Bernoulli at Work

One equation, many uses: it recovers hydrostatics, predicts how fast a tank drains, measures flow in a pipe, powers a perfume spray, and explains why a cricket ball swings.

Hydrostatics as a special case#

Put v₁ = v₂ = 0 in Bernoulli’s equation:

exactly the pressure–depth law of a fluid at rest.

Speed of efflux: Torricelli’s theorem#

A wide tank of cross-section A₁, open at the top, has a small hole of area A₂ a depth h below the free surface. Both the surface and the jet are at atmospheric pressure P₀. Bernoulli between them:

With continuity $A_1v_1 = A_2v_2$ and $A_2 \ll A_1$, the surface barely moves (v₁ ≈ 0), giving

The venturi meter#

A venturi tube has a smooth constriction (the throat). Liquid speeds up in the throat and its pressure drops. Two vertical side tubes, one at the wide part (area A₁) and one at the throat (A₂), show a level difference h, with $P_1 - P_2 = \rho gh$. Bernoulli (same height) gives $P_1 - P_2 = \tfrac12\rho(v_2^2 - v_1^2)$, so

The aspirator (sprayer)#

In a sprayer, air pushed by a piston along a barrel rushes through a narrow nozzle. Its speed there is high, so its pressure is low. A thin tube dips from the nozzle into the liquid below; atmospheric pressure on the liquid’s surface pushes liquid up the tube into the fast airstream, which breaks it into a fine spray. Perfume atomisers, paint sprayers and old car carburettors work this way.

Why a spinning ball swings#

Swing bowlers make a cricket ball drift sideways out of its original plane of motion. Work in the frame of the ball: air streams past it. If the ball spins, it drags the air near it round. On one side the drag adds to the airflow, so the air there moves faster and its pressure drops; on the other side it opposes the airflow, so the air is slower and the pressure higher. The pressure difference pushes the ball sideways, bending its path.