Dynamics and stability of water bells

Abstract

The detailed experimental study conducted by Félix Savart in 1833 has revealed the existence of water bells when a cylindrical jet of diameter D₀ impacts with the velocity U₀ normally on to a disc of diameter D_i. We continue this study with a Newtonian fluid characterized by its density, ρ, kinematic viscosity, v, and surface tension, σ. We first show that for a given Reynolds number, Re ≡ U₀D₀/v, and Weber number, We ≡ ρU²₀D₀/σ, the domain where the bells exist in terms of the diameter ratio, X ≡ D_i/D₀, extends from the minimum value, X₋:

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up to the maximum value, X₊:

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In the domain, X ∈]X₋, X₊[, the liquid film which results from the impact of the jet detaches at the edge of the disc, forming an angle ψ₀ with the direction of the jet. In the non-viscous limit, we show that this angle is determined by the nonlinear equation

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where ψ^max₀ corresponds to the limit of ψ₀ for We >> 1. In that limit, we find that cos (ψ^max₀) ≈ 1 − 0.352X², for X < 1, and cos (ψ^max₀) ≈ 0.1 for X > 1.

The shape of the resulting bell is shown to be a catenary, first analytically described by Joseph Boussinesq in 1869. This shape results from the equilibrium between surface tension and centrifugal acceleration and is characterized by the length L ≡ D₀We/16. This solution holds in the low-gravity limit, gL/U²₀ >> 1, and when the pressure difference, p, across the liquid sheet is small, pL/(2σ) >> 1. Considering the dynamics of formation of that catenary, we show that it is characterized by a quasi-constant velocity along the jet axis.

Finally, we show that these bells are not always stationary and may even undergo self-sustained oscillations. Studying their stability, we derive a general stability criterion and show the sensitivity of the bells to both the pressure difference across the liquid sheet and to the ejection angle. In this latter case, we find a critical angle of ejection above which the bell is periodically destroyed and created. The period of the cycle is shown to scale linearly with the formation time of the bell.