Stagnation zones of ideal flows in long and narrow bands

We investigate stagnation zones of flows of ideal incompressible fluid in narrow and long bands. With the bandwidth being much less than its length, these flows are almost stationary over large subdomains, where their potential functions are almost constant. These subdomains are called s-zones. We estimate the size and the location of these s-zones. 1. Introduction. In this work, we investigate the flows of ideal incompressible fluid in narrow and long bands on surfaces with almost Liouville line element. The investigation is motivated in part by the fact that, with the recent increasing interest in areas such as microelectromechanical systems (MEMS) and nanoscale physiological processes , there is a greater need to improve our understanding of fluid flows in the mi-croscale and nanoscale regimes. When the width of the band is much smaller than the length, the zones inside which the flows are almost stationary, and consequently their potential functions are almost constant, will be of sufficiently large size. Here, we study the size and location of these stagnation zones, which we will call s-zones of ideal flows. At first sight, it seems that the situation is of little interest. However, by remembering that minute change in potential function value occurs over a very long interval, it is clear that a better understanding of such s-zones may allow one to better organize calculations and possible minimize the amount of computation. First, we define the following concept which will be key in this article.