Observations have recently been made of ion cyclotron emission (ICE) that originates from the core plasma in the DIII-D (

In this work, we examine the effect of wall slip for a gravity-driven flow of a Newtonian fluid in a partially filled circular pipe. An analytical solution is available for the no-slip case, while a numerical method is used for the case of flow with wall slip. We note that the partially filled circular pipe flow contains a free surface. The solution to the Navier–Stokes equations in such a case is a symmetry of a pipe flow (with no free surface) with the free surface as the symmetry plane. Therefore, we note that the analytical solution for the partially filled case is also the exact solution for fully filled lens and figure 8 shaped pipes, depending on the fill level. We find that the presence of wall slip increases the optimal fill height for maximum volumetric flow rate, brings the “velocity dip” closer to the free surface, and increases the overall flow rate for any fill. The applications of the work are twofold; the analytical solution may be used to verify numerical schemes for flows with a free surface in partially filled circular pipes, or for pipe flows in lens and figure 8 shaped pipes. Second, the work suggests that flows in pipes, particularly shallow filled pipes, can be greatly enhanced in the presence of wall slip, and optimal fill levels must account for the slip phenomenon when present.

This is a report on the Lovells Springs challenge that was brought to the Mathematics in Industry Study Group at the University of Newcastle, Australia, in January 2020. The design of a furnace that heats steel rods to make them malleable and allow the reshaping of the rods into coiled springs is the challenge.  Mathematical modelling of heat transport in the half-metre long furnace vestibule predicts the effect of vestibule geometry on the temperature of rods entering the furnace, and provides guidelines for deciding on the dimensions of the vestibule for improved energy efficiency of heating. Models considered include treating the rods as equivalent steel sheets, and as discrete steel rods. The relative importance of radiative and convective heat transfer mechanisms is considered. A longer vestibule, with length one or two metres,  is recommended for improved heating efficiency of rods thicker than 25mm.

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