Hydrodynamic and hydromagnetic stability

A thin liquid layer, which is heated from below, has its lower boundary modulated sinusoidally in time with amplitude δ. Weakly‐nonlinear stability theory shows that the modulation produces a range of stable hexagons near the critical Rayleigh number. For small δ the range is O(δ4) in size and decreases with modulation frequency. These hexagons bifurcate subcritically and correspond to downflow at cell centers.

Bénard convection with time‐periodic heating

The convective instability in a plane liquid layer with time-dependent temperature profile is investigated by means of a general method suitable for linear stability analysis of an unsteady basic flow. The method is based on a non-normal approach, and predicts the onset of instability, critical wavenumber and time. The method is applied to transient Rayleigh–Benard–Marangoni convection due to cooling by evaporation. Numerical results as well as theoretical scalings for the critical parameters as function of the Biot number are presented for the limiting cases of purely buoyancy-driven and purely surface-tension-driven convection. Critical parameters from calculations are in good agreement with those from experiments on drying polymer solutions, where the surface cooling is induced by solvent evaporation.

Transient Rayleigh―Bénard―Marangoni convection due to evaporation: a linear non-normal stability analysis

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

The linear stability problem for a fluid in a classical Benard geometry, when the temperature gradient has both a steady and a time‐periodic component, is considered. The modulating effect of the oscillatory gradient on the stability characteristics of the basic configuration is examined. It is found that, in general, there is enhancement of the critical value of a suitably defined Rayleigh number.

G. A. Tanaka

Papers

Modulation of Thermal Convection Instability