Pavlidis M., Dimakopoulos Y., Tsamopoulos J.
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, β ≡ η̃s(η̃s + η̃p) = 0, where η̃p and η̃s are the zero-shear polymer and solvent viscosities, respectively, and the affinity parameter set at ξ = 0; (2) l-PTT or e-PTT model in a planar geometry when β = 0 and ξ ≠ 0; (3) e-PTT model in planar geometry when β = 0 and ξ = 0. The effect of fluid properties, cylinder radius, R̃, and flow rate on the velocity profile, the stress components, and the film thickness, H̃, is determined. On the other hand, the relevant dimensionless numbers, which are the Deborah, De = λ̃Ũ/H̃, and Stokes, St = ρ̃g̃H̃2/ (η̃p + η̃s)Ũ, numbers, depend on H̃ and the average film velocity, Ũ. This makes necessary a trial and error procedure to obtain H̃ a posteriori. We find that increasing De, ξ, or the extensibility parameter ε increases shear thinning resulting in a smaller St. The Stokes number decreases as R̃/H̃ decreases down to zero for a film on the outer cylindrical surface, while it asymptotes to very large values when R̃/H̃ decreases down to unity for a film on the inner surface. When ξ ≠ 0, an upper limit in De exists above which a solution cannot be computed. This critical value increases with ε and decreases with ξ. © Springer-Verlag 2009.
Gravity-driven flow, PTT fluid model, Viscoelastic film flow