Water bells formed on the underside of a horizontal plate. Part 2. Theory

Abstract

In a companion paper (Part 1, Jameson et al. J. Fluid Mech. vol. 649, 2010, 19–43), the discovery of a new type of water bell was reported. When a vertical liquid jet impacts on the underside of a large horizontal plate, the resulting thin film spreads radially along the plate to an unspecified abrupt departure point, from whence it falls away from the plate of its own accord. The departure radius of the fluid from the plate is seen to depend strongly on the volumetric flow rate. The falling liquid may then coalesce to form a water bell. Here we present a theoretical analysis and explanation of this phenomenon. A force balance determining the maximum radial extension of the thin film flow along the plate is considered as a mechanism for fluid departure from the plate, for which an analytical model is developed. This model gives good predictions of the measured radius of departure. When a water bell has been formed, and the flow rate is altered, many interesting shapes are produced that depend on the shapes at previous flow rates. We discuss the origin of this hysteresis, and also present a leading order theory for the bell shape under a regime of changing flow rate. The models are compared with experimental results spanning two orders of magnitude in viscosity.