# Lamb–Oseen vortex

In fluid dynamics, the Lamb–Oseen vortex models a line vortex that decays due to viscosity. This vortex is named after Horace Lamb and Carl Wilhelm Oseen.

The mathematical model for the flow velocity in the circumferential $\theta$ –direction in the Lamb–Oseen vortex is:

$V_{\theta }(r,t)={\frac {\Gamma }{2\pi r}}\left(1-\exp \left(-{\frac {r^{2}}{r_{c}^{2}(t)}}\right)\right),$ with

• $r$ = radius,
• $r_{c}(t)={\sqrt {4\nu t+r_{c}(0)^{2}}}$ = core radius of vortex,
• $\nu$ = viscosity, and
• $\Gamma$ = circulation contained in the vortex.

The radial velocity is equal to zero.

The associated vorticity distribution in the vortex-filament-direction (here ${\hat {z}}$ ) can be found with the curl:

$\omega _{z}(r,t)={\frac {\Gamma }{\pi r_{c}(t)^{2}}}\exp \left(-{\frac {r^{2}}{r_{c}^{2}(t)}}\right),$ An alternative definition is to use the peak tangential velocity of the vortex rather than the total circulation

$V_{\theta }\left(r\right)=V_{\theta \max }\left(1+{\frac {1}{2\alpha }}\right){\frac {r_{\max }}{r}}\left[1-\exp \left(-\alpha {\frac {r^{2}}{r_{\max }^{2}}}\right)\right],$ where $r_{\max }(t)={\sqrt {\alpha }}r_{c}(t)$ is the radius at which $v_{\max }$ is attained, and the number α = 1.25643, see Devenport et al.

The pressure field simply ensures the vortex rotates in the circumferential direction, providing the centripetal force

${\partial p \over \partial r}=\rho {v^{2} \over r},$ where ρ is the constant density