10. Air vortex
Problem Description & Analysis
Official Problem Statement
“Place a small rod-shaped magnet on the bottom of a vessel filled with liquid and set it into rotation using a magnetic stirrer. Above a certain rotation speed, a thin air channel forms, extending from the liquid surface down to the rotating magnet itself. Investigate and explain this phenomenon.”
Vortex Structure and Pressure Distribution
The spinning magnet drags the surrounding liquid into rotation, forming a Rankine vortex with two distinct regions:
Core region (solid-body rotation): for $r leq a$:
$$v_theta(r) = frac{Gamma, r}{2pi a^2} = omega, r$$
Outer region (free vortex): for $r > a$:
$$v_theta(r) = frac{Gamma}{2pi r}$$
where $Gamma$ is the circulation and $a$ is the vortex core radius. The Euler equation in the radial direction for axially symmetric flow:
$$frac{dp}{dr} = frac{rho, v_theta^2}{r}$$
Integrating from $r = infty$ to $r = a$ (outer free vortex) contributes $rhoomega^2a^2/2$, and integrating from $r = a$ to $r = 0$ (solid-body core) adds another $rhoomega^2a^2/2$. The total pressure deficit from ambient to the axis is:
$$Delta p = p_infty – p_{r=0} = rho,omega^2, a^2$$
Critical Condition for Air Channel Formation
An air channel forms when the central pressure deficit exceeds the hydrostatic pressure of the liquid column above the magnet. With liquid depth $h$:
$$Delta p geq rho, g, h$$
Substituting:
$$rho,omega_c^2, a^2 = rho, g, h$$
$$boxed{omega_c = sqrt{frac{g h}{a^2}}}$$
Above $omega_c$, the vortex funnel extends from the free surface all the way down to the magnet, forming a stable, narrow air channel. The shape of the free surface in the outer (free-vortex) region follows from Bernoulli’s equation:
$$z(r) = z_0 – frac{Gamma^2}{8pi^2 g, r^2}qquad (r > a)$$
This inverse-square profile shows that the liquid surface dips sharply near the vortex axis.
Parameters
$omega_c$ — critical angular velocity (rad/s): no air channel forms below this
$h$ — liquid depth (cm): larger $h$ raises $omega_c$ ($omega_c propto sqrt{h}$)
$a$ — vortex core radius ≈ magnet radius (mm): smaller $a$ lowers $omega_c$
$D$ — vessel diameter (cm): walls limit the ambient circulation
$mu$ — liquid dynamic viscosity (mPa·s): affects the velocity profile and the transition sharpness
$rho$ — liquid density (kg/m³): cancels in the critical condition but affects the magnetic torque
Experimental Setup
To measure $omega_c$ as a function of $h$ and $a$:
- Fill a transparent cylindrical vessel with liquid at various depths.
- Increase the magnetic stirrer speed slowly; record the speed at which the air channel suddenly appears.
- Plot $omega_c^2$ versus $h$ — the graph should be linear with slope $g/a^2$.
- Repeat with magnets of different radii to verify the dependence on $a$.
- Side-view photography with a backlight reveals the funnel depth as a function of $omega – omega_c$.
Full Derivation of the Free Surface Shape
The Rankine vortex has two zones that must be solved separately and matched at $r = a$:
Inner region — solid-body rotation ($r leq a$)
Euler equation: $frac{dp}{dr} = rhoomega^2 r$. Integrating and applying free-surface condition ($p = p_text{atm}$):
$$boxed{z_{rm in}(r) = z_0 + frac{omega^2 r^2}{2g}} qquad r leq a$$
An upward-opening paraboloid — the centre is the lowest point.
Outer region — free vortex ($r > a$)
$v_theta = omega a^2/r$. Bernoulli along the free surface (from $rtoinfty$ where $z = h$):
$$boxed{z_{rm out}(r) = h – frac{omega^2 a^4}{2g r^2}} qquad r > a$$
An inverse-square profile that dips sharply toward the axis.
