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Example · Necking of a Falling Water Stream

Problem statement

The smooth stream from a tap visibly narrows ("necks down") as it falls. Measuring the cross-sectional area \(A_0\) at the tap and \(A\) at a distance \(h\) below, determine the volume flow rate \(R_v\) — with a ruler, no flowmeter.

Given information

  • Areas \(A_0\) (top) and \(A\) (bottom), fall height \(h\)
  • Laminar, steady stream; atmospheric pressure all around the jet

Solution strategy

Two conservation laws on the free jet: continuity (what goes in comes out) and Bernoulli (free fall: pressure is atmospheric everywhere on the jet surface, so speed grows like a dropped stone).

Step-by-step solution

  1. Continuity: \(A_0 v_0 = A v \Rightarrow v = \dfrac{A_0 v_0}{A}\)
  2. Bernoulli between the two stations (equal pressure, height drop \(h\)): $\(v^2 = v_0^2 + 2gh\)$
  3. Substitute and solve for \(v_0\): $\(\frac{A_0^2 v_0^2}{A^2} = v_0^2 + 2gh \;\Longrightarrow\; v_0 = \sqrt{\frac{2gh}{\dfrac{A_0^2}{A^2} - 1}}\)$

Final answer

\[\boxed{\;R_v = A_0 v_0 = A_0\sqrt{\frac{2gh}{\left(A_0/A\right)^2 - 1}}\;}\]

(Equivalently \(R_v = A_0 A\sqrt{\dfrac{2gh}{A_0^2 - A^2}}\).)

Key takeaways

  • The stream narrows precisely because it speeds up: constant \(Q = Av\) forces \(A \propto 1/v\). The visible shape of a water stream is the continuity equation.
  • Kitchen metrology: two diameters and a ruler give the flow rate to a few percent.
  • The jet eventually breaks into drops — surface tension (Plateau–Rayleigh), a cousin of the interface instabilities.

Continuity equation · Bernoulli's principle · Tank discharge