(236d) Transverse Migration of Particles and Bubbles in Oscillatory Tube Flow

The transverse
migration of particles in the presence of a shear flow is a phenomenon which
has attracted a great deal of attention ever since the landmark experiments of
Segré and Silberberg.^1,2 Past studies in bounded geometries, which
have been primarily focused on steady flow, demonstrate that the ?tubular
pinch? effect is expected to occur when finite inertial effects exist in a
system (see, for example, Ho and Leal³ or Vassuer and Cox⁴).  The equilibrium position of a non-neutrally buoyant particle in a horizontal tube will be governed by a balance between gravitational and inertial forces.  In gravitational field flow fractionation, for example, such a balance of forces determines the mobility of a particle in the analogous channel flow problem.⁵ It is reasonable to expect that oscillatory flow will affect particles in a similar way, particularly at low frequencies (i.e. low Womersly number).  Bubbles, though fundamentally different, will behave like solid particles if they remain spherical (i.e. low Capillary number).

In this work,
we experimentally examined the dimensionless mobility Dx_p/Dx
of solid polymethylmethacrylate particles and air bubbles which were subjected
to oscillatory flow in a glass capillary tube, oriented both horizontally and
vertically.  In the horizontal orientation, we observed the mobility of both
particles and bubbles to exhibit behavior consistent with well known inertial
lift mechanisms derived for steady flow; that is, the amplitude of motion was a
function of Re_p²/Re_s (the ratio
of inertial to gravitational forces⁶) for a wide range of frequencies. At low Re_p²/Re_s, a transition was observed between the particle rolling with slip along the tube wall and lifting away from the wall.  The mobility then increased rapidly with Re_p²/Re_s, ultimately reaching the neutrally buoyant asymptote as Re_p²/Re_s>> 1.  The mobility was predicted reasonably well by adapting existing steady flow models. 

Bubbles were
found to have a more complex behavior from the competition between two
migration mechanisms: inertial lift and deformation induced lift.  The inertial
lift drives the bubble to an equilibrium position between the tube wall and
center while the deformation induced lift drives the bubbles towards the centerline.⁷  Qualitatively, the mobility of small bubbles was similar to the mobility of particles.

The migration
behavior observed for symmetric oscillatory flow in the horizontally orientated
tube suggested a methodology for extracting or separating particles/bubbles in
confined geometries, such as those commonly found in microfluidic devices.  By
applying a zero-mean asymmetric oscillatory flow, particles and bubbles in a
capillary tube were given a net drift with each stroke.  The direction and
speed of the drift were shown to depend on the ratio of the frequencies in the
forward and reverse direction and corresponded well with displacements predicted
from the experiments performed using symmetric oscillatory flow.  The utility
of asymmetric oscillatory flows in microfluidic
geometries was demonstrated using a direct methanol fuel cell.  An asymmetric
tidal displacement (sin(wt) + 0.25 sin(2wt)) applied to the anode feed
was shown to dramatically reduce instabilities caused by CO₂ bubble
accumulation at low feed rates.

In the
vertical orientation, negatively buoyant particles were observed to travel
opposite gravity for certain symmetric oscillatory flows.  This occurred
because the particle has a different mobility during the upward and downward
stroke.  During the downstroke, a particle will lead the flow and it will be
driven toward the wall; whereas during the upstroke the particle lags the fluid
and migrates to the centerline.  If the migration is fast enough, the upward
velocity will be greater than the downward velocity.  An asymptotic model was
developed which agreed with the observed behavior provided a pseudo
steady-state criterion was satisfied.

(1)        Segre, G.; Silberberg,
A. Journal of Fluid Mechanics 1962, 14, 115-135.

(2)        Segre, G.; Silberberg,
A. Journal of Fluid Mechanics 1962, 14, 136-157.

(3)        Ho, B. P.; Leal, L. G.
Journal of Fluid Mechanics 1974, 65, 365-400.

(4)        Vasseur, P.; Cox, R.
G. Journal of Fluid Mechanics 1976, 78, 385-413.

(5)        Williams, P. S.; Lee,
S. H.; Giddings, J. C. Chemical Engineering Communications 1994, 130,
143-166.

(6)        King, M. R.; Leighton,
D. T. Physics of Fluids 1997, 9, 1248-1255.

(7)        Chan, P. C. H.; Leal,
L. G. Journal of Fluid Mechanics 1979, 92, 131-170.