# Equation of Continuity

Suppose their is a bucket with water inside it and there is a small hole at the bottom of the bucket such that water leaks from the end. Area of cross section of the bucket is A and area of the small hole is a. The velocity with which water is coming out of the hole is v and the velocity with which water is coming before coming out of the hole is dh/dt, where h is the height of water inside the bucket and h is decreasing with time which is obvious.

Now, if we apply equation of continuity , we get A(-dh/dt) = av. My question is, why do we include the minus sign in LHS? Is it because dh/dt by itself is negative and we need to make it positive? But why do we need to make it positive in the first place?

## Answers and Replies

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Chestermiller
Mentor
Rate of accumulation = (Rate of input) - (Rate of output)

Rate of input = 0

Rate of output = av

Rate of accumulation = ##\frac{dV}{dt}=A\frac{dh}{dt}##

So,

$$A\frac{dh}{dt}=0-av$$

Chet

Aniruddha414
Its more easier to think that dh/dt is a decreasing quantity there rate would be of negative u can neglect negative sign if calculating only magnitude

Rate of accumulation = (Rate of input) - (Rate of output)

Rate of input = 0

Rate of output = av

Rate of accumulation = ##\frac{dV}{dt}=A\frac{dh}{dt}##

So,

$$A\frac{dh}{dt}=0-av$$

Chet
Woah woah. I think you're not right on this one. Equation of continuity says:

mass flowing in = mass flowing out
m1 = m2
ρ V1 = ρ V2
ρ A1 v1 Δt = ρ A2 v2 Δt
A1 v1 = A2 v2 =Q=V/t =constant​
But you have said that mass flowing in is 0, which is not correct. It still doesn't explain the negative sign though. Is v velocity or speed?

Chestermiller
Mentor
Woah woah. I think you're not right on this one. Equation of continuity says:

mass flowing in = mass flowing out
m1 = m2
ρ V1 = ρ V2
ρ A1 v1 Δt = ρ A2 v2 Δt
A1 v1 = A2 v2 =Q=V/t =constant​
But you have said that mass flowing in is 0, which is not correct. It still doesn't explain the negative sign though. Is v velocity or speed?
The equation of continuity says mass in = mass out only if there is no accumulation or depletion. I can guarantee you that I'm right on this one.

Chet

The equation of continuity says mass in = mass out only if there is no accumulation or depletion. I can guarantee you that I'm right on this one.

Chet
Why did you take rate of input to be 0? The way I look at it is that we have to include to the minus sign because dh/dt is negative and so that would make the equation wrong, since the direction of velocities should be the same and not opposite. Is dh/dt velocity or speed? I think it is the velocity. Then is equation of continuity a vector equation? And if it is, then V1 and V2 should be in the same direction and not opposite, otherwise it won't make sense. Or is even the area considered as a vector so that the whole equation is a scalar equation, because that would mean the dot product of 2 vectors, namely area and velocity. I am pretty confused with this. :(

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Chestermiller
Mentor
Why did you take rate of input to be 0?
The rate of input is 0 because we are not adding water to the bucket.
The way I look at it is that we have to include to the minus sign because dh/dt is negative and so that would make the equation wrong,
Do you see that the equation I wrote can be re-expressed as A(-dh/dt) = av ?
since the direction of velocities should be the same and not opposite. Is dh/dt velocity or speed? I think it is the velocity. Then is equation of continuity a vector equation? And if it is, then V1 and V2 should be in the same direction and not opposite, otherwise it won't make sense. Or is even the area considered as a vector so that the whole equation is a scalar equation, because that would mean the dot product of 2 vectors, namely area and velocity. I am pretty confused with this. :(
The continuity equation can definitely be expressed in the way that you are describing, using the dot product of the velocity vector with surface area vectors. That's the more advanced version of how it is done. But, before you start working with that kind of development, you need to gain some experience applying it to simpler situations like the present one.

Chet

Is dh/dt in the same direction to v or opposite? And I thought we apply equation of continuity in 2 parts. The velocity of the water inside the bucket and the velocity of the water coming out of the bucket. Why does input (the water we add) even get involved here? It shouldn't be. We are only looking at velocity of water within the bucket and that of the water coming out which certainly doesn't involve input of water at all!

Chestermiller
Mentor
Is dh/dt in the same direction to v or opposite?
dh/dt is the rate at which the depth of liquid in the bucket is increasing with time. Because the depth is actually decreasing with time, dh/dt is negative.

The velocity of the upper surface of the liquid in the bucket is ##\frac{dh}{dt}\vec{i}##, where ##\vec{i}## is the unit vector in the upward direction. Since dh/dt is negative, the velocity vector is actually pointing in the downward direction.

Chet

dh/dt is the rate at which the depth of liquid in the bucket is increasing with time. Because the depth is actually decreasing with time, dh/dt is negative.
Whatever the sign of dh/dt be, it would always represent velocity. Why then do we need to include a negative sign in case dh/dt is negative? Isn't dh/dt by itself suffice to make the equation of continuity? I am unable to comprehend your input-output reasoning because this is not included in my course material. It would be really helpful if you could stick to extent of my curriculum.

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Chestermiller
Mentor
Whatever the sign of dh/dt be, it would always represent velocity. Why then do we need to include a negative sign in case dh/dt is negative?
In my post #2, in which I used input - output = accumulation, there was no negative sign on dh/dt.
Isn't dh/dt by itself suffice to make the equation of continuity?
Yes, if you use the input - output = accumulation approach, it all works out automatically.
I am unable to comprehend your input-output reasoning because this is not included in my course material. It would be really helpful if you could stick to extent of my curriculum.
First of all, I don't know what your curriculum is, so I would have to be a mind reader to do that. Second of all, it is hard to believe that this approach is not intuitively obvious to you. Are you not familiar with the concept of a bank account or checking account?

Chet

In my post #2, in which I used input - output = accumulation, there was no negative sign on dh/dt.

Yes, if you use the input - output = accumulation approach, it all works out automatically.

First of all, I don't know what your curriculum is, so I would have to be a mind reader to do that. Second of all, it is hard to believe that this approach is not intuitively obvious to you. Are you not familiar with the concept of a bank account or checking account?

Chet
The reasoning/explanation I would find understandable and obvious would be that dh/dt doesn't represent velocity but the rate of change of height of water wrt time. Which comes out to be negative since height is decreasing with increase in time. The velocity however has the same magnitude as dh/dt because of obvious reasons (displacement/time). So we need to take the magnitude of dh/dt, which would mean we have to include a minus sign in front of it. If we don't do that, that means the velocity of water in the bucket is opposite in direction to the velocity with which water comes out, which is absolutely wrong.That is the reasoning that I would find viable and correct.

Your approach, although there is nothing visibly wrong with it, doesn't simplify things but in fact complicates things even more (at least to me). So that's why I didn't find that approach very appealing.

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Chestermiller
Mentor
The reasoning/explanation I would find understandable and obvious would be that dh/dt doesn't represent velocity but the rate of change of height of water wrt time. Which comes out to be negative since height is decreasing with increase in time. The velocity however has the same magnitude as dh/dt because of obvious reasons (displacement/time). So we need to take the magnitude of dh/dt, which would mean we have to include a minus sign in front of it. If we don't do that, that means the velocity of water in the bucket is opposite in direction to the velocity with which water comes out, which is absolutely wrong.That is the reasoning that I would find viable and correct.

Your approach, although there is nothing visibly wrong with it, doesn't simplify things but in fact complicates things even more (at least to me). So that's why I didn't find that approach very appealing.
You will some day.

You will some day.
Maybe, but now for now