In almost everything that we have discussed so far, we have assumed, explicitly or implicitly, either that the systems we are looking at are perfectly mixed or that they are not mixed at all. In the former case, concentrations are the same everywhere in the system, so that ordinary differential equations for the evolution of the concentrations in time provide an appropriate description for the system. There are no spatial variables; in terms of geometry, the system is effectively zero-dimensional. At the other extreme, we have unstirred systems. Here, concentrations can vary throughout the system, position is a key independent variable, and diffusion plays an essential role, leading to the development of waves and patterns. Geometrically, the system is three-dimensional, though for mathematical convenience, or because one length is very different from the other two, we may be able to approximate it as one- or two-dimensional. In reality, we hardly ever find either extreme—that of perfect mixing or that of pure, unmixed diffusion. In the laboratory, where experiments in beakers or CSTRs are typically stirred at hundreds of revolutions per minute, we shall see that there is overwhelming evidence that, even if efforts are made to improve the mixing efficiency, significant concentration gradients arise and persist. Increasing the stirring rate helps somewhat, but beyond about 2000 rpm, cavitation (the formation of stirring-induced bubbles in the solution) begins to set in. Even close to this limit, mixing is not perfect. In unstirred aqueous systems, as we have seen in Chapter 9, it is difficult to avoid convective mixing. Preventing small amounts of mechanically induced mixing requires considerable effort in isolating the system from external vibrations, even those caused by the occasional truck making a delivery to the laboratory stockroom. It is possible to suppress the effects of convection and mechanical motion in highly viscous media, such as the gels used in the experiments on Turing patterns as discussed in the previous chapter. There, we can finally study a pure reaction-diffusion system. Systems in nature—the oceans, the atmosphere, a living cell—are important examples in which chemical reactions with nonlinear kinetics occur under conditions of imperfect mixing.
