Welcome back to the video lecture on fluid
mechanics. In the first lecture, we have discussed
the introductory aspects of fluid mechanics
We have discussed about the various fluid
properties, various theories used in fluid
mechanics and how it will be used in our this
video course; we discussed the various fundamental
theories which applicable in the case of fluid
mechanics; we discussed the various flow visualization
techniques, used in fluid mechanics and the
flow lines like stream lines, path lines and
streak lines. Today, in this lecture, first
we will discuss about the classification of
fluids.
As I mentioned earlier, the fluids can be
classified according to various fluid properties,
fluid behavior or the dimensions on which
they will be dealing with the fluids or the
fluid problem which we are dealing. So, mainly
the fluids fluid flow can be classified according
to the rhelogical consideration, spatial dimensions,
dilational tensor, then motion characteristics,
the temporal variations and fluid types. Here,
we will discuss in detail about the various
types of fluid flow according to various aspects.
As I mentioned earlier, the fluids can be
either gases or liquids. The first classification
of fluid flow is whether it is gas gaseous
flow or liquid flow. The first classification,
as you can see in this slide say, the fluid
flow can be either gaseous flow or say liquid
type fluids. Secondly, according to whether
the fluids can be compressed or whether it
cannot be compressed say, accordingly we can
classify the fluids in to compressible fluids
and incompressible fluids; mainly, the gases
which will be generally dealing with will
be compressible in nature and some of the
liquids are may be some compressible depending
up on the type of the fluids and some other
liquids we can consider as incompressible.
Even though sometimes we can compress to every
small level but still it can be considered
as incompressible. For example, water is considered
as incompressible even though, may be to certain
level it can be compressed by applying the
pressure.
As I mentioned, say, the fluid flow or fluid
can be again classified. Fluid flow can be
classified as steady state fluid flow or transient
fluid flow. The fluid, since it is deforming
with respect to space and time, if it is with
respect to time there is no variation at all,
then the fluid flow is said to be steady.
That means with respect to time, there is
no variation of the fluid properties but the
fluid properties like velocity or depth a
flow all this fluid properties are constants
with respect to time and there is no variation
with respect to time.
In the case of UN study flow or transient
flow the fluid property like velocity, pressure
and depth, all these parameters are varying
with respect to time. So, these types of fluids
are called unsteady fluid flow or transient
flow. As we discussed earlier the viscosity
is an important fluid property. So, accordingly,
whether the fluid flow which we are dealing
with has got viscosity or the viscosity is
negligible or there is no viscosity, then
accordingly we can classify the fluids into
viscous fluids or viscous flow and non-viscous
flow or invest flow. So according to the variation
or according to the fluid has viscosity fluid
is called viscous fluid flow and then if viscosity
is not considered then that kind of fluid
is called non-viscous or invest fluid flow.
If there is any circulation or any rotation
aspect as far as the fluid flow is concerned
we can classify the fluid into rotational
fluid or irrotational fluid. In the case of
rotational fluid, there can be circulation
there can be vorticity and all other parameters
but as far as irrotational fluid is concerned
circulation is not there and vorticity is
0. So, those fluids are called irrotational
fluid or irrotational fluid flow.
According to the space we are considering
3 dimension, x, y and z. We can classify the
fluid as 3 dimension fluid flow or 2 dimension
fluid flow or 1 dimension fluid flow. As all
of you know fluid flow is definitely 3 dimensional
in nature as you can see any type of fluid
flow is 3 dimension nature.
But many times, we can consider 3 dimension
fluid flow as 2 dimensions, for example: here
you can see say, a river. We are considering
a river section like this; if you are considering
here as z axis, this as x axis, here say,
y axis, so this is xyz axis. Here you can
see that when we consider the fluid flow with
respect to this river section like this x
verses z, then this other longitude dimension
is not considered. So, in this case, the fluid
flow can be considered as 2 dimension; we
may be interested what is happening with respect
to this x direction and z direction even though
the fluid flow is definitely 3 dimensional
in nature. We are simplifying the 3 dimension
fluid flow into 2 dimension since it will
be very easy to analyze and when interrupt
the results also it will be very easier. So,
depending upon the case, depending upon the
problem and depending upon the fluid flow
problem we can consider a 3 dimensional flow
as 2 dimensional. So that the accuracy of
the results are not much affected.
