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Vorticity and Stokes theorem

  1. Jul 12, 2010 #1
    1. The problem statement, all variables and given/known data
    Consider an imaginary circular disc, of radius R, whose arbitrary orientation is described by the unit vector, [tex] \vec {n} [/tex], perpendicular to the plane of the disc. Define the component, in the direction [tex] \vec {n} [/tex], of the angular velocity, [tex] \vec {\Omega} [/tex], at a point in the fluid by [tex] \vec {\Omega}. \vec {n} = \lim_{R \rightarrow 0}[\frac {1}{2 \pi R^2} \oint_C \vec {u}.dl] [/tex], where C denotes the the boundary (rim) of the disc. Use Stokes' theorem, and the arbitrariness of [tex] \vec {n} [/tex], to show that [tex] \vec {\Omega}= \frac {1}{2} \vec {\omega}[/tex], where [tex] \vec {\omega} = \nabla * \vec {u} [/tex] is the vorticity of the fluid at R=0. [This definition is based on a description applicable to the rotation of solid bodies. Confirm this by considering [tex] \vec {u} = \vec {U} + \vec {\Omega}* \vec{r} [/tex], where [tex] \vec {U} [/tex] is the translational velocity of the body, [tex] \vec {\Omega} [/tex] is its angular velocity and [tex] \vec {r} [/tex] is the position vector of a point relative to a point on the axis of rotation.]


    2. Relevant equations
    Stokes' theorem : [tex] \oint_c u.dl = \iint_S (\nabla * u) .n ds [/tex]




    3. The attempt at a solution
    Answer it gives in back of book is:
    Stokes theorem gives
    [tex]\oint_c u.dl = \iint_S (\nabla * u) .n ds = (\omega.n)\pi a^2 [/tex] (How did they get this !!?)so [tex] \Omega.n = \frac {1}{2} \omega . n[/tex]; but n is arbitrary so [tex] \Omega = \frac {1}{2} \omega. [/tex] NB [tex]\oint_c u.dl = \oint_c U.dl + \oint_c (\Omega * r).dl = \Omega.\oint_c (r*dl) = \Omega.n \int_{0}^{2\pi} a^2 d\theta [/tex]
     
  2. jcsd
  3. Feb 14, 2011 #2
    Re: Vorticity

    [tex] x=\alpha, y=\beta. [/tex]
    [
    i.e. solution of
    ax+by = -e
    cx + dy = -f
    ]

    Therefore if you introduce new variables[tex] \xi=x-\alpha[/tex] and [tex]\eta=y-\beta,[/tex] in the new variables [tex]\xi[/tex] and [tex]\eta [/tex] you will have a system of equations

    [tex]\xi' = a \xi + b \eta[/tex]
    [tex]
    \eta' = c \xi + d \eta[/tex]
     
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