Reduction of a Tri-Modal Lorenz Model of Ferrofluid Convection to a Cubic-Quintic Ginzburg-Landau Equation Using the Center Manifold Theorem

The differential geometric method of the center manifold theorem is applied to the magnetic-Lorenz model of ferrofluid convection in an electrically non-conducting ferrofluid. The analytically intractable tri-modal nonlinear autonomous system (magnetic-Lorenz model) is reduced to an analytically tractable uni-modal cubic-quintic Ginzburg-Landau equation. The inadequacy of the cubic Ginzburg-Landau equation and the need for the cubic-quintic one is shown in the paper. The heat transport is quantified using the solution of the cubic-quintic equation and the effect of ferrofluid parameters on it is demonstrated. The stable and unstable regions in the conductive regime and the conductive-convective regime is depicted using a bifurcation diagram. The noticeable discrepancy between the results of the two models is highlighted and the quintic non-linearity effects are delineated.