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

  • Vikas S. Krishnamurthy (a1), Miles H. Wheeler (a1), Darren G. Crowdy (a2) and Adrian Constantin (a1)


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).


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Constantin, A. & Krishnamurthy, V. S. 2019 Stuart-type vortices on a rotating sphere. J. Fluid Mech. 865, 10721084.
Crowdy, D. G. 1997 General solutions to the 2D Liouville equation. Intl J. Engng Sci. 35, 141149.
Crowdy, D. G. 2003 Polygonal N-vortex arrays: a Stuart model. Phys. Fluids 15, 37103717.
Crowdy, D. G. 2004 Stuart vortices on a sphere. J. Fluid Mech. 498, 381402.
Goodman, J., Hou, T. Y. & Lowengrub, J. 1990 Convergence of the point vortex method for the 2-D Euler equations. Commun. Pure Appl. Maths 43, 415430.
Haslam, M. C. & Mallier, R. 2003 Vortices on a cylinder. Phys. Fluids 15, 20872088.
Loutsenko, I. 2004 Equilibrium of charges and differential equations solved by polynomials. J. Phys. A 37, 13091321.
Mallier, R. 1995 Stuart vortices on a beta-plane. Dyn. Atmos. Oceans 22, 213238.
Mallier, R. & Maslowe, S. A. 1993 A row of counter-rotating vortices. Phys. Fluids A 5, 10741075.
Meiron, D. I., Moore, D. W. & Pullin, D. I. 2000 On steady compressible flows with compact vorticity; the compressible Stuart vortex. J. Fluid Mech. 409, 2949.
Morikawa, G. K. & Swenson, E. V. 1971 Interacting motion of rectilinear geostrophic vortices. Phys. Fluids 14, 10581073.
Newton, P. K. 2001 The N-Vortex Problem: Analytical Techniques. Springer.
O’Neil, K. 2006 Minimal polynomial systems for point vortex equilibria. Physica D 219, 6979.
O’Reilly, G. & Pullin, D. I. 2003 Structure and stability of the compressible Stuart vortex. J. Fluid Mech. 493, 231254.
Saffman, P. G. 1992 Vortex dynamics. Cambridge University Press.
Sakajo, T. 2019 Exact solution to a Liouville equation with Stuart vortex distribution on the surface of a torus. Proc. R. Soc. Lond. A 475, 20180666.
Shusser, M. 2004 Comment on vortices on a cylinder. Phys. Fluids 16, 3506.
Stuart, J. T. 1967 On finite amplitude oscillations in laminar mixing layers. J. Fluid Mech. 29, 417440.
Tur, A. & Yanovsky, V. 2004 Point vortices with a rational necklace: new exact stationary solutions of the two-dimensional Euler equation. Phys. Fluids 16, 28772885.
Tur, A., Yanovsky, V. & Kulik, K. 2011 Vortex structures with complex points singularities in two-dimensional Euler equations. New exact solutions. Physica D 240, 10691079.
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Steady point vortex pair in a field of Stuart-type vorticity

  • Vikas S. Krishnamurthy (a1), Miles H. Wheeler (a1), Darren G. Crowdy (a2) and Adrian Constantin (a1)


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