Eur. Phys. J. E

**6**, 325-348 (2001)

## Quasi-one-dimensional foam drainage

**P. Grassia, J.J. Cilliers, S.J. Neethling and E. Ventura-Medina**

Department of Chemical Engineering, UMIST, P.O. Box 88, Manchester M60 1QD, UK Paul.Grassia@umist.ac.uk

(Received 3 July 2001)

** Abstract **

Foam drainage is considered in a froth flotation cell. Air
flow through the foam is described by a simple two-dimensional
deceleration flow, modelling the foam spilling over a weir. Foam
microstructure is given in terms of the number of channels (Plateau
borders) per unit area, which scales as the inverse square of bubble
size. The Plateau border number density decreases with height in the
foam, and also decreases horizontally as the weir is approached. Foam
drainage equations, applicable in the dry foam limit, are
described. These can be used to determine the average cross-sectional
area of a Plateau border, denoted *A*, as a function of position in
the foam. Quasi-one-dimensional solutions are available in which *A*
only varies vertically, in spite of the two-dimensional nature of the
air flow and Plateau border number density fields. For such
situations the liquid drainage relative to the air flow is purely
vertical. The parametric behaviour of the system is investigated with
respect to a number of dimensionless parameters: *K* (the strength of
capillary suction relative to gravity), (the deceleration of
the air flow), and *n* and *h* (respectively, the horizontal and
vertical variations of the Plateau border number density). The
parameter *K* is small, implying the existence of boundary layer
solutions: capillary suction is negligible except in thin layers near
the bottom boundary. The boundary layer thickness (when converted back
to dimensional variables) is independent of the height of the foam.
The deceleration parameter affects the Plateau border area on
the top boundary: weaker decelerations give larger Plateau border
areas at the surface. For weak decelerations, there is rapid
convergence of the boundary layer solutions at the bottom onto ones
with negligible capillary suction higher up. For strong
decelerations, two branches of solutions for *A* are possible in the
*K*=0 limit: one is smooth, and the other has a distinct kink. The
full system, with small but non-zero capillary suction, lies
relatively close to the kinked solution branch, but convergence from
the lower boundary layer onto this branch is distinctly slow.
Variations in the Plateau border number density (non-zero *n* and *h*)
increase individual Plateau border areas relative to the case of
uniformly sized bubbles. For strong decelerations and negligible
capillarity, solutions closely follow the kinked solution branch if
bubble sizes are only slightly non-uniform. As the extent of
non-uniformity increases, the Plateau border area reaches a maximum
corresponding to no net upward velocity of foam liquid. In the case
of vertical variation of number density, liquid content profiles and
Plateau border area profiles cease to be simply proportional to one
another. Plateau border areas match at the top of the foam independent
of *h*, implying a considerable difference in liquid content for foams
which exhibit different number density profiles.

**PACS**

47.55.Dz - Drops and bubbles.

82.70.Rr - Aerosols and foams.

**©**

*EDP Sciences, Società Italiana di Fisica, Springer-Verlag 2001*