The Water Bell: How a Jet Becomes a Sheet

A jet, a disc, and a dome of water

Point a smooth, slow vertical water jet straight up at a small horizontal disc held a few centimetres above the nozzle. The jet cannot pass through the disc, so it splashes outward: mass conservation forces the water to spread radially across the underside of the disc and leave its rim as a thin, free liquid sheet flying outward in all directions at once. Left to inertia alone, that sheet would sail off horizontally and disintegrate into droplets. Instead, the sheet has two air–water surfaces, and each carries the tension of the liquid’s skin. That tension pulls the sheet inward, bending its outward-flying edge back down and around until the film curves into a closed surface of revolution — the water bell.

The bell is a genuine force balance, not a trick of viewing: the outward momentum of the moving film is turned back by surface tension (helped or hindered by gravity and by the pressure of the air the bell traps). Change the flow rate, the disc size, or the liquid’s surface tension and you smoothly morph the shape — from a flat sheet, to an open crown, to a fully closed “onion.” It is exactly the same physics you see when a kitchen tap hits the back of a spoon and throws up a curved fan of water.

Mass conservation: a film that thins as it flies

Start with the sheet itself. If the volume flow rate delivered by the jet is Q, then every ring of the axisymmetric sheet at radius r must pass the same flux. With sheet thickness h(r) and radial speed u, continuity demands:

• Q = 2π r · h(r) · u, so the film thins as h(r) = Q / (2π r u) — roughly as 1/r.

Near the disc the sheet can be tens of micrometres thick; a few centimetres out it can thin to a few micrometres, which is why a water bell is nearly transparent and shows delicate interference colours where it is thinnest. The radial speed is nearly constant. Surface energy and viscous losses do little work on the fast-moving film, so along a streamline ½u² is conserved except for the slow deceleration gravity imposes over a few centimetres of rise (for a 2 m/s jet, gravity changes the speed by only ~10–15% over a 5 cm bell). To a first approximation, then, u ≈ U, the jet speed, and the film simply gets thinner the farther out it flies. That thinning is the key: it makes surface tension progressively more important as the sheet expands, because a thinner sheet is easier for its own skin to reel back in.

The balance that turns the sheet: inertia versus surface tension

A free liquid sheet is bounded by two surfaces, so its skin exerts a line tension of 2γ (twice the single-surface value) trying to shrink the sheet’s area. The moving fluid resists being turned: to curve the film with radius of curvature R requires a centripetal force per unit area of order ρ h u² / R. The bell shape is where these compete. The clean way to see the turnaround is the Taylor–Culick result (G. I. Taylor, 1959; F. E. C. Culick, 1960): a free film of thickness h retracts at its edge with speed V = √(2γ / ρh). As long as the sheet moves faster than V it keeps expanding; when the flow slows to V, surface tension wins and the film turns back.

Substituting the film’s thickness from continuity, the sheet reaches its maximum radius when u = V, giving a compact prediction for the bell radius:

• R ≈ ρ Q U / (4πγ), equivalently R / a ≈ We / 4, where We = ρ a U² / γ is the jet Weber number (inertia ÷ surface tension) and a is the jet radius.

The numbers are honest. Take a a = 2 mm water jet at U = 2 m/s: We = (1000)(0.002)(2²)/0.072 ≈ 110, so R ≈ (2 mm/4)(110) ≈ 5–6 cm — a bell about the size of a fist, just as observed. Lowering γ (for example by adding a trace of surfactant) makes R larger because the skin pulls more weakly, while a slower or thinner jet shrinks the bell. Below We of a few tens the sheet barely forms; well above ~100 you get a large, easily destabilised bell.

Gravity, trapped air, and the shape of the bell

The full meridian shape is set by an equilibrium equation first written down by Joseph Boussinesq (1869) to explain Savart’s experiments. Balancing the film’s inertia normal to itself against the Laplace-like pull of its two curved surfaces, gravity, and any pressure difference across the sheet gives, locally, roughly ρ h u²/R_m = 2γ(1/R_m + 1/R_φ) − Δp ± ρ h g, where R_m and R_φ are the meridional and azimuthal radii of curvature. Two extra players enter here.

Gravity. For an upward jet, gravity decelerates and thickens the sheet near the top, helping surface tension close it into a teardrop; for a downward configuration it works the other way. The competition between surface tension and gravity is governed by the capillary length ℓ_c = √(γ/ρg) ≈ 2.7 mm for water, and by the Froude number Fr = U²/(gL). Bells much bigger than a few centimetres feel gravity strongly and tend to sag or close.

