We have a two parallel vertical glass plates with gap H between them. The gap is filled with water. At the bottom, the air bubble with diameter D is injected and it starts moving up due to the buoyancy. The question is, for which values of H and D the bubble can be considered as 2D case. The terminal velocity obtained fromexperiment (H=4mm, D=10mm) and numerical simulation (2D potential flow) are very close. Is this just a coincidence?

Comments are welcome!

(1). Hele Shaw(1898); p-216-, Section 4.8, flow fields in which inertia forces are negligible, "An Introduction to Fluid Dynamics", by G.K.Batchelor, Cambridge University Press,1967. (2). In the case of a gas bubble in water between walls, you may have to consider the surface tension , the gap and the surface stability. So, you may have to check other books in this area. I am no expert here.

In general, of course, you can consider the buble to be two-dimensional (with properties that might different from the 3D buble) when the size of the buble is larger than the distance between the two plates. In this case the buble cannot move in the direction normal to the plates.

The best modeling is achieved when the size of the buble is much larger than the distance between the two plates. In this case the effects of the extremeties of the 2D buble and its finite thickness are small. And mostly the buble is FLAT, rather than spherical.

This is true in general for many thin fluid layer problems: if the finite thickness of the flow is much smaller in one dimension than in the two others, then on the scales larger than this thickness the flow is two dimensional. SO when the buble is larger than the thickness of the fluid layer, the buble is mainly two-dimensional; or more precisely it can be modeled with a two dimensional model.

PG