Stability of core-annular flow with a small viscosity ratio

Publisher Website
Google Scholar
Abstract
It is known that the stability problem for core‐annular flow of very viscous crude oil and water is singular, the water annulus appears to be inviscid with boundary layers at the pipe wall and at the interface. In the present paper, this singular problem is treated by the method of matched asymptotic expansions using ε=m/Rα as a small parameter. There are two cases of instability corresponding to different positions of the critical point in the annulus. One case is when the critical point is far away from the interface, the other is when the critical point is close to the interface within a distance of order ε1/3. In both cases, we derive the equations for the eigenvalues, and give the explicit forms for the neutral curves. The stability problem is also treated by the modified finite element code used by Hu and Joseph [J. Fluid Mech. 2 0 5, 359 (1989); Phys. Fluids A 1, 1659 (1989)], taking into account the boundary layers at the pipe wall and at the interface. The results of the two methods agree where they overlap, but the finite element technique goes further.