Dynamics and stability of water bells

Abstract

The detailed experimental study conducted by Felix 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 Di. 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 ≡ Di/D0, extends from the minimum value, X−:formula hereup to the maximum value, X+:formula hereIn 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 equationformula herewhere ψmax0 corresponds to the limit of ψ0 for We >> 1. In that limit, we find that cos (ψmax0) ≈ 1 − 0.352X2, 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.