mass system and sometime close system. And
control volume system sometime depends as
open control mass system goes with control
volume system these are this comparison is
very clear. Close system at it is open system
and one is system this we call system this
we call simply control volume what is the
difference?
In a control mass system or close system,
close system this is the boundary of the system
the restriction is that they are is no mass
flow m is 0. There is no mass in flow or mass
out flow to the system only energy in flow
and energy out flow is their energy may either
come in or energy may either go out. So, system
boundary does not allow any mass to come in
or go out which means that in a close system
it is not only the amount of mass m, but its
identity remain same. That means, the same
mass with the same identity remain same remain
within the close system close system does
not allow the mass transfer. So, mass does
not came in go out to boundary of a close
system may expand may collapse because of
the mechanical work done there is no restriction
for the volume of the close system. But boundary
may expand or collapse without allowing any
mass to come in boundary may be fixed it may
not expand then collapse it may receive only
heat or it may reject heat.
So, any form of energy transfer across the
boundary is possible were as in a control
volume system. So, they are what you see this
is known as close system it is closed it is
not allowing any mass and it is refer to as
system sometime only system. Whereas in a
control volume systems, simply we tell is
that sometimes control volume the mass transfer
is relax. That means, they are may be mass
coming in and they are may be mass going out.
Some portion the mass may come some portion
mass may coming in sorry mass may go out along
with the energy. So, both mass and energy
in flow and out flow takes place, but here
the restriction is that this boundary is reached.
That means, this boundary does not move; that
means, in a control volume the volume is fixed.
Whereas in a control mass or close system
the mass is fixed that is why this is known
as control mass system. And that is why it
is known as control volume system, but in
a control volume mass may come in mass may
go out.
So, therefore, as per as the mass of the system
the control volume characteristic this that
the identity of the mass changes; that means,
at some condition the mass may remain same
in the control volume that amount of mass
coming in exactly equals to that out. But
still if that case control volume system or
open system or control volume differs from
the close system is that it is not only the
mass, but the total identity of the mass remains
you understand. So, here the identity is loss
because the mass is the going out also coming
in mass may or may not remain same depending
upon the balance between in flow and out flow,
but identity is always loss. So, this is the
close system and the control volume or open
system another type of system also we define,
which is isolated system please write isolated
system. So, this system is very simple. So,
this is control mass system or close system
or system. So, sometimes we will refer only
by system and this is control volume system
or simply you will refer by control volume
system or open system.
Isolated system by definition is isolated
in all respect; that means, isolated system
is a Pasic material; that means they are isolated.
That means, there is neither mass interaction
nor energy interaction; that means, then isolated
system does not interact to in this surrounding.
While a cos system interacts with the surrounding
in terms of the energy transfer. A control
volume system interacts with the surrounding
in terms of both mass and energy transfer
this is the surrounding external to the system.
While and isolated system is isolated from
the surrounding in all respect. That means
neither mass transfer nor energy; that means,
total energy and mass, mass even with the
identity remain same in an isolated system
does not change with time. So, these are the
3 terminologies that close system open system
or control volume isolated system, which will
be required after words in describing other
many hydrodynamic parameters and analysis
of many engineering problems.
Now let us see what is continuity equation?
Continuity equation is now first of all we
start conservation of mass, conservation of
mass in fluid flow, conservation of mass in
fluid flow fluid flow. If we apply the conservation
of mass in fluid flow as we know the close
system. If we apply it to a close system,
if we apply it to a close system you can tell
me sir what is great in it? If the mass of
a close system is 0, the conservation of mass
is giving by these expression D n D t is 0.
But this is not true for an open system or
control volume. So, what is then with respect
to an open system, what is this with respect
to an open system or control volume please
tell me. What is this with respect to an open
system or control volume the mass of a close
system is 0 if we have an open system or control
volume, control volume. So, we can simply
tell the conservation of mass like that if
the mass enters to control volume is the mass,
flux entering and the mass, flux leaving.
What is this with respect to an open system
or control volume leaving? What should be
the generous statement for conservation of
mass for a control volume where the mass,
mass flux coming in continuously and mass,
flux is leaving continuously? Can you always
tell that the mass flux leaving is equal to
mass flux entering we cannot always tell.
These we can tell are the particular condition
when the control volume mass will not change.
So, this is the particular condition control
volume may as observe mass that. That means,
may remain with the in the control volume
or control volume may lose mass some mass
may go from the control volume.
So, the most generalize statement deleting
to that is the may rate of increase. Let us
consider increase in mass in the control volume
plus net rate of mass a flux mass a flux from
the control volume is equal to 0 it comes
from basic intuition. So, that net rate of
mass a flux plus net rate of increasing mass
in the control volume is 0; that means, net
rate of increasing mass in the control volume
is the negative of net rate of mass a flux.
