Abstract
Linear and weakly nonlinear properties of magnetoconvection in a sparsely packed porous medium are investigated. We have obtained the values of Takens-Bogdanov bifurcation points and codimension two bifurcation points by plotting graphs of neutral curves corresponding to stationary and oscillatory convection for different values of physical parameters relevant to magnetoconvection in a sparsely packed porous medium near a supercritical pitchfork bifurcation. We have derived a nonlinear two-dimensional Ginzburg-Landau equation with real coefficients by using Newell-Whitehead (1969) method. The effect of the parameter values on the stability mode is investigated and shown the occurrence of secondary instabilities namely, Eckhaus and Zigzag instabilities. We have studied Nessult number contribution at the onset of stationary convection. We have also derived two nonlinear one-dimensional coupled Ginzburg-Landau-type equations with complex coefficients near the onset of oscillatory convection at a supercritical Hopf bifurcation and discussed the stability regions of standing and travelling waves.
Highlights
Magnetoconvection in a porous medium uniformly heated from below is of considerable interest in geophysical fluid dynamics, as this phenomena may occur within the mushy layer of Earth’s outer core
The effect of geomagnetic field on the magnetoconvection instability is of interest in geophysics, particular in the study of Earth’s interior where the molten liquid Iron is electrically conducting, which can become convectively unstable as a result of differential diffusion
Brand and Steinberg [10, 11] investigated convecting instabilities in binary liquid in a porous medium; Plam et al [9] and Brand et al have made use of Darcy’s law (−ν∇2V is replaced by KV where K is the permeability of a porous medium. for nonporous medium K is infinity)
Summary
Magnetoconvection in a porous medium uniformly heated from below is of considerable interest in geophysical fluid dynamics, as this phenomena may occur within the mushy layer of Earth’s outer core. Rudraiah [16] and Rudraiah and Vortmeyer [17] have studied both linear and steady nonlinear magnetoconvection in a sparsely packed porous medium using Brinkman model but they have taken effective viscosity μe same as fluid viscosity μ. By obtaining a one-dimensional Ginzburg-Landau equation in complex amplitude A(X, Y , T) with complex coefficients near a supercritical Hopf bifurcation, we have shown the condition for occurrence of Benjamin-Feir-type instability [21] for travelling and standing waves.
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