Pavlidis M., Dimakopoulos Y., Tsamopoulos J.<\/p>\n
The one-dimensional, gravity-driven film flow of a linear (l) or exponential (e) Phan-Thien and Tanner (PTT) liquid, flowing either on the outer or on the inner surface of a vertical cylinder or over a planar wall, is analyzed. Numerical solution of the governing equations is generally possible. Analytical solutions are derived only for: (1) l-PTT model in cylindrical and planar geometries in the absence of solvent, \u03b2 \u2261 \u03b7\u0303s(\u03b7\u0303s + \u03b7\u0303p) = 0, where \u03b7\u0303p and \u03b7\u0303s are the zero-shear polymer and solvent viscosities, respectively, and the affinity parameter set at \u03be = 0; (2) l-PTT or e-PTT model in a planar geometry when \u03b2 = 0 and \u03be \u2260 0; (3) e-PTT model in planar geometry when \u03b2 = 0 and \u03be = 0. The effect of fluid properties, cylinder radius, R\u0303, and flow rate on the velocity profile, the stress components, and the film thickness, H\u0303, is determined. On the other hand, the relevant dimensionless numbers, which are the Deborah, De = \u03bb\u0303\u0168\/H\u0303, and Stokes, St = \u03c1\u0303g\u0303H\u03032\/ (\u03b7\u0303p + \u03b7\u0303s)\u0168, numbers, depend on H\u0303 and the average film velocity, \u0168. This makes necessary a trial and error procedure to obtain H\u0303 a posteriori. We find that increasing De, \u03be, or the extensibility parameter \u03b5 increases shear thinning resulting in a smaller St. The Stokes number decreases as R\u0303\/H\u0303 decreases down to zero for a film on the outer cylindrical surface, while it asymptotes to very large values when R\u0303\/H\u0303 decreases down to unity for a film on the inner surface. When \u03be \u2260 0, an upper limit in De exists above which a solution cannot be computed. This critical value increases with \u03b5 and decreases with \u03be. \u00a9 Springer-Verlag 2009.<\/p>\n
Gravity-driven flow, PTT fluid model, Viscoelastic film flow<\/p>\n