Asymptotic approximation of the Wiener-Hopf technique as applied to jet atomisation

Abstract

An approximate Wiener-Hopf (WH) technique is developed for solving problems involving fine spatial structure. As an example of the application of this method we investigate the atomisation of a liquid jet. The jet exits from a nozzle into an ambient fluid. Short interfacial waves become unstable and break into small particles. This problem is treated as a potential flow under the influence of capillary effects at the interface and the pressure fluctuation at the nozzle wall. Two simultaneous WH equations are obtained. To solve them, the singular parts in each equation are separated from the regular ones, that leads to a linear system of algebraic equations for the residues. The response-wave amplitudes are evaluated numerically and the instability diagram is presented. It is found that resonance occurs at double roots of the dispersion relation. For a given azimuthal number m, the double roots form two curves parametrised by the Weber number β. They merge at a certain critical point, where an even stronger resonance occurs. This finally selects the dominant modes. By gauging one parameter, namely the velocity ratio U, the theoretical prediction agrees quite well with experimental results of the jet atomisation.