Nonlinear stability analysis of penetrative convection in ferrofluids via internal heating

Abstract

The phenomenon of natural convection driven by internal heating (also known as penetrative convection) has widespread applications in several branches of engineering. The present article investigates the onset of penetrative convection in a horizontal layer of ferrofluids. We assume four different supply functions generate uniform/non-uniform internal heating in the layer. Two sets of realistic boundary conditions (rigid–rigid and rigid-free) are chosen for analysis instead of commonly used artificial boundary conditions of free–free boundaries. The Rayleigh number serves as the eigenvalue of the system of governing equations and is obtained using linear and energy stability analyses. The Chebyshev Tau method is employed to numerically solve the eigenvalue problems obtained through both analyses. It was observed that the energy bounds are lower than the linear ones, indicating the existence of subcritical instabilities for the present problem. The quantitative results obtained for both linear and nonlinear analyses reveal that the Rayleigh number for rigid-free boundaries is always less than that for rigid–rigid boundaries. Moreover, it was also observed that the magnetic parameters advance the onset of penetrative convection.