Aerodynamic effects in the break-up of liquid jets: on the first wind-induced break-up regime

Abstract

We present both numerical and analytical results from a spatial stability analysis of the coupled gas–liquid hydrodynamic equations governing the first wind-induced (FWI) liquid-jet break-up regime. Our study shows that an accurate evaluation of the growth rate of instabilities developing in a liquid jet discharging into a still gaseous atmosphere requires gas viscosity to be included in the stability equations even for low ${\it We}_g$, where ${\it We}_g{=}\rho_gU_l^2R_0/\sigma$, and $\rho_g, U_l, R_0$ and $\sigma$ are the gas density, the liquid injection velocity, the jet radius and the surface tension coefficient, respectively. The numerical results of the complete set of equations, in which the effect of viscosity in the gas perturbations is treated self-consistently for the first time, are in accordance with recently reported experimental growth rates. This permits us to conclude that the simple stability analysis presented here can be used to predict experimental results. Moreover, in order to throw light on the physical role played by the gas viscosity in the liquid-jet break-up process, we have considered the limiting case of very high Reynolds numbers and performed an asymptotic analysis which provides us with a parameter, $\alpha$, that measures the relative importance of viscous effects in the gas perturbations. The criterion $|\alpha|{\ll} 1$, with $\alpha$ computed a priori using only the much simpler inviscid stability results is a guide to assess the accuracy of a stability analysis in which viscous diffusion is neglected. We have also been able to explain the origin of the ad hoc constant 0.175 introduced by Sterling & Sleicher (J. Fluid Mech. vol. 68, 1975, p. 477) to correct the discrepancies between Weber's results (Z. Angew. Math. Mech. vol. 11, 1931, p. 136) and the experimental ones.