On the notion of laminar and weakly turbulent elementary fluid flows: a simple mathematical model

An elementary analytical fluid flow is composed by a geometric domain, a list of analytical constraints and by the function which depends on the physical properties, as Reynolds number, of the considered fluid. For this object, notions of laminar or weakly turbulent behavior are described using a simple mathematical model. 1 Elementary analytical fluid flows Let L a length unit and Ω̄ a closed domain of R, n ≥ 2, formed by a small number (e.g. 2 or 3) of L squares or L cubes. Here we consider the plane case, but the notions are easly extensible to greater dimensions. Let F a fluid entering in Ω from a single edge of a square and flowing out from another single edge. Let s, 0 ≤ s ≤ 1, a parameter which describes the position of a single fluid particle along the inflow edge, so that s0L is the initial position of the generic particle associated to the particular value s0 of the parameter. Let now s 7→ R(s) a function R : [0, 1] → [0, 1] which maps a value of the parameter s to the value identifying the position reached by the particle on the outflow edge, so that this position is identified by the value R(s)L. If x, y is a cartesian coordinates system in the plane of Ω, assume that each streamline, or each particle path in lagrangian view, is described by a parametric curve t 7→ Φ(t) = (x(t), y(t)), with 0 ≤ t ≤ 1. The parameter t could not be, in general, the time variable of the flow. Let this parametric representation be determined by a set A of analytical conditions regarding Φ(t) and Φ̇(t), that is the passage of the particle in some suitable points of Ω and the velocity field direction in some other (or the same) points. An elementary analytical flow is a particular list F = {Ω, R,A}. We propose a