Effect of side wind on the drag of a falling sphere

Question

I just thought of a problem that seems quite fundamental but I couldn't find a clear answer or explanation.

Consider a solid sphere falling in a Newtonian fluid due to gravity. It eventually reaches its terminal vertical velocity. Now let's add a constant side wind: can this induce a net vertical force and hence increase or decrease the vertical speed of the sphere?

At low Reynolds number I believe the answer is no due to the linearity of the Stokes equation. But what about at finite Reynolds number? Even if the object falling is perfectly symmetrical, the flow around is not, possibly dramatically (e.g. recirculation zone, vortex shedding), and so it would not really surprise me if adding side wind to that situation would have an effect on the vertical direction. But intuitively I can't tell if it would indeed have an effect, and if so if it would speed up or slow down the vertical descent.

Answer 1

You can work in the reference frame in which the (constant) crosswind is zero.

In this frame of reference, the speed limit is the same as if there was no crosswind (I am talking only about the speed limit, not how it is reached).

Then, just go back to the original reference frame.

EDIT : Imagine an object falling at a constant speed with no crosswind. Change the frame of reference: you see an object falling at a constant speed with a crosswind. If the solution is unique, it is the right one.

In your example, you talk about a sphere. So orientation in space doesn't matter. You can apply Newton's law to the center of gravity. For any solid, the drag and lift forces depend on the orientation of the solid and it is not obvious that a steady state exists or that the solution is unique. So it's more complicated. Just watch a leaf fall!