Introduction

Flow problems are widely studied in engineering because of their relevance to industrial processes and environmental problems. Such problems belong to the realm of macroscopic phenomena which are formulated in terms of one or more, possibly nonlinear, partial differential equations. If there is no possibility of exploiting symmetry, allowing for separation of variables, these equations are usually solved using finite element or finite difference methods.

The standard problem is the flow of a one-component, isotropic nonpolar liquid, which is described by the Navier–Stokes equations. These equations are based on mass and momentum conservation, and on the assumption of isotropic relaxation towards equilibrium. Finite element methods have been described in Chapter 13 for elasticity; the methods described there may be extended and adapted to develop codes for computational fluid dynamics (CFD), which are widely used by engineers. Such an extension is beyond the scope of this book.

A finite element solution of the Navier–Stokes equations may sometimes become cumbersome when the boundary conditions become exceptionally complicated, as is the case with flow through porous media where the pore diameter becomes very small (and the number of pores very large). Other cases where finite element methods run into problems are multiphase or binary systems, where two different substances or phases exist in separate regions of space. These regions change their shape and size in the course of time. Usually, the finite element points are taken on the system boundaries, but that implies complicated bookkeeping, in particular when the topology of the regions changes, as is the case in coalescence of droplets.