That means, the net rate of mass in flux;
that means, mass in flux minus net rate of
mass a flux means mass a flux minus mass in
flux. So, one can see from very common sense
that rate of mass in flux minus rate of mass
out flux is the net rate of increasing mass
in the control volume.
So, therefore, this is written like that when
the net rate of increase in mass in the control
volume is 0 that control volume does not increase
its mass or decrease its mass by this mass
transfer process. Then net rate of mass a
flux from the control volume is 0 or net rate
of mass in flux which you tell with a positive
or negative sign is 0. That means the rate
of mass in flux than rate of mass a flux is
balance. So, this statement of conservation
of mass apply to a control volume or open
system which is the very basic thing which
comes straight from the common sense in initialize
in deriving an equation. That means, these
constrain of conservation of mass is utilize
with respect to a control volume is deriving
an equation relating the velocity field in
the flow and known as continuity equation
continuity equation. So, continuity equation
is an equation which is derives by using the
conservation of mass with respect to control
volume. Let us derive the continuity equation
all of you have understood?
Now let us consider a x y z coordinate axis
x I have given x in this direction let us
consider this as y and this as j. Now, with
respect to any frame of reference if you like
to define the continuity equation that conservation
of mass apply to a control volume. First type
is that you define a control volume whose
planes are parallel to the co ordinate planes.
That means, here I define the control volume
like this I define a control volume like this
I am not drawing in the spirit of the drawing
actually this, this will be doted. So, this
is a parallely piped. So, if the first step
is to derive or example deriving the continuity
equation and again I am telling with respect
to any frame of reference. First step is to
draw the control volume whose planes are perpendicular
to the co ordinate planes. So, therefore,
in a Cartesian co ordinates system it will
be a parallelepiped. Let us give this name
A let is B C D this is E F G H now we define
a control volume with imaginary boundaries.
Because the boundary of a system or control
volume is not necessary into be a real and
r easy it is an imaginary boundary which one
can choose depending upon the need of the
problem. And let the dimensions of this v
d x in the direct this length along with the
co ordinate direction con convention d y and
let this is d z; that means, this one is d
z well. Now we should find out how the mass
flux is coming in, let us consider the u component
of velocity that is a positive x direction.
V component of velocity in the positive y
direction and z component of velocity in the
positive z direction existing the field.
So, therefore, with this positive x direction
velocity there is a mass flux coming across
this plan A E A H D this plane we call as
x plane. Why because the normal to this plane
is x direction; that means, x plane there
are 2 x planes a e a h d and b what is this
A B C D E F? This is B F G C. So, through
this x plane A E A H D let us consider a mass
flux n dot x is coming. So, due to this typical
velocity field the mass flux n dot x plus
d x we consider these mass flux changes over
a distance d x as n dot x flux d x which is
nothing, but the mass a flux across this x
plane.
That means b f g C which is d x distance apart
from this x plane in the positive direction.
So, what is n x dot that is mass flux coming
in to the control volume through the phase
A E A H D, you know the volume flux in a fluid
flow is giving by the velocity times the area.
So, if the velocity is u what is the area,
area is d y dz dy dz. So, dy dz and volume
flow times the density is the mass flux. So,
mass flux is simply rho u dy dz. That means,
u is the velocity, velocity times the area
is the rate of volume flow times the density
is the rate of mass flow that is coming in…
So, the mass flow going out from the x plane;
that means, from the planes parallel to this
x plane that is B F G C it is parallel to
A E A H D which is going out that will be
m x m dot x plus dx. That means, it will be
m dot x plus del del x of m dot x d x and
since d x is the small we can neglect the
higher order term in the delaxary expansion.
So, simply then we can write this is equal
to rho u d y d z plus del del x y rho u d
y d z d x; that means, del d x of rho u d
x d y d z. So, this is the term extra over
this. Now, therefore, the net rate of a flux
net rate of a flux 
because of this flow through x planes there
are 2 x planes x planes means the plane parallel
to y z plane.
That means, is normally is x is equal to del
del x net rate of a flux means a flux minus
in flux del del x of rho u dx dy dz. So, this
is the net rate of a flux into the control
volume due to the flux is crossing the x plane.
Similarly, if we consider the flux is crossing
the y plane let this is the flux is crossing
the y plane. So, the flux which is coming
into the control volume thorough be y plane;
that means A B C d. So, A B C d is the y plane
through which the flux is coming in because
of the velocity components v existing in this
fashion this is the positive direction of
y. So, let us consider this m dot y similarly,
which is going out through another y plane.
