60th Annual Report on Research 2015

Major goals of the project:

A broad range of industrially relevant fluids, including powders, suspensions, oil sands, and foams, exhibit shear banding: in flow, discontinuities in the velocity gradient develop, splitting the material into a high shear rate band and a low shear rate band.  The processing of shear banding materials is relevant to petroleum production from weakly consolidated reservoirs where shear bands play a critical role in wellbore collapse.

Recent theoretical and experimental work suggests that both interfacial and purely elastic instabilities may occur in these systems. Inertial instabilities are also likely, and these systems thus promise incredibly rich dynamics. In addition, optimal processing of shear banding materials requires an understanding of the scaling of instabilities with these driving forces (interfacial, elastic, and inertial), how these modes of instability interact, and how the boundary conditions affect stability. Wormlike micellar solutions provide a convenient and versatile model system for studies of shear banding systems, and are being used in the present study to gain a better understanding of flow transitions and instabilities in shear banding systems. 

We are using Taylor-Couette (TC) flow, that is, the flow between concentric, rotating cylinders, as the model flow for these experimental studies.  Using a custom-built Taylor-Couette apparatus, which allows independent, computer-controlled rotation of the two cylinders and the establishment of programmable ramping protocols, we are working towards (a) isolating purely elastic and interfacial transitions from inertial ones through the use of outer-cylinder rotation, (b) using counter-rotation of the cylinders to introduce a nodal surface and probe the effects of no-slip versus free boundary conditions for various instability modes, and (c) using a combination of counter-rotation, solution concentration, temperature, and inner cylinder radii to systematically explore the competition between interfacial, inertial, and elastic instabilities and understand how each mode scales with geometric (curvature) and kinematic variables.   

Ultimately, understanding the role of the destabilizing forces, the scaling of the instabilities with curvature, shear rate, normal stresses, etc., understanding the role of boundary conditions, and how the three modes of instability (interfacial, elastic, and inertial) interact, will improve the modeling of and impact processing of complex, shear banding materials. 

Accomplishments to date:   

During this second reporting period, the shear banding system CTAB/NaNO₃ was studied in the Taylor-Couette cell.  We observed the bulk elastic instability in shear banding wormlike micelles and have explored the effect of a variety of parameters on this instability.  Sample visualization results from an experiment in which the inner cylinder rotation speed W is slowly increased are shown in Figure 1 (top).  Figures 1a-1d show flow visualization images of the z-q plane as the purely azimuthal base flow (1a) is replaced by a series of vortex flows of varying wavenumber and complexity.  By taking a single line of pixels from each image and stacking them, a space-time (or, equivalently, z-W) plot is formed, as shown in figure 1(lower half).  For purposes of validation, the wavelengths of the interfacial oscillations were measured and compared with the literature results as a function of shear rate.  We find quantitative agreement with the literature results (Fardin et al., Phys. Rev. Lett., 2009) (Fig.2).

In order to better understand the time scales present in this system, startup of steady shear experiments were performed in the TC cell and four stages for interface formation and evolution were observed, as noted by others (Fardin and Lerouge, EPJE, 2012).  For the interface travel stage in particular, the flat, diffuse interface becomes sharp and moves slightly toward the inner cylinder. The radial position of the interface during the interface travel stage can be related to the stress diffusion coefficient, D, that appears in the diffusive Johnson-Segalman model (dJS) of wormlike micellar solutions.  The dJS model has been used by Fielding and others to describe the dynamics of this system in simulations.

  We have measured the stress diffusion coefficient at different conditions (temperature, shear rate, geometry) using both this interface visualization method and a second method based on superposition rheology.  In the first method, the radial position of the interface is measured over time and an exponential function is fit to the position during the interface travel stage to obtain the diffusion coefficient (Fig.3). We find that the stress diffusion coefficient is independent of the Weissenberg number Wi and increases with increasing temperature. Weissenberg number is defined as the shear rate multiplied by the relaxation time of the shear banding wormlike micelle solution, determined from linear viscoelastic measurements.

We also used superposition rheology to measure the stress diffusion coefficient for the CTAB/NaNO₃ system.  In this method a small amplitude oscillatory shear flow is combined with a steady shear flow and the response of the system is evaluated by measuring the complex viscosity.  For Weissenberg numbers in the shear banding regime, a two fluid model is used to predict the velocity of the interface during the travel stage and therefore, the stress diffusion coefficient [Lettinga et al., J. Rheol, 2007]. We again find that the stress diffusion coefficient increases with increasing temperature in agreement with the interface visualization results.  When results from these two methods are combined, we observe that the diffusion coefficient monotonically increases with increasing gap in the Taylor-Couette cell (Fig. 4).   This is an unexpected result, but explains some inconsistencies in the literature for values of the stress diffusion coefficient.  The strong dependence of the stress diffusion coefficient D on the length scale of the geometry used in the measurements has also been demonstrated in a second system, CPCl/NaSal (data not shown).  We are presently working to understand the origin of this dependence.

In addition, we are also working towards examining the dynamics of wormlike micellar systems as the Elasticity number, El= Wi/Re , approaches 1 and the case of El <1, where both inertia and elastic effects are important.