Two-dimensional Euler flows in slowly deforming domains

Abstract

We consider the evolution of an incompressible two-dimensional perfect fluid as the boundary of its domain is deformed in a prescribed fashion. The flow is taken to be initially steady, and the boundary deformation is assumed to be slow compared to the fluid motion. The Eulerian flow is found to remain approximately steady throughout the evolution. At leading order, the velocity field depends instantaneously on the shape of the domain boundary, and it is determined by the steadiness and vorticity-preservation conditions. This is made explicit by reformulating the problem in terms of an area-preserving diffeomorphism g Λ which transports the vorticity. The first-order correction to the velocity field is linear in the boundary velocity, and we show how it can be computed from the time-derivative of g Λ .The evolution of the Lagrangian position of fluid particles is also examined. Thanks to vorticity conservation, this position can be specified by an angle-like coordinate along vorticity contours. An evolution equation for this angle is derived, and the net change in angle resulting from a cyclic deformation of the domain boundary is calculated. This includes a geometric contribution which can be expressed as the integral of a certain curvature over the interior of the circuit that is traced by the parameters defining the deforming boundary.A perturbation approach using Lie series is developed for the computation of both the Eulerian flow and geometric angle for small deformations of the boundary. Explicit results are presented for the evolution of nearly axisymmetric flows in slightly deformed discs.

Flexibility and rigidity in steady fluid motion

“…As in [26], one may think about these deformations arising dynamically from a slow adiabatic deformation of the boundary, though we do not establish this point here. We remark also that (H1) may not be strictly needed for the Theorem 3.1 provided kernel of L 0 − Λ is very well understood.…”

Relaxed magnetohydrodynamics with ideal Ohm's law constraint

“…Potential applications include reconnection events and the conjectured formation of equilibrium fractal magnetic and fluid flow patterns in 3-D systems. Other potential physical phenomena to investigate in 3-D systems include the linear normal mode spectrum, nonlinear saturation, bifurcations to oscillatory modes and the effect of quasisymmetry (Nührenberg & Zille 1988;Burby, Kallinikos & MacKay 2020;Rodriguez, Helander & Bhattacharjee 2020;Constantin, Drivas & Ginsberg 2021) on 3-D equilibria with flow (Vanneste & Wirosoetisno 2008).…”

Relaxed Magnetohydrodynamics with Ideal Ohm's Law Constraint

“…This regularization should break the frozen-in flux condition of IMHD on small scales and allow interesting behaviour to be simulated, such as reconnection events and the conjectured formation of equilibrium fractal flow patterns in 3-D systems. Other potential physical phenomena to investigate in 3-D systems include the linear normal mode spectrum, nonlinear saturation, bifurcations to oscillatory modes, and the effect of quasisymmetry [Nührenberg & Zille (1988) [Vanneste & Wirosoetisno (2008)].…”