Scholarly Works, Biomedical Engineering and Mechanics

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Browsing Scholarly Works, Biomedical Engineering and Mechanics by Subject "3-vortex motion"

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    Chaotic scattering of two identical point vortex pairs revisited
    Tophoj, L.; Aref, Hassan (American Institute of Physics, 2008-09-01)
    A new numerical exploration suggests that the motion of two vortex pairs, with constituent vortices all of the same absolute circulation, displays chaotic scattering regimes. The mechanisms leading to chaotic scattering are different from the "slingshot effect" identified by Price [Phys. Fluids A 5, 2479 (1993)] and occur in a different region of the four-vortex phase space. They may, in many cases, be understood by appealing to the solutions of the three-vortex problem obtained by merging two like-signed vortices into one of twice the strength and by assuming that the four-vortex problem has unstable periodic solutions similar to those seen in the thereby associated three-vortex problems. The integrals of motion, linear impulse and Hamiltonian are recast in a form appropriate for vortex pair scattering interactions that provides constraints on the parameters characterizing the outgoing vortex pairs in terms of the initial conditions.
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    Stability of relative equilibria of three vortices
    Aref, Hassan (American Institute of Physics, 2009-09-01)
    Three point vortices on the unbounded plane have relative equilibria wherein the vortices either form an equilateral triangle or are collinear. While the stability analysis of the equilateral triangle configurations is straightforward, that of the collinear relative equilibria is considerably more involved. The only comprehensive analysis available in the literature, by Tavantzis and Ting [Phys. Fluids 31, 1392 (1988)], is not easy to follow nor is it very physically intuitive. The symmetry between the three vortices is lost in this analysis. A different analysis is given based on explicit formulas for the three eigenvalues determining the stability, including a new formula for the angular velocity of rotation of a collinear relative equilibrium. A graphical representation of the space of vortex circulations is introduced, and the resultants between various polynomials that enter the problem are used. This approach adds considerable transparency to the solution of the stability problem and provides more physical understanding. The main results are summarized in a diagram that gives both the stability and instability of the various collinear relative equilibria and their sense of rotation.