Intrinsic Route to Melt Fracture in Polymer Extrusion: A Weakly Nonlinear Subcritical Instability of Viscoelastic Poiseuille Flow

As is well known, the extrusion rate of polymers from a cylindrical tube or slit (a “die”) is in practice limited by the appearance of “melt fracture” instabilities which give rise to unwanted distortions or even fracture of the extrudate. We present the results of a weakly nonlinear analysis which gives evidence for an intrinsic generic route to melt fracture via a weakly nonlinear subcritical instability of viscoelastic Poiseuille flow. This instability and the onset of associated melt fracture phenomena appear at a well-defined ratio of the elastic stresses to viscous stresses of the polymer solution.

Figure 1

Illustration of the surface irregularities that occur in the extrusion of a polymer fluid from a capillary tube (the wider structure at the top) in the absence of sharkskin or spurt-flow instabilities. For small speeds, the surface is smooth (a), but beyond some flow speed surface irregularities develop (b),(c). The wavelength is of the order of twice the diameter $d$ of the capillary. After [1].

Figure 2

Summary of our main result for the stability of viscoelastic Poiseuille flow in a cylinder in the zero Reynolds number limit. (a) Semiquantitative sketch of the bifurcation scenario. See text. (b) Our results for the threshold values of the amplitude beyond which the flow is unstable; these curves correspond to the dashed branch in (a). Dots indicate the critical value of the shear rate at the wall (normalized to the shear in the unperturbed case), and squares indicate the critical value of the relative shear stress perturbation, multiplied by 10.

Figure 3

Values of the critical shear stress amplitude ${\tau }_c(k)$, the ratio of the perturbation in the shear stress ${\tau }_{rz}$ (normalized to the value ${\tau }_{rz}^{\mathrm{u}\mathrm{n}\mathrm{p}}$ of the unperturbed state) at the wall above which the perturbation is nonlinearly unstable, as a function of the wave number $k$ of the perturbation.

Figure 4

Plot of the width of the $k$ band where the corresponding modes render the basic Poiseuille flow profile unstable.