Vorticity Diffusioin Homework: Steady State Solution

[steven2006]

Homework Statement

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.

Homework 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.

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?

 

[KataKoniK]

Thank you for your question. In order to understand why the vorticity is almost distributed uniformly throughout the fluid for t≥h^2/ν, we can look at the behavior of the Fourier series solution for u(y,t) as t increases. As t increases, the exponential term in the Fourier series, exp(-n^2π^2νt/h^2), decreases exponentially. This means that for larger values of t, the terms in the series with larger values of n will contribute less to the overall solution. This is because the exponential term will decrease faster for larger values of n. As a result, the solution will be dominated by the terms with smaller values of n, which correspond to larger wavelength modes. This means that as t increases, the solution will become more and more dominated by longer wavelength modes, and will approach a uniform distribution of vorticity throughout the fluid. This is why for t≥h^2/ν, the vorticity is almost distributed uniformly throughout the fluid. I hope this helps to clarify the concept for you. Good luck with your research!

 

[piercas]

I would first clarify the statement "the vorticity is almost distributed uniformly" with the person who wrote it. Vorticity is a measure of the local rotation of a fluid element, and it is not clear how it can be "distributed uniformly" throughout the fluid. It may be that the person meant to say that the vorticity is almost constant with respect to y, but that would require further clarification.

Assuming that is what was meant, then we can look at the solution for u(y,t) given above. As t increases, the exponential term in the sum approaches zero, and the amplitude of the sine term decreases. This means that for large enough t, the sum approaches zero, and u(y,t) becomes a constant, which is the steady state solution. In other words, the flow becomes steady and the velocity profile becomes flat, with u(y,t) = U for all y.

As for the vorticity, it is given by
\omega = \frac{\partial u}{\partial y} = \frac{2U\pi}{h}\sum_{n=1}^{\infty}\exp(-n^2\pi^2\nu t/h)\cos(n\pi y/h).

Note that the amplitude of the cosine term decreases as t increases, which means that the vorticity becomes smaller as t increases. This is why it can be said that the vorticity is almost constant for large enough t, as it is close to zero. However, it is important to note that the vorticity is not exactly zero, and it is not clear what "almost distributed uniformly" really means in this context.