Physic Labs

Fluid mechanics

The continuity equation

Watch an incompressible flow through a narrowing pipe and verify mass conservation A1v1=A2v2A_1v_1=A_2v_2: where the section shrinks, the stream speeds up so the volume flow rate Q=AvQ=Av stays constant.

Undergraduate

Equipment

  • Animated pipe with a constriction; visible flow speed
  • Slider “Tỉ số diện tích A₂/A₁” (area ratio)
  • Slider “Lưu lượng Q” (flow rate)

Procedure

  1. Watch the constriction

    With default settings, follow the flow into the narrow section: the stream accelerates exactly where the pipe pinches. Since the liquid cannot compress, the same volume passes every cross-section per second, Q=Av=constQ=Av=\mathrm{const}.

  2. Check the ratio

    Set the area-ratio slider to A2/A1=0.5A_2/A_1=0.5 and read the speed-up factor at the throat — halving the section must double the speed, v2=v1A1/A2v_2=v_1A_1/A_2. Try 0.25 and confirm the factor of four.

  3. Change the flow rate

    Move the flow-rate slider Q and confirm the ratio of speeds is unchanged — only the overall scale rises. Then reason ahead: the faster flow in the throat has lower pressure by Bernoulli's equation, the effect behind atomizers and the Venturi meter.

Simulation

Experiment history

The continuity idea is older than fluid mechanics itself: Leonardo da Vinci's notebooks already observe that a river runs faster where it narrows. Giovanni Battista Venturi quantified the related pressure drop in constricted pipes around 1797, giving the Venturi effect its name. Daniel Bernoulli's Hydrodynamica (1738) tied the speed change to the pressure change for steady flow — continuity supplying the kinematics, Bernoulli the energetics. Together they explain carburetors, pitot tubes, and how architects use nozzles to control water jets.

Related physicists

Daniel Bernoulli

Related library topics

The continuity equation