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Old Jun 2nd 2014, 02:35 AM   #1
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The incompressible continuity equation

Hi, I was wondering if perhaps someone here had time to check my thinking on the conceptual understanding of some basic fluid dynamics - specifically, the incompressible version of the continuity equation. Mathematically, it's just setting the divergence of the flow field equal to zero and I was thinking about how I would explain this if someone asked me to describe the physical significance of it. I came up with this:

Imagine that it is impossible to squeeze the fluid - in other words, no matter how violently you treat it (how much you ramp up the advection forces) you will not be able to increase the originally given number of molecules per unit volume. Since there is no spontaneous creation of matter going on, the math has to reflect the fact that anything that flows into a certain volume has to continuously push out an equal amount. So inflow equals outflow all over the field and no point experiences any divergence.

Would this be an acceptable conceptual explanation?
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Old Jun 2nd 2014, 04:41 AM   #2
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Just a couple of suggestions:

1. I've never heard the term "advection force" before. I suggest you use something else, such as "compressive force."

2. Since you are trying to describe what divergence = 0 means, don't use the term divergence in the last sentence.

I would suggest that the key point is this: for any given volume that has no sources or sinks of fluid within it, and given the assumption of constant density of fluid at all times and places, the mass of fluid within that volume must be constant, which means the mass flow of fluid (or "flux") into or out of the volume across its surface must be zero. Now if you consider smaller and smaller volumes, in the limit as volume becomes infinitesimal, the flux across the infinitesimal surface is still zero - this is the situation of divergence at all points = 0.

Last edited by ChipB; Jun 2nd 2014 at 04:43 AM.
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Old Jun 2nd 2014, 01:20 PM   #3
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1-D Continuity... Leibnitz's calculus...

Devlan, Hi...

I pondered the same math some years ago. I asked a prof or two "set hand-waving aside", show me the one-dimensional PROOF. None did.

This first item I attach is my start. Needs cleaning up a bit. A tool, "differentiation of an integral with variable limits" is needed.

It is easier to use then to prove (for me).

I always liked this problem.

Good luck with your studies. TSH
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