Dynamics and stability of water bells

The detailed experimental study conducted by Félix Savart in 1833 has revealed the existence of water bells when a cylindrical jet of diameter D_o impacts with the velocity U_o 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 ≡ U_oD_o/v, and Weber number, We ≡ ρU_o²D_o/σ, the domain where the bells exist in terms of the diameter ratio, X ≡ D_i/D_o, extends from the minimum value, X_: X_ = 62/We′ up to the maximum value, X₊: 17.6 X₊/Re^1/3 [1 + 3.51 (X₊/Re^1/3) ³] = We/Re^1/3. In the domain, X ε]X_,X₊[, the liquid film which results from the impact of the jet detaches at the edge of the disc, forming an angle ψ_o with the direction of the jet. In the non-viscous limit, we show that this angle is determined by the nonlinear equation cos (ψ_o) - cos (ψ_o^max) = 8X/We sin (ψ_o), where ψ_o^max corresponds to the limit of ψ_o for We ≫ 1. In that limit, we find that cos (ψ_o^max) ≈ 1 - 0.352X², for X < 1, and cos (ψ_o^max) ≈ 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 ≡ D_oWe/16. This solution holds in the low-gravity limit, gL/U_o² ≪ 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 show to scale linearly with the formation time of the bell.