Mechanical Properties of Fluids

For an incompressible fluid, the same volume must pass every cross-section of a pipe per second, mass can’t pile up anywhere in steady flow. That gives the equation of continuity:

Now apply the work-energy theorem to a slice of fluid moving from a wide section (pressure $P_1$, speed $v_1$, height $h_1$) to a narrow one ($P_2, v_2, h_2$). The pressure difference does work, part of which changes kinetic energy and part gravitational potential energy:

Dividing through by $\Delta V$ and rearranging:

For a horizontal pipe, $h_1=h_2$, and this collapses to exactly what the simulation shows: wherever $v$ is larger, $P$ must be smaller, so their sum stays fixed.

A narrower pipe feels like it should restrict flow. It doesn’t, not the rate. The continuity equation says $Av$ is the same everywhere along a single pipe: the same volume of fluid crosses the wide section and the narrow section every second, the narrow section just makes it move faster to keep up. Nothing is being held back.

The bigger myth sits in aviation. Countless textbooks explain wing lift as: air splits at the leading edge, the path over the curved top is longer, so that air must speed up to “meet its partner” at the trailing edge at the same time, and by Bernoulli, faster air means lower pressure on top, hence lift. The “equal transit time” assumption in the middle of that story is simply false, wind-tunnel measurements show the air on top arrives at the trailing edge well before the air on the bottom, not simultaneously. Bernoulli’s equation is completely valid physics; this particular popular explanation for why the top air is faster in the first place is not.

The more complete picture involves the wing deflecting a large mass of air downward and Newton’s third law, genuinely more involved than one equation, which is exactly why the simpler, wrong version keeps getting repeated.