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Continuity Equation for Cylindrical Coordinates, Fluid Mechanics, Mechanical Engineering, GATE Video Lecture
FAQs on Continuity Equation for Cylindrical Coordinates, Fluid Mechanics, Mechanical Engineering, GATE Video Lecture
| 1. What is the continuity equation in fluid mechanics? |  |
Ans. The continuity equation in fluid mechanics is a mathematical equation that states that the mass of fluid entering a control volume must be equal to the mass of fluid leaving the control volume, assuming there are no sources or sinks of mass within the control volume. It can be expressed as:
$$\frac{{\partial(\rho A)}}{{\partial t}} + \frac{{\partial(\rho A u)}}{{\partial r}} = 0$$
where $\rho$ is the density of the fluid, $A$ is the cross-sectional area, $u$ is the velocity of the fluid, and $r$ is the radial distance from the axis of symmetry in cylindrical coordinates.
| 2. How is the continuity equation derived in cylindrical coordinates? | |
Ans. The continuity equation in cylindrical coordinates can be derived by considering a control volume in the shape of a cylinder. By applying the principle of conservation of mass, the rate of change of mass within the control volume is equal to the net mass flow rate through its boundaries.
The control volume is divided into two parts, the inlet and outlet sections, and the mass flow rates through these sections are calculated. By equating the mass flow rate at the inlet to the mass flow rate at the outlet, we can derive the continuity equation in cylindrical coordinates.
| 3. What is the significance of the continuity equation in fluid mechanics? | |
Ans. The continuity equation is of great significance in fluid mechanics as it represents the conservation of mass. It ensures that the mass of fluid remains constant within a control volume, which is essential for the proper analysis and understanding of fluid flow. The equation helps in predicting the behavior of fluids in various engineering applications, such as pipe flow, pump design, and turbine performance.
| 4. Can the continuity equation be applied to compressible fluids? | |
Ans. The continuity equation, as derived in the given article, assumes incompressible flow, where the density of the fluid remains constant. However, it can be extended to compressible fluids by considering variations in density with respect to time and space. In such cases, the continuity equation is modified to include terms related to compressibility effects, such as changes in density and velocity of the fluid.
| 5. How is the continuity equation used in mechanical engineering? | |
Ans. The continuity equation is widely used in mechanical engineering, particularly in the design and analysis of fluid systems. It helps engineers to determine the flow rates, velocities, and pressure variations in pipes, ducts, and other fluid-carrying systems. By solving the continuity equation along with other governing equations, engineers can optimize the performance of various devices, such as pumps, turbines, and heat exchangers, ensuring efficient and safe operation.
Text Transcript from Video
so in this video total you are going to
learn about how to drive continued
equation for cynical coordinates right
so no more time let's get started make
this diagram it will video that's why
it's terribly are there would not be any
hotchpotch right Phi this is Phi this is
you see inside you will see cuddly
cuddly you see bike ugly the dashing is
our this is still dark and this side I'm
taking you five careful here you Phi
plus curly you for you by curly in this
direction five only are should be here
multiple are Delphi okay this is for the
taiga this is known as Phi nu so this
diagram is most important for you know
for continuous technical courses this is
liquor coordinates we are taking and
just follow my step and keep doing by
yourself also take a pen and paper and
just stop are doing derivation because
only watching derivation cannot help you
to can definitely can help you to
understand it but if you wanted to do
derivation in the domination and you
have to practice you have to drive into
yourself at least twice or thrice then
only you can able to derive in the
examination okay because my videos will
not be there to help you so you will be
there only so you are the one who need
to understand all things right so let's
get started so what happens if I have
seen mass flow rate entering entering
radially okay
little under here radius I am taking as
R is equal to you see ro are curly why
curly Jed okay now if something is
entering and it is a open channel that
definitely something would be leaving so
I can say a mass flow rate leaving
radially at radius R you can see right
is equal to what will happen now here is
the thing you need to understand if
something that is entering until
definitely something is living but here
radius is r plus del R or not only are
because they are small but I am taking
considering here a small part that small
part even L dinner so what will happen
in this clip you will see curly you see
Callie are dinner multiple ro R plus
della curly size on them curling charge
okay now you can clearly see the
differences between this equation and
this equation just because of here and
here okay so this is about the equation
and now what we'll have to find out if a
net net flow rate or net things are
remaining in these are in this control
volume I can say so before going to do
that to multiplication of these things
because here I am getting lot of things
to multiply with the multiplication so
he cooks time and of course takes for
you and it will take time for you to
understand the whole things also right
so first of all what I am doing the
thing about it will get four thumbs here
okay just multiple you see with this one
so you see ro ah
color Delphi and del Z so you're not
look at them
you see Kelly are Dell are the same
thing I am writing here yeah
so the same thing I am writing here then
and both are plus Taylor curly fide
collegiate right now again we have to
multiply will get two terms from here we
