The cross-stream migration of a deformable drop in two-dimensional Hagen–Poiseuille
flow at finite Reynolds numbers is studied numerically. In the limit of a small Reynolds
number (< 1), the motion of the drop depends strongly on the ratio of the viscosity
of the drop fluid to the viscosity of the suspending fluid. For viscosity ratio 0.125 a
drop moves toward the centre of the channel, while for ratio 1.0 it moves away from
the centre until halted by wall repulsion. The rate of migration increases with the
deformability of the drop. At higher Reynolds numbers (5–50), the drop either moves
to an equilibrium lateral position about halfway between the centreline and the wall –
according to the so-called Segre–Silberberg effect or it undergoes oscillatory motion.
The steady-state position depends only weakly on the various physical parameters of
the flow, but the length of the transient oscillations increases as the Reynolds number
is raised, or the density of the drop is increased, or the viscosity of the drop is
decreased. Once the Reynolds number is high enough, the oscillations appear to persist
forever and no steady state is observed. The numerical results are in good agreement
with experimental observations, especially for drops that reach a steady-state lateral
position. Most of the simulations assume that the flow is two-dimensional. A few
simulations of three-dimensional flows for a modest Reynolds number (Re = 10), and
a small computational domain, confirm the behaviour seen in two dimensions. The
equilibrium position of the three-dimensional drop is close to that predicted in the
simulations of two-dimensional flow.
Poiseuille flow of soft glasses in narrow channels: From quiescence to steady state