So, this parallel to this y plane A B C d
that is what A F G H; that means this is these
y plane perpendicular to the y axis that we
can designate m y plus d y; that means, it
is the change of the mass flux over a distance
d y. Then we can write with the similar concept
that m y dot is rho time the volume flux that
is the flow velocity in these direction v
times this area dx dz. And with this similar
rotation m dot y plus d y is m dot y plus
del del y of m dot y d y what we can write?
We can write this is equal to rho v d x d
z plus del del x of sorry del del y of rho
v dx dy dz.
So, therefore, we can write the net rate flux,
net rate flux, net rate of a flux from the
control volume net rate of s flux from the
control volume due to the fluxes parallel
to the y planes is equal to del del y of rho
v d x d y d z. In the similar fashion we can
do for the z plane that means the planes through
which flux is crossing through z planes; that
means, there are 2 z planes. Now, 4 planes
we have consider rest to planes are they are
one is A B A v and D C G i. So, flux is coming
in through this bottom plane because of the
existing velocity component. So, this is m
dot j similarly, flux is going out through
this z plane D C G H which is at a distance
d z above this plane. And we just give it
n value n dot z plus d z and in the similar
way we can write n dot z is rho w d x d y
the area of this plane.
And similarly, we can write n dot z plus d
z as del del z n dot z d z plus n dot z plus
n dot z plus n so z. So, n dot z will come
fast. So, this can be written as rho w d x
d y plus del d z of rho w d x d y d z. So,
d x d y d z I can substitute as d v that is
the elemental volume of this control volume
d v. So, that earlier also I can write it
d v and I can write it d v. Now what I can
write net rate of a flux in the control volume,
in the control volume, in the control volume
due to the flux is across all the phases.
Due to the flux is across all the phases will
be some of these net rate of a flux due to
fluxes across x plane plus net rate of a flux
due to fluxes across y plane plus the net
rate of a flux due to flux is across z plane.
That means some of this, this I am writing
as d v some plus this some of this, this and
that means, this is equal to del del x of
rho u plus del del y of rho v plus del del
z of rho w times d v which is a constant.
Now, what is net rate of increase, because
if we again recall our continuity equation
which came simple from physical common sense.
Net rate of increase in mass in the control
volume net rate of mass a flux from so net
rate of mass a flux from the control volume
we are found out. But what is net rate of
increase of mass in the control volume net
rate of a mass a flux I am net rate of every
time I have writing mass a flux what a flux
mass a flux net rate of am mass a flux. So,
now, net rate of increasing mass, net rate
of increasing mass, net rate of next thing
we have to find out of increasing mass net
rate of increasing mass within the control
volume. What will be is expression within
the control volume? What will be is expression
within the control volume? What will be is
expression, which expression will be the rate
of change of mass within the control volume?
That means, if I define the mass at the control
volume at any in stand what will be that rho
d v? And if change with time control volume
is fixed.
So, if change with time; that means, this
is the mass of the control volume this will
represent this thing that net rate of increasing
mass within the control volume that instant
in as mass of the control volume its rho and
its volume d v. So, volume of the control
volume as I have told by its definition is
in very end with time it will come out. So,
this is very important to know that how we
are v is coming out d v, because of the definition
of the control volume. So, according to this
statement physical statement now I can write
del rho del t plus del del x of rho u. So,
according to the conservation of mass apply
to a control volume in a cartesian co ordinate
system we can write this into d v is equal
to 0. True and this is valid for any value
of d v it is irrespective of the volume of
the control volume which is an arbitrary parameter
which means this quantity has to be 0. Because
d v is not 0 it is valid for any value of
d v any finite volume of the control volume
this quantity has to be valid..
So, finally, therefore, we can write this
has been continuity equation del rho del t
plus del del x of rho u so in a Cartesian
co ordinate system. So, this is the continuity
equation; that means, in a crustacean co ordinate
system the consequence of conservation of
mass at application of conservation of mass
in a control volume gives than equation which
is known as continuity equation. This continuity
equation form this is the continuity equation
in Cartesian coordinate system del rho del
t plus del del x of rho u plus del del y of
rho v plus del del z of rho w is equal to
0.
So, when a special case now we consider that
when the fluid is incompressible. For incompressible
fluid or incompressible flow we are not interested
fluid is incompressible or compressible we
have already recognize the earlier that whether
density changes in the flow or not we are
interested. For the flow at density does not
change it is an incompressible flow were the
Mac number is below 0.33. Then density is
in worried no were in the flow fluid density
changes density is not a function of x y z
and similarly, these becomes 0 and this comes
out. So, therefore, the equation becomes del
u del x plus del v del y. So, it is very important
that for incompressible flow at density is
neither a function of time or nor a function
of x y z. In a Cartesian coordinate system
with u v w are respected velocity components
all x y z direction this is a special case
of continuity equation for an Cartesian. So,
today after this well, next class, we will
discuss again you cannot animation to this.
Thank you.
