It is universally accepted that the Navier-Stokes equations are a good model for the flow of a wide class of fluids. The underlying assumption is a constitutive model which gives a linear relationship between stress and rate-of-strain governed by the liquid's (constant) viscosity.

However, the study of fluids which do not obey this linear relationship is growing in importance. Many liquids such as polymers, multigrade oils and plastics which are used in current industrial processes show non-linear characteristics, as do many naturally occurring liquids such as blood and synovial fluid.

The simplest case of non-linearity is the Generalised Newtonian Model which allows for a variable viscosity depending on rate-of-strain, temperature and pressure.

The more complicated case is that of a viscoelastic fluid which exhibits properties of both liquids and solids, depending on flow conditions. For example, magic putty will flow if left in a container but will bounce if thrown against the floor. In this case the stress and rate-of-strain are related by a differential and/or integral equation which implies that the current stress depends on the past history of strain. The governing equations comprise the conservation of mass, conservation of momentum, and constitutive equations which constitute a mixed elliptic-hyperbolic system of non-linear partial differential equations which must be solved in conjunction with suitable initial and boundary conditions for the velocity and stress fields.

There are many problems still outstanding in the numerical solution of viscoelastic fluid flow. Finite difference and finite element methods may be used but the interest within the School is in the use of spectral methods which are advantageous for some applications, finite volume and wavelet techniques. In addition to the non-Newtonian field, there are several areas for research into the application and efficient implementation of numerical methods in other areas of computational fluid mechanics.