We can consider the river flow in some times
as 1 dimensional flow say for example, if
you are considering the river is flowing like
this. In this case, we are dealing from one
section to another section, say, here 1 1,
2 2, 3 3; we are dealing from what is happening
from section one to section two and section
three; in this case, obviously, the fluid
flow is 3 dimensional in nature. We can consider
the flow as what is happening with respect
to this longitudinal direction as far as this
river flow is concerned. We can consider in
this case, the fluid flow as 1 dimension.
All the fluid flow which we are dealing with
is 3 dimensional in nature but depending upon
the problem to simplify the problem we can
consider the fluid flow as 2 dimension or
whether we can consider sometimes as 1 dimension.
So this simplification makes the problem analyze
much simpler and the interpretation of results
also much easier without losing much across.
Most of the problems even though 3 dimensional
in nature, we may be solving as 2 dimensional
fluid flow problem or 1 dimensional fluid
flow problem.
As we have seen the previous slide, the fluids
can be either gases or liquids based up on
the molecular behavior as you can see in this
slide. Then the second case: the fluids when
we are considering continuum or discrete fluids.
A continuum means individual molecular properties
are negligible so that we are considering
as a continuum. In most of the fluid flow
problems you will be considering as continuum
and in other aspects, we can also consider
discrete fluid; in the case of discrete fluids,
each molecule is treated separately and then
what is happening with respect to that molecule
is studied. So, two aspects are there in fluid
flow analysis: the continuum analysis or discrete
fluid flow analysis. Most of the time we will
be dealing with the continuum fluid flow analysis
and in very rare cases we deal with the discrete
fluid flow analysis.
In third case, we can again say the fluids
can be either perfect your ideal fluids or
the fluids can be real fluid. Real fluids,
as I mentioned earlier does not slip past
a solid wall and most of the fluids which
we are dealing with is real fluids. So, the
third case is perfect verses real fluids.
Te fourth case, as I mentioned is Newtonian
verses non Newtonian fluid. In the Newtonian
fluids we consider the qualification of viscosity
than on coefficient of viscosity mu constant
for a fixed fluid temperature and pressure.
For example, water is considered as Newtonian
fluids and then we have already seen earlier
non Newtonian fluids, when the shear stress
verses shear strain various is not linear
those fluids are non Newtonian fluids and
then the non Newtonian fluids as I mentioned
in the mu varies, say, for example, milk is
a non Newtonian fluid
In the fifth case the fluid can be classified
as compressible or incompressible fluid. As
we have seen earlier in the compressible fluid
the density changes with applied pressure;
if you will apply some pressure to the fluid
the density changes., for example we consider
just some air then we can see in a container
if you put some pressure in this container
the density inside changes, so this is what
is called compressible fluid. In incompressible
fluid, density will not change by external
force, so this kind of fluid is called incompressible
fluid, say, for example, water. When we apply
some pressure the density is not changing
accordingly, so that type of fluid is called
incompressible fluid. The sixth classification
as I mentioned in the fluids can be classified
as that the steady state or the unsteady.
In the steady fluid flow the properties like
velocity, the pressure or independent of time
or in unsteady fluid flow the property is
depending on the time.
We have already discussed the 1 dimensional,
2 dimensional and 3 dimensional flows according
to the spatial variations. In the eighth classification,
we have already seen the rotational verses
irrotational flow. Irrotational flow- there
is no rate of angular deformation of any fluid
particle, for example, potential flow can
be considered as irrotational flow. The rotational
flow-there is a rate of angular deformation
with respect to the fluid flow. These are
some of the important classifications of fluids
and fluid flow used in fluid mechanics. Based
up on these classifications the study of the
fundamental principles also slightly varies.
In the coming lectures, we will be discussing
in detail with respect to various types of
fluids; also how it will be dealing, what
would be the principle, what are the changes
as per the principles compared to the incompressible
real fluids or compressible fluids. That we
will be discussing in later lectures. Now,
finally in this introductory lecture we will
discuss how we can solve a fluid flow problem.
If we get a fluid flow problem how can we
solve the problem? The problem may be to find
out the velocities or the problem may be to
find out the pressure distribution or the
viscosity changes or density changes or any
type of engineering fluid flow problem. If
we get how we can approach the problem and
then how we can solve the problem, these questions
we will be asked generally.
Here this flow chart briefly indicates how
the fluid flow problem can be solved. In general,
as for as fluid flow analysis is concerned
there can be two types of investigations:
first one is it can be a theoretical investigation;
second one is the experimental investigation.
