Introductory description

The modeling and simulation of fluids and solids with significant coupling and thermal effects is an important area of study in applied mathematics and engineering. Necessary for such studies is a fundamental understanding of the basic principles of continuum mechanics and thermodynamics. This course, which will closely follow the text "A first course in continuum mechanics'' by Andrew Stuart, is a clear introduction to these principles.

The outline will be as follows: we will begin with a review of tensor algebra and calculus, followed by mass and force concepts, kinematics, and then balance laws. We will then proceed to derive some commonly used models governing isothermal fluids and solids, consisting of systems of partial differential equations (PDEs). If time permits we will also explore the thermal case.

Module aims

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

• Tensors, their algebra and calculus.
• Forces, kinematics and balance laws.
• Derivation of models for isothermal fluids and solids.
• Solution of the model equations in some special cases.

Learning outcomes

By the end of the module, students should be able to:

• Manipulate tensors using their algebra and calculus
• Explain the physical concepts of mass, force, stress, deformation, displacement and strain as used in continuum modelling
• Apply physical principles including conservation of mass and momentum to derive general PDE balance laws
• Solve the resulting equations in some simple cases, and interpret these results to make physical predictions

Interdisciplinary

This module provides a rigorous mathematical approach to deriving partial differential equation models used in Physics, Engineering and Life Sciences, and so naturally connects with these disciplines.

Subject specific skills

Ability to apply tools of calculus to derive models fluid and solid systems
Ability to convert physical principles into mathematical equations
Ability to interpret mathematical models of the real world

Transferable skills

Ability to translate scientific ideas into mathematical language
Ability to communicate complex ideas and mathematical results clearly