Welcome back to the video course on fluid
mechanics. In the last few lectures we are
discussing about the introduction theories
of fluid mechanics introduction to various
fluid properties, we discussed about fluid
statics its related theories, we have discussed
about buoyancy pressure fluid pressure center
of pressure met center extras. Today we will
discussed the another topics fluid kinematics.
The main object is of the topic on kinematics
of fluid flows to introduce various aspects
of fluid motion without being concerned with
the actual forces necessary to produce the
motion. The other object is of this topic
are to introduce the kinematics of the motion
the velocity and acceleration of the fluid
the description and visualization without
considering the force.
we will introduce the fluid dynamics and finally
to understand the fluid motion through the
kinematics fluid flow.
That is the important object that is of the
section on kinematics of fluid flow. As we
have already discussed the kinematics of fluid
flow means the fluid flow is taking place
due to various forces. But when we consider
the fluid motion here in this kinematics of
fluid flow consider the forces as such but
without considering the force what happens
to a fluid or how the fluid is moving how
the velocity can be calculated how the acceleration
can be calculated, we will go to the various
the principles of fluid mechanics like consideration
of mass consideration of momentum based up
on the fluid kinematics. If you consider the
various examples of fluid motions,
Like smoke emerging from a chimney or the
flow of the atmosphere as indicated by the
motion of clouds or the motion of waves in
a lake. All this fluid motions here generally
when we try to analyze all this fluid motion
there can be consider by inside to this fluid
motion by considering the kinematics or such
flows without the concern with the specific
force that drives them. The smoke emerging
from the chimney there is force which is taking
place which is driving this smoke or the wind
on the atmospheric motion or the motion of
clouds and or the waves in the lake all there
are same forces but we the analyzes be much
simpler when we consider this fluid phenomena
or fluid motion without much concern to the
specific force that derive them.
That is the importance of this kinematics
of fluid flow. we will be considering the
section the kinematics of fluid flow with
respect to this various fluid motions without
giving much attention to the force driving
this the driving force which we are which
due to this the fluid is moving. We considering
the various aspects of kinematics of fluid
flow starting from the velocity field we discuss
the Lagrangian and in concepts we will discuss
about the acceleration fields etcetera. The
fluid parameters,
By field representation as I mentioned earlier
the fluid moment is then you consider the
fluid moment it the moment is in with respect
to space as well as time. If you consider
the space we consider the 3 dimensional motion
of a fluid with respect to xyz axis time t
will be al will be there.
The fluid motion is with respect to space
with respect to time the fluid parameters
are generally represented by the space the
space coordinate xyz time t. for example the
as shown in this figure here the fluid is
moving in a pipe or in an open channel when
the fluid is moving. The motion is with respect
to space with respect to time that we can
represent that show in this figure the velocity
fluid can be represented as v is equal to
U xyzt of I the unit vector plus Vxyzt of
the unit vector j plus wxyzt the unit vector
k.
Since the velocity is it has got a direct
direction as well as magnitude that we represent
a early with respect to ij and k which are
the unit vectors U here uvw are the magnitude
of the velocity moment and finally the velocity
field can be represented as v is equal to
ui plus vj plus wk where, uvw are depend up
on the spatial coordinate xyz and time t.
Finally, the speed of flow we can represent
when we are considering a pipe flow or when
you are considering an open channel flow or
any kind of flow we can represent the speed
of flow as it will be the absolute value of
this v the it can be represent as square root
of u square plus v square plus w square as
shown in this slide. In the speed of flow
is square root of u square, plus v square
plus w square that use the speed of flows.
The velocity field is the generally represented
as the function of spatial coordinate xyz
and time; here you can see that a particle
is moving.
From a position A to another position at time
t position is here and the time t plus delta
t the particle is moving from this position
to the new position. The position vector if
you draw the position vector. This rAt is
the initial position like vector rAt plus
delta t is the present position on the fluid
particle. With respect to this the particle
is located in terms of the position vector
with respect to this only we want to describe
the moment of a fluid particle in any fluid
media. When we represent the velocity here
this position will be with respect to xyz
coordinate and time the current time here,
the new position will be al represented in
terms of the new position xyz time t, the
particle location in terms of the position
vector.