In theoretical investigations, what we are
doing is based up on the available theory
or based up on the fundamental principles
or based up on the available literature published
papers we are trying to approach the problem
and we are trying to solve it theoretically;
it may be theoretical background or some exact
solutions are available or analytical solutions
are available or it may be some numerical
metrologies which we can directly apply for
the problem. This is actually a mathematical
way of approach since we are trying to solve
the problem theoretically.
In case of excremental investigations, what
we are doing? We are trying to replicate the
problem in the laboratory to certain scale
so a scaling is to be done; most of the real
field problem will be very large in nature;
we cannot replicate the same in the laboratories.
What we can do is that we can reduce the dimensions;
we can change it to a certain scale so that
the problem will be small in nature and we
can easily deal in the laboratory. So, experimental
investigation is very much used in fluid mechanics.
In the last two hundred to three hundred years
experimental fluid mechanics is very much
developed in the solution of various problems.
As far as the experimental investigations
are concerned there are certain limitations.
The limitations are: it is very expensive
in nature and we have to do the scaling; through
the scaling certain aspects of the problems
may not be able to replicate in the laboratory.
So certain level of the reality of the problem
may be last in the replications of the laboratory
and then running the model in the laboratory,
the accuracy may be reduced. So, experimental
investigations have certain limitations but
if theoretical investigations or analytical
solutions are not available then it has got
some limitations, the accuracy again may be
reduced. Now, as far as theoretical investigations
are concerned how are we approaching the problem?
If the engineering fluid flow problem is given
then first we can understand the problem in
a physical nature, what is the problem, what
are the inputs data available and what are
the outputs expected what the physical principles
are based up on the problem which we have
to investigate. So this analysis is called
the physical analysis.
The physical analysis can be done based up
on the representation of the problem by a
client or so; we can go to the real problems
for example, if we are constructing a bridge
across a river then 
we have to construct pillars and then we have
to understand the flow behavior across the
river section so that we have to do fluid
flow analysis as far as the section is concerned
where the bridge is constructed. We may go
to the field and see how the river is flowing,
what are the important parameters which we
have to deal here. We have to analyze and
we have to understand the problem physically.
So we may have to go to the field or we have
to analyze in our office depending up on the
data available given by the client.
For example when we are going to construct
a bridge across the river, we have to see
depending up on the monsoon or depending upon
the rain fall through out the year how the
water level will be varying, how it is increasing
or how it is decreasing and then what will
be the flow velocity at the particular location.
We are going to construct the bridge the bridge
and then the bank of the river has any motion
problems or all these things we have to analyze
physically. This is called physical analysis.
In the case of fluid mechanics analysis this
physical analysis can be based up on the force
concept or energy concept. The force concept
means we will analyze what are the forces
acting on the particular problem which we
have dealing with and then how it is going
to affect the fluid flow properties or the
particular variable which we are going to
understand and the second concept is called
energy concept, how energy variations are
like here as they would be the energy creations;
we will be analyzing how it will be affecting
the particular problem so the physical analysis
is very important in the solution of any fluid
flow problem. Physical analysis generally
can be there upon the force concept or based
up on the energy concept. After this physical
analysis is over we can understand the problem
in which way we have to approach the problem
and we can try to make a mathematical formulation
of the problem. After this physical analysis
next step is the mathematical analysis. In
the mathematical analysis with respect to
the physical analysis we already know what
will be the domain of the problem which we
will be dealing.
As I mentioned, we are going to construct
a bridge across the river. So when the pillars
will be constructed what will be the flow
conditions? All these things will be analyzed.
We know the domain of the problem and then
also we can easily identify the data. For
example, flow of the variations at various
seasons with respect to throughout the days
of the year how the flow depth will be varying.
All these data are available; then the boundary
conditions with the problem is concerned at
particular location of the domain say the
boundary, then how the flow will be the conditions
like depth variations or velocity variation
can be identified and then based upon the
particular problem we can get the mathematical
equations; mathematical equations can be derived
based upon the theoretical analysis based
upon the first concept or the energy concept.
So, once the mathematical equations are derived
and then the boundary conditions are required
for unsteady state problem or transient problems.
But in the case of transients problems or
unsteady state problems you have to also prescribe
the initial conditions, time t is equal to
0 but the conditions may be the depth, velocity
or pressure all these parameters will be varying.
