## Abstract

An exact solution of the Navier-Stokes equations of incompressible flow, which represents the interaction of a diffusing line vortex and a linear shear flow aligned so that initially the streamlines in the shear flow are parallel to the line vortex, is presented. If $\Gamma$ is the circulation of the line vortex and v the kinematic viscosity then, when Re = $\Gamma/2\pi$v is large, the vorticity of the shear flow is expelled from the circular cylinder $0 < r \simeq(vt)^\frac{1}{2} \operatorname{Re}^\frac{1}{3}$, where r is the distance from the axis of the diffusing line vortex and t the time since initiation of the flow. At larger radii a peak vorticity 0.903$\Omega$ Re$^\frac{1}{3}$ is found at a radial distance 1.26(vt)$^\frac{1}{2}$ Re$^\frac{1}{3}$, where $\Omega$ is the initial uniform vorticity in the shear flow. The vortex filament is embedded in a growing cylinder from which vorticity has been expelled, the cylinder itself being bounded by an annular region of thickness of order Re$^\frac{1}{3}$ (vt)$^\frac{1}{2}$ in which the vorticity is of order $\Omega$ Re$^\frac{1}{3}$.

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