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.
Mathematical description Oseen looked for a solution for the Navier–Stokes equations in cylindrical coordinates ( r , θ , z ) {\displaystyle (r,\theta ,z)} with velocity components ( v r , v θ , v z ) {\displaystyle (v_{r},v_{\theta },v_{z})} of the form
v r = 0 , v θ = Γ 2 π r g ( r , t ) , v z = 0. {\displaystyle v_{r}=0,\quad v_{\theta }={\frac {\Gamma }{2\pi r}}g(r,t),\quad v_{z}=0.}
where Γ {\displaystyle \Gamma } is the circulation of the vortex core. Navier–Stokes equations lead to
∂ g ∂ t = ν ( ∂ 2 g ∂ r 2 − 1 r ∂ g ∂ r ) {\displaystyle {\frac {\partial g}{\partial t}}=\nu \left({\frac {\partial ^{2}g}{\partial r^{2}}}-{\frac {1}{r}}{\frac {\partial g}{\partial r}}\right)}
which, subject to the conditions that it is regular at r = 0 {\displaystyle r=0} and becomes unity as r → ∞ {\displaystyle r\rightarrow \infty } , leads to
g ( r , t ) = 1 − e − r 2 / 4 ν t , {\displaystyle g(r,t)=1-\mathrm {e} ^{-r^{2}/4\nu t},}
where ν {\displaystyle \nu } is the kinematic viscosity of the fluid. At t = 0 {\displaystyle t=0} , we have a potential vortex (free or irrotational vortex) with concentrated vorticity at the z {\displaystyle z} -axis; and this vorticity diffuses away as time passes. The only non-zero vorticity component is in the z {\displaystyle z} -direction, given by
ω z ( r , t ) = Γ 4 π ν t e − r 2 / 4 ν t . {\displaystyle \omega _{z}(r,t)={\frac {\Gamma }{4\pi \nu t}}\mathrm {e} ^{-r^{2}/4\nu t}.}
The pressure field simply ensures the vortex rotates in the circumferential direction, providing the centripetal force
∂ p ∂ r = ρ v 2 r , {\displaystyle {\partial p \over \partial r}=\rho {v^{2} \over r},}
where ρ {\displaystyle \rho } is the constant density. Below are two methods by which Oseen's vortex can be derived.
Generalized Oseen vortex The generalized Oseen vortex may be obtained by looking for solutions of the form
v r = − γ ( t ) r , v θ = Γ 2 π r g ( r , t ) , v z = 2 γ ( t ) z {\displaystyle v_{r}=-\gamma (t)r,\quad v_{\theta }={\frac {\Gamma }{2\pi r}}g(r,t),\quad v_{z}=2\gamma (t)z}
that leads to the equation
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