What is the equation of continuity?

In summary, the equation of continuity relates to the conservation of mass principle by expressing the volume conservation in terms of mathematical equations. The equation states that the mass in minus the mass out is equal to the rate of accumulation of mass. This concept is similar to Newton's Law of Gravity, which states that what goes up must come down. The equation can be written in different forms, such as with or without the nabla operator, but it still captures the principle of mass conservation.
  • #1
avito009
184
4
OP warned about asking questions that can easily be Googled.
How does equation of continuity relate to conservation of mass principle? Laymans terms please.
 
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  • #2
The first hit I got on Google: http://theory.uwinnipeg.ca/mod_tech/node65.html

The ``continuity equation'' is a direct consequence of the rather trivial fact that what goes into the hose must come out. The volume of water flowing through the hose per unit time (i.e. the flow rate) at the left must be equal to the flow rate at the right or in fact anywhere along the hose. Moreover, the flow rate at [any] point in the hose is equal to the area of the hose at that point times the speed with which the fluid is moving:
 
  • #3
The idea that the flow rate is identical at every point along the hose traces back to mass conservation, along with the fact that water is incompressible and that the hose does not change diameter over time.
 
  • #4
avito009 said:
How does equation of continuity relate to conservation of mass principle? Laymans terms please.
Equation of continuity (in layman's terms):
What goes in, must come out. :wink:

This is analogous to Newton's Law of Gravity (also in layman's terms):
What goes up, must come down. :wink:
 
  • #5
The equation of continuity translates the principle of conservation of mass into the language of mathematics. It says that mass in minus mass out is equal to the rate of accumulation of mass. What mathematical form of the equation of continuity are you trying to understand? If you provide that equation, I will tell you how each term relates to mass in, mass out, and rate of accumulation.

Chet
 
  • #6
avito009 said:
How does equation of continuity relate to conservation of mass principle? Laymans terms please.
The equation of continuity reflects the volume concervation:
[itex]Q=\frac{V}{t}=\int_{0}^{S}w(s)ds[/itex]
where w is velocity and S is the cross section area,
and mass is related to volume as it is integral of dencity over the volume:
[itex]m=\int_{0}^{V}\rho (v)dv[/itex]
when
[itex]\rho =const(v)[/itex]
water at no very high pressure ie,
[itex]m=\int_{0}^{V}\rho (v)dv = \rho\cdot\int_{0}^{V}dv = \rho\cdot V[/itex]
the mass per time:
[itex]G=\frac{dm}{dt}= \rho\cdot Q[/itex]
and Q is shown at the 3-rd line of the post.
 
  • #7
I don't think that what has been discussed in most of the posts in this thread reflect what the OP was thinking of when he asked about the continuity equation. I think that the form of the continuity equation he was referring to was:
$$\frac{\partial ρ}{\partial t}+\vec{∇}\centerdot (ρ\vec{v})=0$$
The rest of the posts in the thread are just specific examples of how this equation can be applied in practice.

Chet
 
  • #8
Chestermiller said:
I think that the form of the continuity equation he was referring to was
I'm interested too what OP mean. But I see that all who tried to answer tried to ralate the "continuity equation" and the "conservation of mass principle". And all tried to follow the paradigm "Laymans terms".
Is the nabla operator following that paradigm?:smile:
 
  • #9
IgorIGP said:
I'm interested too what OP mean. But I see that all who tried to answer tried to ralate the "continuity equation" and the "conservation of mass principle". And all tried to follow the paradigm "Laymans terms".
Is the nabla operator following that paradigm?:smile:
My understanding was what that he was asking to explain what the equation was saying in layman's terms. So I guess the nabla operator is not inconsistent with that paradigm, if the equation he was referring to contained the nabla parameter.
 
  • #10
Chestermiller said:
My understanding was what that he was asking to explain what the equation was saying in layman's terms. .
I see that if you do not to see the:
avito009 said:
How does equation of continuity relate to conservation of mass principle
you will not see that what ever all we do.
Chestermiller said:
if the equation he was referring to contained the nabla parameter
How much books it need to link showing that equation without the nabla symbol exist? That equation may be written even in no differential form as
$$v_i \cdot s_i = const $$
To be honest it needs to say that the v is the velocity that is the output-averaging by the cross section (do not sure on how it must be said english).
And since that averaging is based on the law of conservation of mass, here, too, there is an opportunity to show this relationship "laymans terms".
 
  • #11
IgorIGP said:
I see that if you do not to see the:

you will not see that what ever all we do.

How much books it need to link showing that equation without the nabla symbol exist? That equation may be written even in no differential form as
$$v_i \cdot s_i = const $$
To be honest it needs to say that the v is the velocity that is the output-averaging by the cross section (do not sure on how it must be said english).
I have no idea what you are saying. The form of the continuity equation I wrote appears in every book on fluid mechanics and transport phenomena (exactly as I wrote it), and captures mathematically the principle of conservation of mass.
 
  • #12
Chestermiller said:
I have no idea what you are saying.
I see. I have a lot of sources but they are not english. That's why I show the first-google-paged links:
http://theory.uwinnipeg.ca/mod_tech/node65.html
http://www.fsl.orst.edu/geowater/FX3/help/8_Hydraulic_Reference/Continuity_Equation.htm
and so on. Such links can be found by request equation of continuity.
I am sorry that our doalog is so strange but I do not see the sense to continue. We have about a one hour after midnight now. Good by mr Chestermiller.:smile:
 
  • #13
IgorIGP said:
I see. I have a lot of sources but they are not english. That's why I show the first-google-paged links:
http://theory.uwinnipeg.ca/mod_tech/node65.html
http://www.fsl.orst.edu/geowater/FX3/help/8_Hydraulic_Reference/Continuity_Equation.htm
and so on. Such links can be found by request equation of continuity.
I am sorry that our doalog is so strange but I do not see the sense to continue. We have about a one hour after midnight now. Good by mr Chestermiller.:smile:
I don't even know what we are arguing about, or even if we are arguing. Are you saying that the equation that I wrote is incorrect, or are you just saying that you don't think (based on some sort of sixth sense) that the standard continuity equation I wrote is what the OP was referring to?

My Warning finger is getting itchy.

Chet
 

What is the equation of continuity?

The equation of continuity is a fundamental principle in fluid mechanics that expresses the relationship between the velocity, density, and cross-sectional area of a fluid flow. It states that the mass flow rate at any point along a streamline must remain constant.

How is the equation of continuity derived?

The equation of continuity can be derived from the principle of conservation of mass, which states that mass cannot be created or destroyed. By considering a small volume of fluid flowing through a pipe with varying cross-sectional areas, the equation of continuity can be derived through the application of the conservation of mass principle.

Why is the equation of continuity important?

The equation of continuity is important because it allows us to understand and analyze fluid flow in various systems. It is a fundamental concept in fluid mechanics and is used in many applications, such as in the design of pipes, pumps, and turbines.

What are the limitations of the equation of continuity?

The equation of continuity assumes that the fluid is incompressible and that there is no loss or gain of mass from the system. This means that it is not applicable to compressible fluids, such as gases, and systems with significant changes in pressure or temperature.

Can the equation of continuity be applied to non-uniform flows?

Yes, the equation of continuity can be applied to non-uniform flows as long as the principle of conservation of mass is still valid. This means that the mass flow rate must remain constant at any point in the flow. However, the equation may become more complex and require additional factors to be considered in non-uniform flow situations.

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