Linear and weakly nonlinear analysis of a ferrofluid layer for an LTNE model with variable gravity and internal heat source

A ferrofluid saturated porous layer convection problem is studied in the variable gravitational field for a local thermal nonequilibrium (LTNE) model. Internal heating and variation (increasing or decreasing) in gravity with distance through the layer affected the stability of the convective system. The Darcy model is employed for the momentum equation and the LTNE model for the energy equation. The boundaries are considered to be rigid-isothermal and paramagnetic. For the linear stability analysis of the three-dimensional problem, the normal mode has been applied and the eigenvalue problem is solved numerically using Chebyshev pseudospectral method, while weakly nonlinear analysis is carried out with a truncated Fourier series. The effect of different dimensionless parameters on the Rayleigh number has also been studied. We found that the system becomes unstable on the increasing value of nonlinearity index of magnetization (M3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_3$$\end{document}), porosity-modified conductivity ratio (β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document}), and internal heat parameter (ξ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\xi $$\end{document}). It is observed that the system is stabilized by increasing the value of the Langevin parameter (αL\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _\mathrm{{L}}$$\end{document}), variable gravity coefficient (δ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta $$\end{document}), and effective heat transfer parameter (H1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_1$$\end{document}). Runge–Kutta–Gill method has been used for solving the finite-amplitude equations to study the transient behavior of the Nusselt number. Streamlines and isotherms patterns are determined for the steady case and are presented graphically.