# What is the equation of continuity?

How does equation of continuity relate to conservation of mass principle? Laymans terms please.

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DrClaude
Mentor
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:

jbriggs444
Homework Helper
2019 Award
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.

SteamKing
Staff Emeritus
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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.

This is analogous to Newton's Law of Gravity (also in layman's terms):
What goes up, must come down.

Chestermiller
Mentor
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

How does equation of continuity relate to conservation of mass principle? Laymans terms please.
The equation of continuity reflects the volume concervation:
$Q=\frac{V}{t}=\int_{0}^{S}w(s)ds$
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:
$m=\int_{0}^{V}\rho (v)dv$
when
$\rho =const(v)$
water at no very high pressure ie,
$m=\int_{0}^{V}\rho (v)dv = \rho\cdot\int_{0}^{V}dv = \rho\cdot V$
the mass per time:
$G=\frac{dm}{dt}= \rho\cdot Q$
and Q is shown at the 3-rd line of the post.

Chestermiller
Mentor
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

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?

Chestermiller
Mentor
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?
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.

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:
How does equation of continuity relate to conservation of mass principle
you will not see that what ever all we do.
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 writen 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".

Chestermiller
Mentor
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 writen 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.

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 sence to continue. We have about a one hour after midnight now. Good by mr Chestermiller.

Chestermiller
Mentor
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 sence to continue. We have about a one hour after midnight now. Good by mr Chestermiller.
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