Weakly nonlinear instability of a ferromagnetic fluid rotating about a vertical axis

Abstract We study the thermoconvective instability in a rotating ferromagnetic fluid confined between two parallel infinite plates with temperature modulation at the boundaries. We use weakly nonlinear stability theory to analyze the stationary convection in terms of critical Rayleigh numbers. The influence of parameters such as the Taylor number, the ratio of the magnetic force to the buoyancy force and the magnetization on the flow behaviour and structure are investigated. The heat transfer coefficient is analyzed for both the in-phase and the out-of-phase modulations. A truncated Fourier series is used to obtain a set of ordinary differential equations for the time evolution of the amplitude of convection for the ferromagnetic fluid flow. The system of differential equations is solved using a recent multi-domain spectral collocation method that has not been fully tested on such systems before. The solutions sets are presented as sets of trajectories in the phase plane. For some supercritical values of the Rayleigh number, spiralling trajectories that transition to chaotic solutions are obtained. Additional results are presented in terms of streamlines and isotherms for various Rayleigh numbers.

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Nonlinear instability of a contact line driven by gravity

Journal of Fluid Mechanics ◽

10.1017/s0022112000008508 ◽

2000 ◽

Vol 413 ◽

pp. 355-378 ◽

Cited By ~ 34

Author(s):

SERAFIM KALLIADASIS

Keyword(s):

Thin Film ◽

Differential Equation ◽

Partial Differential Equation ◽

Dynamical Systems ◽

Contact Line ◽

Nonlinear Instability ◽

Inclined Plane ◽

Translational Invariance ◽

Weakly Nonlinear ◽

Fingering Instability

A thin liquid mass of fixed volume spreading under the action of gravity on an inclined plane develops a fingering instability at the front. In this study we consider the motion of a viscous sheet down a pre-wetted plane with a large inclination angle. We demonstrate that the instability is a phase instability associated with the translational invariance of the system in the direction of flow and we analyse the weakly nonlinear regime of the instability by utilizing methods from dynamical systems theory. It is shown that the evolution of the fingers is governed by a Kuramoto–Sivashinsky-type partial differential equation with solution a saw-tooth pattern when the inclined plane is pre-wetted with a thin film, while the presence of a thick film suppresses fingering.

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A Comparison of the Weakly Nonlinear Instability of Westerly and Easterly Jets in a Two-Layer Beta-Plane Model

Journal of the Atmospheric Sciences ◽

10.1175/1520-0469(1991)048<1701:acotwn>2.0.co;2 ◽

1991 ◽

Vol 48 (15) ◽

pp. 1701-1717 ◽

Cited By ~ 2

Author(s):

Steven B. Feldstein

Keyword(s):

Nonlinear Instability ◽

Plane Model ◽

Weakly Nonlinear ◽

Beta Plane

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Weakly nonlinear oscillatory convection in a rotating fluid

Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences ◽

10.1098/rspa.1992.0003 ◽

1992 ◽

Vol 436 (1896) ◽

pp. 33-54 ◽

Cited By ~ 9

Keyword(s):

Standing Wave ◽

Perturbation Technique ◽

Vertical Axis ◽

Travelling Wave ◽

Base Flow ◽

Two Dimensional ◽

Oscillatory Convection ◽

Weakly Nonlinear ◽

Wave Disturbances ◽

The Stability

The problem of weakly nonlinear two- and three-dimensional oscillatory convection in the form of standing waves is studied for a horizontal layer of fluid heated from below and rotating about a vertical axis. The solutions to the nonlinear problem are determined by a perturbation technique and the stability of all the base flow solutions is investigated with respect to both standing wave and travelling wave disturbances. The results of the stability and the nonlinear analyses for various values of the rotation parameter τ and the Prandtl number P (0 ≼ P < 0.677) indicate that there is no subcritical instability and that all the base flow solutions are unstable. Disturbances with highest growth rates are found to be some particular disturbances superimposed on two-dimensional base flow. Particular standing wave disturbances parallel to two-dimensional base flow are the most unstable ones either for sufficiently small P or for intermediate values of P with τ below some critical value τ *. Travelling wave disturbances inclined at an angle of about 45° to the wave vector of two-dimensional base flow are the most unstable disturbances either for P sufficiently close to its upper limit or for intermediate values of P with τ ≽ τ *. The dependence on P and τ of the nonlinear effect on the frequency and of the heat flux are also discussed.

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