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Thermomagnetic convection in a differentially heated square cavity with an infinitely long third dimension is numerically simulated using the single relaxation time lattice Boltzmann method (LBM). This problem is of considerable interest when dealing with cooling of microelectronic devices, in situations where natural convection does not meet the cooling requirements, and forced convection is not viable due to the difficulties associated with pumping a ferrofluid. Therefore, circulation is achieved by imposing a magnetic field, which is created and controlled by placing a dipole at the bottom of the enclosure. The magnitude of the magnetic force is controlled by changing the electrical current through the dipole. In this study, the effects of combined natural convection and magnetic convection, which is commonly known as “thermomagnetic convection,” are analysed in terms of the flow modes and heat transfer characteristics of a magnetic fluid.

In micro-and nano-flows, since the molecular mean free path is comparable to the system's characteristic length, the effect of rarefaction should be considered. In this category of flows, the continuum assumption is no longer valid; therefore, heat transfer, velocity profile and pressure drop markedly and depart from those which are common to macro-flows. Rarefaction may occur in various applications, such as in zero gravity flights and vacuum devices. This phenomenon can occur at the wall surfaces when the Mach number is sufficiently small, which is the case of interest in this study. In these conditions, at the solid-fluid interfaces, the fluid has a slip velocity and a temperature jump is also present. Three-dimensional, thermally developing incompressible laminar flows in a square microchannel with constant temperature walls are studied numerically using the lattice Boltzmann method. The effect of rarefaction on the velocity and temperature distributions and on the Nusselt number is analysed. The LBM predictions are compared with those obtained using the Navier-Stokes equations with the slip condition for the same configuration. The results indicate that an increase of the slip velocity yields an increasing Nusselt number, whereas an increase of the temperature jump has the opposite effect on the Nusselt number values; therefore, the combined result of these two competing effects may yield an increase or decrease on the Nusselt number.

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