Couette Flow in a Tank | Solving Boundary Conditions

[maxtor101]

Homework Statement

Consider a closed tank filled with water, with depth h. A plate on the surface moves horizontally with constant velocity U


(Excuse my poor diagram)

The Attempt at a Solution

Velocity field:

$$\mathbf{v} = w(z) \mathbf{i}$$

$$\rho \frac{\partial w}{\partial x} = -\frac{\partial p}{\partial x} + \mu \frac{\partial^2w}{\partial z^2}$$

Since flow is due to moving plane and not pressure gradient p=const

Also since velocity is constant

$$\frac{\partial w}{\partial t} = 0$$

Hence

$$\frac{\partial^2w}{\partial z^2}$$

So w will be of the form

$$w(z) = Az + B$$


This is where I'm stuck, I'm not sure what boundary conditions to impose

I have the obvious one first

$$w(h) = U$$

Therefore

$$B = U - Ah$$

Any help would be greatly appreciated!

 

[mark726]

Thank you for your post. I would like to provide some clarification and suggestions for your problem.

Firstly, your approach to solving the velocity field using the Navier-Stokes equation is correct. However, there are a few points that need to be addressed.

1. The velocity field should be in the form of \mathbf{v} = w(z) \mathbf{i} + u(x,y) \mathbf{j} + v(x,y) \mathbf{k}, where u and v are the velocities in the x and y directions, respectively. In this case, since the plate is moving horizontally, u will be constant and v will be zero. So the final velocity field will be \mathbf{v} = w(z) \mathbf{i} + u \mathbf{j}.

2. The pressure term in the Navier-Stokes equation, -\frac{\partial p}{\partial x}, should not be neglected. It is true that the flow is mainly driven by the moving plate, but there will still be a pressure gradient due to the water's weight and the boundary conditions at the tank's walls.

3. You are correct in assuming that the time derivative of w is zero since the plate's velocity is constant. However, the time derivative of u and v should also be zero since there is no change in the velocity in the x and y directions.

4. As for the boundary conditions, you have correctly identified that w(h) = U. But there are two more conditions that we can impose: u = 0 at all points since there is no motion in the x direction, and v = 0 at the tank's walls since there is no flow through the walls.

I hope this helps you in solving the problem. Best of luck!