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.