Here in the slide we can see that velocity
field is for example, in a river there is
a flow type space here you can see that the
flow is with respect to various flow conditions
the flow can be this direction of the other
direction. The flow velocity vectors the velocity
vector is we can represent with respect to
spatial coordinate xyz and also with respect
to time. Here, this figure shows the actually
this figure here theme contamination or theme
glom is introduced in this river system how
it is moving with respect to that generally,
we will be solving the for the this kind of
problem solving first the hydro dynamics the
velocity we will be determining then with
respect to the that the velocity. We are describing
the contaminant moment or the glow moment
will describe with respect to the velocity
vector.
It is very important that we should note the
velocity field that means with respect to
space and with respect to time how the velocity
is changing that is very important that is
actually the major problem. As for as a fluid
dynamics or the fluid flow is concerned, the
velocity field of the velocity determination
with space and time is very important. As
you can see in this slide also the velocity
field we can with respect to article float
race.
We can trace as shown in this figure we can
trace the particle how it is moving with respect
to the path line or with respect to either
stream line we can represent the velocity
the flow how it takes place we can represent
as shown here this is actually a river system
joining an. How it is behaving with respect
to the velocity field we can represent with
respect to the path lines then stream lines
and as shown in the previous slide we can
also use the velocity vectors to represent
the velocity field. As I mentioned here we
will just discussing brief example problem
how to determine the velocity field, as I
mentioned since the velocity varying with
respect to xyz direction. That we will determine
the velocity field with respect to the xyz
coordinate as well as the time the problem
here is the example problem here is.
The velocity field the velocity in a flow
field is expressed as v bar is equal to the
velocity vector is equal to 7 plus 2 xy I
the unit vector I plus minus 2 yz minus 4
t unit vector j plus minus yz plus z square
by 8 unit vector k. This is the velocity field
is represented by this function given with
respect to I j and k. The magnitude is magnitude
xyz directions are represented by this the
quantities the x direction that means velocity
component u if is given by 7 t plus 2 xy and
the velocity component is implied and direction
v is given by minus 2 yz minus 4 t and the
velocity component in z direction w is represented
here as minus yz plus z square by 8. We want
to determine the magnitude of velocity at
a point q the position of q is x equal to
3 y is equal to minus 1 z is equal to 2 and
at time t is equal to 3. The coordinate system
s xyz is given here and time al given we want
to determine the magnitude of the velocity
at a point x equal to 3 y equal to minus 1
z is equal to 2 a time t is equal to 3. Here
this function the velocity field is represented
as already given by this v is equal to this
function.
Do this we can represent as a vector we have
discussed the velocity can represented as
the velocity vector uvw and with respect to
the ijk as ui plus vj plus wk where, ijk are
the unit vectors in a xyz direction and uvw
are the velocity in xyz direction. Here v
is represented as I mentioned u is here with
respect to the given equation for velocity
for u is 7 t plus 2 xy and v is minus 2 yz
minus 4 t and w the velocity z direction is
minus yz plus z square by 8.
The position of the point is given x equal
to 3 y is equal to minus 1 and z is equal
to 2 and time t is equal to 3 that we can
determine the magnitude of velocity in x direction
uq we will substitute values here in for u.
We can write uq is equal to that means the
velocity component u at point q is 7 into
time 3 t is equal to 3 7 into 3 plus 2 into
3 into minus 1. that will give 15 as the result
here uq is equal to 15 vq the velocity in
y direction at point q is given as vq is equal
to minus here the values minus 2 yz minus
4 t 2 into minus 1 since y is minus 1 in time
is t z is 2 minus 2 into minus 1 2 plus time
is 3, 4, 3 that will give the q as minus 8
similarly we can determine wq. wq is given
as minus yz plus z square by 8 that is if
you substitute for y and z y is minus 1 minus
of minus 1 into 2, z is 2 plus 2 square by
8 that will give the value of 2.5.
Finally, as we have seen the velocity can
be the magnitude of the velocity at the point
q can be determined by taking square root
of uq square plus vq square plus wq square.
