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Derivation of Bernoulli's Equation, Fluid Mechanics, Mechanical Engineering, GATE Video Lecture
FAQs on Derivation of Bernoulli's Equation, Fluid Mechanics, Mechanical Engineering, GATE Video Lecture
| 1. What is Bernoulli's equation? |  |
Ans. Bernoulli's equation is a fundamental equation in fluid mechanics that relates the pressure, velocity, and elevation of a fluid at any point along a streamline. It states that the sum of the pressure energy, kinetic energy, and potential energy per unit volume of a fluid remains constant along a streamline, assuming no energy losses due to friction or external work.
| 2. How is Bernoulli's equation derived? | |
Ans. Bernoulli's equation can be derived by applying the principle of conservation of energy to a fluid flowing along a streamline. The derivation involves considering the work done by pressure forces, gravitational forces, and the change in kinetic energy of the fluid. By equating the initial and final energy states, Bernoulli's equation can be obtained.
| 3. What are the assumptions made in deriving Bernoulli's equation? | |
Ans. The derivation of Bernoulli's equation requires the following assumptions:
- Steady flow: The flow conditions do not change with time.
- Incompressible fluid: The density of the fluid is assumed constant.
- Non-viscous fluid: There are no internal frictional forces within the fluid.
- Negligible external work: The work done by external forces, such as pumps or turbines, is assumed to be negligible.
- Along a streamline: The equation is valid only along a particular path of fluid particles known as a streamline.
| 4. What are the applications of Bernoulli's equation in engineering? | |
Ans. Bernoulli's equation has several applications in engineering, including:
- Venturi meters and orifice plates: These devices use Bernoulli's equation to measure the flow rate of fluids.
- Aircraft aerodynamics: Bernoulli's equation helps explain the lift generated by an aircraft wing.
- Pipe flow: It is used to analyze the flow of fluids in pipes, including the design of water distribution systems.
- Pump and turbine design: Bernoulli's equation is used in the design and analysis of pumps and turbines.
- Wind turbine design: It helps in understanding the fluid flow around wind turbine blades.
| 5. What are the limitations of Bernoulli's equation? | |
Ans. While Bernoulli's equation is a useful tool in fluid mechanics, it has certain limitations. These include:
- It applies only to steady, incompressible, and non-viscous flow conditions.
- It neglects the effects of friction and viscosity, which can be significant in real-world scenarios.
- It assumes no energy losses due to external work or heat transfer.
- It is valid only along a streamline, and cannot be directly applied to situations involving flow separation or turbulence.
Text Transcript from Video
here this is Virgil commands here and in
this video tutorial I am going to share
with you how to derive Bernoulli's
equation right so probably this total
would be quite lengthy but yes yeah this
is really going to help you to
understand about it so let's get started
okay so first of all we are I am
assuming few things just like Delta e is
cross sectional area cross sectional
area and you I must say U is a fluid
velocity and of course what exactly once
in other thing that is a heavy mean that
is we are going to derive this equation
will be the string Y is this direction
and we are considering it as key okay so
these things we are assuming here in
this derivation now we go for the
diagram which is really important for
the derivation point of view
okay and this is its center line suppose
that field is falling through this right
so this pipe you might say that so let's
say this is Theta and okay a B and C
this is a triangle let's say this is you
previously you and beside u Plus D u by
DT into Delta G okay so and this one is
Delta K this is Delta H so this thing
these things you need to remember for
this derivation right now what exactly
we know we know all simple things
against that is pressure pressure is
equal to force by area right first
impulse formula we are going to use in
this derivation always one is 10p
density is equal to mass by volume we
also use this formula
and of course another thing we'll use
that is integration integration so three
most important and basic things we are
going to use in this valuation I hope
you understand this right now let's go
for it so if we can say that for
pressure is equal to force by area then
we can say force is equal to patient
multiple area isn't that so this is like
simple now what we can say
net force F is equal to special Delta in
cross sectional area we have already
considered minus P plus DP by DT into
Delta G's multiple Delta a minus mg
sine-theta
okay so one thing I would like to share
with you if you remember this and then
think done
I mean you don't need to fall L don't
need to be in order to be create any
Parliament thing as a free members case
equation and this is really simple to
remember right love do little bit
syncopation P Delta I'll multiple this P
Delta e minus DP by D K Delta K into
Delta a minus mg sine theta so this one
this one get cancel so what we got minus
DP by P K Delta G multiple Delta a is
minus mg sine theta so this is the
equation we got from here now what we
can we know basically I have already
share with you density is equal to mass
white volume so we can say we can say
mass is equal to density into volume so
density is Rho and volume stream Delta
cake multiple area cross section area
okay now we know another thing
acceleration acceleration is equal to
velocity divided by time therefore
velocity is what here you see you buy BK
multiple Delta P minus u divided by
delta T time now this do comment just to
calculation this one and this one get
cancelled so what we got
bu-bye de multiple Delta K by delta T so
what we can write from here that is you
multiple you multiple do you by DT so
this is the thing we can write from here
because what is you basically because U
