On Stable Plane Vortex Flows of an Ideal Fluid

2D-flows of an ideal incompressible fluid are treated in a rectangular. If analytical (resolved in series of powers of coordinates), the stationary flows are uniquely determined with the inflow vorticity. When excluded vortices of a spectral origin, such flows prove to be stable.


Vorticies of 1883 and 1941
Apparently, the idea of hydrodynamic stability was born in 1883 from "particular cause" of O. Reynolds [1] and "disturbed equilibrium" of J.W.S. Rayleigh [2] to give rise soon to velocity pulsations and correlations of W. Thomson [3] referred now to as turbulent stresses. The phenomenon involved had initially been found closely related to circulations as vortices spontaneously originated in a fluid motion. However, some of them continue stubbornly to reveal their non-linear stability (by A.M. Lyapunov [4]), as follows.
Counter-flow circulations. In 1941, P.A.M. Dirac as a participant of the first atomic project (Tube Alloys) had unexpectedly developed an artificial idea (of a chemical origin from R.S. Mulliken [5] and H.C. Urey [6]) on mass separation in the uranium hexafluoride. In the almost invisible rotating gas rarefied to centre and adhering to the boundary of the rotor (cylinder) of the future industrial centrifuge, he had seen [7] a natural cause required for the desired fractionating of the isotope mixture (consisting basically of U-235 and U-238 [8]). The cause proved to be the counter-flow circulation [9] in the form of a vortex ring [10][11][12] twisted round the axis as shown in the Fig. 1.
The rotation of an air mass (mesocyclone) in the atmospheric swirl (tornado) is accompanied with the same circulation (see Fig. 1). It could hastily be taken for a secondary flow originating in a rotating medium owing to the loss of stability as with Taylor vortices taken to be the counter-flows of the Couette flow between cylinders that arise inevitably as` bifurcations [13][14][15] due to the Reynolds number [1]. Meanwhile, the absence of the internal rigid wall (in the form of the cylinder) in a region around air swirl or hexafluoride gas center (of which the first one has always the pressure drop [16, § 1] while the second turns out to be a rarefied gas [8,9,17]) seems to play a crucial part in the forming of their true flow pattern that is reduced not to a secondary velocity field (as with Taylor vortices) and proved to be a basic (non-disturbed) flux (as with the Couette flow).
As we shall see below, a plane vortex in a channel delivers such an example of a basic (or undisturbed) flow related directly to the counter-flow circulation.
Rip-flow vortices. In the same year (1941), G. I. Taylor had exactly estimated parameters of what that had been subsequently declassified [18] and realized in the atomic mushroom [19]. For small scales, it had evidenced on the non-linear stadium of RTI, or the hydrodynamic instability come from the first (linear) considerations of Rayleigh [2] and Taylor [20] validated first by D.J. Lewis [21].
In the case of accelerations produced with a shock wave, RTI takes the form of RMI (the Richtmayer-Meshkov Instability) [22][23][24] where an array of compact mushrooms arise first behind the wave front as shown in the Fig. 2. Related RTI-phenomena are known as well [25].  s µ (microseconds) behind the shock front crossing normally the disturbed boundary from Argon to Xenon in RMI [23,24].
Meanwhile, the discontinuity of the normal velocity component in the contact boundary seems to be a not less significant cause for RM-type instabilities then the difference in densities of two media contacted: familiar sea water waves striking a board of a ship in a calm harbor produce similar vortex mushrooms originating (after impact) and resembling that usually produced with breakers crossing sand bars off the shore when the water has to travel back out to sea through a gap in the sand bar, creating a fast (and dangerous) rip current.
and excluding some of spectral points Each of the treated problems (with the inflow vorticity prescribed) have the unique non-stationary smooth solution (for any values of parameters involved) by Yuodivich theorem [27]. Existence and uniqueness of the corresponding stationary solution exclusively in the class of analytical functions are proved in [12,16,29,44].
The paper is aimed to study the stability of PF-, BF-and RF-flows for different values of parameters (4). Such a study is possible for the known non-stationary inflow vorticity theorem (of existence and uniqueness) [27] saying that for any smooth initial data

50
On Stable Plane Vortex Flows of an Ideal Fluid The another known restriction in the classical stability [4] is the necessary uniqueness of the corresponding stationary flow , u u * * . According to the so called stationary inflow (or outflow) vorticity theorem from [12, § 5], [16, § 7] or [29, § 1], for any problems PF, BF or RF there is a unique stationary smooth Euler flow , u u * * (for a pressure p * ) provided that two following conditions (i) and (ii) are fulfilled: (i) velocity components u * and u * are to be analytical, Conditions (i) and (ii) are used below.

Zonal Potentials
As is well known, for a pair of smooth real-valued functions given in a one-connected region like a rectangular V , the absence of the vorticity or the existence of the classical potential [30]. For a rectangular with periodical walls 0, y h = (forming a topological cylinder, or a ring on a plane with the Euclidian metric) we have the following for a zonal potential of the form and boundary conditions V Uy

Spaces with a Volume
In terms of Lie groups the vortex equation in (12) describes rotations y (i.e. angular velocities) of an ideal the Poison's brackets. On the other hand, following [38], instead of Lie groups we shall treat the vortex equation (12) (or (15)) in terms of Lie algebras (the latter are generally being introduced independently of the former). To be more precisely, we shall consider (15)

Stationary Solutions
Let us construct stationary flows in Fig. 3 (13) (18) is given by (14). For flows in Fig. 3, 4 it is constant and the corresponding flows are determined by two following functions: and boundary conditions ( ) Proof. The analyticity and the symmetry of Φ and Γ are provided by the classical formulae [39,40]. Convex-concave properties noted follow from maximum principle, lemma on normal derivative [41] and some extended versions [12,16] where X x h = and Y y h = for β −∞ < < ∞ (19) and ( ) which completes the proof. Statement 3. Critical and spectral points as corresponding to topological bifurcations in Fig. 3-5 and spectral reconstructions in Fig. 5 are (7) and m β + (or m α + ) from (8) where the maximum principle is violated (or Bifurcations and reconstructions are repeated as shown in Fig. 5.

Rip-flow Stability
we take the familiar energy identity Multiplying (28) by λ and substituting from (29) we come to the energy defect inequality: As in [38], for rip-flows, we introduce the scalar square The fact that D is dense in 1 M [40] with respect to the norm , respectively [44][45][46][47].
This work was supported by Russian Science Foundation, project no. 141100719.