Here the uq the magnitude of the velocity
equal to square root of 15 square plus minus
8 square plus 2.5 square that is equal to
square root of 295.25 and that is equal to
17.18. Its corresponding units can be put
here the meter per second or a numeral second
depending up on the problem. like this the
velocity can be represented in xy with respect
to the space xyz time, if you want to determine
the new position of the velocity if the function
the velocity variation as represented in this
figure is known then we can determine the
magnitude of the velocity. This is the velocity
field here the velocity can be represented
with respect to space and time. Like this
the velocity field is determined.
Now we will discuss about the flow field description.
In the introductory lecture we have seen the
various aspects of how a flow field can be
represented as there we discussed about two
types of fluid flow field description. First
one is eulerian description second one was
legrangian description. Here further we will
discuss since we are discussing the kinematics
of fluid flow further you will discuss the
two description fluid flow field description
the first one is the eulerian description.
As we have seen earlier the eulerian description
uses the field concept as we have already
seen the velocity field or the acceleration
field or the pressure field whatever it is.
Here the eulerian description uses the field
concept and fluid motion is given by completely
prescribing the necessary properties such
as the fluid property such as pressure density
velocity extra as functions of space and time.
As we have already seen the velocity here
is represented as a function of xyz coordinates
and time t. In the eulerian description the
fluid motion is given by prescribing the necessary
properties which can be velocity or pressure
or density whatever it is as a function of
this and time. if you use the eulerian description
from this method you obtain the information
about the flow in terms of what happens at
fixed points in space at the fluid flows pass
those points. What will be doing same if you
consider the fluid motion in an open channel
or pipe flow for example let us consider river
flow here.
This is a river flow what we are doing here
is with respect to flow is taking place all
the times. We will be considering particular
sections here one or section two and between
the sections one and two to what happen to
this various fluid flow properties like velocity
how it is changing or how the pressure is
changing or the various other fluid flow a
parameter are changing. That is what we are
generally describing in eulerian description
generally using most of the fluid flow problems
as we are discussed earlier we will be generally
using eulerian description. From this we are
getting the information as the flow progresses
in terms of what happens fixed point here
the fixed points which are considering this
section one.
Or even it can be a fixed point or it can
be a section like this with respect to this
happens for the fluid properties when the
fluid flow is moving from one section to another
or at that particular section with respect
to time what happens to the fluid flow. That
is the way which we will be describing the
fluid flow in the eulerian and eulerian description.
Here you can see in this figure here a river
flow takes place and.
The concentration higher concentration or
the contaminant glom is used at his position
we want to see in the eulerian description
what we can d is there can be a acceleration
point as shown in this time there can be observation
point like this. Then what will be describing
is when the river flow takes place with respect
to time will be taking what happens whether
the contaminant the concentration how it is
vary with respect this particular section
with respect to time how it is varying. If
you brought this for you have particular position
then you have see the contamination the concentration
will be reducing as the flow proceeds like
this graph shows how it will be behaving.
This shows the eulerian description in a fluid
flow as we have discussed earlier. Generally
the eulerian description is very much useful
in fluid mechanics and most of our fluid flow
analysis will be based up on the eulerian
description.
The second method which we have discussed
earlier is the lagrangian method. The lagrangian
method as I have mentioned in the lagrangian
method we will be flow into the fluid particle
what will if fluid flow in a open channel
or in a river flow or in a pipe flow then
that is what we are discussing is same if
this is the flow river flow then we will be
chasing a particular particle. them with respect
to this t is equal to zero then t is equal
to t1 or t is equal to t2 or then t is equal
to t3.
Like that what happens to this particular
fluid particle that is the way which we are
chasing out in the lagrangian method? The
lagrangian method flow the individual fluid
particles as they move under the determined
how the fluid properties asciated with this
particles change as a function of time. This
is mainly with respect to time what happens
and position is anywhere with respect to the
fluid particle moment it is always changing
the particular fluid particles are traced
what happens those particles. It is what we
are doing the lagrangian method here the fluid
particles are tagged or identified ask time
progresses happens to those particles. That
e are what we are describing and since as
you can see that fluid motion or fluid flow
is concern it is very difficult to track the
individual particles like this see what happens
to those particles.
