Is Bernoulli's Equation related to the Conservation of Mechanical Energy?

[nuclearfireball_42]

So the Bernoulli's Equation..

My question : Are the terms on the left hand side equal to the total mechanical energy? So can I rewrite this equation as

?

 

[mfig]

Yes, the Bernoulli equation is very much related to the conservation of mechanical energy. Recall that pressure is Energy per unit Volume, and this is easy to see.

It is important when interpreting/using Bernoulli to always keep in mind the assumptions used to derive it:

1. Along a streamline.
2. Inviscid flow (no frictional/shear dissipation).
3. Steady flow.
4. Constant density.

The terms as you have written them do not have units of energy, so you cannot simply write that the equation means E = constant. However, they do have units of energy per volume, so that by integrating over a volume along a streamline you can write that energy is conserved within.

 

[Cem]

The equation

is only true for a fluid with constant density $\rho$. In general, if the fluid is ideal, we have $\frac{1}{2} v^2+w+gz=\rm{const}.$ along a streamline for a steady flow where $w=u+p/q$ is the enthalpy per unit mass and $u$ is the internal energy per unit mass. So we have a term related to the internal energy of the fluid which can still be interpreted as mechanical if you consider it as the sum of mechanical energies of the molecules, but it is not related to the macroscopic motion of the fluid.