Introduction

Mechanical engineers design new products for consumer use: engines for automobiles, airplanes, and other devices; cars; air-conditioners; heat pumps; compressors; fans; hair dryers; and all sorts of other products. Increasingly, mechanical and chemical engineers are involved in the design of biomedical devices for drug delivery systems, biochemical sensing, and rapid molecular analysis. In this chapter, the basic numerical techniques used to provide design and performance criteria of these devices are described. A simplified view of a general design process is depicted in Figure 10.1.

In this chapter, we shift gears a bit and discuss some basic concepts associated with numerical methods.1 These methods are required when no simple analytical solution is possible. What is meant by the term analytical is that no solution can be found in terms of simple functional forms such as the polynomial, trigonometric, exponential, logarithmic, or hyperbolic functions. In the following, much attention is focused on the basic methods required to solve a nonlinear ordinary differential equation. This requires several different capabilities:

• Numerical differentiation

• Solving sets of linear(ized) equations

• Numerical integration

There are many situations for which numerical methods are required in micro- and nanofluidics. For example, determining the dependence of the ζ potential on pH in Chapter 7 requires a numerical zero-finding technique. The non-linear Poisson equation for the electrical potential discussed in the preceding chapter requires a numerical solution for the potential. In general, the solution of the potential equation for multicomponent and multivalent mistures requires a numerical solution.