From time t equal to 0, transient analysis
or unsteady state analysis of the fluid flow
problem is starting. This initial condition
is very much important in a fluid flow analysis.
The given equations are prescribed; we have
already prescribed the boundary conditions
of the problem; with respect to this the mathematical
analysis is complete. Again with respect to
a figure here I describe, for example, if
you consider the potential flow problems,
it can flow in a homogeneous isotropic axis
which is given by del square phi is equal
to 0, the Laplace equations. If you consider
a rectangle domain like this then we know
the length and the breath. So now the dimensions
of the domain are known and the domain nature
is known. Now, for example we deal with the
potential problem, the genuine equations is
del square phi is equal to 0, the Laplace
equation. This is the steady state problem.
So, the boundary conditions can be the potential
phi is prescribed as boundary phi is equal
to phi1 here and phi is equal to phi2 here.
So the boundary condition on this way in this
domain is ABCD. In the particular case, the
boundary conditions on AD is phi equal to
phi1and similarly, on BC we can prescribe
phi is equal to phi2. Now, the given equations
are known and the boundary conditions are
known. On boundary AB, for example, if we
know the variation or the flux, del phi by
del n which is equal to 0 on the boundary
AB and CD, here del phi by del n is equal
to 0. That means no flow boundary condition
on AB and CD. So there cannot be flux and
cannot cross that means this side AB and CD
are imperil in nature.
These boundary conditions are called nature
boundary conditions that mean, what exists
in nature. So this flow cannot cross this
can imperil the boundary this side AB and
on this side CD, so these conditions are called
nature conditions or and the boundary conditions
which we prescribed on AD and BC are called
divisional boundary conditions or direct boundary
conditions.
So as far as this potential flow problem is
concerned the mathematical analysis or mathematical
statement is over since we have only described
the domain, we have already described the
boundary condition and we have already described
the given equations. Now, with respect to
this, we can solve this problem either we
can get analytical solution depending upon
the problem or we can solve the problem numerically
so depending upon the case.
This is about the mathematical analysis and
as I mentioned in the experimental investigation
we have to replicate the real problem in the
laboratory and then with respect to certain
scale we will be running the experiment what
is happening in the field that will be done
in the laboratory. Then we will be taking
some readings sometimes it can be say either
the velocity measurements are the depth variations
or pressure measurements or this measurements
will be done experimentally in the laboratory.
Then to have some specific relationships we
can give some dimensional analysis as mentioned
in this slide here. Dimensional analysis is
also very important in experimental investigation
as well as mathematical investigations. So,
dimensions with respect to the bucking phi
theorem or various theorems, we will discuss
about this dimensional analysis later.
In summary, in this introductory lecture on
fluid mechanics, we have discussed the various
aspects of fluid flow and the various fluid
properties. We have discussed the fundamental
principles which will be used in the fluid
flow analysis like the mass consideration
of momentum, consideration of energy and then
we also discussed the various fluid visualization
techniques like using dies, using smoke or
other metrologies. We have discussed various
metrologies of representing of a fluid flow
like a stream lines, potential line, streak
lines path lines will have discussed and then
we have discussed based up on the various
fluid properties we have classified the flows
with respect to space, one D flow, two D flow
and three D flow or with respect to the viscosity,
that is, viscous flow or non-viscous flow
and then finally, we have discussed about
how we can approach a fluid flow problem or
solution to a fluid flow problem.
As I mentioned, we can approach the fluid
flow problem theoretically or experimentally
and then in theoretical investigation we can
derive a mathematical model by describing
the domain, then given equation in boundary
condition. So this is about the introductory
lecture on this video course on fluid mechanics
In this second chapter on fluid mechanics
video course, we will be discussing mainly
on fluid statics.
The main objectives in this section is to
introduce the concepts of fluid pressure,
forces on solid surfaces, buoyant forces and
related theories. Secondly, we shall see the
determination of fluid pressure and forces
for various cases. Then we will emphasis in
the importance of fluid statics as far as
fluid mechanics is concerned.
Now, as I mentioned earlier, static fluids
means fluid is not moving that means fluid
is in rest. So, you can see there is some
water in a small basin. Here fluid is now
at rest; there is no movement as far as the
fluid is concerned; it is as far as the boundary
is concerned the fluid is at rest so that
we can say the fluid is in statics conditions
and the various theories on statics will be
applicable as far as a static fluid is concerned.
