AJEE Online

ISSN 1324-5821

ABSTRACTS

A non-linear dynamical system: Flow past a sluice gate

KEYWORDS: Free-surface flow; potential flow; Korteweg-de Vries equation; dynamical systems.

ABSTRACT: We consider the steady flow of water past a sluice gate to find the upstream and downstream water heights for different parameters and gate geometries. This leads to a study of a two-dimensional non-linear dynamical system suitable for lecturers to use in second- and third-year courses on differential equations. A thorough analysis in the phase space of the problem enables us to establish under what conditions solutions for straight and curved sluice gates exist.

REFERENCE: Binder, B. J. 2009, “A non-linear dynamical system: Flow past a sluice gate”, Australasian Journal of Engineering Education, Vol. 15, No. 2, pp. 27-34.

Tissue growth and the Pólya distribution

KEYWORDS: Tissue growth; discrete models; continuous models; Pólya distribution.

ABSTRACT: In tissue growth, cells divide and so increase in number. This produces tissue elongation. We describe a discrete model for proliferative tissue growth. This model produces an interesting distribution for the displacement of the original cells in the tissue. We take the reader on our journey to determine this distribution – it turns out to be the well-known Pólya distribution associated with drawing coloured balls from bags. Connections between this model and a simple differential equation model are also investigated. The exercises outlined in this paper can be incorporated into undergraduate courses on probability and differential equations, and the unanswered research problems investigated in a student project.

REFERENCE: Binder, B. J. & Landman, K. A. 2009, “Tissue growth and the Pólya distribution”, Australasian Journal of Engineering Education, Vol. 15, No. 2, pp. 35-42.

Ship stability and parametric rolling

KEYWORDS: Differential equations; Mathieu equation; resonance; parametric rolling; stability; modelling; variable coefficients.

ABSTRACT: Ships, and therefore ship stability, is of vital importance for the transportation of humans and livestock, as well as providing the only means of transporting heavy cargoes between the continents. We present a simple model for ship stability based on the well known Mathieu equation, a second-order differential equation with periodic coefficients, which describes the phenomenon of parametric rolling. Using MATLAB, this model can be used in a classroom setting to introduce students to an important class of differential equations that are not ordinarily taught in the undergraduate engineering curriculum.

REFERENCE: Jovanoski, Z. & G Robinson, G. 2009, “Ship stability and parametric rolling”, Australasian Journal of Engineering Education, Vol. 15, No. 2, pp. 43-50.

Piped water cooling of concrete: An exercise in scaling

KEYWORDS: Scaling; mathematical modelling; industrial mathematics; continuum modelling.

ABSTRACT: Continuum modelling in a industrial engineering context is as much an art as a science. “Keep it simple” is the best modelling advice one can offer while the correct scaling of the problem is generally the key technical tool. Here, a dam construction problem that arose out of a South African Mathematics in Industry Study Group meeting provides a useful foil for illustrating these principles to students in their later undergraduate years.

REFERENCE: Fowkes, N. D. & Bassom, A. P. 2009, “Piped water cooling of concrete: An exercise in scaling”, Australasian Journal of Engineering Education, Vol. 15, No. 2, pp. 51-58.

Diffusing populations: Ghosts or folks?

KEYWORDS: Diffusion; motility; simulation; partial differential equation; multi-scale modelling.

ABSTRACT: Random walk phenomena abound in engineering contexts, from pedestrian traffic to cell motility in tissue engineering. We contrast two random walk models. The ghost model involves individuals who pass through each other unhindered. The folks model involves agents that interact by refusing to share the same location. Simple simulations reveal behaviour consistent with classical diffusion ideas. Using intuitive arguments, we demonstrate how the models are naturally associated with partial differential equations. Seductive opportunities for the misinterpretation of experimental data are discussed.

REFERENCE: Simpson, M. J., Hughes, B. D. & Landman, K. A. 2009, “Diffusing populations: Ghosts or folks?”, Australasian Journal of Engineering Education, Vol. 15, No. 2, pp. 59-68.