get two terms from here so what exactly
we're getting row you see right are
curly five collegiate plus Rho u C del R
calif i college ad plus curly you see
Callias del are in delphi candle jet in
row in our plus you can say curly you
see Callias ella ella del phi del jet
whoa and definitely they're not the are
because there is only one so now think
about it here all these small terms
because of this small terms you for
multiple point one multiple point one
then this will become more smaller right
so because of this thumb is quietly
small this is really small that's why we
are neglecting this term and we are
considering all these freedoms by
getting my point because this term it
quite in a much quite a smaller class
one now if you are thinking about net
net mas big living in the radial
direction so we have to find a much
stronger space right so we have to find
out that mass rate leaving in de
radial directions okay so we have to
find out this only so here is it
you see Rho del R del Phi and then said
plus Scully you see by Kelly arm through
R del v del R and then jet soda fizzing
have a technology now hot how I'm
kidding using that is what exactly I'm
doing and just the vegetable equation is
equation two and this is equation one
specifically what I am doing I am doing
equation 2 minus equation one after
doing this I am getting this thing are
you getting my point
now this is really important ready to be
really important this thing don't
require is going to what aim to do me to
subtract now now look at the common
terms we are getting here we are getting
common terms here as this and this right
and we can see calif i Callie are and
collegiate is believing okay total
volume is incident so just right through
a living you see couriers row are yes
this is assume we are getting now we can
get outside what we can get these things
coming on some so row curly be you will
see Colleen simple curly R into R so
this is the thing we get in case of what
direction in case of direction that is
the radial tension so radial direction
we are getting right now thinking of
yourself we have done we have call we
have done equation for only one
direction that is radial direction now
what we have to do now we have to do in
only for hydration as well as chanted
chat direction we'll have to do equation
for other both equations so let's go for
it
similarly similarly Phi Direction what
will you get for production we are
getting here u Phi plus curly u5y
are 35 then our alpha and after multiple
this way row row curly are and
collegiate - EU v curly del R del Z so
what is that we are getting here one by
our curly yo5 curly Phi Rho del V okay
the color this one and this one get
cancel so we'll get this one already now
I can say for that direction what
exactly we are getting who gets a
direction attention we are getting here
use that plus curly you check Wycherley
bad days at night and here row
stellar multiples are alpha minus u Z
row L are multiple are multiple Delphi
now but a supplicating see like this
like this here what happened this one
and this one get canceled here also this
one and this one it canceled so what
will get here curly uture for collegiate
Rho Delta V why this happening because I
have already share with you this thing
we are considering here right that means
del Phi del R + del Z is equal to Del V
so because of that which way are we have
done this here also and here also we
have done this now what we have to find
out Lester f ik flux across such little
details control volume
gee because we know that so we can see
net reflects across da across top
cylindrical Linda can control volume is
soy is G so what we can write
you see why our curly you see by curly
are Rho del v1 by are are you hi boy
curly Phi Rho del V Plus curly user by
curling sad Rho del P is equal to 0 so
we can sell em as you can get coming up
and that size of aquatic 8 u uu c by r
plus curly you see by curly r plus 1 by
r curly u phi by curly Phi plus can we
use that boy that is that is equal to 0
so this is a situation we call continues
equation for the case of physical
coordinate the TSO cylindrical calling
it a fourth I have share with you
already how to find out a continuity
equation for Cartesian coordinates in my
my videos and definitely that one is
what's important this one strand Raquel
coordinate so if you are preparing for
University solution then for his
physical coordinates is also important
and of course is required time for
deriving the equation I hope you
understand and it will help you to get
good marks in the exhibition Vista plus
for information thank you very much for
watching and thanks a lot bye-bye
learn about how to drive continued
equation for cynical coordinates right
so no more time let's get started make
this diagram it will video that's why
it's terribly are there would not be any
hotchpotch right Phi this is Phi this is
you see inside you will see cuddly
cuddly you see bike ugly the dashing is
our this is still dark and this side I'm
taking you five careful here you Phi
plus curly you for you by curly in this
direction five only are should be here
multiple are Delphi okay this is for the
taiga this is known as Phi nu so this
diagram is most important for you know
for continuous technical courses this is
liquor coordinates we are taking and
just follow my step and keep doing by
yourself also take a pen and paper and
just stop are doing derivation because
only watching derivation cannot help you
to can definitely can help you to
understand it but if you wanted to do
derivation in the domination and you
have to practice you have to drive into
yourself at least twice or thrice then
only you can able to derive in the
examination okay because my videos will
not be there to help you so you will be
there only so you are the one who need
to understand all things right so let's
get started so what happens if I have
seen mass flow rate entering entering
radially okay
little under here radius I am taking as
R is equal to you see ro are curly why
curly Jed okay now if something is
entering and it is a open channel that
definitely something would be leaving so
