The velocity of a liquid slug falling in a capillary tube is lower than
predicted for Poiseuille flow due to presence of menisci, whose shapes
are determined by the complex interplay of capillary, viscous, and
gravitational forces. Due to the presence of menisci, a capillary
pressure proportional to surface curvature acts on the slug and
streamlines are bent close to the interface, resulting in enhanced
viscous dissipation at the wedges. To determine the origin of
drag-force increase relative to Poiseuille flow, we compute the force
resultant acting on the slug by integrating Navier-Stokes equations
over the liquid volume. Invoking relationships from differential
geometry we demonstrate that the additional drag is due to viscous
forces only and that no capillary drag of hydrodynamic origin exists
(i.e., due to hydrodynamic deformation of the interface). Requiring
that the force resultant is zero, we derive scaling laws for the steady
velocity in the limit of small capillary numbers by estimating the
leading order viscous dissipation in the different regions of the slug
(i.e., the unperturbed Poiseuille-like bulk, the static menisci close
to the tube axis and the dynamic regions close to the contact lines).
Considering both partial and complete wetting, we find that the
relationship between dimensionless velocity and weight is, in general,
nonlinear. Whereas the relationship obtained for complete-wetting
conditions is found in agreement with the experimental data of Bico and
Quere [J. Bico and D. Quere, J. Colloid Interface Sci. 243, 262
(2001)], the scaling law under partial-wetting conditions is validated
by numerical simulations performed with the Volume of Fluid method. The
simulated steady velocities agree with the behavior predicted by the
theoretical scaling laws in presence and in absence of static contact
angle hysteresis. The numerical simulations suggest that wedge-flow
dissipation alone cannot account for the entire additional drag and
that the non-Poiseuille dissipation in the static menisci (not
considered in previous studies) has to be considered for large contact
angles.