Equilibrium configuration of a rectangular obstacle immersed in a channel flow

Abstract

Fluid flows around an obstacle generate vortices which, in turn, generate lift forces on the obstacle. Therefore, even in a perfectly symmetric framework equilibrium positions may be asymmetric. We show that this is not the case for a Poiseuille flow in an unbounded 2D channel, at least for small Reynolds number and flow rate. We consider both the cases of vertically moving obstacles and obstacles rotating around a fixed pin.

On the steady motion of Navier-Stokes flows past a fixed obstacle in a three-dimensional channel under mixed boundary conditions

“…The purpose of this section is to provide explicit lower bounds for the constants S p and J p , for p ∈ [2,6].…”

“…By following closely the proofs given by Fursikov & Rannacher in [15], one discovers that, in order to yield such explicit threshold, several tools of functional analysis must be carefully studied. Therefore, in Section (2.1) we firstly provide lower bounds for the Sobolev constant of the embedding H 1 (Ω) ⊂ L p (Ω), p ∈ [2,6], involving functions that vanish only on Γ W or on Γ I ∪ Γ W . Thus, symmetrization techniques as in [20,21,22] can only be applied after properly reflecting Ω with respect to the planes x = ±L, and performing a suitable even extension of the functions considered, as in [32,Chapter 2].…”

“…This paper is organized as follows. In Section 2 we exploit some techniques of functional analysis that are needed for the study of (1.3), such as: inequalities associated to the immersion H 1 (Ω) ⊂ L p (Ω), for p ∈ [2,6] (regarding functions that vanish on a proper part of ∂Ω), and a priori bounds for the solutions of (1.5). In Section 3 we begin by providing estimates for the (linear) Stokes operator under mixed boundary conditions in the presence of an external force, which are essential to establish the well-posedness of (1.3) and to provide the explicit estimates given in the main results (Theorem 3.3 and Corollary 3.1).…”

“…The main goal of this section is to provide explicit upper bounds on the drag and lift exerted over K, and to illustrate our results through a concrete example. Firstly we derive a more handy formula for the drag and lift, in the spirit of [6,Lemma 3]. Proposition 4.1.…”