I can say a mass flow rate leaving
radially at radius R you can see right
is equal to what will happen now here is
the thing you need to understand if
something that is entering until
definitely something is living but here
radius is r plus del R or not only are
because they are small but I am taking
considering here a small part that small
part even L dinner so what will happen
in this clip you will see curly you see
Callie are dinner multiple ro R plus
della curly size on them curling charge
okay now you can clearly see the
differences between this equation and
this equation just because of here and
here okay so this is about the equation
and now what we'll have to find out if a
net net flow rate or net things are
remaining in these are in this control
volume I can say so before going to do
that to multiplication of these things
because here I am getting lot of things
to multiply with the multiplication so
he cooks time and of course takes for
you and it will take time for you to
understand the whole things also right
so first of all what I am doing the
thing about it will get four thumbs here
okay just multiple you see with this one
so you see ro ah
color Delphi and del Z so you're not
look at them
you see Kelly are Dell are the same
thing I am writing here yeah
so the same thing I am writing here then
and both are plus Taylor curly fide
collegiate right now again we have to
multiply will get two terms from here we
get two terms from here so what exactly
we're getting row you see right are
curly five collegiate plus Rho u C del R
calif i college ad plus curly you see
Callias del are in delphi candle jet in
row in our plus you can say curly you
see Callias ella ella del phi del jet
whoa and definitely they're not the are
because there is only one so now think
about it here all these small terms
because of this small terms you for
multiple point one multiple point one
then this will become more smaller right
so because of this thumb is quietly
small this is really small that's why we
are neglecting this term and we are
considering all these freedoms by
getting my point because this term it
quite in a much quite a smaller class
one now if you are thinking about net
net mas big living in the radial
direction so we have to find a much
stronger space right so we have to find
out that mass rate leaving in de
radial directions okay so we have to
find out this only so here is it
you see Rho del R del Phi and then said
plus Scully you see by Kelly arm through
R del v del R and then jet soda fizzing
have a technology now hot how I'm
kidding using that is what exactly I'm
doing and just the vegetable equation is
equation two and this is equation one
specifically what I am doing I am doing
equation 2 minus equation one after
doing this I am getting this thing are
you getting my point
now this is really important ready to be
really important this thing don't
require is going to what aim to do me to
subtract now now look at the common
terms we are getting here we are getting
common terms here as this and this right
and we can see calif i Callie are and
collegiate is believing okay total
volume is incident so just right through
a living you see couriers row are yes
this is assume we are getting now we can
get outside what we can get these things
coming on some so row curly be you will
see Colleen simple curly R into R so
this is the thing we get in case of what
direction in case of direction that is
the radial tension so radial direction
we are getting right now thinking of
yourself we have done we have call we
have done equation for only one
direction that is radial direction now
what we have to do now we have to do in
only for hydration as well as chanted
chat direction we'll have to do equation
for other both equations so let's go for
it
similarly similarly Phi Direction what
will you get for production we are
getting here u Phi plus curly u5y
are 35 then our alpha and after multiple
this way row row curly are and
collegiate - EU v curly del R del Z so
what is that we are getting here one by
our curly yo5 curly Phi Rho del V okay
the color this one and this one get
cancel so we'll get this one already now
I can say for that direction what
exactly we are getting who gets a
direction attention we are getting here
use that plus curly you check Wycherley
bad days at night and here row
stellar multiples are alpha minus u Z
row L are multiple are multiple Delphi
now but a supplicating see like this
like this here what happened this one
and this one get canceled here also this
one and this one it canceled so what
will get here curly uture for collegiate
Rho Delta V why this happening because I
have already share with you this thing
we are considering here right that means
del Phi del R + del Z is equal to Del V
so because of that which way are we have
done this here also and here also we
have done this now what we have to find
out Lester f ik flux across such little
details control volume
gee because we know that so we can see
net reflects across da across top
cylindrical Linda can control volume is
soy is G so what we can write
you see why our curly you see by curly
are Rho del v1 by are are you hi boy
curly Phi Rho del V Plus curly user by
curling sad Rho del P is equal to 0 so
we can sell em as you can get coming up
and that size of aquatic 8 u uu c by r
plus curly you see by curly r plus 1 by
r curly u phi by curly Phi plus can we
use that boy that is that is equal to 0
so this is a situation we call continues
equation for the case of physical
coordinate the TSO cylindrical calling
it a fourth I have share with you
already how to find out a continuity
equation for Cartesian coordinates in my
my videos and definitely that one is
what's important this one strand Raquel
coordinate so if you are preparing for
University solution then for his
physical coordinates is also important
and of course is required time for
deriving the equation I hope you
understand and it will help you to get
good marks in the exhibition Vista plus
for information thank you very much for
watching and thanks a lot bye-bye
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