The Continuum Model

Most of the modelling introduced in the following chapters uses the continuum approach. In this introductory chapter, we list the commonly used equations: those describing diffusion, convection, radiation, and fluid and solid mechanics. We do not attempt to give an even partially rigorous derivation of any of these equations; our purpose is to provide a ready resource, and to indicate source books of wider scope. Above all, this section should be seen as answering the question “why did they start from those equations?”.

Let us approach the continuum model by considering the example of diffusion. Diffusion is a molecular process. Consider the diffusion of heat: the diffusion happens because a hot region of a material has molecules of higher energy than those in cooler parts. Energy equalisation therefore takes place by molecular interaction – and the heat is said to “diffuse”. To enable us to view this at the continuum level, local averages (for example, over many molecules) are taken: the molecular picture is smeared. The concept of heat as a variable having a value only at molecular sites is replaced by a framework in which heat is regarded as a variable that has continuous values. The laws governing changes in the continuous functions to describe heat transfer are treated “phenomenologically”. Models are created at both levels; their usefulness, success, and relevance depends on the application.

Conservation Laws

Physical phenomena expressed in the continuum paradigm are usually phrased in terms of conservation laws. Let us introduce this in a generic fashion: we consider a substance, A, distributed continuously.