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.     

Flow past a cylinder: shear layer instability and drag crisis

Flow past a circular cylinder for Re=100 to 10⁷ is studied numerically by solving the unsteady incompressible two‐dimensional Navier–Stokes equations via a stabilized finite element formulation. It is well known that beyond Re ～ 200 the flow develops significant three‐dimensional features. Therefore, two‐dimensional computations are expected to fall well short of predicting the flow accurately at high Re. It is fairly well accepted that the shear layer instability is primarily a two‐dimensional phenomenon. The frequency of the shear layer vortices, from the present computations, agree quite well with the Re^0.67 variation observed by other researchers from experimental measurements. The main objective of this paper is to investigate a possible relationship between the drag crisis (sudden loss of drag at Re ～ 2 × 10⁵) and the instability of the separated shear layer. As Re is increased the transition point of shear layer, beyond which it is unstable, moves upstream. At the critical Reynolds number the transition point is located very close to the point of flow separation. As a result, the shear layer eddies cause mixing of the flow in the boundary layer. This energizes the boundary layer and leads to its reattachment. The delay in flow separation is associated with narrowing of wake, increase in Reynolds shear stress near the shoulder of the cylinder and a significant reduction in the drag and base suction coefficients. The spatial and temporal power spectra for the kinetic energy of the Re=10⁶ flow are computed. As in two‐dimensional isotropic turbulence, E(k) varies as k^?5/3 for wavenumbers higher than energy injection scale and as k^?3 for lower wavenumbers. The present computations suggest that the shear layer vortices play a major role in the transition of boundary layer from laminar to turbulent state. Copyright © 2004 John Wiley & Sons, Ltd.     

Dynamics of spatio-temporal defects in the Taylor-Dean system

The dynamics of defects in a pattern of traveling inclined rolls has been investigated. Two regimes were identified in the neighborhood of defects: a diffusive regime, with a negative phase diffusion coefficient, and a coalescence regime in which the phase gradient diverges in time following a power law behavior. The observed periodic nucleation of defects is related to the frequency inhomogeneity induced by the disymmetry of the wave amplitude. Amplitude holes have been observed in the secondary modulated pattern. Received 23 February 1999 and Received in final form 26 May 1999     

Experimental and numerical investigations on the pure material instability of an Oldroyd's 3-constant model

Received March 25, 2001 / Published online November 15, 2001     

Fluid-elastic instability in tube arrays subjected to air-water and steam-water cross-flow

Flow induced vibrations in heat exchanger tubes have led to numerous accidents and economic losses in the past. Efforts have been made to systematically study the cause of these vibrations and develop remedial design criteria for their avoidance. In this research, experiments were systematically carried out with air-water and steam-water cross-flow over horizontal tubes. A normal square tube array of pitch-to-diameter ratio of 1.4 was used in the experiments. The tubes were suspended from piano wires and strain gauges were used to measure the vibrations. Tubes made of aluminum; stainless steel and brass were systematically tested by maintaining approximately the same stiffness in the tube-wire systems. Instability was clearly seen in single phase and two-phase flow and the critical flow velocity was found to be proportional to tube mass. The present study shows that fully flexible arrays become unstable at a lower flow velocity when compared to a single flexible tube surrounded by rigid tubes. It is also found that tubes are more stable in steam-water flow as compared to air-water flow. Nucleate boiling on the tube surface is also found to have a stabilizing effect on fluid-elastic instability.     

ORBITAL INSTABILITY OF STANDING WAVES FOR THE COUPLED NONLINEAR KLEIN-GORDON EQUATIONS

This paper deals with a type of standing waves for the coupled nonlinear Klein-Gordon equations in three space dimensions. First we construct a suitable constrained variational problem and obtain the existence of the standing waves with ground state by using variational argument. Then we prove the orbital instability of the standing waves by defining invariant sets and applying some priori estimates.