June 15, 2001 Processing Puzzle

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 P₁(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 P₁(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 P₁(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 P₁(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.