Choosing boundary conditions for mathematical model

[fiontie]

Greetings, everyone!

The problem below is actually a task on Numerical Methods. But I have difficulties making a mathematical model.

Homework Statement

Let us have a longitudinally homogeneous system of a pipe of radius [tex]R[/tex] and a propeller of nearly the same radius inside it (we shall ignore the gap between the wall and propeller). The propeller rotates at the angular velocity [tex]\omega[/tex] and spins the water that flows though (the operating mode is assumed to be steady state). The problem is to find the field of velocities.

2. The attempt at a solution
Naturally, we shall consider a transverse section of the system and make use of polar coordinates (velocity [tex]\vec u = (u_r, u_{\phi})[/tex]). So that the mode can be considered steady, [tex]\omega[/tex] must be high enough. Then the motion of water is given by the steady Navier-Stokes equations coupled with the continuity equation.

Now to the boundary conditions: the first one is obviously [tex]u_r|_{r=R} = 0[/tex]. The second one, I believe, is a no-slip condition which concerns the velocity's tangential component on the propeller ([tex]u_{\phi}|_{?} = \omega r[/tex]). And that's what makes me in trouble, given the circumstance that the equations are time-independent.

The question is, how do I write out this condition if the propeller's position is not fixed? By fixing it in any way, we'll get selected directions. I am probably missing something but the statement of the problem doesn't seem completely correct to me.

I'd appreciate any advice. Thanks in advance!

 

[lippyka]

The boundary condition is a no-slip boundary condition, which states that the velocity of the fluid at the boundary (in this case the propeller) is equal to the velocity of the boundary (in this case the angular velocity of the propeller). This means that u_φ|_{r=R_p} = ω*R_p, where R_p is the radius of the propeller. So you don't need to fix the position of the propeller; the boundary condition is simply that the tangential velocity at the boundary must be equal to the angular velocity of the propeller.