is equal to no be GU PI D K so TK by ET
this is Q flute band with AD sling it's
case stream wise Direction stream and
this is time so distance by time is one
in this basement by time is known as
Spano City so this is about you right so
what we got from here we caught mask
week on acceleration okay so these two
things we caught from here now let's use
another basic thing that is Newton's
Newton's second law
so what later second law to us that
force is equal to mass into acceleration
we know that force is equal to mass into
acceleration and mass what we we have
used here mass that is density Delta P
multiple Delta acceleration you do by DK
we have used this okay do by degrees now
what is force we got from the diffusion
okay
this is force we got from this equation
forces so we have to just write it down
you know okay so what I got basically DP
by D K Delta K Delta Connection ba-dee
ah yeah it's Delta k DP by D K Delta K
da and must say that so d a minus mg
multiple Delta R sorry to Delta P Delta
is V sine theta we have written all
these right so this is about this is
about all about the notification right
so I hope you understand it right so
let's go for the next steps that is here
only now actually what exactly doing
with 330 if you know math into mg what
is MV basically we have already written
what is MV here mg okay so MD please is
here M what is M basically and M it must
so we can say density is equal to mass
or volume and mass is equal to density
into volume what is density volume is
here so in case of MV we can write this
one so to simplify if I'll simplify this
what I can write this again that is also
important that is you know - DP
bar-b-que Delta K Delta a okay all term
minus Rho Delta K Delta a into G Delta H
into G multiple multiple a sine theta is
equal to Rho Delta G Delta e you bu-bye
thinking okay got it so till now what we
have done we have just done
simplification okay now from this time
let's see the triangle ABC
so from triangle ABC what we can say
sine theta is equal to P by H so py pass
means here P is C V by C okay so C V is
Delta exact and C is Delta T's so here
in place of sine theta what we can write
basically Delta set by delta T this is
the thing we can write here okay now
what we have to do basically that is
also important so what is bad
let me share with you so what's that
admission with you so what we do we'll
just put this value first here and Doda
distance education so minus DP by D K
Delta X Delta Y minus Rho Delta P Delta
P Delta J Delta Rho is equal to Rho
Delta K Delta u do you find DT now we do
just be reason all the all members on
both sides we will divide both side we
will divide by Delta P and it is what
we'll get minus DP by DT minus Rho G
Delta sad file so Delta got which is
known Delta that we can write that these
at five PT right is equal to Rho let's
you know this we can check so Rho u tu
by BK in the simple it is really simple
right now what we can do basically we
can send this number to this side so
what will happen DP 1 DK plus Rosie
DZ by DT is equal to o plus Rho u du u
by DT is equal to u if we'll do it in
simplification here then this is known
as Euler's equation and it will do
little simplification here then this is
known as linear equation okay now simple
we will after in creation of this
equation is known as Bernoulli's
equation I just need to integrate this
and to fiddle with simplification then
what will happen this equation become
Bernoulli's equation so that is the
interesting part right so let's go for
it
those integration first so what exactly
we are going to do we are going to do
integration here that is DP by DK plus
Rho G DZ y DK plus Rho u D u by DP is it
continue after integration what we get P
plus Rho GZ plus Rho u square by 2 is
equal to C ok now exactly one thing I
have to do that is you know before this
integration I can do I can divide okay
after integration we can do because this
is constant okay there is no values of
this constant so we can do anything with
space because this constant having any
number of values and this is also known
as arbitrary constant right so we have
to be midis number with whole number by
Rho checked so what we get P by Rho G
Plus u is square by 2g
Tolga cancel Plus said is equal to
constant
and you know this is known as
Bernoulli's equation I hope you
understand each and every thing of this
equation and probably after watching
this video will not give any problem to
derive Bernoulli's equation well thanks
for watching see you soon in my next
video bye
this video tutorial I am going to share
with you how to derive Bernoulli's
equation right so probably this total
would be quite lengthy but yes yeah this
is really going to help you to
understand about it so let's get started
okay so first of all we are I am
assuming few things just like Delta e is
cross sectional area cross sectional
area and you I must say U is a fluid
velocity and of course what exactly once
in other thing that is a heavy mean that
is we are going to derive this equation
will be the string Y is this direction
and we are considering it as key okay so
these things we are assuming here in
this derivation now we go for the
diagram which is really important for
the derivation point of view
okay and this is its center line suppose
that field is falling through this right
so this pipe you might say that so let's
say this is Theta and okay a B and C
this is a triangle let's say this is you
previously you and beside u Plus D u by
DT into Delta G okay so and this one is
Delta K this is Delta H so this thing
these things you need to remember for
this derivation right now what exactly
we know we know all simple things
against that is pressure pressure is
equal to force by area right first
impulse formula we are going to use in
this derivation always one is 10p
density is equal to mass by volume we
also use this formula
and of course another thing we'll use
that is integration integration so three
most important and basic things we are
going to use in this valuation I hope
you understand this right now let's go
for it so if we can say that for
pressure is equal to force by area then
