The film is assumed to be geometrically described in the 
- 
plane. Let 
stand for the Cauchy stress tensor, and 
for the velocity vector. On the upper and lower free sides of the
film, zero traction conditions apply. In view of the small film thickness, it is
assumed that this is true across the film thickness. From the vertical momentum
equation, the pressure, 
, can be found:
 | (18–1) |
That is,
 | (18–2) |
where 
is the extra-stress component that obeys the fluid constitutive
equation. Because the divergence of the velocity is equal to zero, the following
equation holds:
 | (18–3) |
where 
is the coordinate in the film-thickness direction.
Let 
be the thickness of the film, equal to the value of
at the upper side of the film. Mass conservation on a film element
of thickness 
yields
 | (18–4) |
where the term involving the partial derivative of the thickness with respect to
time is omitted in steady or evolution calculations. This partial differential
equation (Equation 18–4) is of the
hyperbolic type. From a mathematical viewpoint, a boundary condition for the
thickness 
must be specified along a line crossing all flow trajectories at
their entrance into the computational domain. In the geometric model, this line
corresponds to the die exit, and hence to the film entrance. Note that the thickness
must be specified only along this line.
When solving a film problem, the thickness-averaged velocity field, which is a
function of 
and 
, will be calculated. As in any flow problem, flow boundary
conditions must be specified. General information on boundary conditions can be
found in Boundary Conditions. When a
nonisothermal film flow is solved, an average temperature equation will be
calculated as well. This field obeys the energy equation. Boundary conditions
must therefore be specified for the temperature field.
For the particular case of the DCPP model used within a film casting flow simulation, care may be required when imposing the boundary conditions for the viscoelastic variables at the inlet of the calculation domain. The inlet is identified as the border where the thickness condition is imposed. Here, instead of imposing values for all components of the orientation tensor, you will be asked about anisotropy described in terms of orientation and factor, as well as stretching amplitude. Imposing boundary conditions for the viscoelastic unknowns (orientation and stretching) is optional. If conditions are imposed, you will need to provide the following information:
the direction of anisotropy of the orientation tensor (
or 
) the anisotropy factor ranging between 0 (isotropic) and 1 (anisotropic)
the inlet stretching