Draining Bathtub Vortex Dynamics

• Draining bathtub vortex is a free-surface vortex generated as fluid drains through an outlet, intensifying pre-existing circulation into a central depression.
• The vortex exhibits logarithmic spiral streamlines, where an idealized sink-plus-circulation model elucidates its formation and analogy to rotating acoustic black holes.
• Laboratory and theoretical studies extend the model to include quantum, superfluid, and shallow-water regimes, revealing insights into wave scattering, superradiance, and analogue gravity.

The draining bathtub vortex is a free-surface vortex generated when fluid with nonzero angular momentum drains through a bottom outlet, so that radial convergence toward the drain intensifies azimuthal motion and produces a central surface depression or funnel. In fluid mechanics, the term denotes both the familiar laboratory or household free-surface vortex and an idealized two-dimensional sink-plus-circulation flow; in analogue gravity, that idealization becomes the canonical rotating acoustic black-hole model, with an ergoregion and a horizon for wave propagation (Falahatpisheh et al., 2012, Dolan et al., 2012).

1. Formation as a free-surface flow

A direct laboratory visualization of vortex formation was reported in a rectangular tank filled with tap water at $25^\circ\mathrm{C}$, with water occupying $25\times 25\times 10\ \mathrm{cm}^3$, the free surface open to atmospheric pressure, and drainage through a $5\ \mathrm{mm}$ ball valve at the bottom. The initially stagnant water was stirred until “all the points in the water” were rotating at $2\pi/\mathrm{sec}$, after which the drain was opened and the formation process was recorded by a Y3 high-speed camera at $200$ fps and $352\times 824$ pixels; the recorded $3.9\ \mathrm{s}$ interval was exported as a $160\ \mathrm{s}$ slow-motion video at $5$ fps and $1280\times 720$ resolution. An important observation was that “the height of the water remains almost unchanged while acquiring the images,” so the sequence isolates local vortex intensification rather than gross emptying of the tank (Falahatpisheh et al., 2012).

The observed sequence is the standard one for a rotating draining flow. Before drainage, the water behaves approximately like a forced vortex, with externally imposed bulk rotation. Once the bottom valve is opened, the flow acquires an inward radial component; as fluid parcels spiral toward the outlet, decreasing radius intensifies angular velocity and tangential speed, sharpening the free surface into a visible central depression and then a concentrated vortex core. Standard interpretive background writes the forced-vortex stage as $25\times 25\times 10\ \mathrm{cm}^3$0, the free-vortex stage as $25\times 25\times 10\ \mathrm{cm}^3$1, and the intermediate intensification as the approximate conservation law $25\times 25\times 10\ \mathrm{cm}^3$2. This suggests that the bathtub vortex is best viewed as a drain-induced concentration of pre-existing circulation rather than merely a late-stage geometrical consequence of shallow water.

The same experiment also indicates the limits of purely visual evidence. It does not provide velocity fields, pressure measurements, circulation estimates, Reynolds or Froude numbers, surface-profile data, or core-radius evolution. It therefore documents formation dynamics clearly but does not by itself constitute a complete hydrodynamic theory of the phenomenon.

2. Idealized planar kinematics and logarithmic spirals

A minimal mathematical model treats the visible surface flow as a two-dimensional vector field with a single singular point at the drain, no other sources or whirls away from the origin, rotational invariance, and vanishing circulation and flux around closed curves that do not enclose the origin. Under these assumptions, the velocity field must take the form

$25\times 25\times 10\ \mathrm{cm}^3$3

where $25\times 25\times 10\ \mathrm{cm}^3$4 controls the sink or source part and $25\times 25\times 10\ \mathrm{cm}^3$5 the vortex or circulation part. For a draining bathtub, the relevant case is $25\times 25\times 10\ \mathrm{cm}^3$6 and $25\times 25\times 10\ \mathrm{cm}^3$7 (Boyadzhiev, 2021).

