Planar Limits of Three-Dimensional Incompressible Flows with Helical Symmetry

Abstract

ABSTRACT. Helical symmetry is invariance under a one-dimensional group of rigid motions generated by a simultaneous rotation around a fixed axis and translation along the same axis. The key parameter in helical symmetry is the step or pitch, the magnitude of the translation after rotating one full turn around the symmetry axis. In this article we study the limits of three-dimensional helical viscous and inviscid incompressible flows in an infinite circular pipe, with respectively no-slip and no-penetration boundary conditions, as the step approaches infinity. We show that, as the step becomes large, the three-dimensional helical flow approaches a planar flow, which is governed by the so-called two-and-half NavierStokes and Euler equations, respectively. MSC Subject Classifications: 35Q35, 65M70.

The limit of vanishing viscosity for the incompressible 3D Navier–Stokes equations with helical symmetry

“…Conversely, if v and p are defined through (2.11) for some w = w(y 1 , y 2 ), q = q(y 1 , y 2 ), then v is a helical vector field and p is a helical scalar function. This is precisely Proposition 2.1 in [11], in the case σ = 2π, to which we refer the reader for the proof.…”

“…Next we recall the relation between three-dimensional helical vector fields and their two-dimensional traces on "slices" z = constant, as discussed in [11]. Recall that we are assuming κ = 1 so, in the notation of [11], σ = 2π. Lemma 2.3.…”

Dissipation enhancement of planar helical flows and applications to three-dimensional Kuramoto-Sivashinsky and Keller-Segel equations

“…People have studied helical flows with diffrerent definitions. For instance in the work [14,28], they defined helical flows based on some helical symmetry. In our definition, we call it " planar helical" because the trajectory of the flow wraps around the torus.…”

Estabilidade não/linear de soluções estacionárias das equações de Euler incompressíveis com simetria helicoidal

“…Como visto neste trabalho, escoamentos sob o regime de simetria helicoidal têm, em certo sentido, características de escoamentos bidimensionais (veja [35], [43] e [44] para outros detalhes). Se, por um lado, pesquisas com Ćuxos bidimensionais podem ser abundantemente encontradas na literatura desde décadas passadas, em contrapartida, apenas recentemente alguns resultados com Ćuxos helicoidais têm sido apresentados na comunidade cientíĄca.…”

“…Nesta, o campo de velocidades, quando escrito em coordenadas cilíndricas, depende da combinação (𝑟, 𝜃 + 𝑧). Em [13], [24], [27], [35], [43] e [44] encontramos teoremas de existência de soluções globais e de limites evanescentes para as equações de Euler e de Navier-Stokes helicoidais. Na primeira parte deste trabalho, temos como resultado principal a existência de soluções estacionárias e estáveis para as equações de Euler helicoidais.…”