<p>We consider the evolution of a viscous vortex dipole in <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathbb {R}^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> originating from a pair of point vortices with opposite circulations. At high Reynolds number <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\textrm{Re}\gg 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>Re</mtext> <mo>≫</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, the dipole can travel a very long way, compared to the distance between the vortex centers, before being slowed down and eventually destroyed by diffusion. In this regime we construct an accurate approximation of the solution in the form of a two-parameter asymptotic expansion involving the aspect ratio of the dipole and the inverse Reynolds number. We then show that the exact solution of the Navier–Stokes equations remains close to the approximation on a time interval of length <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathcal {O}(\textrm{Re}^\sigma )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">O</mi> <mo stretchy="false">(</mo> <msup> <mtext>Re</mtext> <mi>σ</mi> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\sigma < 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>σ</mi> <mo><</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> is arbitrary. This improves upon previous results which were essentially restricted to <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\sigma = 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>σ</mi> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. As an application, we provide a rigorous justification of an existing formula which gives the leading order correction to the translation speed of the dipole due to finite size effects.</p>

The Long Way of a Viscous Vortex Dipole

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发表日期：

2026年3月22日

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Michele Dolce,
Thierry Gallay

摘要

We consider the evolution of a viscous vortex dipole in $\mathbb {R}^2$ $R^{2}$ originating from a pair of point vortices with opposite circulations. At high Reynolds number $\textrm{Re}\gg 1$ $Re ≫ 1$ , the dipole can travel a very long way, compared to the distance between the vortex centers, before being slowed down and eventually destroyed by diffusion. In this regime we construct an accurate approximation of the solution in the form of a two-parameter asymptotic expansion involving the aspect ratio of the dipole and the inverse Reynolds number. We then show that the exact solution of the Navier–Stokes equations remains close to the approximation on a time interval of length $\mathcal {O}(\textrm{Re}^\sigma )$ $O ({Re}^{σ})$ , where $\sigma < 1$ $σ < 1$ is arbitrary. This improves upon previous results which were essentially restricted to $\sigma = 0$ $σ = 0$ . As an application, we provide a rigorous justification of an existing formula which gives the leading order correction to the translation speed of the dipole due to finite size effects.