Robert McKibbin

The current paper examines some novel and interesting aspects of the mathematical modelling of fluid flow and forced convection heat transfer from a plate stack embedded in a homogeneous porous media. Time-dependent forced convection heat transfer from some cylinders embedded in groundwater are explored numerically using finite difference methods. An algorithm was developed about rectangular heated solid objects with the same size placed in an in-line arrangement in homogeneous porous media. Central and one-sided finite difference formulae are used to discretize the domain. Applying the mathematical model, continuity, momentum, and energy equations are solved by the finite difference methods. Generally, analytical solutions are not possible for such problems, so most of the work is done numerically. The numerical results for the two-dimensional velocity potential and stream function have been calculated and the resulting fluid and temperature contours are represented graphically for various values of the parameters involved (e.g. length of the entrance, the fluid flow rates, coefficient of thermal diffusion and advection in the observed long-time plume behaviour, effect of time and temperature on heat transfer, etc.). It is observed that fluid and heat fluxes and fluid velocity depend upon the size, shape, burial depth, position of heated objects, and positions of the entrance and exit of the homogeneous porous media. Moreover, the hydraulic conductivity of the porous medium affects the pressure drops, fluid velocity as well as the heat transfer rates.

To predict the trajectory of projectiles such as bullets and mortar shells, we require knowledge of the acting forces, including drag. For the most part, the drag coefficient (which is dependent on the local Mach number) is well-understood for subsonic and supersonic velocities. However, there is often a rapid and unintuitive change in the drag behaviour near the speed of sound. The transonic behaviour of the projectiles is addressed in two major ways. First we explain the underlying physics of drag in the three major regimes. The appearance of shock waves alters the drag forces dramatically. In some situations, the physical models are simplified to directly obtain the drag coefficient profile. Then we tackle an inverse problem, where firing table data gives the drag coefficient profile. The drag profile obtained by both point-by-point optimisation and by parametrising a suitable family of functions. Finally, transonic data is difficult to obtain in wind-tunnel experiments, so based on our understanding of the physics, alternative experiments are suggested. This research was undertaken as part of the 2017 Mathematics in Industry Study Group (Adelaide) with industry partner, the Defence Science and Technology group.

The self-thinning rule and plant population modelling with resource constraints

Modelling microbial pollutant loads associated with surface water run-off in water supply catchments

This is a review of progress made since [R. McKibbin, “An attempt at modelling hydrothermal eruptions”, Proc. 11th New Zealand Geothermal Workshop 1989 (University of Auckland, 1989), 267–273] began development of a mathematical model for progressive hydrothermal eruptions (as distinct from “blasts”). Early work concentrated on modelling the underground process, while lately some attempts have been made to model the eruption jet and the flight and deposit of ejected material. Conceptually, the model is that of a boiling and expanding two-phase fluid rising through porous rock near the ground surface, with a vertical high-speed jet, dominated volumetrically by the gas phase, ejecting rock particles that are then deposited on the ground near the eruption site. Field observations of eruptions in progress and experimental results from a laboratory-sized model have confirmed the conceptual model. The quantitative models for all parts of the process are based on the fundamental conservation equations of motion and thermodynamics, using a continuum approximation for each of the components.