Unsteady extrusion of a viscoelastic annular film: II. Linearized model and its analytical solution

Abstract

The unsteady extrusion of a viscoelastic film from an annular and axisymmetric die is examined, when the gravitational and the capillary forces in the film are small relative to the viscous forces. The Oldroyd-B constitutive equation is employed. This moving boundary problem is solved by mapping the inner and outer liquid/air interfaces of the extruded film 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 and is used as the small parameter, ε, in a regular perturbation expansion of the governing equations. It is shown that when the St and Ca −1 numbers are of appropriately small magnitude, the base state in the perturbation scheme is a uniformly falling film. The effect of these dimensionless numbers is demonstrated by analytically calculating the next order solution in a Taylor expansion in the Reynolds number. It is found that the present results agree very well with the numerical ones calculated by solving a large nonlinear equation set in [K. Housiadas, J. Tsamopoulos, Unsteady extrusion of a viscoelastic film I. General model and its numerical solution, J. Non-Newtonian Fluid Mech. 1999, in press], when the dimensionless numbers are as small as required by the present analysis. The present analysis also shows where and which auxiliary conditions should be applied in this problem.