Rigid-Surfaced Collapsing Wedge Model

The simplest model was a theoretical version of the collapsing film wedge. As discussed previously (Section 5.3.3.1), the collapsing wedge geometry was imposed on the analysis of the experimental data to generate instantaneous film drainage rates. Therefore, it seemed reasonable to use this same shape in the development of the first theoretical model [51]. This led to the condition that the shape of the film was invariant with 

Within the wedge, the momentum in the vertical direction is conserved by that component of the Navier-Stokes equations  

To simplify matters, it is assumed that the flow is so slow and gradually changing that the left hand side of the equation can be neglected. The flat sides of the wedge along with the slow flow assumption allow the pressure gradient term to be neglected. The differential equation for the velocity profile in the film is then obtained  

The solution to this equation is subject to dwidx = 0 in the centre of the film and (0 = 0 on the surface of the film, x = k (z, t). The velocity profile is given by 

This equation clearly indicates that drainage of the film is accelerated by increasing density and inhibited by increasing viscosity. 

---