A core–annular flow, the concurrent axial flow of two immiscible fluids in a circular
tube or pore with one fluid in the core and the other in the wetting annular region,
is frequently used to model technologically important flows, e.g. in liquid–liquid displacements in secondary oil recovery. Most of the existing literature assumes that the
pores in which such flows occur are uniform circular cylinders, and examine the interfacial stability of such systems as a function of fluid and interfacial properties. Since
real rock pores possess a more complex geometry, the companion paper examined the
linear stability of core–annular flows in axisymmetric, corrugated pores in the limit
of asymptotically weak corrugation. It found that short-wave disturbances that were
stable in straight tubes could couple to the wall's periodicity to excite unstable long
waves. In this paper, we follow the evolution of the axisymmetric, linearly unstable
waves for fluids of equal densities in a corrugated tube into the weakly nonlinear
regime. Here, we ask whether this continual generation of new disturbances by the
coupling to the wall's periodicity can overcome the nonlinear saturation mechanism
that relies on the nonlinear (kinematic-condition-derived) wave steepening of the
Kuramoto–Sivashinsky (KS) equation. If it cannot, and the unstable waves still saturate, then do these additional excited waves make the KS solutions more likely
to be chaotic, or does the dispersion introduced into the growth rate correction by
capillarity serve to regularize otherwise chaotic motions?We find that in the usual strong surface tension limit, the saturation mechanism of
the KS mechanism remains able to saturate all disturbances. Moreover, an additional
capillary-derived nonlinear term seems to favour regular travelling waves over chaos,
and corrugation adds a temporal periodicity to the waves associated with their
periodical traversing of the wall's crests and troughs. For even larger surface tensions,
capillarity dominates over convection and a weakly nonlinear version of Hammond's
no-flow equation results; this equation, with or without corrugation, suggests further
growth. Finally, for a weaker surface tension, the leading-order base flow interface
follows the wall's shape. The corrugation-derived excited waves appear able to push
an otherwise regular travelling wave solution to KS to become chaotic, whereas
its dispersive properties in this limit seem insufficiently strong to regularize chaotic
motions.
Linear instability, nonlinear instability and ligament dynamics in three-dimensional laminar two-layer liquid–liquid flows
