Why a tap stream crosses itself
No spout is perfectly round. Surface tension pulls the water toward a circle. Inertia carries it past, so an oval flips 90° and back, again and again.
The water falls while it flips. Each moment of the wobble lands at its own height, so the wobble stands still in the air. The light lines bend one way, then the other, and cross: X.
Lower down the water is faster and thinner, so the X's stretch, until the stream breaks into drops.
Numbers for this stream
Equations
Falling and thinning (z is distance below the spout):
Shape of the surface, n = 2 oval, 3 three-cornered, 4 square:
Rayleigh's frequency of a wobbling liquid column (x = ka):
The pattern stands still, so the wave number solves k·U = ω(k) at each height. Bohr's finite-amplitude correction for the oval:
Viscous damping and stretching of the wobble:
Breakup into drops (Rayleigh–Plateau), ripples carried down at fixed frequency w, k = w/U:
Every pixel is a ray traced through this surface: Snell refraction, Fresnel reflection, n = 1.333, dye absorbed along the path inside the water.
Constants
| Gravity g | 9.81 m/s² |
| Surface tension σ (18 °C) | 73.0 mN/m · IAPWS 2014 |
| Viscosity μ (18 °C) | 1.05 mPa·s · Vogel fit |
| Density ρ (18 °C) | 998.6 kg/m³ · Thiesen fit |
| Refractive index | 1.333 |
| Spout bore | 8 mm |
| Flows: thread · trickle · gentle · steady · brisk | 0.25 · 0.6 · 1.0 · 1.6 · 2.4 L/min |
What is estimated
- A real tap's spout shape is unknown. The five spouts are typical guesses; the icons draw them three times exaggerated.
- The spout releases a plug flow, with no vena contracta and no velocity-profile relaxation.
- The jet is assumed smooth (laminar). Above Reynolds ≈ 4000 a real spout may run rough and blur the X.
- Bohr's 37/24 coefficient is quoted from the oscillating-jet literature, not re-derived.
- Damping and stretching laws are leading-order estimates. Here they roughly cancel over 30 cm.
- Breakup ripples start from an assumed tap vibration of 10⁻⁴·³ of the radius. The beading point moves a few cm per factor of 10 in that guess. Beads past pinch-off are drawn schematically.
- At the Thread flow a real 8 mm spout may cling and drip; the jet is modelled as leaving cleanly.
- The dye and the morning light are invented, to make clear water visible. The shape and optics are computed.
- The gentle motion is an invented tap flicker: the wobble's strength varies by ±10 % at 1.1 Hz and ±5 % at 5.5 Hz, carried down at the water's own speed.
Sources
- Lord Rayleigh, "On the capillary phenomena of jets", Proc. R. Soc. Lond. 29 (1879).
- N. Bohr, "Determination of the surface-tension of water by the method of jet vibration", Phil. Trans. R. Soc. A 209 (1909).
- Experimental observation of Bohr's nonlinear fluidic surface oscillation, Sci. Rep. 6, 19805 (2016).
- J. Eggers & E. Villermaux, "Physics of liquid jets", Rep. Prog. Phys. 71 (2008).
- IAPWS, Revised release on the surface tension of ordinary water substance (2014).
- Observation: r/askscience, "What causes X shaped lines on the stream?"