Abstract
Abstract
Highlights
Invariant solutions of the Navier–Stokes equations play a key role for the dynamics of transitional shear flows (Kawahara, Uhlmann & van Veen 2012)
Discrete symmetries of the specific equation control whether such homoclinic orbits in space generically exist and whether an equation supports homoclinic snaking (Burke, Houghton & Knobloch 2009). In both the Navier–Stokes solutions and in solutions of the Swift–Hohenberg equation with cubic-quintic nonlinearity, the snakes-and-ladders bifurcation structure is composed of two pairs of intertwined snaking branches along which the solutions are invariant under discrete symmetries that are part of the equivariance group of the respective system
The snaking solutions found by Schneider et al (2010a) in Plane Couette flow (PCF) are shown in figure 3 together with their bifurcation diagram
Summary
Invariant solutions of the Navier–Stokes equations play a key role for the dynamics of transitional shear flows (Kawahara, Uhlmann & van Veen 2012). Discrete symmetries of the specific equation control whether such homoclinic orbits in space generically exist and whether an equation supports homoclinic snaking (Burke, Houghton & Knobloch 2009) In both the Navier–Stokes solutions and in solutions of the Swift–Hohenberg equation with cubic-quintic nonlinearity, the snakes-and-ladders bifurcation structure is composed of two pairs of intertwined snaking branches along which the solutions are invariant under discrete symmetries that are part of the equivariance group of the respective system. In the Swift–Hohenberg equation with cubic-quintic nonlinearity, breaking a specific discrete symmetry destroys the snakes-and-ladders bifurcation structure that solutions of Navier–Stokes in the plane Couette geometry remarkably well resemble. Both in the one-dimensional model system and in the full three-dimensional Navier–Stokes problem, the initial breakdown of the snakes-and-ladders structure can be explained in terms of symmetry breaking This manuscript is organised as follows: in § 2, we introduce the plane Couette system with wall-normal suction and discuss its symmetry properties.
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