Steady point vortex pair in a field of Stuart-type vorticity

Abstract

A new family of exact solutions to the two-dimensional steady incompressible Euler equation is presented. The solutions provide a class of hybrid equilibria comprising two point vortices of unit circulation – a point vortex pair – embedded in a smooth sea of non-zero vorticity of ‘Stuart-type’ so that the vorticity $\unicode[STIX]{x1D714}$ and the stream function $\unicode[STIX]{x1D713}$ are related by $\unicode[STIX]{x1D714}=a\text{e}^{b\unicode[STIX]{x1D713}}-\unicode[STIX]{x1D6FF}(\boldsymbol{x}-\boldsymbol{x}_{0})-\unicode[STIX]{x1D6FF}(\boldsymbol{x}+\boldsymbol{x}_{0})$ , where $a$ and $b$ are constants. We also examine limits of these new Stuart-embedded point vortex equilibria where the Stuart-type vorticity becomes localized into additional point vortices. One such limit results in a two-real-parameter family of smoothly deformable point vortex equilibria in an otherwise irrotational flow. The new class of hybrid equilibria can be viewed as continuously interpolating between the limiting pure point vortex equilibria. At the same time the new solutions continuously extrapolate a similar class of hybrid equilibria identified by Crowdy (Phys. Fluids, vol. 15, 2003, pp. 3710–3717).

Free rings of bouncing droplets: stability and dynamics

“…2003; Celli, Lacomba & Pérez-Chavela 2011) and generalized vortices of finite area (Crowdy 1999; Crowdy & Cloke 2002; Crowdy 2003; Krishnamurthy et al. 2019). Vortex arrays have also been observed in magnetized non-neutral plasmas (Schecter et al.…”

“…In fluid mechanics, rings composed of multiple vortices are of particular interest and have been observed over a broad range of scales, from baths of superfluid helium (Yarmchuk, Gordon & Packard 1979) to hurricane eyewalls (Kossin & Schubert 2004). Several theoretical studies have characterized the stability of polygonal vortex arrays composed of both point vortices (Thomson 1883;Havelock 1931;Aref et al 2003;Celli, Lacomba & Pérez-Chavela 2011) and generalized vortices of finite area (Crowdy 1999;Crowdy & Cloke 2002;Crowdy 2003;Krishnamurthy et al 2019). Vortex arrays have also been observed in magnetized non-neutral plasmas (Schecter et al 1999;Durkin & Fajans 2000) and in both bosonic and fermionic systems, where vortices play a critical role in superconductivity (Saarikoski et al 2010).…”

Liouville chains: new hybrid vortex equilibria of the two-dimensional Euler equation

“…These ideas can be extended to obtain Stuart vortex solutions on a torus (Sakajo 2019) and on a hyperbolic sphere (Yoon, Yim & Kim 2020). The introduction in Krishnamurthy et al (2019) discusses other applications and extensions of Stuart vortices.…”

“…In these cases, invoking symmetry arguments is sufficient to ensure that the point vortices are stationary, and the solutions obtained are therefore steady. In a recent paper, Krishnamurthy et al (2019) showed the existence of an asymmetric family of hybrid vortex equilibria (although the background field is referred to there as 'Stuart-type' vorticity in deference to Stuart 1967). They showed that a colinear three point vortex equilibrium, which is a limiting case of the N = 2 hybrid equilibrium discussed in Crowdy (2003), can be continuously deformed into another non-trivial family of hybrid equilibria comprising a point vortex pair in equilibrium in an ambient field of Liouville-type vorticity.…”

“…The hybrid solutions of Crowdy (2003) correspond toM = 1 in (1.3) and invoking symmetry is sufficient to satisfy (1.4). The family studied by Krishnamurthy et al (2019) corresponds to the caseM = 2. These families of hybrid equilibria and their point vortex limits signal the possibility that certain point vortex equilibria can be 'connected' by families of hybrid equilibria.…”