Chebyshev Spectral Collocation Method Approximation to Thermally Coupled MHD Equations

Abstract

In this study, a Chebyshev spectral collocation method (CSCM) approximationis proposed for solving the full magnetohydrodynamics (MHD) equations coupledwith energy equation. The MHD flow is two-dimensional, unsteady, laminar and incompressible,and the heat transfer is considered using the Boussinesq approximation forthermal coupling. The flow takes place in a square cavity which is subjected to a verticallyapplied external magnetic field, and the presence of the induced magnetic field is alsotaken into account due to the electrical conductivity of the fluid. The governing equationsgiven in terms of stream function, vorticity, temperature, magnetic stream function, andcurrent density, are solved iteratively using CSCM for the spatial discretisation, and anunconditionally stable backward difference scheme for the time integration. The inducedmagnetic field is obtained by means of its relation to the magnetic stream function. Thebehaviours of the flow and the heat transfer are investigated for varying values of Reynolds(Re), magnetic Reynolds (Rem), Rayleigh (Ra) and Hartmann (Ha) numbers.