(adsbygoogle = window.adsbygoogle || []).push({}); 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?

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# Homework Help: Vorticity diffusioin

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