618-342 Industrial and Applied Mathematics

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Objectives:

On completion of this subject, students should:

Comprehend:

• the basic principles governing the flow of continuous media and transport processes within continuous media;

• the apparatus needed to formulate these principles mathematically (including vector and tensor methods);

• the concept of a constitutive equation.

Have developed:

• the ability to select a constitutive equation and correctly pose relevant boundary-value problems;

• skill in solving transport and flow problems in simple geometries;

• insight into the validity of approximate analyses;

• the ability to interpret solutions in physical terms.

Appreciate:

• the potential for mathematical modelling of flow and transport processes which arise in manufacturing, mineral exploitation and other areas of science and technology;

• the intimate connection between continuum mechanical problems and fundamental mathematical problems.

Content:

Introduction to continuum mechanics: dyadic realization of tensor analysis, Eulerian and Lagrangian viewpoints, transport theorem, linear and angular momentum equations, Cauchy stress principle, existence of the stress tensor, constitutive equations. Incompressible ideal fluids: potential flow, conformal mappings, Bernoulli's theorem, Kelvin's Theorem, persistence of irrotationality, d'Alembert's paradox. Compressible ideal fluids: isentropic flow, acoustics, subsonic and supersonic flow. Incompressible viscous fluids: Navier-Stokes equations, survey of exact solutions, dynamical similarity, low Reynolds number flow, high Reynolds number flow, unstable flows, introduction to turbulence. Extensions and applications selected from: interfacial instabilities, porous media flows, colloid science, linear elasticity, and other areas.

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