Steady Stokes Flow in Domains with Unbounded Boundaries

Abstract

So far, with the exception of the half-space, we have considered flows occurring in domains with a compact boundary. Nevertheless, from the point of view of the applications it is very important to consider flows in domains Ω having an unbounded boundary, such as channels or pipes of possibly varying cross section. In studying these problems, however, due to the particular geometry of the region of flow, completely new features, which we are going to explain, appear. To this end, assume Ω to be an unbounded domain of ℝⁿ with m > 1 “exits” to infinity, of the type (see Section III.4.3)

$$\Omega = \bigcup\limits_{i = 0}^m {{\Omega _i}} ,$$

where Ω₀ is a smooth compact subset of Ω while Ω _i , i = 1,..., m, are disjoint domains which, in possibly different coordinate systems (depending on Ω _i ) have the form

$${\Omega _i} = \left\{ {x \in {R^n}:{x_n} > 0,x' \equiv \left( {{x_1},...,{x_{n - 1}}} \right) \in \sum\nolimits_i {{x_n}} } \right\}$$

Nel dritto mezzo del campo malign vaneggia un pozzo assai largo e profondo di cul euo loco dicerö l’ordlgno.

DANTE, Inferno XVIII, vv. 4–6