Prof Craig SimmonsProf Raja HuilgolDr Vincent Post , Associate Prof Adrian Werner

Galileo Galilei is credited with the quotation Mathematics is the language with which God has written the universe. More than four centuries later we find that most of the important biophysical processes are modelled with help of systems of equations sometimes referred to as governing equations. Since many of the key variables modeled evolve both in time and space, partial differential equations and their modern solution methods such as the finite element method often play an important role. While classical applications such as those to the transfer of heat, fluid mechanics and shock waves continue to influence new developments, in recent years modern applications in areas such as the spread of pollution through air, water and soil as well as problems related to climate change have gained prominence.

Consequently, the work of Flinders' Continuum Modelling and Environmental Applications research group is currently devoted to development of innovative mathematical models of problems related to flows of viscoplastic fluids, maintaining the quality of ground water, salinity intrusion and sustainability, in general.

In particular, a lot mathematical modelling and simulation work is carried out under the leadership of Craig Simmons who is Professor of Hydrogeology at Flinders University and Director of the $60M National Centre for Groundwater Research and Training, which is a co-funded Centre of Excellence of the Australian Research Council and National Water Commission. Craig is one of Australia’s foremost groundwater researchers. A sample of just two, out of a much larger collection, of Craig’s research themes is included below.

Professional and community engagement

In 2011 Craig Simmons was elected Fellow of the Royal Society of South Australia.

Jerzy Filar and Craig Simmons are both editors of Springer's international journal Environmental Modeling and Assessment where Jerzy serves as Editor-in-Chief.

Operator-splitting methods and flows of viscoplastic fluids

Raja Huilgol has been working on the flows of viscoplastic fluids including inertia and thermal effects in order to model the flows of relevance to injection moulding of fibre reinforced plastics. Since these fluids possess yield stress, it has been found that the best way to solve the flow problems is to split the equations of motion along with the continuity and energy equations into a set of four separate problems. This approach is known as the operator-splitting method and leads to a set of partial differential equations which can be solved by the finite element method.


Three recently accepted publications in this area are as follows:


R. R. Huilgol: Fluid Mechanics and Viscoplasticity. Springer-Verlag Berlin Heidelberg, 2015. (
R. R. Huilgol and G. H. R. Kefayati: Natural Convection Problem in a Bingham Fluid using the Operator-splitting Method. Journal of Non-Newtonian Fluid Mechanics, 220:22-32, 2015. (
R. R. Huilgol and G. H. R. Kefayati: From mesoscopic models to continuum mechanics: Newtonian and non-Newtonian Fluids. Journal of Non-Newtonian Fluid Mechanics (


Applicability of classical dimensionless numbers in the geosciences

In simple settings, classical dimensionless Rayleigh numbers (Ra) have been widely used for nearly a century in classical fluid mechanics to assess the ratio of buoyancy forces (which drive convection) to dispersive forces (which dissipate convection). They have been used to determine whether a variable density system that is potentially unstable (dense fluid overlying less dense fluid) will remain stable (no movement of fluid and transport occurring by conduction and diffusion only for heat and solute cases respectively) or will become unstable (free convection occurs when the Rayleigh number exceeds some critical value). A series of papers by Simmons and his colleagues [Simmons et al., 2001; Prasad and Simmons, 2003; Simmons, 2005] have clearly demonstrated the problems and assumptions made in applying the Rayleigh number to the analysis of variable density flow problems. These include steady state flow assumptions, averaging of spatially distributed properties for use in dimensionless number computation, an inability to accurately quantify both time-dependent non-dimensionalising length scales and dispersion in plume problems and a limited knowledge of the actual critical Rayleigh numbers (Rac) for the onset of convection in real field settings. In the latter, the critical transition regions in flow and transport behaviour in groundwater systems are rarely known and it cannot be assumed that they are the same as those defined in earlier fluid mechanics for extremely simple boundary and layer conditions. Moreover, Simmons et al., [2001], Prasad and Simmons [2003] and Simmons et al., [in press] have clearly shown that the classical Rayleigh number used to predict the occurrence of instabilities fails in most cases when highly heterogeneous conditions prevail. This is a major contribution, since the Rayleigh number (based on mean quantities) has also been widely and erroneously applied in highly heterogeneous geologic systems by using homogeneous averaging for important aquifer geologic hydraulic properties.

Mathematical model simulation and benchmarking of variable density flow phenomena

A large number of numerical codes exist to simulate variable density flow phenomena. However, there are emerging difficulties and inconsistencies in recent literature that clearly show variable density flow simulation can be problematic at even low to moderate density contrasts. Professor Simmons’ work has made major contributions in this area. His work has involved the rigorous testing of variable density numerical computer models and the development of a widely used new test case “The Salt Lake Problem” [Simmons et al., 1999] leading to the more accurate simulation of these phenomena. Significantly, it has brought a better understanding of the two most frequently used mathematical methods of solving Darcy's Law for convective flows”. Professor Simmons and his colleagues have also made major contributions on the rigorous evaluation and extension of classic benchmark problems for testing variable density simulators which allowed for much better testing and evaluation of the numerical codes [Weatherill et al., 2004; Voss et al., 2010]. One of the major problems in the simulation of free convective phenomena has been that there has been a complex interplay between real physical processes associated with these processes (e.g., multiple steady state “bifurcation” type solutions associated with the inherent semi-chaotic behavior, or indeed highly oscillatory transient regimes in which no steady state exists at all) and numerical aspects of the simulation process (e.g., numerical perturbations triggering unrealistic fingering processes, numerical dispersion smearing out fingering and so on). Professor Simmons and his colleagues have now resolved this extremely important problem, after more than 20 years of ambiguity and confusion in the literature. To achieve this, Professor Simmons joined forces with experts on the pseudospectral technique at Imperial College London. The resultant breakthrough paper by Van Reeuwijk et al., [2009] aimed to clarify and circumvent the issue of multiple steady state solutions in the Elder problem. The body of research by Simmons and his colleagues has led to a much deeper understanding of the simulation of free convection processes using computer based mathematical models.

Production and abatement time scales in sustainable development

Jointly with J. Krawczyk (Victoria University of Wellington) and Manju Agarwal (UniSA), Jerzy Filar has been exploring a time scales' conjecture, which postulates that “reasonable" notions of sustainability must include a suitable synchronisation of time scales of both the processes of human development and those of the natural environment. We perform our analysis within a coarse, five variable, model of man-nature interactions expressed as a system of differential equations where production and human capital are coupled with both renewable and non-renewable natural resource. We demonstrate a phenomenon that we name the “sustainability screw" that describes a spiral like trajectory of the three key variables: non-renewable and renewable resources as well as the production capital. Under many plausible scenarios, this spiral tends unacceptably fast to an undesirable equilibrium. However, we also show that by adjusting the ratio of 'intensity of production effort' and 'intensity of abatement effort', parameters of the relative time scales of production and natural recovery processes can be altered in a manner that produces, arguably, more sustainable trajectories.

Sustainability Screw