# Vorticity diffusioin

1. Jul 24, 2008

### steven2006

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
$$\mathbf{u}=(u(y,t),0,0)$$

$$\frac{\partial u}{\partial t} = \nu \frac{\partial^2 u}{\partial y^2}$$
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
$$u(y,t)=U(1-y/h)-\frac{2U}{\pi}\sum_{n=1}^{\infty}\frac{1}{n}\exp(-n^2\pi^2\nu t/h^2)\sin(n\pi y/h).$$

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

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