Module Catalogues, Xi'an Jiaotong-Liverpool University   
Module Code: MTH308
Module Title: Mathematical Models of Solids and Fluids
Module Level: Level 3
Module Credits: 5.00
Academic Year: 2019/20
Semester: SEM2
Originating Department: Mathematical Sciences
Pre-requisites: MTH207
To provide an introduction to the mathematical theory of viscous fluid flows and solid elastic materials.
Cartesian tensors are first introduced.

This is followed by modelling of the mechanics of continuous media.

The module includes particular examples of the flow of a viscous fluid as well as a variety of problems of linear elasticity.
Learning outcomes 
A. Work out the mathematical form of a given model in different coordinate systems, using the transformation properties of tensors.

B. Model simple systems using the basic concepts of continuum mechanics such as continuum hypothesis, conservation laws, stress, deformation and constitutive relations.

C. Build a model to predict deformation of elastic solids under sufficiently small applied forces.

D. Solve the Navier-Stokes equation for incompressible viscous fluids in simple cases, and estimate viscosity of fluids from observations.

E. Critically analyse the predictions of a quantitative model to validate or reject the assumptions.

Method of teaching and learning 
This module will be delivered by a combination of formal lectures and tutorials.
Cartesian tensors: Transformation of components, symmetry and skew symmetry.

Isotropic tensors.

Kinematics: Transformation of line elements, deformation gradient, Green strain.

Displacements, flow, linearization in the case of small displacements.

Stress. Cauchy stress tensor. Material laws. Hooke’s law.

Global balance laws, statically admissible stress tensors.

Equations of motion, boundary conditions.

Modelling of surface tension: estimation of the surface tension of water, competition between gravity and area minimization.

Modelling of viscosity from Stokes’ law for the terminal velocity of a sphere falling in a fluid.

Euler and Navier-Stokes equations, vorticity.

Newtonian fluids: Constitutive law, uniform flow, Poiseuille flow, flow between rotating cylinders.
Measurements and estimates of viscosity.

Dimensional analysis and physical interpretation of the associated numbers (for instance Reynolds number and the notion of turbulence).

Linear elasticity. Field equations.

Young's modulus, Poisson's ratio.

Some simple problems of elastostatics.

Expansion of a spherical or cylindrical shell.

Delivery Hours  
Lectures Seminars Tutorials Lab/Prcaticals Fieldwork / Placement Other(Private study) Total
Hours/Semester 26     26      98  150 


Sequence Method % of Final Mark
1 Final Exam 80.00
2 Class Test 10.00
3 Class Test 10.00

Module Catalogue generated from SITS CUT-OFF: 8/20/2019 6:21:53 PM