we can say force is equal to patient
multiple area isn't that so this is like
simple now what we can say
net force F is equal to special Delta in
cross sectional area we have already
considered minus P plus DP by DT into
Delta G's multiple Delta a minus mg
sine-theta
okay so one thing I would like to share
with you if you remember this and then
think done
I mean you don't need to fall L don't
need to be in order to be create any
Parliament thing as a free members case
equation and this is really simple to
remember right love do little bit
syncopation P Delta I'll multiple this P
Delta e minus DP by D K Delta K into
Delta a minus mg sine theta so this one
this one get cancel so what we got minus
DP by P K Delta G multiple Delta a is
minus mg sine theta so this is the
equation we got from here now what we
can we know basically I have already
share with you density is equal to mass
white volume so we can say we can say
mass is equal to density into volume so
density is Rho and volume stream Delta
cake multiple area cross section area
okay now we know another thing
acceleration acceleration is equal to
velocity divided by time therefore
velocity is what here you see you buy BK
multiple Delta P minus u divided by
delta T time now this do comment just to
calculation this one and this one get
cancelled so what we got
bu-bye de multiple Delta K by delta T so
what we can write from here that is you
multiple you multiple do you by DT so
this is the thing we can write from here
because what is you basically because U
is equal to no be GU PI D K so TK by ET
this is Q flute band with AD sling it's
case stream wise Direction stream and
this is time so distance by time is one
in this basement by time is known as
Spano City so this is about you right so
what we got from here we caught mask
week on acceleration okay so these two
things we caught from here now let's use
another basic thing that is Newton's
Newton's second law
so what later second law to us that
force is equal to mass into acceleration
we know that force is equal to mass into
acceleration and mass what we we have
used here mass that is density Delta P
multiple Delta acceleration you do by DK
we have used this okay do by degrees now
what is force we got from the diffusion
okay
this is force we got from this equation
forces so we have to just write it down
you know okay so what I got basically DP
by D K Delta K Delta Connection ba-dee
ah yeah it's Delta k DP by D K Delta K
da and must say that so d a minus mg
multiple Delta R sorry to Delta P Delta
is V sine theta we have written all
these right so this is about this is
about all about the notification right
so I hope you understand it right so
let's go for the next steps that is here
only now actually what exactly doing
with 330 if you know math into mg what
is MV basically we have already written
what is MV here mg okay so MD please is
here M what is M basically and M it must
so we can say density is equal to mass
or volume and mass is equal to density
into volume what is density volume is
here so in case of MV we can write this
one so to simplify if I'll simplify this
what I can write this again that is also
important that is you know - DP
bar-b-que Delta K Delta a okay all term
minus Rho Delta K Delta a into G Delta H
into G multiple multiple a sine theta is
equal to Rho Delta G Delta e you bu-bye
thinking okay got it so till now what we
have done we have just done
simplification okay now from this time
let's see the triangle ABC
so from triangle ABC what we can say
sine theta is equal to P by H so py pass
means here P is C V by C okay so C V is
Delta exact and C is Delta T's so here
in place of sine theta what we can write
basically Delta set by delta T this is
the thing we can write here okay now
what we have to do basically that is
also important so what is bad
let me share with you so what's that
admission with you so what we do we'll
just put this value first here and Doda
distance education so minus DP by D K
Delta X Delta Y minus Rho Delta P Delta
P Delta J Delta Rho is equal to Rho
Delta K Delta u do you find DT now we do
just be reason all the all members on
both sides we will divide both side we
will divide by Delta P and it is what
we'll get minus DP by DT minus Rho G
Delta sad file so Delta got which is
known Delta that we can write that these
at five PT right is equal to Rho let's
you know this we can check so Rho u tu
by BK in the simple it is really simple
right now what we can do basically we
can send this number to this side so
what will happen DP 1 DK plus Rosie
DZ by DT is equal to o plus Rho u du u
by DT is equal to u if we'll do it in
simplification here then this is known
as Euler's equation and it will do
little simplification here then this is
known as linear equation okay now simple
we will after in creation of this
equation is known as Bernoulli's
equation I just need to integrate this
and to fiddle with simplification then
what will happen this equation become
Bernoulli's equation so that is the
interesting part right so let's go for
it
those integration first so what exactly
we are going to do we are going to do
integration here that is DP by DK plus
Rho G DZ y DK plus Rho u D u by DP is it
continue after integration what we get P
plus Rho GZ plus Rho u square by 2 is
equal to C ok now exactly one thing I
have to do that is you know before this
integration I can do I can divide okay
after integration we can do because this
is constant okay there is no values of
this constant so we can do anything with
space because this constant having any
number of values and this is also known
as arbitrary constant right so we have
to be midis number with whole number by
Rho checked so what we get P by Rho G
Plus u is square by 2g
Tolga cancel Plus said is equal to
constant
and you know this is known as
Bernoulli's equation I hope you
understand each and every thing of this
equation and probably after watching
this video will not give any problem to
derive Bernoulli's equation well thanks
for watching see you soon in my next
video bye
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