The streamline geometry then follows directly. In Cartesian form,

$25\times 25\times 10\ \mathrm{cm}^3$8

and the corresponding phase curves are logarithmic, or equiangular, spirals,

$25\times 25\times 10\ \mathrm{cm}^3$9

Equivalently, if one writes the polar components as $5\ \mathrm{mm}$0 and $5\ \mathrm{mm}$1, then

$5\ \mathrm{mm}$2

which integrates to the same spiral law. In this framework, a pure sink gives radial lines, a pure vortex gives circles, and a sink plus vortex gives logarithmic spirals. The visible spiral pattern above a drain is therefore the geometric signature of simultaneous inward transport and azimuthal circulation (Boyadzhiev, 2021).

This model is deliberately restricted. It explains planar streamline or pathline geometry, especially the surface pattern traced by bubbles, dye, or floating debris, but it does not describe the three-dimensional free-surface funnel, vertical velocity structure, boundary layers, viscous vorticity generation, or the detailed near-drain core. It is therefore exacting as a statement about symmetry and streamline shape, but incomplete as a full model of the physical bathtub vortex.

3. The standard draining-bathtub analogue spacetime

In analogue-gravity treatments, the draining bathtub vortex is usually formulated as an inviscid, barotropic, locally irrotational background flow in $5\ \mathrm{mm}$3 dimensions with velocity

$5\ \mathrm{mm}$4

or equivalently

$5\ \mathrm{mm}$5

depending on notation. Under the acoustic analogy, linear perturbations satisfy a massless Klein–Gordon equation on an effective spacetime. One common line element is

$5\ \mathrm{mm}$6

which can be rewritten in Boyer–Lindquist-like form as

$5\ \mathrm{mm}$7

with

$5\ \mathrm{mm}$8

The resulting geometry has an acoustic horizon

$5\ \mathrm{mm}$9

and an ergosurface

$2\pi/\mathrm{sec}$0

or, in the $2\pi/\mathrm{sec}$1 notation,

$2\pi/\mathrm{sec}$2

This is the standard rotating acoustic-hole geometry and is explicitly presented as the analogue of the Kerr metric (Dolan et al., 2012, Lemos, 2013).

Mode analysis proceeds by separating perturbations as azimuthal harmonics and introducing a tortoise coordinate. In one standard form,

$2\pi/\mathrm{sec}$3

so that the horizon is mapped to $2\pi/\mathrm{sec}$4. The horizon-frame frequency shift appears as

$2\pi/\mathrm{sec}$5

or equivalently $2\pi/\mathrm{sec}$6, with $2\pi/\mathrm{sec}$7. This structure reproduces the kinematics of horizon trapping, ergoregion physics, and rotational mode splitting (Dolan et al., 2011, Oliveira et al., 2020).

The physical meaning of the analogy is precise but limited. The draining bathtub reproduces a large class of black-hole kinematical effects—wave propagation, horizons, ergoregions, superradiance, quasinormal ringing—but it is not a solution of Einstein’s equations. It is an effective spacetime for perturbations in a moving medium.

4. Scattering, superradiance, absorption, and resonances

The low-frequency scattering problem reveals a modified analogue Aharonov–Bohm structure. With

$2\pi/\mathrm{sec}$8

the large-$2\pi/\mathrm{sec}$9 phase shifts satisfy

$200$0

while the axisymmetric mode obeys

$200$1

The corresponding low-frequency scattering length is

$200$2

Circulation alone gives the familiar AB-type $200$3 behavior; drainage adds absorption and left-right asymmetry. At high frequency, the same system exhibits orbiting oscillations from interference between co- and counter-orbiting rays, with angular fringe spacing

$200$4

and a rotation-dependent offset of the pattern (Dolan et al., 2012).

Superradiant, or superresonant, amplification follows from the horizon-frame frequency shift. In one standard scattering relation,

$200$5

so the reflected wave is amplified when

$200$6

or, in rescaled notation, $200$7. The same geometry also supports quasinormal ringing, late-time tails, and mirror-driven instability. For the rotating draining bathtub, late-time decay behaves as

$200$8

and with a reflecting mirror the system develops boxed quasinormal modes; this is the acoustic analogue of the black-hole bomb, described in that literature as the “sonic bomb,” with the alternative energy-storage interpretation “sonic plant” (Lemos, 2013, Dolan et al., 2011).

