Viscoelastic instabilities are of practical importance, and are the subject of growing interest. Reviewed here are the fresh developments as well as earlier work in this area, organized into the following categories: instabilities in Taylor-Couette flows, instabilities in cone-and-plate and plate-and-plate flows, instabilities in parallel shear flows, extrudate distortions and fracture, instabilities in shear flows with interfaces, instabilities in extensional flows, instabilities in multidimensional flows, and thermohydrodynamic instabilities. Emphasized in the review are comparisons between theory and experiment and suggested directions for future work.

Instabilities in viscoelastic flows

Viscosity regularisation methods are probably the most popular current method for computing visco-plastic fluid flows. They are however generally used in an ad hoc manner. Here we examine convergence of regularised solutions to those of the corresponding exact models, in both mathematical and physical senses. Mathematically, the aim is to give practical guidance as to the order of error that one might expect for different regularisations and for different types of flow. Our theoretical results are illustrated with a number of computed example flows showing the orders of error predicted. Physically, the question is whether or not the regularised solutions behave in the same way as the exact solutions, qualitatively as well as quantitatively. We show that there are flows for which regularisation methods will generate their maximum errors, e.g. lubrication-type flows. In this context, we also consider the effects of regularisation on problems of hydrodynamic stability. For broad classes of problems, stability characteristics of the flow are incorrectly predicted by the use of viscosity regularisation methods.

On the usage of viscosity regularisation methods for visco-plastic fluid flow computation

The linear stability of a two-layer Couette flow of upper convected Maxwell liquids is considered. The fluids have different densities, viscosities, and elasticities, with surface tension at the interface. At low speeds, the interfacial mode may become unstable, while other modes stay stable. The shortwave asymptotics of the interfacial mode is analyzed. It is found that an elasticity difference can stabilize or destabilize the flow even in the absence of a viscosity difference. As the viscosity difference increases, the range of elasticities for which there is shortwave stability widens. A linearly stable arrangement can be achieved by placing the less viscous fluid in a thin layer to stabilize longwaves and using elasticities to stabilize shortwaves. Such an arrangement can be stable even when the density stratification is adverse.

Stability of the interface in two-layer couette flow of upper convected maxwell liquids

The interfacial stability of coextruded polypropylene (PP) and high-density polyethylene (HDPE) has been examined experimentally in a slit-die geometry. Our experimental apparatus makes use of a novel system for introducing temporally regular disturbances with a controllable amplitude and frequency. The growth or decay of the disturbance can be observed via optical windows mounted in the test die and characterized with the aid of digital image-processing techniques. This approach has allowed us to construct experimentally determined stability and growth-rate plots as a function of disturbance wavenumber and layer-depth ratio. Our experimental results are compared to theoretical results based on the linear stability analysis using a modified Oldroyd-B equation. We have found this model to be quantitatively correct in its prediction of the critical layer-depth ratio and the existence of a maximum spatial growth rate in the vicinity of a dimensionless wavenumber of one. The theoretically predicted growth rates agree with our experimental data at low wavenumbers but fail to agree at higher wavenumbers. Overall, our findings provide evidence that in order to achieve quantitative agreement between theoretical and experimental results, a constitutive model should be utilized that accounts for the shear-rate dependence of first and second normal stresses and a spectrum of relaxation times.

An experimental investigation of interfacial instabilities in multilayer flow of viscoelastic fluids: Part I. Incompatible polymer systems

