Continuity Equation - Why do the flow rates have to be equal?

[Shackleford]

I'm reading my fluids chapter in my University Physics textbook. We actually didn't go over this in my University Physics I course. :uhh:

At any rate, I'm looking at the equation of continuity. In explaining it, it says the flow rates through two areas have to be the same because there is no flow in or out. I googled another explanation; it says the masses have to be the same - conservation of mass. This elucidates it a bit better for me. If you choose an arbitrary amount of mass through an area, of course, if there is no exchange of mass with the surroundings the mass will eventually pass through another area. Without doing experimentation, how could you intuitively conclude that the flow rates would have to be the same?

 

[ideasrule]

I'm reading my fluids chapter in my University Physics textbook. We actually didn't go over this in my University Physics I course. :uhh:

At any rate, I'm looking at the equation of continuity. In explaining it, it says the flow rates through two areas have to be the same because there is no flow in or out. I googled another explanation; it says the masses have to be the same - conservation of mass. This elucidates it a bit better for me. If you choose an arbitrary amount of mass through an area, of course, if there is no exchange of mass with the surroundings the mass will eventually pass through another area. Without doing experimentation, how could you intuitively conclude that the flow rates would have to be the same?

Let's consider a pipe and try to prove that the flow rate for two points on the pipe--call them A and B--is the same. Think about the volume between A and B. If water flows into this volume from point A more quickly than it flows out at point B, the volume would have to grow. That's not possible, so flow rate has to be the same at A and B.

 

[Andy Resnick]

<snip>

If you choose an arbitrary amount of mass through an area, of course, if there is no exchange of mass with the surroundings the mass will eventually pass through another area. Without doing experimentation, how could you intuitively conclude that the flow rates would have to be the same?

I wonder if you are having confusion between mass flow and volumetric flow. Otherwise, how could you conclude differently?

 

[HallsofIvy]

You are, of course, assuming there is no "sink" or "source". If there is no source at a point, there can be no increase of volume or mass so the amount going in cannot be greater than the amount going out. If there is no sin, there can be no decrease of volume or mass so the amount going in cannot be less than the amount going out.

 

[BobbyBear]

the equation of continuity is:

$\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{v}) = 0$

You can think of the divergence of ($$\rho$$v) as the mass flow rate 'out of a point' (or an infinitesimally small volume surrounding the point). Only under stationary conditions must the divergence at each point be zero.

The continuity equation is a statement of the conservation of mass principle. Think about where, in what form (ie, integrated in an open control volume?), and under what conditions (stationary? constant density?) you are applying it for you to be able to deduce that flow rate through two areas (which areas?) have to be the same.