Some fluid flow problems are sufficiently simple that they can be treated mathematically in a straightforward way, making use of definitions of physical quantities, and taking into account initial and boundary conditions. For example, our derivation of the average velocity in a conduit with parallel walls (Example Problem 3.7.1) was obtained in a straightforward way once we specified the geometry of the problem, then made use of the definition of a Newtonian fluid and the no-slip condition. Whereas this type of analysis may work for some problems, it would be misleading to think that such direct approaches to solving problems are, in principle, always possible, hinging only on one’s mathematical skills and adeptness in specifying the geometry of a problem. Herein arise two noteworthy points. First, when initially examining a problem, one can sometimes obtain a clear idea of the desired solution before attempting a formal mathematical analysis. The means to do this, as we shall see below, is supplied by dimensional analysis, and it is a strategy that ought to be adopted in many circumstances. In fact, it is worth noting that dimensional analysis underlies many of the problems presented in this text. The advantage of knowing the form of a desired solution, of course, is that one has a clear target to guide the subsequent mathematical analysis. Indeed, this is the vantage point from which many classic problems, for example Stokes’s law for settling spheres, were initially examined. Second, a complete mathematical formulation of a problem may not be possible, due to the complexity of the problem, or due to absence of information required to constrain the mathematics of the problem. As a simple example, suppose that we were unaware of the no-slip condition in our analysis of the conduit-flow problem (Example Problem 3.7.1). Our analysis in this case would have essentially ended with (3.70), with the constant of integration C undetermined. Nevertheless, we could get close to our result (3.75) for the average velocity by another way.
Dimensional analysis in the venous system