Hydrodynamic and hydromagnetic stability

The stability of a horizontal fluid layer when the thermal (or concentration) gradient is not uniform is examined by means of linear stability analysis. Both buoyancy and surface-tension effects are considered, and the analogous problem for a porous medium is also treated. Attention is focused on the situation where the critical Rayleigh number (or Marangoni number) is less than that for a linear thermal gradient, and the convection is not (in general) maintained. The case of constant-flux boundary conditions is examined because then a simple application of the Galerkin method gives useful results and general basic temperature profiles are readily treated. Numerical results are obtained for special cases, and some general conclusions about the destabilizing effects, with respect to disturbances of infinitely long wavelength, of various basic temperature profiles are presented. If the basic temperature gradient (considered positive, for a fluid which expands on heating, if the temperature decreases upwards) is nowhere negative, then the profile which leads to the smallest critical Rayleigh (or Marangoni) number is one in which the temperature changes stepwise (at the level at which the velocity, if motion were to occur, would be vertical) but is otherwise uniform. If, as well as being non-negative, the temperature gradient is a monotonic function of the depth, then the most unstable temperature profile is one for which the temperature gradient is a step function of the depth.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112075002662

The onset of transient convective instability

The method of energy is used to discuss the stability of time-dependent diffusive temperature profiles in fluid layers subject to impulsive changes in surface temperature.Bounds for the ratio of disturbance energy production to dissipation are found to be parametric functions of time because the basic temperature develops through diffusion. This time dependence leads to the demarcation of regions of stability in a Rayleigh number-time plane and the interpretation of these regions is given. Numerical results are presented for the cases of impulsive heating and cooling of initialty isothermal fluid layers. New global stability results which give the Rayleigh number below which the diffusive solution to the Boussinesq equations is unique are reported for these cases.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S002211207300008X

Global stability of time-dependent flows: impulsively heated or cooled fluid layers

This study is devoted to the onset of convection in differentially heated cylinders under gravity modulation. It specifically concerns the case of a vertical cylinder of infinite length, when a negative temperature gradient is maintained in the upward direction. The effect of modulation on the stability limits given by linear theory in the standard steady case is analysed. A method based on Floquet theory is proposed in the case of small values of the modulation amplitude e, for a fixed value of the frequency ω. A general technique, called matrix method, which can easily be adapted to various kinds of geometries and boundary conditions, has been developed. Analytical approaches have been derived in some cases. Finally, an asymptotic analysis is presented for large ω, under very general boundary conditions and periodic constraints, for finite e. An asymptotic relation is established for the onset of convection under periodic gravity modulation for large ω values, when e [Lt ] ω; the mathematical and physical foundations of this inequality are discussed.

Natural convection in a long vertical cylinder under gravity modulation

Onset of instability in a fluid layer heated sinusoidally from below

The formation of convective cells in a fluid between two horizontal rigid boundaries with time-periodic temperature distribution is studied by the use of the Floquet theory. Numerical results for the critical Rayleigh number are given for a Prandtl number of 0·73 (air) and for various values of the frequency and magnitude of the primary temperature oscillation. Some numerical results for a Prandtl number of 7·0 (water) are also given. The most striking feature of the results is that the disturbances (or convection cells) oscillate either synchronously or with half frequency.

Chin-Hsiu Li

Papers

Instability of unsteady flows or configurations. Part 2. Convective instability