The concentric, two-phase flow of two immiscible fluids in a tube of sinusoidally
varying cross-section is studied. This geometry is often used as a model to study the
onset of different flow regimes in packed beds. Neglecting gravitational effects, the
model equations depend on five dimensionless parameters: the Reynolds and Weber
numbers, and the ratios of density, viscosity and volume of the two fluids. Two
more dimensionless numbers describe the shape of the solid wall: the constriction
ratio and the ratio of its maximum radius to its period. In addition to the effect of
the Weber number, which depends on both the fluid and the flow, the effect of the
Ohnesorge number J has been examined as it characterizes the fluid alone. The governing
equations are approximated using the pseudo-spectral methodology while the
Arnoldi algorithm has been implemented for computing the most critical eigenvalues
that correspond to axisymmetric disturbances. Stationary solutions are obtained for a
wide parameter range, which may exhibit flow recirculation at the expanding portion
of the tube. Extensive calculations are made for the dependence of the neutral stability
boundaries on the various parameters. In most cases where the steady solution
becomes unstable it does so through a Hopf bifurcation. Exceptions to this are cases
where the viscosity ratio is O(10−3) and, then, the most unstable eigenvalue remains
real. Generally, steady core–annular flow in this geometry is more susceptible to
instability than in a straight tube and, in similar ranges of the parameters, it may
be generated by different mechanisms. Decreasing the thickness of the annular fluid,
inverse Weber number or the Ohnesorge number or the density of the core fluid stabilizes
the flow. For stability reasons, the viscosity ratio must remain strictly below unity
and it has an optimum value that maximizes the range of allowed Reynolds numbers.
Dynamics of retracting surfactant-laden ligaments at intermediate Ohnesorge number