In solid mechanics, most of the problems we
will be analyzing are in that condition. So,
most of the theories in solid mechanics are
applicable for fluids at rest or static fluid.
In static fluids you can see that since fluid
is not moving it is at rest here, so that
there is no shearing force. Whenever the fluid
is trying to move only between the layers
there will be shearing force but as far as
fluid is at rest or static fluid there is
no shearing force. We need not have to worry
about the shearing force and now before going
to more details of fluid statics we will just
review some aspects of forces.
We can classify the forces into body forces
and surface forces. A body force acts through
a distance, for example, when there is a gravitational
force the force of gravity is not directly
acting, but it is acting from a distance so
the action is through the distance and also
if you consider the electro magnetic force
most of the time it will be acting at a distance.
This kind of forces like gravity force and
electromagnetic force acting on fluids and
that kind of study which we are dealing will
be related to the body force. The second kind
of force is called the surface force; it is
due to the virtue of direct or in between
the molecules. What is happening as far as
the fluid is concerned is if you apply a force
here on the fluids, then between the virtue
and direct contact between the fluid molecules
only there is a force. That force is called
the surface force and as the stress is concerned
we can also classify, as we have discussed
earlier, stresses as normal stresses and shear
stresses. Normal stress means when we are
considering direct normal force on the body
that kind of stress is called in a normal
stress and if a shearing force is applied
that kind of stress is called shear stress.
We will be dealing with normal stresses and
shear stresses in fluid mechanics and also
we will be dealing with xyz directions.
Here, this is x this is y this is z. So the
normal stress can be in x direction, y direction
or z direction or with respect to x and y,
with respect to x and z or with respect to
y and z. Like that the stresses will be varying,
for example, if sigma is the stress in normal
stress then if we consider a body like this
in 3 dimension then if you consider x, y and
z the normal stress we would be considering
if it is acting from x direction, the normal
stress can be sigma x and it is acting from
the y direction it can be sigma y and from
the z direction it can be sigma z. Then with
respect to x y plane if it is acting, it can
be sigma xy or sigma yx or if it is with respect
to same x and z axis, then it can be sigma
xz or sigma zx; then if it is with respect
to y and z it can be sigma yz or sigma zy.
Like that we can have stress tensor. So, with
respect to xyz direction we can classify the
stresses in x direction, y direction and z
direction or xz xy or yz direction.
Stress tensor is very much used in many of
the fluid analysis. In a very similar way,
the shear stress is concerned for example
tau is the shear stress; if it is depending
upon the problem if you consider again a cube
of three D then the shear stress also will
be acting if a shear force is applied in the
other direction and the shear force is applied
either so this other face also. According
to this we can have the shear stress in xy
plane or tau yx then tau yz or tau zy or tau
xz or tau zx.
According to the sheer force acting it will
be varying with respect to xyz. The stress
tensor for normal stresses and shear stresses
will be utilizing the shears stress tensor
with respect to xyz direction and then as
far as force is concerned, force will be especially
if static fluid is concerned force will be
acting upon externally. So, the force between
fluid and boundaries for example if you consider
it acting always at right angle boundary,
so here in this slide you can see there is
a boundary here and then say the force is
acting. With respect to the boundary generally,
the force will be acting at right angle so
you can see that in the fluids that is the
fluid is at rest. When we are applying a force
we will be considering this acting at right
angles and all the fluid static analysis will
be accordingly. So as I mentioned, most of
the theories which have been developed in
the cases fluid mechanics are also applicable
in the case of static fluid or fluid at rest.
Fluid at rest, we can consider each element
for example, this basin water is at rest so
when we consider the fluid statics or static
fluid here. If we consider each element then
the fluid element should be in equilibrium;
there is no moment and the fluid is at rest.
So, if there are n forces acting then finally
each fluid element should be in equilibrium.
If you consider any solid with respect to
solid face, if the solid is at rest and in
equilibrium then say we can consider with
respect to, for example, say this a solid
and if it at rest and if some forces are acting,
we know that the body is in equilibrium then
in solid mechanics we use the equilibrium
condition. For example, if the forces in x
directions sigma fx should be equal to 0;
this is one of the condition which will be
generally used in solid mechanics and then
sigma fy is the forces in y direction should
be 0 and then sigma fz should be 0.
