Stationary equilibrium singularity distributions in the plane

ISSN： 0951-7715

Source： Nonlinearity, Vol.25, Iss.2, 2012-02, pp. : 495-511

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Abstract

We characterize all stationary equilibrium point singularity distributions in the plane of logarithmic type, allowing for real, imaginary or complex singularity strengths. The dynamical system follows from the assumption that each of the N singularities moves according to the flow field generated by all the others at that point. For strength vector &vec{Gamma} in {Bbb R}^N ;, the dynamical system is the classical point vortex system obtained from a singular discrete representation of the vorticity field from ideal, incompressible fluid flow. When &vec{Gamma} in Im ;, it corresponds to a system of sources and sinks, whereas when &vec{Gamma} in {Bbb C}^N ; the system consists of spiral sources and sinks discussed in Kochin et al (1964 Theoretical Hydromechanics 1 (London: Interscience)). We formulate the equilibrium problem as one in linear algebra, &A vec{Gamma} = 0 ;, &A in {Bbb C}^{N times N} ;, &vec{Gamma} in {Bbb C}^N ;, where A is a N × N complex skew-symmetric configuration matrix which encodes the geometry of the system of interacting singularities. For an equilibrium to exist, A must have a kernel and &vec{Gamma} ; must be an element of the nullspace of A. We prove that when N is odd, A always has a kernel, hence there is a choice of &vec{Gamma} ; for which the system is a stationary equilibrium. When N is even, there may or may not be a non-trivial nullspace of A, depending on the relative position of the points in the plane. We provide examples of evenly and randomly distributed points on curves such as circles, figure eights, flower-petal configurations and spirals. We then show how to classify the stationary equilibria in terms of the singular spectrum of A.