ABSTRACT
Depending on the collapse occurring at the top of the foam, part of the gas resides in the foam, whereas the rest escapes from the top due to collapse. Because we have defined the reference plane to be such that all the escaped gas resides between it and the foam/gas interface (i.e., between z=0 and z=z1), a mass balance for the bubbling gas can be written as:
(125)
In Eq. (125), A is the cross-sectional area and G is the superficial gas velocity. The first term represents the gas in the foam, whereas the second term corresponds to the gas escaped from the collapsed bubbles. Equation (125) can be simplified to obtain:
(126)
Applying Leibnitz’s rule and using the fact that:
(127)
we have:
(128)
Eliminating dz1/dt using Eq. (62) yields:
(129)
When the gas supply is shut off, G=0 and Eq. (129) reduces to which is the same as Eq. (71). As long as there is no collapse at the top during generation, the motion of the foam/gas interface is determined by the bulk movement of the foam. However, as soon as the thickness of the liquid films at the top decreases to a critical value, the foam begins to collapse and a downward component is superimposed on the bulk movement of the foam. As a result, the net upward velocity of the foam/gas interface decreases. With less stable foams, it is possible to arrive at a steady state when the rate of foam collapse at the top becomes equal to the rate of foam generation (i.e., dz1/dt=dz2/dt) and the foam length (z2−z1) does not change with time. This steady state has been used to characterize the stability of short-lived foams.
