steven2006
Jul24-08, 03:10 PM
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?
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?