Generally, this lagrangian method is very
rarely use in fluid flow problem due to it’s
the difficulty to track the particular particle
to track particular particle what happens
to that particular particle it is very difficult
to do this kind of as analysis. Generally,
for most of the fluid mechanics problem eulerian
method is used since it is simpler we get
the information what we are looking for especially
at particular section with respect to time
we are getting. That is the most of the time
we will be using in eulerian method in the
fluid mechanics of fluid flow analysis. Finally,
in this slide here this slide show how the
eulerian approach and lagrangian approach
is used as for as fluid flow analysis is concern.
Here you can see that the fluid is moving
from a section a channel like this. here in
the eulerian approach we are taking consider
a particular point at a particular section
here like this the point is that is from this
x at the distance of x0 and y0 we are considering
a particular point the location is t is equal
to t x0 y0 t. What happens to fluid flow as
for as the particular point is consider that
is we will the studying in the eulerian approach
this will be generally with aspect to time
how the fluid is behaving but as far as lagrangian
approach is concerned here you can it is mainly
with respect to this time.
We are we have already tracked on particular
particle and this position to this position
how it was work like that we are chasing on
the particular particle, we are describing
the flow property with respect to the properties
for this particular particle is concerned
and that is the lagrangian approach. Even
though we rarely used lagrangian approach
but once the fluid flow parameters are known
we can convert to eulerian description or
from the eulerian description to the lagrangian
description al the conversion is possible.
Just let us discuss a small example how we
can utilize this eulerian approach lagrangian
approach as for as fluid flow field description
is concerned here the problem which we are
discussing s here the flow field description.
In an experiment of flow description were
conducted in laboratory flow. the 2 dimensional
flow in a lagrangian system is already obtained
as x is equal to x1 into e to the power lambda
t plus y 1 into 1 minus e to the power 2 lambda
t. in the lambda is a constant and t is time
x is the x axis here with respect to x with
respect to y time this is a two dimensional
problems the fluid flow properties are changing
with respect to the space xy and xy direction
and time t. Here already the two-dimensional
flow in the lagrangian system is given as
x is equal to x1 into e to the power lambda
t plus y 1 into 1 minus e to the power 2 lambda
t and also for y direction is concern is given
as y is equal to y 1 into e to the power lambda
t. This y1 and x1 gets what is the initial
positions this is x1 y1 the variation with
respect to x and y are given with respect
to time t by these equation number 1 equation
number two. We want to find an expression
for path line of the particle with respect
to this lagrangian system given we want to
get the corresponding equation eulerian system
velocity components we want to determine.
Problem is we want an expression for path
line of the particle eulerian system velocity
components these we can solve first the first
problem is we want to find path line for the
particles. Path line means it should be an
expression without any times we will use this
equation number one and two.
We will eliminate it t the time component
to get path line from the second equation
from second equation you will get e to the
power lambda t is equal to y by y1 equation
is given y is equal to y1 in e to the power
lambda t. The from that we can write e to
the power lambda t is equal to y by y1 this
e to the power lambda t is already there in
equation number one here. you will substitute
that for this e to the power lambda t in equation
number one here for the expression for x we
will get x is equal to x1 into y1 by y since
into the power lambda t already obtain as
y by y1 e to the power minus lambda t y1 by
y. x is equal to x1 into y1 by y plus y1 into
1 minus y1 by y whole square. This gives the
expression for the path line description of
the particle in the lagrangian system.
here there is no time component we want al
gives the expression for path line of the
particle the second part of the problem is
u1 to get eulerian system velocity component
as for as this with respect to the lagrangian
system given. The eulerian system the velocity
component we can write as since the eulerian
system which we are discuss the as here you
can see that at a particular position what
happen that gives the velocity component.
