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.
相似文献