Oscillatory Modes Of Instability Research Articles

Direct Resonance in Double‐Diffusive Systems

In thermohaline convection and a number of other systems, both direct and oscillatory modes of instability are possible. Should these modes coalesce at some point in parameter space, nonlinear instabilities are likely to be more dramatic in the neighborhood of such a point than elsewhere. A finite‐amplitude evolution equation describing such events is derived here by the method of multiple scales. The results are compared with those obtained previously for model systems and by other methods. The direct resonance point of view is found to provide some new insights. A generalization to allow for slow spatial modulation of the amplitude is given and, among other possibilities, solitary‐wave solutions are obtainable.

ON THE OVERSTABILITY CONVECTION

The oscillatory mode of instability in a rotating fluid layer heated from below is theoret-ically investigated with a combination of Eulerian and Lagrangian analyses. An exact analysisis not only nmde for the case of two free bounding surfaces, but for the case of one free andone rigid bounding surfaces. The physics of overstability convection is attributable to the dual role of the thermaldiffusion in this rotating fluid Thermal diffusion ordinarily only destroys the buoyancy offluid parcels. But if the rotation rate is sufficiently large and if the viscous dissipation iscomparatively weak, the heat exchanges alternately produce overcooling and overheating of theparcels and thereby give rise to a net positive feedback mechanism for the instability. The parcels near a rigid bounding surface (they are different from those near a freebounding surface), however, do not have a net horizontal drift and penetrate more and moreinto the interior of the layer, and they just progressively drift towards the rigid surface. Andthe vertical velocity near the rigid surface layer lags behind that in the bulk of the interiorlayer by a quarter of the oscillation period. These differences stem from the dynamical effectsof the frictionally induced secondary circulation known as Ekman pumping and suction. The spatially averaged energetics of oscillatory unstable modes are examined and discussed. On the basis of the theoretical results, it is suggested that the observod oscillatory vortexin a laboratory simulation of the flow over a heated Qinghai-Xizang model plateau (Researchon Experimental Simulation, 1978) may be interpreted in terms of the notion of overstabilityconvection.

Surface Tension Driven Instability in a Horizontal Liquid Layer with a Deformable Free Surface. II. Overstability

The theory described in the first paper of this series [M. Takashima: J. Phys. Soc. Jpn. 50 (1981) 2745] is extended to include the possibility that surface tension driven instability in a horizontal thin liquid layer confined between a solid wall and a deformable free surface can set in as purely oscillatory motions. Linear stability theory is applied to derive a time-dependent eigenvalue relationship. Numerical computations are carried, and it is found that whichever side of the liquid layer is a free surface oscillatory modes of instability can occur when and only when the temperature of the solid wall is lower than that of the air phase.

Electrohydrodynamic Instability in a Viscoelastic Liquid Layer

The problem of the onset of instability in a horizontal layer of viscoelastic dielectric liquid under the simultaneous action of a vertical ac electric field and a vertical temperature gradient is analyzed. Applying linear stability theory, an equation of eighth order is derived. Under somewhat artificial boundary conditions, this equation can be solved exactly to yield the required eigenvalue relationship from which various critical values are determined in detail. It is shown that oscillatory modes of instability exist only when the thickness of the liquid layer is smaller than about 0.5 mm and for such a thin layer the force of electrical origin is much more important than buoyancy force.

Marangoni instability with an evaporating interface

The hydrodynamic stability of an evaporating fluid layer is analyzed using a linear model which takes explicit account of the uniform velocity induced by evaporation. From this model a characteristic equation is derived and results of a series of numerical calculations are presented graphically. Translation due to evaporation tends to stabilize the system with respect to surface tension driven convection. Unstable conditions are characterized by much smaller length scales and faster growth rates than when interface translation is absent. No oscillatory modes of instability are found.