The axisymmetric long-wave instability of a concentric two-phase pipe flow

The long-wave instability in a confined flow system is studied by considering a vertical pipe containing two immiscible fluids arranged in a concentric geometry. Each fluid has a different density, but otherwise their properties are identical. Gravity and an imposed pressure gradient drive a flow in this system, which is susceptible to an axisymmetric long-wavelength instability for any density ratio. When the density ratio is not too different from 1 we find that stable flows are usually downward flows when the lighter fluid is in the center and are usually upward flows when the heavier fluid is in the center. However, there are some parameter ranges in which the opposite is true. We explain this behavior in terms of a physical mechanism for the long-wave instability.