Dynamics of liquid jets revisited

Abstract

In this paper we investigate the long-wavelength approximations of the equations governing the motion of an inviscid liquid jet. Using a formal perturbation expansion it will be shown that the one-dimensional equations presented by Lee (1974) are inconsistent. The inconsistency arises from the fact that terms which have been retained in the boundary conditions should have been rejected in view of the approximations made in the momentum equations. With the correct equations a number of anomalies between Lee's model and other models are eliminated. An explicit periodic solution to the nonlinear evolution equations we have derived is presented. However, it turns out that the wavenumbers for which this solution is valid lie outside the range in which the long-wavelength approximations are applicable. In addition we present numerical solutions to the nonlinear equations we have derived. In the unstable regime we find that, as disturbances grow, the characteristic axial lengthscales of the major features are typically of the order of the radius of the jet. This casts some doubt on the validity of the long-wavelength approximations in the study of nonlinear liquid jet dynamics.