Nonlinear analysis of the deformation and breakup of viscous microjets using the method of lines

We present a numerical model for predicting the instability and breakup of viscous microjets of Newtonian fluid. We adopt a one‐dimensional slender‐jet approximation and obtain the equations of motion in the form of a pair of coupled nonlinear partial differential equations (PDEs). We solve these equations using the method of lines, wherein the PDEs are transformed to a system of ordinary differential equations for the nodal values of the jet variables on a uniform staggered grid. We use the model to predict the instability and satellite formation in infinite microthreads of fluid and continuous microjets that emanate from an orifice. For the microthread analysis, we take into account arbitrary initial perturbations of the free‐surface and jet velocity, as well as Marangoni instability that is due to an arbitrary variation in the surface tension. For the continuous nozzle‐driven jet analysis, we take into account arbitrary time‐dependent perturbations of the free‐surface, velocity and/or surface tension as boundary conditions at the nozzle orifice. We validate the model using established computational data, as well as axisymmetric, volume of fluid (VOF) computational fluid dynamic (CFD) simulations. The key advantages of the model are its ease of implementation and speed of computation, which is several orders of magnitude faster than the VOF CFD simulations. The model enables rapid parametric analysis of jet breakup and satellite formation as a function of jet dimensions, modulation parameters, and fluid rheology. Copyright © 2010 John Wiley & Sons, Ltd.     

A simple proof of the linear instability of rotating liquid drops

We give a simplified proof of the linear instability of equilibrium figures of rotating liquid based on energy estimates.      

Unstable plasma characteristics in mirror field electron cyclotron resonance microwave ion source

Electron cyclotron plasma reactor are prone to instabilities in specific input power [3–7] region (150–450 watts). In this region power absorption by gas molecules in the cavity is very poor and enhanced input power gets reflected substantially without increasing ion density. There are abrupt changes in plasma characteristics when input power was decreased from maximum to minimum, it was observed that reflected power changed from <2% to ∼50%. Minimum two jumps in reflected power were noticed in this specific power region and these appear to be highly sensitive to three stub tuner position in the waveguide for this particular input power zone. Unstable plasma region of this source is found to be dependent upon the magnetic field strength. Some changes in reflected power are also noticed with pressure, flow and bias and they are random in nature.     

Study of the membrane effect on turbulent mixing measurements in shock tubes

The effect of the thin membrane on the time evolution of the shock wave induced turbulent mixing between the two gases initially separated by it is investigated using two different sets of experiments. In the first set, in which a single-mode large-amplitude initial perturbation was studied, two gas combinations (air/SF and air/air) and two membrane thicknesses were used. The main conclusion of these experiments was that the tested membrane has a negligible effect on the evolution of the mixing zone, which evolves as predicted theoretically. In the second set, in which similar gas combinations and membrane thicknesses were used, small amplitude random-mode initial perturbation, caused by the membrane rupture, rather than the large amplitude single-mode initial perturbation used in the first set, was studied. The conclusions of these experiments were: (1) The membrane has a significant effect on the mixing zone during the initial stages of its growth. This has also been observed in the air/air experiment where theoretically no growth should exist. (2) The membrane effect on the late time evolution, where the mixing zone width has reached a relatively large-amplitude, was relatively small and in good agreement with full numerical simulations. The main conclusion from the present experiments is that the effect of the membrane is important only during the initial stages of the evolution (before the re-shock), when the perturbations have very small amplitudes, and is negligible when the perturbations reach relatively large amplitudes. Received 29 August 1998 / Accepted 25 December 1998     

Transverse nonlinear instability of solitary waves for some Hamiltonian PDE's

We present a general result of transverse nonlinear instability of 1d solitary waves for Hamiltonian PDE's for both periodic or localized transverse perturbations. Our main structural assumption is that the linear part of the 1-d model and the transverse perturbation “have the same sign”. Our result applies to the generalized KP-I equation, the Nonlinear Schrödinger equation, the generalized Boussinesq system and the Zakharov–Kuznetsov equation and we hope that it may be useful in other contexts.     

Orbital Instability of Standing Waves for the Klein-Gordon-Zakharov System

O Introduction We consider the orbital instability of standing waves for the Klein-Gordon-Zakharov system with different propagation speeds in three space dimensions     

Aspect-ratio dependent oscillatory thermocapillary convection in the floating half zone

The dependency of the critical Marangoni number on the geometrical aspect ratio of the floating half zone is essential to predict the onset of oscillatory thermocapillary convection.The experimental studies in the microgravity conditions on floating half zones of several centimeters in diameter have predicted that the critical Marangoni number increases with the increasing aspect ratio,and the terrestrial experimental studies have predicted the contradictory conclusion for floating half zones of several mil...     

Instability analysis of saddle points by a local minimax method

The objective of this work is to develop some tools for local instability analysis of multiple critical points, which can be computationally carried out. The Morse index can be used to measure local instability of a nondegenerate saddle point. However, it is very expensive to compute numerically and is ineffective for degenerate critical points. A local (weak) linking index can also be defined to measure local instability of a (degenerate) saddle point. But it is still too difficult to compute. In this paper, a local instability index, called a local minimax index, is defined by using a local minimax method. This new instability index is known beforehand and can help in finding a saddle point numerically. Relations between the local minimax index and other local instability indices are established. Those relations also provide ways to numerically compute the Morse, local linking indices. In particular, the local minimax index can be used to define a local instability index of a saddle point relative to a reference (trivial) critical point even in a Banach space while others failed to do so.

     

Stability of droplets and channels on homogeneous and structured surfaces

Wetting of structured or imprinted surfaces which leads to a variety of different morphologies such as droplets, channels or thin films is studied theoretically using the general framework of surface or interface thermodynamics. The first variation of the interfacial free energy leads to the well-known Laplace equation and a generalized Young equation which involves spatially dependent interfacial tensions. Furthermore, we perform the second variation of the free energy for arbitrary surface patterns and arbitrary shape of the wetting morphology in order to derive a new and general stability criterion. The latter criterion is then applied to cylindrical segments or channels on homogeneous and structured surfaces. Received 4 August 1999     

Application of the sequential gradient-restoration algorithm to thermal convective instability problems

The problem of the thermal stability of a horizontal incompressible fluid layer with linear and nonlinear temperature distributions is solved by using the sequential gradient-restoration algorithm developed for optimal control problems. The hydrodynamic boundary conditions for the layer include a rigid or free upper surface and a rigid lower surface. The resulting disturbing equations are solved as a Bolza problem in the calculus of variations. The results of the study are compared with the existing works in the literature.The authors acknowledge valuable discussions with Dr. A. Miele.