7. Sinking funnel

7. Sinking Funnel

When a long-stemmed and sufficiently heavy funnel is released vertically into water with the stem pointing downward, it undergoes a series of damped vertical oscillations while gradually sinking. Investigate the motion and stability of the sinking funnel and the conditions under which the funnel tips over.

Place a plastic funnel open-end down on a water surface — air is trapped inside. As the trapped air compresses (Boyle’s law), the funnel gradually sinks. What controls the rate and dynamics of descent?

Funnel Geometry

• Base radius R, half-angle θ, cone height H = R/tan θ
• Cone volume: V_cone = πR²H/3
• Stem radius r_s, length L_s → V_stem = πr_s²L_s
• Initial trapped air volume: V₀ ≈ V_cone

Physics: Boyle’s Law + Equation of Motion

Let z be the depth of the air pocket below the water surface. Boyle’s law (isothermal) gives:

V_air(z) = P₀ · V₀ / (P₀ + ρ_w · g · z)

Equation of motion (z positive downward):

m · z̈ = (m − ρ_w · [V_funnel + V_air(z)]) · g − ½ C_D ρ_w A v|v|

Two Behavioural Regimes

1. Initially floating: If m < ρ_w(V_funnel + V₀), the funnel floats. As depth increases, V_air shrinks until the net force turns downward — a tipping point leading to accelerating descent.
2. Immediately sinking: If m > ρ_w(V_funnel + V₀), the funnel sinks at once with increasing acceleration (positive feedback from shrinking V_air).

Two-Phase Analysis: From Stem Entry to Cone Submersion

The funnel’s descent proceeds in two distinct phases that must be treated separately:

Ratchet Mechanism — Why Oscillations Drive Gradual Sinking

The problem statement says the funnel undergoes damped vertical oscillations while gradually sinking. These two phenomena are coupled through a ratchet (one-way cycle) mechanism:

$$Delta V_text{ref,cycle} = -int_0^T Q(z(t)),dt < 0 quadtext{(one-way)}$$

The chain of consequences:

Boyle’s law positive feedback on every downstroke

In addition to the long-term ratchet, every downstroke contains an instantaneous positive feedback loop:

This positive feedback means the depth equilibrium is unstable: the funnel oscillates as long as the initial buoyancy ($m < rho_w(V_f + V_0)$) permits. As the ratchet drains the air, this buoyancy is lost and the funnel transitions to monotonic sinking.

Oscillation amplitude also decreases because the restoring buoyancy force weakens as air escapes — this is the “damping” of the oscillations.

Air Escape Through Stem (Advanced Model)

If air escapes via Hagen–Poiseuille flow through the stem:

dV_air/dt = −(π r_s⁴ / 8μ L_s) · ΔP(z)

where ΔP(z) = ρ_w·g·z is the overpressure and μ is air viscosity. A narrower stem → slower air escape → slower, more controlled descent.

Terminal Velocity

At large depths V_air → 0, so:

v_terminal = √(2(m − ρ_w V_funnel)g / C_D ρ_w A)

Advanced Analysis: Volumetric Flow Rate and Reynolds Number

Volumetric Flow Rate Q

The Hagen–Poiseuille law gives the volumetric flow rate of air escaping through the stem:

$$Q(z) = frac{pi r_s^4}{8,mu_{mathrm{air}},L_s},Delta P = frac{pi r_s^4,rho_w g}{8,mu_{mathrm{air}},L_s},z$$

Fourth-power law: doubling the stem radius → 16× the flow rate → 16× faster sinking.

Reynolds Number — Validating the Laminar-Flow Assumption

Hagen–Poiseuille is valid only for Re < 2300 (laminar flow). The mean air velocity and Reynolds number in the stem are:

$$v_{mathrm{avg}} = frac{Q}{pi r_s^2} = frac{r_s^2,rho_w g z}{8,mu_{mathrm{air}},L_s}$$

$$mathrm{Re} = frac{rho_{mathrm{air}},v_{mathrm{avg}},(2r_s)}{mu_{mathrm{air}}} = frac{rho_{mathrm{air}},r_s^3,rho_w g z}{4,mu_{mathrm{air}}^2,L_s}$$

For stems with r_s > 2 mm the flow may become turbulent. Testing whether the t_sink ∝ r_s⁻⁴ scaling still holds in that regime is one of the interesting experimental questions.

Experimental Setup

1. Basic Apparatus

• Glass aquarium 30×20×20 cm with depth marks every 1 cm on the side wall
• Plastic lab funnels — several sizes for systematic comparison

2. Controlling Stem Radius r_s

• Nail varnish: several coats inside the stem — each coat reduces r_s by ~0.1 mm
• Wax: melt paraffin into stem, then bore a central hole with a heated wire
• Metal wire: insert a wire of known diameter, seal with silicone around it
• 3-D printing: print funnel with precise r_s (±0.1 mm) — best control

3. Preventing Tip-Over ️

During descent the buoyancy force acts at the centre of buoyancy (top of the cone) while gravity acts at the centre of mass (lower). This couple tends to capsize the funnel:

• Guide rail (recommended): a vertical rod from tank bottom to top + a sliding ring on the funnel — prevents rotation without significant friction
• Annular weight: a heavy ring clipped to the funnel rim — lowers the centre of mass and makes the funnel self-righting

4. Measurements

• Camera at 120 fps + Tracker (free) to extract position from video frames
• Direct flow measurement: a U-tube manometer attached to the stem outlet tracks V_air(t) directly
• Pressure sensor (advanced): MPX5010DP + Arduino inside the funnel

5. Controlled Experiment Design

• Independent variable: r_s — change only this, keep everything else fixed
• Five repetitions per condition → mean ± standard deviation
• Plot t_sink vs 1/r_s⁴ — should be linear through the origin if H–P holds
• Log–log plot of t_sink vs r_s — slope should be ≈ −4 in the laminar regime

Interactive Python Simulation — Sinking Funnel Dynamics

This simulation numerically solves the sinking-funnel ODE system in three state variables: depth $z$, velocity $v$, and reference air volume $V_text{ref}$ (volume at atmospheric pressure).

References

• Denny, M. W. (1993). Air and Water. Princeton University Press. — buoyancy and air-compression dynamics.
• White, F. M. (2015). Fluid Mechanics, 8th ed. McGraw-Hill. — Hagen–Poiseuille derivation and laminar-to-turbulent transition.
• Munson, B. R. et al. (2012). Fundamentals of Fluid Mechanics, 7th ed. Wiley. — drag coefficients for bluff bodies.

Technical Glossary