4.1 Introduction

Taylor series often do not work because the functions involved are not analytic at the point of interest. For example, this is common if the behavior of interest is at large time, or for large values of some other parameter, like the Reynolds number Re of a flow. (In fact, for flows in infinite domains it normally also occurs for small Reynolds numbers.

Finding nonanalytic limits is then needed. Applications are very similar to those of Taylor series:

• Because it is needed.
• To reduce effort.
• To increase accuracy.
• For making estimates of how importants something is.
• To get more insight. For example, consider the laminar flow past a flat plate if the plate is aligned with the incoming flow velocity $U$. For finite Reynolds numbers, there is little more you can say than that the flow velocity will be zero at the plate, and $U$ far away from the plate. To get more insight than that, you can ask: “What is the limit of the velocity for infinite Reynolds number Re, assuming that you keep the streamwise location $x$ fixed, as well as keep the ratio $\eta$ $\vphantom0\raisebox{1.5pt}{$=$}$ $y\sqrt{\rm Re}/x$ fixed, where $y$ the distance from the wall?” The answer is $Uf'(\eta)$ where $f'$ is the Blasius function tabulated in any real book on fluid mechanics. (If you instead keep $y$ itself fixed at a nonzero value, the limit is $U$, which is not very interesting.) Most of my theoretical research in fluids (as opposed to in numerical methods) really simply finds limits like this.
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