Technical Details
Since polyethylene is a polymer, it has a highly non-newtonian flow
pattern. This means that its viscosity varies with the shear stress that
is applied to it. Since polyethylene is a polymer, its molecules act as
bunches of rubber bands, tangling and stretching, giving it both viscous
and elastic characteristics.
In Chem E 475 we modeled polyethylene for
two cases. The first case was simplified by assuming that polyethylene
was a newtonian fluid. The second case removed the simplifications by
using the carreau equation for a non-newtonian fluid. This allowed the
fluid to be modeled in the regions where polyethylene behaves as a
newtonian fluid, a power law fluid, and the area in between those flow
regimes. In both cases solutions were obtained by programming Matlab
to solve for the solution.
For simplicity here, I will explain the solution of the newtonian
flow. Below are the differential equations governing the flow of
polyethylene as a newtonian fluid. Y = 0 is taken as the midpoint
between the two plates, where velocity(u) is maximum.
This is a Boundary Value Problem. The shooting method was used to
modifiy this to be an Initial value problem. This was done by stating
that velocity at point zero(the center of the flow region) is alpha. By
doing this we modify the equation so that we have an initial value problem
with four equations and four unknowns. Those equations are seen
below.
The discussion of the non-newtonian solutions is much more
complicated, and it uses the carreau method to estimate the viscosity
throughout the regime. It is beyond the scope of this web site.
