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with the corresponding boundary conditions
ʸ
(ʾ
=
0
)
=
0
,ʸ ʾ
=
1
)
=
ʸ
w
(11)
In order to obtain approximate solutions of these differential equations the numerical
method used is based on orthogonal collocation. The solution is achieved through
a series of known trial functions; these series of functions are called test functions,
which are studied in the differential equation and the result is called the residual. We
employ orthogonal collocation using the Jacobi polynomials for the flow problem
considered in this work. Therefore by applying the method of placement it is required
that:
N
+
2
i
N
+
2
i
ʸ
=
B
ji
ʸ
i
,ˆ
=
A
li
ˆ
i
,
(12)
=
1
=
1
where
j
=
1
,
2
,...,
N
+
2
.
Quantities
B
ji
and
A
li
are the components of the matrix formed by the Jacobi
polynomials which correspond to the discretized forms of the corresponding deriv-
atives (Villadsen and Michelsen
1978
). Convergence is accomplished by successive
calculations and the use of collocation points which are similar to the mesh points
or nodes in finite difference methods.
3 Results and Discussion
In this section, the numerical predictions for the PLA processing in a single screw
extruder are presented. The model was solved using the programming language
FORTRAN.
The proposed model is validated by comparing typical curves for the Newtonian
case published in the literature, in this case the Couette flow according to Tadmor and
Gogos (
2006
). Figure
2
shows only the contribution of the drag flow, a straight line
Fig. 2
Dimensionless
simple shear flow
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