The influence of surface tension upon trapped waves and hydraulic falls

We consider steady two-dimensional free-surface flows past submerged obstructions on the bottom of a channel. The flow is assumed to be irrotational, and the fluid inviscid and incompressible. Both the effects of gravity and surface tension are considered. Critical flow solutions with subcritical flow upstream and supercritical flow downstream are sought using fully nonlinear boundary integral equation techniques based on the Cauchy integral formula. When a second submerged obstruction is included further upstream in the flow configuration in the absence of surface tension, solutions which have a train of waves trapped between the two obstacles before the critical flow have already been found (Dias and Vanden-Broeck 2004). We extend this work by including the effects of surface tension. Trapped wave solutions are found upstream for small values of the Bond number, for some values of the Froude number. Other types of trapped waves are found for stronger tension when the second obstruction is placed downstream of the hydraulic fall generated by the first obstacle

Nonlinear flexural waves in fluid–filled elastic channels

Nonlinear waves on liquid sheets between thin infinite elastic plates are studied analytically and numerically. Linear and nonlinear models are used for the elastic plates coupled to the Euler equations for the fluid. One-dimensional time dependent equations are derived based on a long-wavelength approximation. Inertia of the elastic plates is neglected, so linear perturbations are stable. Symmetric and mixed-mode travelling waves are found with the linear plate model and symmetric travelling waves are found for the nonlinear case. Numerical simulations are employed to study the evolution in time of initial disturbances and to compare the different models used. Nonlinear effects are found to decrease the travelling wave speed compared with linear models. At sufficiently large amplitude of initial disturbances, higher order temporal oscillations induced by non-linearity can lead to thickness of the liquid sheet approaching zero

The trajectory and stability of a spiralling liquid jet:Viscous theory

AbstractWe examine a spiralling slender viscous jet emerging from a rapidly rotating orifice, extending Wallwork et al. [I.M. Wallwork, S.P. Decent, A.C. King, R.M.S.M. Schulkes, The trajectory and stability of a spiralling liquid jet. Part 1. Inviscid theory, J. Fluid Mech. 459 (2002) 43–65] by incorporating viscosity. The effects of viscosity on the trajectory of the jet and its linear instability are determined using a mixture of computational and asymptotic methods, and verified using experiments. A non-monotonic relationship between break-up length and rotation rate is demonstrated with the trend varying with viscosity. The sizes of the droplets produced by this instability are determined by considering the most unstable wave mode. It is also found that there is a non-monotonic relationship between droplet size and viscosity. Satellite droplet formation is also considered by analysing very short wavelength modes. The effects of long wavelength modes are examined, and a wave which propagates down the trajectory of the jet is identified for the highly viscous case. A comparison between theoretical and experimental results is made, with favourable agreement. In particular, a quantitative comparison is made between droplet sizes predicted from the theory with experimental observations, with encouraging agreement obtained. Four different types of break-up are identified in our experiments. The experimentally observed break-up mechanisms are discussed in light of our theory