Table of contents

The generation of mean flows by trains of traveling, inviscid water waves is investigated. All ambiguities associated with the velocity potential are resolved by treating uniform wave trains as limits of wave packets, and appealing to the conservation of mass and momentum. This formulation leads to a systematic multiple-scales description of weakly nonlinear wave trains and the associated mean flows. The results are compared with the amplitude equation formulation of Davey and Stewartson and radiation stress formulation of Longuet-Higgins and Stewart which do not conserve mass. The momentum of the wave train can be uniquely specified only by an analysis of the wave generation mechanism. The present theory is sufficiently general that mean flows arising from different generation mechanisms can be included, and shows that a recently proposed singularity associated with mean flows is absent.

The present paper deals with the numerical computation of the viscous fingering phenomenon occurring when gas is pushed into a thin layer of liquid between two parallel flat plates. The phenomenon is formulated mathematically as a gas-liquid two-phase flow problem. The moving interface between gas and liquid is not treated as a part of boundaries of a solution domain, but is recognized as the surface of density discontinuity in the domain. The density is treated as one of primary field variables of the problem. The three-dimensional flow equations, namely the momentum, continuity and energy equations, are integrated and averaged in the direction of thickness of the layer to yield the two-dimensional governing equations expressed in terms of velocity, pressure and density. The equations thus derived are discretized in space by the finite element method and in time by the finite difference method. Three characteristic growth patterns called spreading, splitting and shielding and their combined phenomenon have been calculated. Encouraging results have been obtained.

The influence of a horizontal top plate on self-sustaining oscillation over a rectangular cavity is investigated experimentally. From both the mean and the spectral analysis of the cavity shear flow, the presence of the top plate does provide a powerful external perturbation to modify the oscillatory characteristics of the cavity shear layer. While the leading-edge of the top plate is located right above the most sensitive region (or the point of flow separation) of the cavity shear layer, the self-sustaining oscillation of the shear layer is promoted most effectively. However, the effectiveness decreases as the leading-edge of the top plate moves downstream away from the most sensitive region. Insertion of the horizontal top plate not only produces a favorable streamwise pressure gradient across the cavity mouth, it also creates a stronger feedback effect from the downstream edge of the cavity. Significant flow acceleration near the most sensitive region of the cavity shear-layer reduces the local momentum thickness and enhances the receptivity of the separated shear layer to the feedback effect. Therefore, stronger feedback from the downstream edge of the cavity and high sensitivity (or receptivity) to the feedback at the most sensitive region of the cavity shear layer are the key mechanisms to promote the oscillation of the shear-layer across the cavity mouth while the horizontal top plate is inserted.

The stability against small disturbances of the plane laminar motion of an electrically conducting fluid between parallel plates in relative motion under a transverse magnetic field is investigated. Assuming that the outer regions adjacent to the fluid layer are electrically non-conducting and non-ferromagnetic, the appropriate boundary conditions on the magnetic field perturbations are presented. The Chebyshev collocation method is adopted to obtain the eigenvalue equation, which is then solved numerically. The critical Reynolds number R_c, the critical wavenumber α_c and the critical wave speed c_c are obtained for wide ranges of the magnetic Prandtl number P_m and the Hartmann number M. It is found that there exists a stationary mode of instability in addition to a travelling-wave mode of instability, and that except for the case when P_m is sufficiently small, the fluid flow becomes more unstable to the stationary mode as P_m increases.