Non-autonomous Ginzburg Landau Equation Research Articles

Synchronous and asynchronous boundary temperature modulations of Bénard–Darcy convection

A theoretical analysis of thermo-convective instability in a densely packed porous medium is carried out when the boundary temperatures vary with time in a sinusoidal manner. By performing a weakly non-linear stability analysis, the Nusselt number is obtained as a function of amplitude of convection which is governed by a non-autonomous Ginzburg–Landau equation derived for the stationary mode of convection. The paper succeeds in unifying the modulated Bénard–Darcy, Bénard–Rayleigh, Bénard–Brinkman and Bénard–Chandrasekhar convection problems and hence precludes the study of these individual problems in isolation. A new result that shows that asynchronous temperature modulation may be effectively used to either enhance or reduce heat transport by suitably adjusting the frequency and phase-difference of the modulated temperature is presented.

Study of Heat Transport in a Porous Medium Under G-jitter and Internal Heating Effects

In this article we study the combined effect of internal heating and time-periodic gravity modulation on thermal instability in a closely packed anisotropic porous medium, heated from below and cooled from above. The time-periodic gravity modulation, considered in this problem can be realized by vertically oscillating the porous medium. A weak non-linear stability analysis has been performed by using power series expansion in terms of the amplitude of gravity modulation, which is assumed to be small. The Nusselt number has been obtained in terms of the amplitude of convection which is governed by the non-autonomous Ginzburg–Landau equation derived for the stationary mode of convection. The effects of various parameters such as; internal Rayleigh number, amplitude and frequency of gravity modulation, thermo-mechanical anisotropies, and Vadasz number on heat transport has been analyzed. It is found that the response of the convective system to the internal Rayleigh number is destabilizing. Further it is found that the heat transport can also be controlled by suitably adjusting the external parameters of the system.

Non-autonomous Ginzburg-Landau equation and its attractors

The behaviour as of solutions of the non-autonomous Ginzburg-Landau (G.-L.) equation is studied. The main attention is focused on the case when the dispersion coefficient in this equation satisfies the inequality for , where is an unbounded subset of . In this case the uniqueness theorem for the G.-L. equation is not proved. The trajectory attractor for this equation is constructed. If the coefficients and the exciting force are almost periodic (a.p.) in time and the uniqueness condition fails, then the trajectory attractor is proved to consist precisely of the solutions of the G.-L. equation that admit a bounded extension as solutions of this equation onto the entire time axis . The behaviour as of solutions of a perturbed G.-L. equation with coefficients and the exciting force that are sums of a.p. functions and functions approaching zero in the weak sense as is also studied.