Influence of a white noise at channel inlet on the parallel and wavy convective instabilities of Poiseuille-RayleighBénard flows Phys. Fluids 24, 084101 (2012) The role of boundaries in the magnetorotational instability Phys. Fluids 24, 074109 (2012) The onset of steady vortices in Taylor-Couette flow: The role of approximate symmetry Phys. Fluids 24, 064102 (2012) Transient growth in Rayleigh-Bénard-Poiseuille/Couette convection Phys. Fluids 24, 044103 (2012) Additional information on Phys. Fluids

Davey, Di Prima & Stuart's (1968) double amplitude expansion for disturbances in flow between concentric cylinders is formulated in matrix notation. The stability of the secondary equilibrium (Taylor-vortex) flow is calculated using fifth-order terms in amplitude, and using the full equations rather than the small-gap approximation. Qualitative confirmation is found of instabilities to the Taylor-vortex flow to non-a.xisymmetric disturbances at about 10 % above the first critical Taylor number.

A set of Jeffery–Hamel profiles (for radial, viscous, incompressible flow) have been shown by Fraenkel (1962, 1963) to approximate to profiles in certain two-dimensional divergent channels. The stability of a family of these profiles is investigated by a numerical solution of the Orr-Sommerfeld problem. Neutralstability curves are calculated in the (R,k)-planes (where R is the Reynolds number of the basic flow and k is the wave-number of the disturbance), and fairly low critical Reynolds numbers are found. For those profiles that have regions of reversed flow, negative wave velocities are found on the lower branch of the neutral curve, and also it is found that Rk tends to a finite limit as R → ∞ on the lower branch. These unexpected results are further discussed and verified by independent methods. The relation of the calculations to some experiments of Patterson (1934, 1935) is discussed.

Link to this article: http://journals.cambridge.org/abstract_S0022112075001425How to cite this article: P. M. Eagles and M. A. Weissman (1975). On the stability of slowly varying flow: the divergent The linear stability of a slowly varying flow, the flow in a diverging straightwalled channel, is studied using a modification of the 'WKB' or 'ray' method.It is shown that 'quasi-parallel' theory, the usual method for handling such flows, gives the formally correct lowest-order growth rate; however, this growth rate can be substantially in error if its magnitude is comparable to that of the rate of change of the basic state. The method used clearly demonstrates the dependence of the growth rate, wavenumber, neutral curves, etc., on the crossstream variable and on the flow quantity under consideration. When applied to the divergent channel, the method yields a much wider 'unstable' region and a much lower 'critical ' Reynolds number (depending on the flow quantity used) than those predicted by quasi-parallel theory. The determination of the downstream development of waves of constant frequency shows that waves of all frequencies eventually decay.

A theory for steady-state thin-film flow on a curved bed is presented and disturbances of Taylor or Goertler type are calculated, with reasonable agreement with experimental work.

For flow of a viscous fluid in a divergent channel of small angle it is shown that small disturbances to the basic Jeffery-Hamel flow may grow, according to nonlinear theory, to produce a secondary (supercritical) flow, in which the main flow winds from side to side in the channel and vortices form, with the whole pattern moving slowly downstream.

Considered below are the high Reynolds number flows of an incompressible fluid, and their nonparallel stability properties, in certain plane channels whose widths vary slowly in the streamwise direction. The first approximation to the steady-state flow is governed by the classical boundary layer equations but with the pressure unknown. Solutions, some including separation and reattachment, to this are obtained numerically for a range of the parameters involved, and the stability of the resulting flows is considered using fixed frequency disturbances and taking into account the nonparallel nature of the basic flow by use of a W.K.B. method. The calculations yield critical Reynolds numbers, below which all the disturbances decay downstream, and for various Reynolds numbers above the critical values the growths of the small disturbances are c~culated. The results are specialized to a particular class of channels which are straight far upstream and far downstream but vary in between. So the predictions should be much more easily amenable to experimental investigation and comparisons than those of the more idealized, diverging channel, flow problem studied by Eagles & Weissman [1 ] and the only other basic flow seriously studied from the viewpoint of nonparallel flow stability theory, the Blasius boundary layer. The results also represent the first application of both quasiparallel and slightly non-parallel stability theory to channels involving slowly varying but finite changes in width.mental investigations have also been made on the stability of nonparallel flows but mainly for

A tube of circular cross section whose radius is a function of a slow variable Z = (1/R)z, where z is the co-ordinate in the axial direction and R is a large streamwise Reynolds number, may be designated a slender tube. An elementary approximation to the flow in such,tubes is obtained and results compared with an approximation based on the profiles obtained by Daniels and Eagles [7] for exponential slender tubes.