Two spheres falling through a pipe

Imagine a pipe of radius $$a$$ perpendicular to the ground. A liquid flows in the pipe. Now you throw (instantaneously) two balls with radius $$b$$ (where $$b << a$$) through the pipe: one through the center and the other really close to the pipe wall. Assume the viscosity of the liquid is uniform.

If we were to regard just the forces acting on the vertical direction (gravitational force $$F_g$$, friction force ball-water $$f_{bw}$$ and friction force water-pipe wall $$f_{pw}$$) I'd say the ball falling through the center of the pipe would touch the ground before the ball falling close to the pipe wall.

Why?

$$f_{pw} > f_{bw}$$

Is this a correct idea?

I did not use fluid dynamics in my thought experiment but please feel free to do so in your response.

• I feel like we need to consider how the fluid is flowing - is it just falling or is the pressure difference between the top and bottom such that the fluid is actually being propelled through the pipe? In any event, If the flow is laminar, the liquid in the middle flows faster and your analysis seems basically correct to me. – levitopher May 7 at 17:37
• The idea is to set the pressure equal in both sides of the pipe. However, this seems to be too idealistic. – JD_PM May 16 at 12:47
• I don't think you mean equal pressure, that's actually relatively hard to achieve in practice. For example, a vertical pipe standing out in the rain would have approximately equal pressure. You probably want something like a pipe coming out of a reservoir, where the pressure is just due to the liquid above the pipe - or any other situation in which the pressure different is relatively low. – levitopher May 17 at 13:47