Imagine an infinitely long flow channel filled with water flowing at a constant speed everywhere parallel to the infinitely long dimension of this channel. Furthermore, imagine we place a body (say a sphere of some radius) in that channel with zero speed (the body is supposed to float in a way that it is in the center of the flow channel and would stay there if the surrounding water was not moving).

  1. Is there an easy way to describe how the body is accelerated?
  2. Is it correct that the speed of the body will approach the speed of the fluid as time goes to infinity?

The situation is exactly the same as if the water was stationary and the body was moving (well, assuming you're far enough from the walls for edge effects not to matter). In that case the equation of motion for the body will be:

$$ \frac{dv}{dt} = A(v) $$

where $A$ will be given by something like the quadratic drag equation:

$$ mA = \frac{1}{2} \rho \space C_d A \space v^2 $$

at high Reynolds numbers, and the Stokes equation:

$$ mA = 6\pi \mu R v $$

at low Reynold's numbers. Solving this will give you the velocity relative to the water, then just add the velocity of the water to get the velocity relative to a stationary observer.

The Stokes equation will give you a velocity dependance of the form:

$$ v(t) = Ae^{-Bt} $$

so the body will, in principle, take an infinite time to match speed with the water, but in practice exponential decay is sufficiently fast that the difference between the body and water velocities will quickly become insignificant.


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