Bernoulli and the Venturi Effect

Steady incompressible flow through a pipe with a constriction. Mass conservation forces $Q = A(x)\,v(x) = \text{const}$, so the fluid speeds up where the pipe narrows; Bernoulli's theorem $p + \tfrac12\rho v^2 = \text{const}$ (horizontal pipe) then forces the static pressure down exactly where the speed is up. The piezometer columns make this visible: they are tall in the wide sections and dip at the throat, the Venturi effect. Tracer particles accelerate through the throat ($v \propto 1/A$). The same principle is the cartoon airfoil: faster flow over the curved top means lower pressure and net upward lift. The headline readouts are the two invariants, the relative spread of the Bernoulli constant and of the flux $A v$ along the pipe, both held below $10^{-3}$ because the model is the exact algebra, not a simulation.