Absorption can be treated exactly in terms of confluent Heun functions. For the radial equation

$200$9

the exact solution can be written in terms of $352\times 824$0, and the low-frequency total absorption length approaches the horizon perimeter,

$352\times 824$1

This is the $352\times 824$2-dimensional counterpart of the usual low-frequency black-hole absorption law (Oliveira et al., 2020).

5. Vorticity, deformations, and nonstandard extensions

Several extensions depart from the ideal irrotational $352\times 824$3 swirl while preserving the draining-bathtub framework. A particularly important one replaces the azimuthal profile by a smooth rotational core,

$352\times 824$4

so that the core is vortical and the exterior remains asymptotically $352\times 824$5. In this case the perturbation equation acquires an additional term $352\times 824$6, where $352\times 824$7, so the field behaves as if it had a spatially localized effective mass. The corresponding effective potential can develop a well between barriers, supporting long-lived quasibound states in addition to ordinary quasinormal modes (Patrick et al., 2018).

Field-theoretic deformations of the acoustic metric alter the same structures in different ways. In the higher-derivative Abelian Higgs model, the effective geometry is controlled by

$352\times 824$8

and the horizon radius depends on both draining and circulation. In that setting the superresonant frequency window is modified, the differential scattering becomes asymmetric even in the non-draining vortex limit, and the low-frequency absorption is

$352\times 824$9

with the striking feature that it does not vanish as $3.9\ \mathrm{s}$0 in the approximation used there (Anacleto et al., 2018).

Noncommutative and Lorentz-violating extensions likewise deform the draining-bathtub metric and the analogue AB effect. In those models, the phase shift can persist even when the circulation and draining parameters vanish, because the limiting geometry becomes a conical defect rather than flat space. The modified scattering is therefore attributed not only to the fluid circulation but also to the residual geometry induced by the deformation parameters (Anacleto et al., 2012, Anacleto et al., 2012).

A recent vortical generalization studies total transmission modes rather than quasinormal modes. There the azimuthal velocity is again smoothed as

$3.9\ \mathrm{s}$1

and the boundary conditions for right TTMs are ingoing at both the horizon and infinity. Numerical computations with the Chebyshev–Lobatto pseudospectral method show that these TTM spectra can have either positive or negative imaginary parts depending on parameters, and that higher overtones display pronounced spectral mobility under changes in $3.9\ \mathrm{s}$2 and $3.9\ \mathrm{s}$3 (Yu et al., 5 May 2026).

6. Shallow-water experiments, spectroscopy, and topological wave effects

The shallow-water version of the draining bathtub vortex has moved beyond idealized theory into controlled laboratory measurements. A large-scale water-tank program reported the first experimental verification in an analogue rotating-black-hole system of superradiance, quasi-normal or light-ring ringing, and backreaction. In a rectangular tank about $3.9\ \mathrm{s}$4 long and $3.9\ \mathrm{s}$5 wide, with continuous pumping and central drainage through a hole of radius $3.9\ \mathrm{s}$6, the background flow was characterized by PIV with fitted parameters

$3.9\ \mathrm{s}$7

and co-rotating modes were measured with reflection coefficients

$3.9\ \mathrm{s}$8

The same program also interpreted the relaxation spectrum in terms of light-ring modes and used it for “Analogue Black Hole Spectroscopy,” while wave-induced changes in the mean water level were identified as backreaction on the draining flow (Patrick, 2020).