Here with respect to this slide here we will
get the velocity component UX equal to dx
by dt and uy will be dy by dt that means velocity
y direction will be dy by dt and velocity
next direction will be dx by dt. We will just
differentiate this equation number one here
give here that you differentiate that will
be velocity component the x direction. we
will differentiate the equation x equal to
x1 into the power minus lambda t plus y 1
into 1 minus e to the power 2 lambda t if
you differentiate ux equal to dx by dt that
is equal to d by dt of x1 into e to the power
minus lambda t plus y 1 into 1 minus t to
the power 2 lambda t. If you simplify if you
differentiate and simplify you will get an
expression for velocity that is the eulerian
systems. ux is equal to minus lambda x plus
lambda y into e to the power minus lambda
t plus e to the power minus 3 lambda t.
Here this after differentiation we will simplify.
Finally, you will get the expression for the
velocity as ux is equal to minus lambda x
plus lambda y into e to the power minus lambda
t plus e to the power minus 3 lambda t. Similarly,
the velocity component in y direction you
will get uy is dy by dt that we can just differentiate
since the initial equation the expression
for y is given in the lagrangian system y
is equal to y1 in e to the power lambda t.
if you differentiate you will get uy is equal
to dy by dt that is equal to d by dt of y1
into e to the power lambda t. that is uy is
equal to lambda 1 into y1 into lambda into
e to the power lambda t. That will give this
can be written as since y is y1 into e to
the power lambda t uy can be written as lambda
y that gives the expression for the velocity
component in y direction for the eulerian
system.
This is the velocity field or in flow particular
fluid flow parameters is known in one system
either lagrangian system or in eulerian system
convert to other system; with respect to the
various mathematical relationship available
with respect to both the system. We have seen
how we describe the various fluid flow properties
with respect to discussion the eulerian description
and the lagrangian description as I mentioned.
Generally we will be using the eulerian description
using the eulerian description since it is
much easier and it is easily the analysis
must simpler and we can get results especially
since fluid flow much bother about what happens
at particular section than just tracing some
fluid particle.
But some cases also some problems will be
taken in the lagrangian approach by tracking
particular particle. That also we will discuss
later with respect to this the field description
the fluid flow as we have discussed earlier
can be described briefly again review this
fluid flow descriptions., space wise as I
mentioned the flow can be either one-dimensional
two-dimensional or three-dimensional flows
as I mentioned. Generally the all the flows
are three dimensions in nature it is varying
with respect to xy and z and al time. But
many of the problems for example as I mentioned
when we discussing a river flow if you want
to know a particular with respect to the longitudinal
direction l here or here this x direction.
If you want to know with respect to this longitudinal
direction what happens to the flow properties
like velocities or the head or the pressure
or the parameters then it is better that we
consider the fluid flow as one-dimension or
when we need more details the same problem
we will be describing as two dimensions that
this x and y both component we will be describing.
That the velocity in the lateral direction
this in u and v will be al considered and
al correspondingly the time the other parameters
will be constant two-dimensions and otherwise
if you are looking for accurate the analysis
if you are looking for a problem in three
dimensions. That we have to consider for example
the flow in a river is concern here we are
to consider the depth wise also, what happens
with respect to z direction not only the longitudinal
the lateral we have concerned the depth.
Also depending up on the problem most of the
fluid flow is concerned this three dimensional
nature but we will be simplify the problem
into one-dimensional two-dimensional three-dimensional
problem. As you can see here it is a river
flow.
Here this slide, shows what happens with respect
to the flow here the analysis is various section
here considering analysis with respect to
this one dimension what happens. Space is
one dimension with respect to time the variation
is with respect to this the length of the
river. Here with respect to x time we will
be analyzing this is special in one-dimensional
flow is concerned analyzing with respect to
what is happening in one d. this is fluid
flow in 1 dimension when we analysis as 1
dimension the next here.
This slide shows this space two d shows this
what will this happening with respect to two
dimensions. That is why all this here in this
blue with respect to what happens in y direction
latterly also we have considered that is why
here the space wise this is two-dimension
flow which we are concerned the flow is the
definitely three-dimension that we are simplifying
flow to two-dimension we are analyzing as
a two-dimension flow next here you can see
the space three dimension.
