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Homework Help: Vorticity diffusioin

  1. Jul 24, 2008 #1
    1. The problem statement, all variables and given/known data
    Viscous flow between two rigid plates in which a lower rigid boundary y=0 is suddenly moved with speed U, which an upper rigid boundary to the fluid, y=h, is held at rest.

    2. Relevant equations
    [tex]\mathbf{u}=(u(y,t),0,0)[/tex]

    [tex]\frac{\partial u}{\partial t} = \nu \frac{\partial^2 u}{\partial y^2} [/tex]
    The initial condition: u(y,0)=0, 0<y<h;
    and boundary conditions: u(t,0)=U and u(h,t)=0 for t>0.

    3. The attempt at a solution
    By the separation of variables and Fourier series, the solution is
    [tex]u(y,t)=U(1-y/h)-\frac{2U}{\pi}\sum_{n=1}^{\infty}\frac{1}{n}\exp(-n^2\pi^2\nu t/h^2)\sin(n\pi y/h).[/tex]

    I don't understand why
    "For time [tex]t\geq h^2/\nu[/tex] the flow has almost reached its steady state, and the vorticity is almost distributed uniformly throughout the fluid."

    It seems that I should show [tex]\frac{\partial u}{\partial t}=0[/tex] for [tex]t\geq h^2/\nu[/tex]. But how can
    [tex]\frac{\partial u}{\partial t}=\frac{2U\pi\nu}{h^2}\sum_{n=1}^{\infty}n\exp(-n^2\pi^2\nu t/h)\sin(n\pi y/h)[/tex]
    approach zero?
     
  2. jcsd
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