Instability of channel flows of elastic liquids having continuously stratified properties

Abstract

This paper investigates inertialess channel flow of elastic liquids having continuously stratified constitutive properties. We find that an Oldroyd-B fluid having a sufficiently rapid normal stress variation shows instability. The mechanism is the same as for the two-fluid co-extrusion instability that arises when elasticity varies discontinuously. We find, using numerical and asymptotic methods, that this mechanism is opposed by convective effects, so that as the scale over which the elastic properties vary is increased, the growth rate is reduced, and finally disappears. A physical explanation for the stabilisation is given.Regarding an Oldroyd-B fluid as a suspension of Hookean dumbbells, we show that a sufficiently steep variation in dumbbell concentration (with attendant rapid changes in both viscosity and elasticity) will provide an instability of the same kind.Finally we show that Lagrangian convection of material properties (either polymer concentration or relaxation time) is crucial to the instability mechanism. A White±Metzner fluid having identical velocity and stress profiles in a channel flow is found to be stable. The implications for extrudate distortion, and constitutive modelling are briefly discussed. #

Direct numerical simulations of two-layer viscosity-stratified flow

“…Broadly speaking we have replaced the sharp interface separating the phases by a region of sharp variation in properties, which has a finite thickness, approximately 4Dx (Sarkar and Schowalter, 2001a). Wilson and Rallison (1999) found for channel flow of elastic liquids that as the thickness of the layer over which the elastic properties vary is increased, the instability mechanism is countered by convective effects and the growth rate is reduced. A similar effect could apply in our simulations as well.…”

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

“…2 For a varicose mode however, the perturbation to the shear rate, Á , proportional to 2 HH k 2 2, vanishes where y 0, 2 It would, of course, be possible to treat sinuous modes by applying a numerical patch near y 0, but the results for 3 would then depend unsatisfactorily upon the size of the patch. Alternatively, as noted in Section 1, a Bird±Carreau regularisation of the viscosity could be employed.…”

“…In an earlier paper [2] we discussed pure elastic (i.e. inertialess) co-extrusion instabilities of a class of Oldroyd-B fluids in channel flow.…”

“…We demonstrated that two ingredients are required to sustain such an instability: first, there must be a sufficiently steep gradient in normal stress within the flow, and second, this gradient must result from a rapid change in a material property that is convected by the fluid. The mechanism for this instability is elucidated in [2,3]; it relies on the tilting of base flow streamlines to expose the rapid N 1 variation, but by itself this is insufficient. For a particular choice of flow parameters a WM fluid having rapid variation of relaxation time with shear rate is shown in [2] to remain stable, while an Oldroyd-B fluid with the same base state velocity and stress profiles is unstable.…”

Instability of a free-shear layer in the vicinity of a viscosity-stratified layer

“…In these and several other studies, a viscosity jump was prescribed across the interface. Wilson & Rallison (1999) asked what happens when viscosity is stratified across a thin layer rather than as a step function. They found that a smearing out of the interface has a large stabilizing effect.…”