Highly Accurate Solutions of a Laminar Square Duct Flow in a Transverse Magnetic Field With Heat Transfer Using Spectral Method

A liquid metal forced-convection fully developed laminar flow inside a square duct, whose surfaces are electrically insulated and subjected to a constant temperature in a transverse magnetic field, is solved numerically using the spectral method. The axial momentum, induction, and nonlinear energy equations are solved by expanding the axial velocity, magnetic field, and temperature in double Chebyshev series and are collocated at Gauss points. The resulting system of equations is solved numerically by Gauss elimination for the expansion coefficients. The velocity and the magnetic field coefficients are directly solved for, while the temperature coefficients are solved for iteratively. Results show that the velocity profile is flattened in the direction of the magnetic field, but it is more round in the direction normal to it, in a similar fashion to the case of circular tube studied previously. The powerful spectral method resolves the sharp velocity gradient near the duct walls very well leading to accurate calculation of friction factor and Nusselt number. These parameters increase with the strength of the magnetic field due to the increasing flatness of the velocity profile. Comparison with the results for the circular tube shows that the effect of magnetic field on square duct flow is slightly lower from that one for circular pipe flow.