Matching condition at $r = a$
Requiring $z_{rm in}(a) = z_{rm out}(a)$:
$$z_0 = h – frac{omega^2 a^2}{g}$$
Complete free-surface profile:
Axis depth: $z(0) = h – omega^2 a^2/g$
Critical condition ($z(0) = 0$): $omega_c = sqrt{gh}/a$
Air Channel Radius Above $omega_c$
For $omega > omega_c$ the axis depth would go negative — instead an air channel of radius $r_c$ opens where $z(r_c) = 0$. Two regimes:
Solving $z_{rm in}(r_c) = 0$:
$$r_c = asqrt{2!left(1 – frac{omega_c^2}{omega^2}right)}$$
At $omega = omega_c$: $r_c = 0$ — channel just opens
At $omega = sqrt{2},omega_c$: $r_c = a$ — channel reaches core edge
Why the Channel Reaches All the Way Down to the Magnet
The problem explicitly states the channel extends from the surface down to the magnet itself. Three reasons:
- Uniform pressure distribution with depth: The Rankine flow is the same at every horizontal cross-section. The pressure minimum is always on the axis ($r=0$) at every height. So if the channel opens at the surface, the opening condition is satisfied at all depths too.
- Topological stability: A closed air column inside liquid — if it narrows in the middle, the Bernoulli effect narrows it further (unstable constriction). A straight cylinder from top to bottom is the stable configuration.
- Magnet as anchor: The magnet spins at the bottom and imparts the maximum torque on the nearest liquid. The vortex core is centred directly above the magnet — the lowest-pressure point is right at the magnet’s location.
The Role of Surface Tension — Why “Thin”?
Surface tension $sigma$ at the air–liquid interface creates an additional pressure across a cylindrical channel of radius $r_c$:
$$Delta p_{rm ST} = frac{sigma}{r_c}$$
The true critical condition becomes:
$$rhoomega_c^2 a^2 = rho g h + frac{sigma}{r_c}$$
As $r_c to 0$, $sigma/r_c to infty$ — surface tension resists infinitely thin channels. The minimum observable channel radius is set by the capillary length:
$$lambda_c = sqrt{frac{sigma}{rho g}} approx 2.7,text{mm} quadtext{(water)}$$
$$We = frac{rhoomega^2 a^2, r_c}{sigma}$$
“Thin” channel means $We sim 1$: vortex pressure and surface tension are in balance. For $We gg 1$ the channel is wide and stable. Using a low-$sigma$ liquid (soapy water) makes the threshold channel thinner; a high-$sigma$ liquid makes it appear wider and more abruptly.
Dimensional Analysis — Why Does Density Cancel?
The critical condition $omega_c = sqrt{gh}/a$ contains no $rho$ or $mu$. The physical reason:
Both sides scale with $rho$, so it cancels. Strong experimental prediction: $omega_c$ should be identical for water, glycerol, and oil (density ratio up to 3×). This is one of the model’s sharpest tests.
Viscosity $mu$ does not appear because the Rankine model is inviscid. In reality $mu$ changes the core radius $a$ and the transition sharpness but not $omega_c$ directly.
Interactive Python Simulation
Four panels: (1) Rankine velocity profile, (2) free surface shape $z(r)$, (3) channel radius $r_c$ vs $omega/omega_c$, (4) critical speed vs liquid depth.
- Panel 1: Rankine profile — linear increase inside core, 1/r decay outside; maximum velocity exactly at r = a.
- Panel 2: Below ω_c the surface dips but stays above the bottom; at ω = ω_c the surface just touches the axis; above ω_c an air channel forms.
- Panel 3: ω_c ∝ √h / a — to verify experimentally, plot ω_c² vs h; the slope should equal g/a².
References
- Tritton, D. J., Physical Fluid Dynamics, 2nd ed., Oxford, 1988 — Chapter on vortices and rotating flows.
- Acheson, D. J., Elementary Fluid Dynamics, Oxford, 1990 — Rankine vortex and free surface.
- Kundu, P. K., Cohen, I. M., Fluid Mechanics, 6th ed., Academic Press, 2015 — Chapter on rotational flow.
- Vatistas, G. H., “A note on liquid vortex sloshing and Kelvin’s equilibria,” J. Fluid Mech. 217 (1990).
- Halász, G., Gyure, B., Jánosi, I. M., Szabó, K. G., Tél, T., “Vortex flow generated by a magnetic stirrer,” American Journal of Physics 75(12), 1092–1098 (2007). [arXiv: physics/0702199] — Most directly relevant: same experiment (magnetic stirrer in cylindrical container), PIV measurements, Rankine velocity profile characterisation.
- Escudier, M. P., “Observations of vortex breakdown in an open cylindrical container with a rotating bottom,” Experiments in Fluids 2, 189–196 (1984). [DOI: 10.1007/BF00196990] — Landmark experimental map of critical conditions (aspect ratio H/R vs. Re) for air-core formation in the equivalent geometry.
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