A distinct shallow-water experiment emphasizes circulation as a topological control parameter. In a $3.9\ \mathrm{s}$9 tank with water height $160\ \mathrm{s}$0, a controlled draining-bathtub vortex was combined with traveling and standing surface waves. Outside the core the background flow was taken as

$160\ \mathrm{s}$1

with dimensionless circulation parameter

$160\ \mathrm{s}$2

Traveling waves exhibited wavefront dislocations associated with the hydrodynamic Aharonov–Bohm effect, while standing-wave nodal lines rotated rigidly with

$160\ \mathrm{s}$3

providing a direct hydrodynamic analogue of Lense–Thirring frame dragging. Circulation was measured by PIV and the surface pattern by a caustic-imaging technique (Singh et al., 23 Apr 2026).

Shallow-water theory also shows that the pure effective-metric picture can be incomplete when perturbations carry potential vorticity. In the long-wave, linear regime, nonzero PV obeys an advection law and simultaneously appears as a source term in the gravity-wave equation. The result is that outgoing waves may contain both reflected incident gravity waves and gravity waves emitted by the PV disturbance itself. This makes the hydrodynamic draining bathtub richer than a purely potential acoustic spacetime and suggests caution in interpreting measured outgoing wave amplitudes (Churilov et al., 2018).

7. Quantum and superfluid versions

In superfluids, the draining bathtub vortex becomes a problem of quantized vorticity rather than continuous rotational cores. A Gross–Pitaevskii model of a superfluid drain vortex at $160\ \mathrm{s}$4 shows that the rotational core is realized as a bundle of quantized vortex lines above a localized sink. Because the lower part of the bundle is pulled inward more strongly than the upper part, the local angular velocity

$160\ \mathrm{s}$5

increases near the bottom, producing helical twist. The paper’s central conclusion is that this twisted bundle strengthens the downward axial flow into the drain, making the quantum drain vortex the superfluid analogue of the classical bathtub vortex (Ruffenach et al., 2022).

For superfluid $160\ \mathrm{s}$6He, a rotor-induced suction flow has been interpreted as a three-region bathtub vortex: a giant-vortex-like cavity near the free surface, a bulk bundle of singly quantized vortices beneath it, and a bottom region influenced by the vessel boundary. In the bulk, the authors prescribe a Rankine-vortex-like normal-fluid field with a confined downflow tube and derive a morphological criterion

$160\ \mathrm{s}$7

for straight axial bundle formation rather than a helically polarized cylindrical vortex layer. They further suggest that the steady-state bundle structure could be inferred by measuring the diffusion constant after the macroscopic normal flow is switched off, estimating

$160\ \mathrm{s}$8

for an untwisted bundle in the Osaka City University parameter range (Inui et al., 2020).

A related quasi-two-dimensional Bose-fluid study treats a multiply quantized central vortex embedded in a localized sink, explicitly calling the configuration the quantum analogue of the classical draining bathtub flow. There the drain suppresses the primary splitting instability of the multiply quantized vortex and can completely quench it for strong enough flows, but a reflective circular trap introduces a secondary superradiant instability analogous to the black-hole bomb. Its nonlinear endpoint is a shock wave propagating around the boundary and nucleating additional vortices (Patrick, 2024).

The same rotating-draining geometry also appears in Bose–Einstein-condensate analogue gravity. In the optical-superradiance-induced vortex phase of a condensate, a simplifying draining-bathtub model with

$160\ \mathrm{s}$9

leads to the acoustic superradiance condition

$5$0

linking the onset of a rotating condensate vortex to the same amplification mechanism that governs classical draining-bathtub scattering (Ghazanfari et al., 2014).

Taken together, these developments show that the draining bathtub vortex is not a single model but a hierarchy of related objects: a three-dimensional free-surface vortex in ordinary fluids, a symmetry-reduced sink-plus-circulation flow with logarithmic spirals at the surface, a canonical rotating acoustic spacetime for wave mechanics, and a quantum or superfluid structure in which the rotational core is carried by quantized vortex bundles. The shared core of all these formulations is the same: inward transport coupled to azimuthal circulation. The differences lie in which degrees of freedom are retained—free surface, viscosity, vorticity diffusion, boundary layers, dispersion, or quantized vortex filaments—and therefore in which aspects of the draining bathtub vortex become visible.