Here the depth is also considered in a flow
with respect to the length of the channel
later direction. It is totally three-dimension
depending upon the necessity depending up
on the type of problem which we are dealing
the fluid flow analysis can be one-dimension
two-dimension or three dimensions. Also as
we have discussed earlier the time wise the
flow can be most of time the flow will varying
with respect to time. But many times we can
also that once there is not much variation
with respect to time then we can consider
there is no variation with respect to time
for the fluid properties then you can consider
as study state flow with respect to time we
can consider the fluid flow steady state flow
and unsteady state flows. In unsteady state
flow definitely the fluid flow properties
are varying with respect to time.
We have to see the spatial variation as well
as the time variation but as far as steady
state flows as. We will be discussing with
respect to spatial direction time wise it
is steady state. Also we have discussed generally
way to describe the fluid flow we will be
using various fluid visualization flow visualization
techniques and also certain lines like a stream
lines streak lines and a path lines are generally
used for to describe the fluid flow. This
we have already seen in the previous slide
how we are using this kind of analysis as
for as the fluid flow descriptions concern.
You can see the path lines are used and stream
lines are used for the velocity description
and here the stream line the velocity vectors
are used and with respect to the velocity
vector we can find out the stream line. The
stream lines path lines steak lines which
we have already described discussed in the
interacting chapter streak lines or these
lines are used for the fluid flow description
and with respect to that only will be describing
what happens to the fluid flow.
Flow description is we can use this various
techniques and in steady state flow the stream
lines and path lines are the same. It will
be generally used as for as the stream lines
will be used for unsteady state flow analysis.
We have already seen the velocity field we
have seen how we can describe the fluid flow
with respect to eulerian description are with
respect to legrangian description. The next
topic and also we have see how we can describe
the fluid flow whether it can be fluid flow
can be considering one-dimensional two-dimensional
three-dimensional or steady state or unsteady
state like that. We will discuss the acceleration
field.
Acceleration is most of the flow takes place
due to acceleration due to gravity of the
applied acceleration. Acceleration field is
also another important acceleration is al
another important property as for as fluid
flow is concerned. In the kinematics of fluid
flow here you will discuss the acceleration
field. The particle acceleration which is
generally here you can utilize this Newton
second law which is given as force is equal
to mass into acceleration. We have seen two
methodologies of description of fluid flow
one is the lagrangian method.
As for as lagrangian method is concerned,
generally the fluid flow is described with
respect to time only the lagrangian method
the acceleration is described as a varying
of with respect to time. That means particle
is we are looking in to that and how it is
moving or how it is behaving with respect
to time. Here the space is not the major issues
you are tracking the particle what happens
for that particle with respect to time.
As for as acceleration field is al concern
here we will be describing the acceleration
is with respect to time the variation is with
respect to time, but as far as the eulerian
method which we have seen earlier is in the
eulerian description we are considering a
particular sections. We will be discussing
what happens to fluid flow properties at the
particular section. Here the acceleration
field is concerned in the eulerian description
we will be describing a, is equal to a xyz
and t time the acceleration field is described
with respect to the space xyz and time t and
as we know the acceleration time write of
change velocity for a given particle the lagrangian
method. We can write acceleration a, is equal
to as a functional type and the eulerian description
you can write as a functional space and time
here this.
Figure shows if you consider a particular
path of a particular fluid particle. For particle
a here you can see that the particle path
is like this. Particle a, at time t this is
here it is described with respect to xyz time
t. Its position vector is represented like
this with respect to the velocity componen
t xyz directions. S ua the velocity component
x direction the VA the velocity component
this VA is the magnitude the final velocity
and this shows the velocity on the y direction
and this is the velocity component in the
z direction.
Particle A the velocity is described as VA
is a position vector of r and t as described
here VA is the function of rA and t and that
can be put as VA xAt YAt and zAt as shown
in this slide.
That will be represented like this with respect
to xy and z and spatial direction and time
t equation of acceleration is concerned.
In the legrangian description than we will
be writing with respect to u is equal to del
XA by del t the velocity described as here
we will consider the equation for acceleration.
The velocity component in x direction u is
equal to del XA by del t and v is del YA by
del t and w is del zA by del t as shown in
this figure.
