Water Waves and Related Free-Surface Flows

Nonlinear travelling waves on a spiralling liquid jet

“…We see that as δ increases, the most unstable wavenumber increases from 0.697 which is the limiting value for δ → 0. 2 Obviously, this method neglected the equation (12) which is used only to calculate the steady (initial) solutions. For T = 0 the equation (12) is satisfied identically, but for T > 0 only approximately.…”

“…This graph can be viewed as the error which resulted from the neglect of this equation as the time evolves. If the translational velocity terms v 0 and w 0 had been retained in the equations of motion, as well as allowing X 0t = 0 and Z 0t = 0, then these terms would appear on the right-hand side of (12). Hence this error is a measure of the size of the translational velocity components v 0 and w 0 , and hence the size of this temporal translational velocity of the jet's centreline.…”

“…In section V an equation (26) is derived for E. It can be seen that E satisfies a first order inhomogenous partial differential equation and the inhomogenous part of that equation is given by the left hand-side of (12). Therefore Figure 4 shows the magnitude of the forcing imposed on the error E. Since E is proportional to the translational velocity components v 0 and w 0 , E should be measured in relation to the mean jet speed u 0 , which is initially close to one and grows as S and T grow.…”

“…It is found that (12) and (15) are satisfied until break up when our theory fails. The steady solutions for X and Z are found from (12) and (15).…”

“…An extensive review of the work on straight axisymmetric liquid jets is given by Eggers 11 and Vanden-Broeck. 12 Curved liquid jets and sheets have been studied before by Weber 13 20 , Yarin et al 21 Hohman et al 22 and others. Here we extend these works and Wallwork et al 2 to develop a long wavelength nonlinear theory for slender inviscid jets subjected to rotational forces.…”