Bernoulli's Equation (Two Points)

Daniel Bernoulli published this in 1738, and every one of its three terms is an energy per unit volume: static pressure, the kinetic term $\tfrac{1}{2}\rho v^2$, and the potential term $\rho g z$. Their sum is fixed along a streamline, so any one of them can only grow at another's expense. Push water from 2 m/s to 8 m/s in a level pipe and the pressure must fall by $\tfrac{1}{2}\times 1000 \times (64 - 4) = 30\ \text{kPa}$, which is the entire working principle of a Venturi, a carburettor, a laboratory aspirator and an aircraft wing.

The famous field mistake is applying it where its assumptions have quietly died. Bernoulli assumes no friction, no pump, no heat and constant density. Real pipe runs bleed head to friction, so a designer writes the extended form with an $h_f$ term on the downstream side and gets that number from Darcy-Weisbach. Gases obey it only while Mach number stays under about 0.3, above which density stops being constant and compressible relations take over.

Here is the part that catches people out: the equation holds along a streamline, not across a flow field, unless the flow is irrotational. Two points in the same pipe cross-section can carry genuinely different totals, which is why a pitot traverse across a duct reads a curve rather than a plateau. And note that velocity enters squared, so this page's velocity brains return the positive root only. If your flow actually runs from 2 back to 1, swap the two ends rather than expecting a negative answer.