So, these three conditions as far as forces
are concerned, considering the solid equilibrium
and then we consider each element in equilibrium,
so that we can say all the sum of the forces
in x directions is equal to 0; sigma fx is
equal to 0; sigma fy is equal to 0 and sigma
fz is equal to 0.
So these three equations as far as the all
the brace, some of the forces sigma fx, sigma
fy sigma fz are used in the solid mechanics.
These equations, the solid at rest, are applicable
for static fluid or fluid statics. We can
use these equations sigma fx is equal to 0,
sigma fy is equal to 0 and sigma fz is equal
to 0. Also in solid mechanics, we know that
for a solid at rest but in equilibrium condition,
when there is a force acting then the sum
of the moments of force at any point must
be 0 as far as the particular solid is concerned.
This equation is also very much used in a
fluid mechanics for fluid at rest. If we consider
that the sum of the moments, if you apply
any force and then its moment with respect
to that force acting upon the fluid at rest
and then with respect to about any particular
point sigma mx that means the sum of the moments
of force should be equal to 0.
These fundamental equations or sigma fx is
equal to 0 and sigma fy is equal to 0 sigma
fz is equal to 0, sigma Mx or the sum of the
moment of forces about any point must be 0.
These four equations are very much valid in
the case of static fluid also. These equations
generally, we use in solid mechanics and directly
these equations are applicable in the case
of static fluid. Now, another important aspect
as far as fluid statics is concerned is the
pressure.
Pressure, as I mentioned earlier, pressure
is the average of normal stresses. So if there
is force acting, force per unit area is termed
as pressure and generally we will be considering
the average of normal stresses as bulk stresses
and that term is defined as pressure. It is
the force per unit area and the general used
unit in the system international is one Pascal
which is equal to Newton per meter square
and its dimension is M L to the power minus
1 and T to the power minus 2 as shown in this
slide and say, for example, another unit which
is for bar used is 1 bar is equal to 10 to
the power 5 Newton per meter square. This
pressure is also very important as far as
fluids are concerned. We will be discussing
further about the pressure shortly for fluid
in static state and another important aspect
for fluid statics is equilibrium. Equilibrium
condition in solid mechanics which we generally
use is stable equilibrium for example, this
cylinder here is said to be stable if it is
place like this. So that it is stable; there
is no change; so it is when if we apply more
force it will come back to its normal proportion
with respect to small forces. That is called
stable equilibrium but if you consider this
cylinder if you put it in the other way like
this then you can see with a small force it
is quit unstable, so this solid in this position
it is said to be in unstable equilibrium;
then third equilibrium condition used is called
neutral equilibrium. So when it is lying like
this on this flow it is a neutral equilibrium;
whatever force is applied it comes back to
the same condition. So it is said to be neutral
equilibrium. Equilibrium conditions namely
stable equilibrium, unstable equilibrium and
neutral equilibrium are very much solid mechanics.
These equilibrium conditions are also very
much valid in the case of statics fluid which
will be discussing later.
Now, we will be discussing about the fluid
pressure. One of the important theories is
developed in seventeenth century called the
Pascal law for pressure at any point. This
law has been developed by balky Pascal by
various experiments. Here, if you consider
the fluid at rest and now we are considering
this slide fluid element and then with respect
to the fluid elements, Pascal offset with
respect to various experiment that the pressure
at any point is same in old directions.
We consider a fluid element like this; the
pressure at any point is same in all directions.
For example, if you consider the water in
this basin, if you consider the pressure at
any point in this; since the fluid is statics
there is no moment; if you consider the pressure
the pressure will be the same old directions.
This is the law derived by Pascal and it is
called a Pascal’s law. Here you can see
a fluid element in this slide. If you consider
fluid element with respect to this phase,
the pressure will be py in this direction
py into delta x into delta z; in this direction
this is the force and the other direction
it is delta x into delta y and this direction
it is x direction which we are dealing is
px into delta x into delta z.
So, if you consider fluid element like this,
then according to the Pascal law when a certain
pressure is applied at any point in a fluid
at rest the pressure is equally transmitted
in all direction and to every other points
in the fluid. If you consider this with respect
to the fluid element which we consider sigma
fy can be written as sigma fy is equal to
py into delta x in to delta z minus ps in
to delta x delta x sin theta, that is, equal
to rho into delta x delta y delta z by 2 into
ay, where rho is the density of the fluid
and ay is the acceleration of the fluid element
which we are considering. Here, we are using
the Newton’s second law: the total force
is equal to mass into acceleration.
