Session E36: Instability: Viscoelastic Effects

Absolute Instability of a Variable Visccosity Jet

The linear stability of an incompressible jet issuing into an ambient of higher viscosity is examined. Motivated by experimental results, only axisymmetric disturbances are considered. It is shown that for a constant-density jet with a prescribed velocity profile and Reynolds number, there exists a critical viscosity ratio between the jet centerline and far-field values, at which the jet transitions from convective to absolute instability. The axial disturbance corresponding to absolute instability is similar to the ``column'' mode found in the absolute instability of low-density jets. The boundary between absolute and convective instability is tracked as a function of viscosity ratio, Reynolds number, jet shear layer thickness, and density ratio. Shadowgraph and schlieren visualization is performed for a hot water jets issuing into a cold medium, over an experimental parameter range suggested by linear theory. A sudden increase in the jet spreading angle is interpreted as the onset of a global mode; hot film anemometry measurements corroborate the hypothesis. Global modes are observed for sufficiently large viscosity ratios. The onset and disappearance of global modes qualitatively matches the predictions of linear theory.   

Instability of Newtonian and Viscoelastic Submerged Jets

We study the behavior of high speed submerged liquid jets using a novel experimental setup equipped with strobe imaging flow visualization. Visualizations for Newtonian liquids at high Reynolds numbers (Re $\sim$ 150) show that for large wave-numbers a varicose mode dominate and the nonlinear growth of instability leads to the appearance of axisymmetric bags that roll up and encapsulate the central jet. At lower wave-numbers the varicose mode initially starts to grow close to the nozzle but is overwhelmed by the sinuous mode as the jet moves downstream. Due to the difference in the wave speeds for these two different modes of instability, the varicose waves slowly pile up into continuously growing sinuous waves leading to some unique concertina or chevron like morphologies. Tests with different viscoelastic model solutions show that by increasing the fluid elasticity the disturbance growth of both modes can be substantially inhibited due to streamline tension. These observations are the first experimental validation of theoretical predictions obtained from an elastic Rayleigh equation [1]. \\[4pt] [1] J. M. Rallison and E. J. Hinch, ``Instability of a high-speed submerged elastic jet,'' J. Fluid Mech., vol. 288, pp. 311-324, 1995.   

Interfacial instability of wormlike micellar solutions sheared in a Taylor-Couette cell

We report experiments on wormlike micellar solutions sheared in a custom-made Taylor-Couette (TC) cell. The computer controlled TC cell allows us to rotate both cylinders independently. Wormlike micellar solutions containing water, CTAB, and NaNo3 with different compositions are highly elastic and exhibit shear banding. We visualized the flow field in the $\theta$-z as well as r-z planes, using multiple cameras. When subject to low shear rates, the flow is stable and azimuthal, but becomes unstable above a certain threshold shear rate. This shear rate coincides with the onset of shear banding. Visualizing the $\theta$-z plane shows that this instability is characterized by stationary bands equally spaced in the z direction. Increasing the shear rate results to larger wave lengths. Above a critical shear rate, experiments reveal a chaotic behavior reminiscent of elastic turbulence. We also studied the effect of ramp speed on the onset of instability and report an acceleration below which the critical Weissenberg number for onset of instability is unaffected. Moreover, visualizations in the r-z direction reveals that the interface between the two bands undulates with shear bands evolving towards the outer cylinder regardless of which cylinder is rotating.   

Non-Modal Stability Analysis of High Strain-Rate Plastic Shear Flow

High-speed oblique impact of two metal plates results in the development of an intense shear region at their interface, which leads to interfacial profile distortion and interatomic bonding. If the relative velocity is sufficient, a wavy pattern with a well-defined wavelength and amplitude is observed. The wavy structure has similarities to shear instabilities observed in fluid dynamics and predicted by hydrodynamic stability theories. However, since the impact is a short-time transient dynamical phenomenon, non-modal stability analysis presumably is more relevant than conventional eigenvalue analysis. Here, a non-modal shear flow stability analysis of a perfectly plastic material is performed to investigate the transient growth of disturbances and to assess if a connection exists with the corresponding predictions obtained from modal analysis.   

Instabilities around a rotating ellipsoid in a stratified fluid

Geosismic observations have revealed the stacking of horizontal layers of water with different densities in the ocean, particularly above and beneath lens-shaped eddies. We present a simplified model together with an experimental setup to reproduce and identify the mechanism responsible for this layering phenomenon: we consider the stably stratified flow around a rotating, solid ellipsoid. Experimentally, a flat oblate rotating ellipsoid reproduces faithfully the boundary condition of an oceanic eddy, whereas the case of a rotating sphere provides an analytically tractable base flow, suitable for a numerical linear analysis. Two instabilities are witnessed experimentally and numerically. The first instability is the classical, inviscid, strato-inertial instability that tends to develop at the equator of the ellipsoid independently of the value of the Schmidt number. The second instability is localised in the vicinity of the poles and appears only if the Schmidt number differs from one. Hence, this instability is reminiscent of the double-diffusive McIntyre instability, a valuable candidate to explain layering in oceanic eddies.   

Analysis of the onset of elastic instabilities in a homogenous stagnation point flow using dilute polymer solutions

We compare numerical and experimental results for viscoelastic flows in the optimized cross-slot extensional rheometer - OSCER (Haward et al., Phys Rev Lett 109:128301, 2012) up to the onset of elastically-driven flow instabilities. Model polymer solutions with almost constant shear viscosity are used in the experiments, and the FENE-CR constitutive model is used in the 2D numerical simulations together with an in-house finite-volume viscoelastic flow solver. We match the model parameters to the rheology of the fluids used in the experiments, and the simulations are conducted for a wide range of flow rates, ranging from Newtonian-like flow at low Weissenberg numbers (Wi) up to the onset of time-dependent elastic instabilities at high Wi. We test the applicability of a dimensionless stability criterion (McKinley et al., J Non-Newt Fluid Mech 67:19-47, 1996) for predicting the onset of flow instability for both the experimental and computational data sets, using a spatially-resolved procedure to locally compute the stability criterion in the vicinity of the stagnation point. By evaluating this dimensionless criterion on a pointwise basis we are able to clearly distinguish the OSCER flow geometry from the archetypal cross-slot geometry.