Trapped air. A closed bell seals a pocket of air, and the pressure difference Δp = p_inside − p_atm across the thin film pushes on it directly. Inflate the interior slightly and the bell balloons outward; apply gentle suction and it collapses inward. This is why a tiny hole matters: a pinhole (or a fine needle) lets air in or out so Δp relaxes to zero, giving the “free” bell whose shape is set by surface tension and gravity alone. Seal the interior and the fast sheet slowly drags (entrains) the trapped air away, lowering p_inside, which sucks the bell tighter and can make it snap shut. Christophe Clanet exploited exactly this in his 2000s experiments, connecting the bell’s interior to a controlled reservoir to open, close, and reshape the same bell at will — turning Δp into an experimental dial.

From flat sheet to open crown to closed onion

Because the shape is a competition of a few clean parameters — flow rate, disc size, surface tension, gravity, and internal pressure — you can walk continuously through a family of forms. At high Weber number (fast jet or large disc) inertia dominates and the film flies out almost flat, its rim beading up and flinging off droplets before tension can turn it: a flat radial sheet. Reduce the flow and you get an open bell or crown, the sheet curving downward but still gaping at the bottom. Reduce it further, shrink the disc, or lower the surface tension’s reach and the sheet folds all the way back to the axis into a closed onion, where the converging film meets on the centreline and merges into a single downward jet.

Where the sheets meet they do not simply vanish. Two colliding liquid sheets rebound into a new sheet at right angles, and a rim collecting fluid can pinch into a chain of fluid links — the fluid chains and “fishbone” patterns catalogued by Bush and Hasha (2004). The flat Savart sheet (a jet striking a small disc and spreading into a flat circular sheet bounded by a beaded rim that sheds drops) is the standard laboratory system for studying the same surface-tension-versus-inertia balance in a flat sheet rather than a closed bell.

Instabilities: flapping, rims, and atomization

Water bells are only stable in a window. Three failure modes bound it.

• Rim (Rayleigh–Plateau) break-up. Surface tension collects the sheet’s edge into a thickened rim, and that rim — effectively a curved cylinder of liquid — is unstable to the same varicose instability that breaks a jet into drops, shedding a fringe of droplets from the bell’s lower edge.
• Sheet flapping. As the jet speed rises, the surrounding air couples to the thin film through a Kelvin–Helmholtz-type shear instability. Taylor (1959) and H. B. Squire (1953) showed that a fast sheet develops growing flapping waves — the film waves like a flag — and a smooth, glassy bell gives way to a fluttering, noisy one that ultimately tears.
• Atomization. Push further and the flapping sheet shatters into a fine spray. This is not a nuisance everywhere: the sheet-to-droplet route is precisely how many fuel injectors and spray nozzles are designed to atomise liquid, deliberately operating past the bell’s stable regime.

A clean bell therefore needs a laminar, well-conditioned jet; a turbulent or wobbling jet (high jet Reynolds number, Re = ρUa/μ) prints its disturbances onto the sheet and prevents a smooth closed surface from ever forming.

How we know it: Savart to Clanet, and where it matters

The phenomenon has a long pedigree. Félix Savart (1833) published the first systematic study of a jet striking a circular plate, mapping how sheets and bells form and when they turn from smooth to agitated — remarkably, decades before the surface-tension theory of these shapes was worked out. Boussinesq (1869) then derived the bell’s shape from surface tension, and Georges Bouasse documented a menagerie of bells and sheets in the early twentieth century. The modern quantitative picture — the shape equations, the role of trapped-air pressure, and the stability boundaries — comes from G. I. Taylor’s thin-sheet papers (1959), Culick’s retraction velocity (1960), and the careful experiments of Christophe Clanet and collaborators, summarised in his Annual Review of Fluid Mechanics article “Waterbells and Liquid Sheets” (2007). We measure bells by high-speed imaging, by thin-film interference to read the micrometre-scale thickness, and by tapping the interior to control Δp.

Beyond its beauty, the water bell is a working model for engineering free-surface flows: curtain coating (the stable liquid curtains that lay down photographic film and paper coatings live or die by the same sheet stability), spray and fuel-injector atomization, agricultural nozzles, and decorative fountains. Above all it is a vivid, quantitative demonstration that surface tension — a force of only ~0.07 N per metre — can seize a fast-moving liquid and shape it into a standing structure in the air.

Morphing between liquid-sheet regimes as flow rate, disc size, and surface tension are varied

Regime	Typical control	Sheet trajectory	Role of surface tension
Flat radial sheet	High We (fast jet or large disc)	Flies out nearly horizontal; edge frays into drops	Too weak to turn the sheet before it fragments
Open bell / crown	Moderate We	Curves downward but stays open at the base	Bends the film inward, but not enough to close it
Closed bell (onion)	Low We (slow jet, small disc, or higher surface tension γ)	Sheets converge on the axis and merge into a single jet	Pulls the whole film back to the axis and seals it
Flapping / atomizing sheet	Very high We with strong air coupling	Sheet flaps like a flag and shatters into spray	Overwhelmed by aerodynamic (Kelvin–Helmholtz) flapping