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

Abstract

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