Bernoulli equation

[moffy]

I had often heard of the Bernoulli effect, where increasing the speed of a liquid reduces the pressure, why planes fly etc. but I must admit to have never looked at the equation(to my recollection, maybe somewhere in the dim past university days I may have). So curiosity got the better  and I looked it up on Wikipedia: https://en.wikipedia.org/wiki/Bernoulli%27s_principle#:~:text=Bernoulli%20equation%20for%20incompressible%20fluids&text=Define%20a%20parcel%20of%20fluid,its%20volume%20m%20%3D%20ρA%20dx.

There are lots of complex forms but the simple form: v*v/2 + gz + P/d = Constant. Where v=velocity of stream, d = density of stream, P = pressure at a point, g = gravity and z = elevation of the point above the reference plane.
If we eliminate the 'gz' term, keep everything level, and multiply both sides by d we get: (d*Constant - P) = (d*v*v)/2 which looks awfully like the kinetic energy of a moving object: E = (m*v*v)/2. If we put the gz term back in we get: (d*Constant - P) = (d*v*v)/2 + d*g*z, which adds potential energy to the equation. You would have to multiply 'd' by a volume to get the true energy, but it is nice to see the symmetry in physics.

[T3sl4co1l]

Yup, simple mass conservation, among other things.

The same symmetry gives credence to tau being perhaps more fundamental than pi: circle area is tau r^2 / 2 for example.

Tim

[moffy]

Yup, simple mass conservation, among other things.

The same symmetry gives credence to tau being perhaps more fundamental than pi: circle area is tau r^2 / 2 for example.

Tim

Don't know about 'tau' first I've heard of it apart from being a letter of the Greek alphabet. But I look at the Bernoulli equation as more of an energy conservation equation rather than mass, as mass is always conserved except in nuclear reactions where it can be converted to energy. Though being an incompressible fluid the mass flow at any cross section is always the same(I am assuming flow through a pipe), but the Bernoulli equation lets us calculate what that flow is.

[RoGeorge]

When I was in school, the Bernoulli's law was in the IX-th grade (the 9th year of school, first year of high school), in the Physics book, i.e. eq. (9.19) at page 125 of 162 in this pdf scan of the manual (in Romanian, but the equations and diagrams are self-explanatory):  http://manualul.info/Fizica_IX_1988/Fiz_IX_1988.pdf

\[ p + \frac{1}{2} \rho v^2 + \rho g z = constant = C   \tag{9.19} 
\label{eq:bernoulli}\]
In our physics manual, \eqref{eq:bernoulli} was called the Equation of Bernoulli (1700-1782), and each of the 3 terms is a pressure, so "static pressure" + "dynamic pressure" + "position/potential pressure" is always a constant for the flowing current lines inside an ideal fluid of density \$\rho\$, at speed \$v\$, in a gravitational field of acceleration \$g\$, and along a height difference of \$z\$. 

[moffy]

Thanks RoGeorge, that form makes so much sense and correlates much better to: E = (m*v^2)/2 + mgh.

[DavidKo]

It is the same equation as yours. Just multiply it with d. Since d is constant than you const*d is only new constant.

[moffy]

It is the same equation as yours. Just multiply it with d. Since d is constant than you const*d is only new constant.

I did of course realise that, what I was talking about was the difference in form to the Wikipedia article, the Romanian form is more recognisable to the general energy equation whereas, as you rightly mention, the Wikipedia form needs both sides multiplied by d, density to equal RoGeorge's. Sorry if I wasn't very clear previously.