# What was the first continuity equation?

1. Aug 7, 2013

### copernicus1

Does anyone happen to know who wrote down the first continuity equation and with regard to what? I know it shows up everywhere but was it originally a fluid dynamics equation? I've been trying to research this but I'm not coming up with much history on it.

Thanks!

2. Aug 7, 2013

### jfizzix

Continuity equations are also known as local conservation laws, and appear throughout physics (as you know).

The continuity equation for fluids was I believe first published by Euler in 1757, and considering that the math of differential equations hadn't been around much earlier than that, I think we can credit him with being the first to write down a continuity equation of any kind.

This reference might be useful:
http://en.wikipedia.org/wiki/Euler_equations_(fluid_dynamics)#cite_note-2

Conservation laws had been known before then, though they weren't written in terms of continuity equations until well after their discovery:
Conservation of Momentum (1670s)
Conservation of angular momentum (statement made by Euler in Euler's reformulation of the laws of motion, though cross products weren't invented until the 1880s by Josiah Willard Gibbs)

Hope this helps:)

3. Aug 7, 2013

### Jolb

I would doubt that it was Euler. The Bernoullis were dealing with hydrodynamical equations before Euler, and maybe Newton also did. Johann Bernoulli was Euler's teacher, and his nephew (?) Daniel Bernoulli (still older than Euler) is credited with the more advanced Bernoulli equation. But not everything got published back in those days--the Bernoullis were known to be highly secretive of their results.

Also, cross products were around before Gibbs. He probably just introduced new notation. For example, I don't see how Green's theorem could have been proved without a "cross product" in one form or another (Green died in 1841.) Surely the arithmetic of quaternions involves a "cross product" of sorts, and Hamilton died in 1865. (Quaternions are what Maxwell used to formulate electrodynamics, and his equations were published in 1862. How can you do EM without cross products???) Anyway, it was Heaviside who introduced our modern vector calculus notation.

Last edited: Aug 7, 2013
4. Aug 7, 2013

### jfizzix

Whether Euler or the Bernoullis wrote down the first instance of the continuity equation is debatable, and I'm no historian of science. The earliest record of its official publication I know of is attributed to Euler in 1757.

http://www.math.dartmouth.edu/~euler/pages/E226.html

Daniel Bernoulli's "Hydrodynamica", published in 1738 has among other things, the first instance of the conservation of energy in fluids (bernoulli's principle). Unfortunately, I don't have an english copy on me, so I couldn't say if that book also discusses the continuity equation, but it would be well worth checking out in your university library.

Angular momentum had the same fundamental expression in Euler's day as it does now. The only thing that's changed is that we have a (much) simpler notation.

5. Aug 8, 2013

### copernicus1

Thanks, the Euler publication is interesting! That's the oldest published version I've seen yet.

6. Aug 8, 2013

### jfizzix

No problem:)

It's always a surprise to me how much Euler contributed (explicitly) to physics.

7. Aug 9, 2013

### copernicus1

I've been trying to decide if Bernoulli's equation is more fundamental than the continuity equation, but I'm not actually able to derive the latter from the former (so far), so I'm gathering that maybe Bernoulli's equation is less general---maybe Bernoulli's equation is applicable in the realm of incompressible flow while the continuity equation is more general?

8. Aug 9, 2013

### jfizzix

Bernoulli's equation

$\frac{1}{2}\rho v^{2} + \rho g h + P = \text{const.}$

Is true for incompressible fluids, though there are other equations that hold for compressible fluids.

The continuity equation in differential form
$\frac{d \rho}{d t} = -\nabla\cdot (\rho \vec{v})$
or integral form
$\frac{d M}{d t} = -\oint\rho \vec{v}\cdot d\vec{a}$
Where M is the total mass of fluid inside a closed surface,

applies to all fluids, compressible and incompressible alike. They can't necessarily be derived from one another, though if there were a continuity equation for energy (which there almost certainly is, but I'm not a fluids expert), I bet you could get Bernoulli's equation out of that.

So I guess in a sense the continuity equation is more fundamental than Bernoulli's equation, since if you could find one for the energy of a fluid, you would get Bernoulli's equation out of it with the right constraints.

9. Aug 9, 2013

### copernicus1

Awesome thanks.