Short wave instability of co-extruded elastic liquids with matched viscosities

Abstract

The stability of channel flow of coextruded elastic liquids having matched viscosities but a jump in elastic properties is studied. Inertia and surface tension are neglected. A short wave disturbance is found, confined near the interface, whose growth rate is independent of wavelength. For dilute Oldroyd-B fluids this disturbance is unstable for any non-zero jump in normal stresses, and has a maximum growth rate for intermediate levels of elasticity in the two fluids. When one or other fluid is highly elastic the growth rate falls. In the concentrated limit (a UCM fluid), the disturbance is unstable only for a finite range of normal stress jumps, and is restabilised if one fluid is much more elastic than the other. Because the short wave disturbance is localised, the results apply for any steadily sheared interface across which the normal stress jumps. The results are confirmed for moderate parameter values by means of a full numerical solution for a three-layer planar flow.

Instability of channel flow of a shear-thinning White–Metzner fluid

“…We have seen too that if n is very small, several branches of stable and unstable modes exist. This behaviour is very different from the co-extrusion instability [12], and we seek here to identify the small-n mechanism for a mode having growth rate of order unity. A full explanation would involve obtaining a solution of the problem in the singular n 3 0 limit by matched asymptotic expansions; we have not succeeded in obtaining such a solution, nevertheless, some key features may be identified.…”

Elastic instability in stratified core annular flow

“…The almost ubiquitous ingredient of such an elastic instability is the curvature of streamlines: polymers that have been extended along curved streamlines are taken by fluctuations across shear rate gradient in the unperturbed state which, in turn, couples the hoop stresses acting along the curved streamlines to the radial and axial flows and amplifies the perturbation [2,3]. Flat interfaces between two fluids with different viscoelastic properties can also become unstable [4][5][6] due to the normal stress imbalance across the interface. These instabilities often occur in coextrusion where different polymers are melted in separate screw extruders and then flown simultaneously in the extrusion nozzle.…”

Instability of channel flows of elastic liquids having continuously stratified properties

“…Chen and Joseph [11] suggest that the high stresses near the wall can cause polymer molecules to migrate inwards, creating a depleted wall region. They then propose that the interface dividing this region from the bulk of the fluid would be subject to a short-wave interfacial instability of the type discussed in [7,8]. Our analysis shows that a thin outer layer with lower viscosity than the bulk will be stable to long waves in the absence of inertia, and that a small amount of`blurring' of the interface will stabilise short waves.…”

“…We note that the disturbance does not convect with the interface, as it does for the matched-viscosity case of [7] and Section 3. At this leading order, the fluids are Newtonian so we may check this result by comparing with the flow (in the same geometry) of two Newtonian fluids with different viscosities 3 The growth rate, which appears at order k 2 , is…”

“…These conditions become continuity at y of 2, 2 H U H , ' 12 ÀikAE 11 and ' 22 . It is the appearance of the U H term here, involving the discontinuity of U H , that differs from the flows considered in [2,3,7,8].…”