An axisymmetric vortex, for which the azimuthal velocity uθuθu_\theta is proportional to r−βr^{-\beta}. What are the values for β\beta so that the circulation (Γ(r))\Gamma(r)) is finite

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1 Answer

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Around a circular contour of radius rr:

Γ(r)=∫R∇×u⋅dA=∮Cu⋅dl=∫2π0uθrdθ=2πruθ.\Gamma(r) = \int_R \nabla \times \mathbb{u}\cdot d\mathbf

{A} = \oint_C\mathbf{u} \cdot d\mathbf{l} = \int_0^{2\pi}u_\theta r \, d\theta = 2\pi r u_\theta.

With uθu_\theta proportional to r−βr^{-\beta}, say uθ=Cr−βu_\theta = Cr^{-\beta}, we have Γ(r)=2πCr1−β\Gamma(r) = 2\pi Cr^{1-\beta} and the circulation is finite as r→∞r \to \infty if β⩾1\beta \geqslant 1.

I’m sorry i should have put that I knew that part already, if uθu_\theta is r−βr^-\beta then the circulation as a function of r is 2πruθ2\pi r u_\theta ?

– Abigail

yesterday

sorry 2πr−2β2\pi r^{-2\beta}

– Abigail

yesterday

Do I just take the limit as beta approaches infinity?

– Abigail

yesterday

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@Abigail. See above. Just note that the circulation is proportional to r1−βr^{1-\beta}, not r−2βr^{-2\beta}.

– RRL

yesterday

why does the exponent have to be less than or equal to zero? I’m having a hard time conceptually understanding what is happening in this problem

– Abigail

yesterday