Numerical study of magnetohydrodynamic natural convection flow in a rectangular cavity with heat generation

Abstract

Natural convection flow and heat transfer in a rectangular cavity in the presence of magnetic field and heat generation is numerically studied. The physical problems are represented mathematically by the governing equations along with the corresponding boundary conditions. Two-dimensional steady state Navier-Stokes equations, energy equation and continuity equation are modified to account for the presence of thermal buoyancy, magnetic field and the heat generation effects. With proper choice of the dimensionless variables the equations are transformed to non-dimensional form. The dimensionless parameters appeared in the equations are the external Rayleigh number, internal Rayleigh number, Hartmann number and Prandtl number. The equations are solved numerically by finite element method using Galerkin weighted residual approach. The nonlinear algebraic equations obtained from the differential equation are solved by Newton-Raphson method. The effect of external Rayleigh number, internal Rayleigh number, Hartmann number and Prandtl number on isotherm and stream function has been studied. The rate of heat transfer from the constant temperature wall, variable temperature wall and cold wall of the cavity has been investigated for different internal Rayleigh number and Hartmann number. The numerical results obtained are displayed in tabular form and graphically. In this study all four boundaries of the cavity are considered to be rigid. The left wall is maintained at high temperature while the right wall is held at low temperature. The top wall is assumed to be adiabatic. For the bottom wall three types of heating conditions, namely (i) linear (ii) parabolic and (iii) sinusoidal temperature profiles are used. Three different problems are studied having these three different temperature profiles at the bottom wall. Finally a comparative study of the effect of these three different temperatures on streamlines, isotherms and heat transfer is made. The results indicate that the increase in the values of external Rayleigh number leads to the increase in the size of the primary cell until it occupies the whole cavity space. Also the circulation gets stronger and consequently the temperature contours are moved towards the right corner of the cavity. Rate of heat transfer from the heated left wall and bottom wall increases due to the increase of external Rayleigh number. The temperature of the fluid in the cavity increases due to the increase of the internal Rayleigh number, consequently the rate of heat transfer from the left wall and bottom wall decreases. Increase in the value of internal Rayleigh number leads to develop a secondary cell on the left corner of the cavity and its size increases until it occupies almost half of the cavity. The significant effect of the magnetic field is observed in the heat transfer mechanisms and flow characteristics inside the cavity. Strong suppression of the convective current can be obtained by applying strong magnetic field. This is why average Nusselt number reduces as the strength of the applied magnetic field increases. The obtained results revealed that the temperature, heat transfer and fluid flow characteristics in the cavity strongly depend on the relevant dimensionless parameters. When the external Rayleigh number and the internal Rayleigh number is same, it is seen that the secondary cell in case of parabolic temperature profile is larger than that of linear temperature profile and heat transfer in case of linear temperature profile is more than that of parabolic temperature profile. Further it is observed that when the external Rayleigh number and the internal Rayleigh number is same the secondary cell in case of sinusoidal temperature profile is larger than that of parabolic temperature profile. The heat transfer in case of parabolic temperature profile is more than that of sinusoidal temperature profile.