Acceleration is concerned the variation is
with respect to xyz and t that is why the
partial is used here. Finally, the acceleration
can be written as acceleration is the derivative
of the velocity generally the acceleration
is represented as a is represented as dv by
dt. If we consider the fluid flow at particular
position point A, the acceleration can be
written aA t is equal to dvA by dt which is
the total derivative. That with respect to
this uvw description we can write del vA by
del t plus del vA by del x that means the
acceleration is not total represent with respect
to local acceleration as well as convert the
action.
Here the acceleration for this particular
fluid particle we will be considering with
respect to the time variation with respect
to local acceleration conductive acceleration
with respect to the fluid flow how it is behaving.
That is what we are describing here the acceleration
is represent as the total derivative of the
velocity vector with respect to time the A
is equal to dva by dt. That can be represent
as a local acceleration than VA by del t plus
del VA by del x
dxA by dt this is with respect to the velocity
component in x direction del VA by del y plus
into dyA by dt this is with respect to the
velocity component y direction plus del VA
by del z into del ZA by dzA by dt. This gives
the total acceleration as for as the fluid
movement is concerned this one part is the
local acceleration and other part is the conductive
acceleration as indicated here in this figure.
Here this dxa by dt can be represent as u
and dyA by dt can be represent as v and dzA
by dt can be represent as w. Finally, if you
substitute the total acceleration can be represented
as shown in the slide.
As A is equal to del v by del t with local
acceleration plus conductive acceleration
is given as u into del v by del x plus v into
del v by del y plus w into del v by del z.
This is the general the acceleration as a
vector representation as acceleration since
it is varies with respect to space and time
the direction and the magnitude is that. Acceleration
is can be represented in terms of the x direction
y direction and z direction. with respect
to general description of the acceleration
we can write ax is equal to del u by del t
plus u into del u by del x plus v into del
u by del y plus w into del u by del z this
expression is obtained directly from the general
expression for acceleration. This is x direction
acceleration ax is equal to to del u by del
t plus u into del u by del x plus v into del
u by del y plus w into del u by del z. Similarly
we can write the acceleration in x direction
y direction and z direction.
As I mentioned this term here del u by del
t is the local acceleration that means with
respect to the x component the acceleration
in x direction a axis with respect to the
velocity in x direction what happens the local
acceleration with respect to time, the terms
are called the convective acceleration that
means with respect to the velocity uvw how
the acceleration takes place. These three
terms are called convective acceleration and
the first term is called the local acceleration.
Similarly, we can write the acceleration in
y direction.
As ay is equal to del v by del t plus u into
del v by del x plus v into del v by del y
plus w into del v by del z acceleration in
z direction can be written as del w by del
t plus u del w by del x plus v into del w
by del y plus w into del w by del z.
The acceleration in xyz direction is represented
as local acceleration terms plus conductive
terms this is the general methodology used
for the determination of acceleration. This
ax ay and az are called the components of
acceleration ax ay and az are called the components
of acceleration. Here you can see that this
entire problem which we have seen for with
the analysis the fluid flow analysis which
we have discussed is mainly.
In terms of the without concentric the force
which drives the flow we are not concerned
with the force. But later this is beginning
as for as this topic is concerned later we
will discuss that also here the operator which
we discussed the total derivative is generally
in the previous slide we have seen here the
acceleration we have seen in with respect
to the xyz components. With respect to this
the operator we defined the total derivative
as D by dt is equivalent to the local del
by del t that means with respect to time u
into del off by del x plus v into del by del
y plus w into del off del by del z. this depends
up on the property if it is x direction here
will be putting u y direction will be putting
v and z direction will be putting w.
This terms are called sat the termed as material
derivative. This are termed as material derivative
this describes the time rates of change for
given particle. The time rate for change for
given particle is given by this total derivative
termed as material derivative. This will be
using in most of our derivation later stages
the rate of change of for given particle are
termed as material derivative the total derivative
will be described with respect to the local
term plus the conductive term here uv and
w.
Further we will be describing the applications
of the kinematics fluid flow kinematics further
derivations are derived various equation as
for as fluid flow kinematics concerned. Finally,
that will be with respect to that proceeding
to the fluid flow dynamics.
