Fully developed flow of a viscoelastic film down a vertical cylindrical or planar wall

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, b o (h)\tilde]_s/((h)\tilde]_s +(h)\tilde]_p)=0\beta\equiv {\tilde{\eta}_s}/\left({\tilde{\eta}_s +\tilde{\eta}_p}\right)=0, where (h)\tilde]_p\widetilde{\eta}_p and (h)\tilde]_s\widetilde{\eta}_s 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 x 1 0\xi \ne 0; (3) e-PTT model in planar geometry when β = 0 and ξ = 0. The effect of fluid properties, cylinder radius, (R)\tilde]\tilde{R}, and flow rate on the velocity profile, the stress components, and the film thickness, (H)\tilde]\tilde{H}, is determined. On the other hand, the relevant dimensionless numbers, which are the Deborah, De=(l)\tilde](U)\tilde]/(H)\tilde]De={\tilde{\lambda}\tilde{U}}/{\tilde{H}}, and Stokes, St=(r)\tilde](g)\tilde](H)\tilde]²/((h)\tilde]_p +(h)\tilde]_s )(U)\tilde]St=\tilde{\rho}\tilde{g}\tilde{\rm H}^{2}/\left({\tilde{\eta}_p +\tilde{\eta}_s} \right)\tilde{U}, numbers, depend on (H)\tilde]\tilde{H} and the average film velocity, (U)\tilde]\widetilde{U}. This makes necessary a trial and error procedure to obtain (H)\tilde]\tilde{H} 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 (R)\tilde]/(H)\tilde]{\tilde{R}}/{\tilde{H}} decreases down to zero for a film on the outer cylindrical surface, while it asymptotes to very large values when (R)\tilde]/(H)\tilde]{\tilde{R}}/{\tilde{H}} decreases down to unity for a film on the inner surface. When x 1 0\xi \ne 0, an upper limit in De exists above which a solution cannot be computed. This critical value increases with ε and decreases with ξ.