Inflation of a Cylindrical Parison
Perspective Meter:
Blow molding is a shaping operation commonly used with amorphous
polymers and rapidly crystallizing semicrystalline polymers.
A hot cylindrical parison is inflated against a cold mold in order
to form a hollow object such as a bottle. In the central
region of the object, the process can often be accurately
approximated as the inflation of a cylindrical shell.
Consider the inflation of a cylindrical shell of polymer melt,
as illustrated in the figure below. The cylindrical shell
is initially of radius R_{o} and wall thickness
H_{o}. Starting at time zero, it is inflated by the application
of air pressure P_{1}(t) at the center of the
cylindrical "bubble." Subsequently, the radius R(t) increases as a function
of time and the wall thickness H(t) decreases.
Throughout the process, assume that the cylindrical shape is
maintained and the length of the cylinder is constant at L. At all times
the radius is much larger than the wall thickness, R(t)>>H(t);
in addition, the length of the cylinder is much larger than
the radius, L>>R(t). The pressure on the outside of the cylindrical
shell is constant at P_{atm}. The polymer
contacts the mold wall at the final radius R_{f}.
The density of the polymer melt is constant.
As the cylindrical shell is inflated, the pressure P_{1}(t) is
adjusted so that the inflation velocity dR(t)/dt is constant
throughout at a value of V_{o}.
What is R(t)?
What is H(t)?
What is the velocity distribution within the expanding cylindrical
shell of polymer?
What type of flow is this?
What is the deformation rate as a function of time?
Assume that the polymer melt is a Newtonian fluid of constant
viscosity.
What is the blowing pressure P_{1}(t) required to
produce the specified deformation?
{Hint: inertial effects and interfacial tension are negligible.}
Suppose instead that the polymer is a shear thinning
fluid or a shear thickening fluid.
How would you expect P_{1}(t) for these types of
material to compare to that for a Newtonian fluid?
Check your work carefully before taking a look at
the answer.
