Industrial Mathematics III

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Description

In this course a number of real-world industrial case studies are presented. These case studies lead to mathematical models which involve differential equations. The differential equations are solved and the solutions analysed in order to provide an understanding of the particular industrial process under examination.

Objective

To introduce students to the art of mathematical modelling, whereby an industrial problem is represented by mathematical equations (the model) to be solved, and to extend their knowledge of mathematical methods for solving mathematical models. To introduce students to the art of mathematical modelling, whereby an industrial problem is represented by mathematical equations (the model) to be solved, and to extend their knowledge of mathematical methods for solving mathematical models. At the end of this subject, students should be able to: understand the important role of physical conservation laws in deriving mathematical models; appreciate the need to make assumptions in order to render an industrial problem tractable; understand the physical processes of diffusion and advection; derive from first principles the one-dimensional diffusion, advection and advection-diffusion equations, Fick's and Fourier's laws relating mass/heat flux to a concentration/temperature gradient, simple Stefan conditions for moving boundaries; use some or all of the following mathematical techniques: basic dimensional analysis to simplify a problem, stretching transformations to reduce a PDE to an ODE, perturbation expansions, bifurcation analysis, nonlinear transformations of nonlinear PDEs; interpret results with regard for the assumptions made; present a mathematical argument to peers (via tutorial presentations).

Content

The importance of mathematical modelling. Diffusion and advection and the equations that derive these physical processes. Fick's and Fourier's laws. Common boundary conditions for advection-diffusion problems. Newton cooling. Scaling of equations and basic dimensional analysis. A selection of case studies from the following. Each introduces a different concept or mathematical method. Continuous casting of sheet steel. This involves a Stefan condition for a moving boundary and the reduction of variables by the Boltzmann transformation to give the Boltzmann similarity solution of the heat equation. Water filtration by reverse osmosis, introducing invariance of equations and the method of stretching transformations to reduce a PDE problem to an ODE problem. Laser drilling. A Stefan condition is required for the moving boundary and the method of regular perturbations is introduced. Factory fires due to spontaneous ignition. This introduces bifurcation analysis. Furrow irrigation. The non-linear PDE describing the problem is transformed to a linear PDE by the Kirchoff transformation. Fourier series are used to solve the linear PDE.

David Clements Lecturer for this course

Delivery

5 one-hour lectures and 1 one-hour tutorial every two weeks

Assessment

10% Assignments, 90% Final Examination. In addition, students are expected to participate in tutorial presentations.

Graduate attributes

• Applying mathematical knowledge (1)
• Interpreting data and drawing conclusions (2)
• Formulating mathematical problems (3)
• Problem solving skills (4)
• Flexibility in working environment (5)
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• Communication skills (6)
• Team participation and leadership (7)

Linkage past

Prerequisite is MATHS 1007A/B >Mathematics I (Pass Div 1) or MATHS 2004 Mathematics IIM (Pass Div 1). Assumed knowledge is APP MTH 2007 Differential Equations II or APP MTH 2000 Differential Equations and Fourier Series II or equivalent.

Linkage present

This subject forms part of a Mechanics stream, which could also include courses such as APP MTH 3000 Computational Mathematics III, APP MTH 3013 Differential Equations III, APP MTH 3017 Waves III and APP MTH 3002 Fluid Mechanics III. However, the techniques learned are applicable to modelling in all fields of mathematics.

Linkage future

This subject introduces techniques that may be dealt with more fully in the Level IV courses in the Mechanics stream. It will be valuable for those wanting a research or industry career involving mathematical modelling.

Restrictions

None.

Recommended text

Fulford, G.R. & Broadbridge, P. (2001). Industrial Mathematics: Case Studies in the Diffusion of Heat and Matter. Australian Mathematical Society Lecture Series 16, Cambridge University Press. Copies are available in the library.