Some reduced-dimension models based on numerical methods

In fluid flow problems, understanding frequently is attained by studying very simplified models. This paper presents a systematic way of generating such models; the techniques involved are inspired by numerical analysis. First we illustrate the process by constructing a simple model of Rayleigh-Benard convection which gives a remarkably good estimate of the critical Rayleigh number and the corresponding roll size. We also examine a model which helps explain the formation of shocks in creaming or sedimenting colloids. Finally we discuss two models of droplet break-off which are in excellent agreement with experiment and have played a role in the study of cascading instabilities which can occur near snap-off.