Unsteady flow of an axisymmetric annular film under gravity


Housiadas K., Tsamopoulos J.


The unsteady flow of an annular and axisymmetric film under gravity is examined. This moving boundary problem is solved by mapping the inner and the outer interface of the film in the radial direction onto fixed ones and by transforming the governing equations accordingly. The ratio of the film thickness to its inner radius at the exit of the die is small in relevant processes with polymer melts. This ratio, ε, is used as the small parameter in a perturbation expansion of the general Navier-Stokes equations. Forces applied on the film include gravity, surface tension, inertia, and viscous forces. Their ratios give rise to three dimensionless numbers, St, Ca, and Re. When these dimensionless numbers are up to order one, the base state is quite deformed and it is calculated numerically by simultaneously solving three nonlinear partial differential equations in time and the axial direction. Intuitively it is expected that when the dimensionless numbers are small the base state in the perturbation scheme is a uniformly falling film. This is confirmed by analysis and the two next orders in the perturbation scheme are calculated analytically. In both cases, it was found that increasing the St number (i) accelerates the downward motion of the film, (ii) deflects its inner and outer surfaces towards its axis of symmetry, and (iii) decreases its thickness around the middle of its length. The latter effect may lead to breakup of the film in two parts. It was also found that increasing the Ca number deflects these two interfaces towards its axis of symmetry and increases its thickness monotonically with time and the axial distance. Increasing the Re number from zero, but to not very large values, generally decelerates the film and decreases its deflection from the vertical. Given typical fluid properties and process conditions the St number is up to O(ε0), i.e., much larger than the other two dimensionless numbers, and affects the film shape more significantly. © 1998 American Institute of Physics.