The stability of the interface between a gel of thickness HR and a Newtonian fluid of thickness R subjected to a linear shear flow is studied in the limit where inertial effects are negligible. The shear stress for the gel contains an elastic part that depends on the local displacement field and a viscous component that depends on the velocity field. The shear flow at the surface tends to destabilize the surface fluctuations, and the critical strain rate γc, which is the minimum strain rate required for unstable fluctuations, is determined as a function of the dimensionless quantities H, ηr = (ηg/ηf), and T = (Γ/ER). Here ηg and ηf are the gel and fluid viscosities, E is the gel elasticity, Γ is the surface tension of the gel — fluid interface and the strain rate γ is scaled by (E/ηf). In the limit H ↦∞, we find that γc decreases proportional to H-1 independent of ηr and T. But at finite H, γc is strongly dependent on ηr and T. For ηr ≥1, the interface is stable for all values of the strain rate for $H \sqrt{\eta_{\rm r}}$. For ηr = 1 and H ↦1, we find that γc ∝(H - 1)1/2 for T=0 and γc ∝(H - 1)3/4T1/4 for T ≠0. For ηr > 1, the analysis indicates that $\gamma_{\rm c} \propto(H-\sqrt{\eta_{\rm r}})^{-1}$ independent of T for $H \to \sqrt{\eta_{\rm r}}$. For ηr < 1, the onset of instability depends strongly on the parameter T. For T = 0, the critical strain rate is finite in the limit H ↦0, while for T ≠0 the critical strain rate diverges at a finite value of H. This minimum H decreases proportional to ηr for large T. The instability is caused by the energy transfer from the mean flow to the fluctuations due to the work done by the mean flow at the interface.

https://hal.archives-ouvertes.fr/jpa-00248015/document

Flow induced instability of the interface between a fluid and a gel at low Reynolds number

Consideration is given to the stability of the interface between two Oldroyd liquids with shear-dependent viscosities, flowing in distinct layers while undergoing plane Couette flow. Results are presented as regions of stability in the plane determined by the logarithms of the viscosity and depth ratios. The work of previous authors for two Newtonian, power-law and constant-viscosity Oldroyd liquids is revealingly presented in a similar fashion. It is found that the dependence of the viscosities on shear-rate can drastically affect the regions of interfacial stability in a way over and above that due to just a change in the effective viscosity ratio. It is also found that for the Oldroyd liquids this viscosity variation affects the stability when it is present in the less-viscous layer.

The stability of two stratified non-newtonian liquids in couette flow

Consideration is given to the stability of the flow of two ”power-law“ liquids between two infinite parallel planes when one of the planes moves with constant velocity in its own plane. It is found that the ratios of the power-law parameters for each layer have a dramatic effect and can be chosen to destabilize the flow.

The stability of two stratified ”power-law“ liquids in couette flow

The problem of the drainage of an elastic liquid with a shear dependent viscosity from a vertical surface is solved numerically by using a time-marching finite difference procedure. Shear-thinning is found to have a damping effect on any elastic oscillations during start-up in all but very thin films, and is confirmed as the dominant non-Newtonian property at later times.

The numerical simulation of an oldroyd liquid draining down a vertical surface

The oscillatory drainage of liquids simulated by the Oldroyd four-constant model is investigated using a numerical time-marching technique. The drainage of constant-viscosity elastic liquids is not significantly affected by the oscillatory motion whether applied parallel or orthogonal to the direction of the main flow. During the oscillatory drainage of shear-dependent elastic liquids, there is evidence of drainage enhancement and elastic recoil on applying the vibration parallel to the direction of the main flow. The recoil has the effect of thickening parts of the liquid film. On applying the vibration orthogonal to the direction of the main flow, for such liquids, enhancement is obtained for all film thicknesses.

The effect of oscillation on the drainage of an elastico-viscous liquid

In this, the first of a series of papers on the drainage of a thin film of non-Newtonian liquid down a vertical surface, a parametric solution is obtained for liquids with a shear-rate dependent viscosity and a prescribed initial profile. Experimental measurement of the mass of fluid remaining on a surface as a function of drainage time agrees with the theory.

N.D. Waters

Papers

The stability of two stratified non-newtonian liquids in couette flow

The stability of two stratified ”power-law“ liquids in couette flow

The numerical simulation of an oldroyd liquid draining down a vertical surface

The effect of oscillation on the drainage of an elastico-viscous liquid

Draining thin films—Part 1