Modelling sediment drainage from roads to rivers

KEYWORDS: Sediment drainage; drainage; mathematical modelling.

ABSTRACT: Rainfall on unsealed roads causes a sediment flux that proceeds along the road, through drains, and then down to river systems. Optimal placement of drains can minimise the amount of sediment polluting the river. Two methods of minimising this sediment are considered and compared. The methods provide excellent examples of mathematical modelling, the solution of systems of non-linear equations, function minimisation and Taylor series.

REFERENCE: Barry, S. I. & Thompson, C. J. 2009, “Modelling sediment drainage from roads to rivers”, Australasian Journal of Engineering Education, Vol. 15, No. 2, pp. 69-76.

Critical times in one- and two-layered diffusions

KEYWORDS: Diffusion; critical time; multilayer diffusion.

ABSTRACT: The study of diffusion is commonplace in engineering mathematics courses as a classic example of partial differential equations and separation of variables. However, many textbooks stop at a derived solution without going further to explore what the solution means. The analysis of the critical diffusion time is used here to demonstrate how the solutions obtained can be used to explore additional useful results for diffusion through a single layer of material. We also show how consideration of diffusion through two layers gives rise to some surprising new results. This problem was motivated by analysing the annealing of steel coils, where knowledge of the time to heat a system of air and steel layers is critical in the manufacturing process.

REFERENCE: Hickson, R. I., Barry, S. I. & Sidhu, H. S. 2009, “Critical times in one- and two-layered diffusion”, Australasian Journal of Engineering Education, Vol. 15, No. 2, pp. 77-84.

Taming the complexity of granular materials with vector calculus

KEYWORDS: Vector calculus; tensors; divergence; grad; curl; granular materials.

ABSTRACT: Granular materials, which occur widely in nature and industry, exhibit a vast range of complex behaviour: self-organised pattern formation and multiphase behaviour that defies conventional solid/fluid/gas classification. Arguably the most important source of this complexity is the immense number of degrees of freedom in the system, an aspect that presents serious challenges to the modeller. In particular, the fundamental continuum mechanics concept of strain, the mathematical quantity used to describe how a material deforms, cannot adequately describe the motions of even the smallest particle cluster. In this paper, we demonstrate how key concepts from vector calculus can be used to formulate and model complex particle motions that hold the key to understanding the deformation, failure and flow of granular materials. We present a number of examples to facilitate the implementation of these concepts in lectures. A code in Excel has been written to enable an exploration of a wide range of particle motions, and to provide an invaluable tool for designing problems for students to work on in assignments.

REFERENCE: Tordesillas, A., Kirszenblat, D. & Mangelsdorf, C. 2009, “Taming the complexity of granular materials with vector calculus”, Australasian Journal of Engineering Education, Vol. 15, No. 2, pp. 85-94.

Application of origin-destination matrices to the design of train services

KEYWORDS: Transportation; integer programming; information theory.

ABSTRACT: We consider two related problems in the design of train services on a linear rail network. In the first case, for a prescribed set of practical stopping plans, we determine the number of train services with each allowable stopping pattern that best meets the known demand. We establish fundamental results to define the concept of a maximal origin-destination demand matrix, and use this insight to formulate and solve an integer program that finds the best collection of train services. In the second case we discuss demand estimation from a collection of observed traffic counts. Our aim is to outline the fundamental procedures proposed in a celebrated paper by Van Zuylen & Willumsen (1980). These two problems arose during an Australian Mathematical Sciences Institute (AMSI) industry internship sponsored by Sydney-based company TTG Transportation Technology. These problems are well-suited as a basis of a senior level project-based mathematics course in which students build research skills and develop real-world technical experience through the study of industrial problems. The instructor may use the problems to motivate the study of deterministic mathematical programming and stochastic optimisation, and to introduce undergraduate mathematics students to important techniques in modern applied mathematics.

REFERENCE: Albrecht, A. R., Howlett, P. G. & Coleman, D. 2009, “Application of origin-destination matrices to the design of train services”, Australasian Journal of Engineering Education, Vol. 15, No. 2, pp. 95-104.