Similarly, in z direction, sigma fz is equal
to pz into delta x into delta y minus ps in
to delta x delta x cos theta minus gamma delta
x delta y delta z by 2 since the weight of
the element is considering this term which
is equal to the mass into the acceleration
in z direction. So, here mass is rho into
delta x into delta y into delta z by 2 and
az is the acceleration.
Now, if you simplify the force, sigma fy the
total occupied force in y direction which
is equated to the mass into acceleration and
sigma fz which is equated to mass into acceleration
in dz direction. So if you simplify this you
will get py minus ps is equal to rho into
ay delta y by 2. Since, delta y can be written
as delta x cos theta and delta z can be written
as delta x sin theta, the second equation
can be written as pz minus ps is equal to
rho into az plus gamma into delta z by 2.
Now, with respect to the Pascal law if you
consider the fluid at a particular point then
you can see that delta x, delta y and delta
z will be tending to 0. So that we can see
py is equal to pz is equal to ps as shown
in this slide. That means the pressure at
any particular point is same in all directions.
So, here, if you consider the pressure at
particular point at all directions it will
be same according to the Pascal’s law
After discussing the Pascal’s law, we will
be discussing the pressure variations at various
vertical direction are shown in a direction
and then we trying to derive general equations
which can be applied in all the cases.
So, if we consider the vertical element cylinder
of fluid like this, on this phase the pressure
is even and the area is a, here the pressure
is p2 and the area of cross section is A and
the fluid density is rho and from the datum
this distance is z1 to this phase and z2 is
to the second phase; you can see that the
pressure decreases with height in vertical
direction. We can write the equation p1 into
A minus p2 into A minus rho g A into z2 minus
z1 that is equal to 0. So this gives the pressure
variation in the vertical direction.
Similarly, if you consider the pressure variation
in horizontal direction here a horizontal
fluid element, say, cylinder of fluid is considered
here and the area of cross section is same
A; the fluid density is rho and on this left-hand
side phase the pressure is p1 and the pleft
multiplied by A is equal to pright A. Since
it will be in horizontal direction on this
position in the total pressure the total force
is equal to this position, so that pressure
in horizontal direction is constant; accordingly,
the pressure variation in horizontal direction
is constant
So, with respect to this, if we consider the
pressure intensity of fluid, say, for various
shapes you can see that with respect to in
this slide the shapes are varying from the
rectangular, then the circular, then say conical
differentiates. Now, if you consider an excess
line this is at a height x, so here the fluid
pressure will be same as the surface of the
water; it is ps and then if you consider this
excess line for a fluid with specific gamma
you can see that the fluid pressure density
p is equal to gamma into h plus the fluid
surface pressure ps. So, this will be same
at all the points in this line excess. So,
this is the pressure intensity of fluid at
this particular section excess.
So, these principles we can utilize in many
areas. Now, if you consider a piston type
mechanism here, force verses area. If we consider
a small piston here and for small cylinder
here and larger cylinder which is interconnected
through in the mechanism like this, the pressure
density is same at level p and q. So, the
pressure intensity is at same the level p
and q. Pp is equal to Pq. This is the small
cylinder and this is the large cylinder of
area A2. Here, area of small cylinder area
force section is A1 and pressure intensity
is same p at this point and at this point
on lead of the cylinder and then you can see
that since the area of force section A1 is
smaller here and here the area of force section
is larger, if you get the total force f1 will
be equal to p into A1 on this phase here and
here it will be f2 is equal to p into A2.
Since the pressure intensity is same we can
transmit large quantity force. Here f2 is
equal to p into A2 is much larger compare
to f1. So, the transmission of pressure through
stationary fluid is possible with respect
to this mechanism since the pressure intensity
is same and area of force section A2 is much
larger compare to A1.
So, for larger area we apply larger force
f2. This principle is very much applicable
in many of the mechanisms like hydraulics
utilizing these principles. We can transmit
the larger pressure by applying more force
here and we can obtain larger force here.
This principle is used in lifts, hydraulic
jacks, pressure extra. What we are doing here
is we are applying a small force here f1 that
will give correspondingly large force; input
will be small and larger force output is produced
here which is used in lifts hydraulics jacks
pressures and many other hydraulics equipments.
So this principle is very much very important
in fluid statics which is based upon the theory
that the transmission of pressure through
stationary fluid.
