Kinematic instabilities in two-layer eccentric annular flows, part 2: shear-thinning and yield-stress effects

The original publication is available at www.springerlink.comThis paper investigates the possibility of kinematic interfacial instabilities occurring during the industrial process of primary cementing of oil and gas wells. This process involves flows in narrow eccentric annuli that are modelled via a Hele-Shaw approach. The fluids present in primary cementing are strongly non-Newtonian, usually exhibiting shear-thinning behaviour and often with a yield stress. The study is a sequel to Moyers-Gonzalez and Frigaard (J Eng Math, DOI 10.1007/s10665-007-9178-y, 2007), in which the base analysis has been developed for the case of two Newtonian fluids. The occurrence of static mud channels in primary cementing has been known of since the 1960s, (see McLean et al. 1966; SPE 1488), and is a major cause of process failure. This phenomenon is quantified, which provides a simple semi-analytic expression for the maximal volume of residual fluid left behind in the annulus, f (static), and illustrate the dependency of f (static) on its five dimensionless parameters. It is shown that three of the four different types of static channel flows are linearly stable. Via dimensional analysis, it is shown that the base flows depend on a minimal set of eight dimensionless parameters and the stability problem depends on an additional two dimensionless parameters. This large dimensional parameter space precludes use of the full numerical solution to the stability problem as a predictive tool or for studying the various stability regimes. Instead a semi-analytical approach has been developed based on solution of the long-wavelength limit. This prediction of instability can be evaluated via simple quadrature from the base flow and is suitable for use in process optimisation

Yield stress effects on Rayleigh-Bénard convection

We examine the effects of a fluid yield stress on the classical Rayleigh-Bénard instability between heated parallel plates. The focus is on a qualitative characterization of these flows, by theoretical and computational means. In contrast to Newtonian fluids, we show that these flows are linearly stable at all Rayleigh numbers, Ra, although the usual linear modal stability analysis cannot be performed. Below the critical Rayleigh number for energy stability of a Newtonian fluid, RaE, the Bingham fluid is also globally asymptotically stable. Above RaE, we provide stability bounds that are conditional on Ra - RaE, as well as on the Bingham number Pr, the Prandtl number Ra, and the magnitude of the initial perturbation. The stability characteristics therefore differ considerably from those for a Newtonian fluid. A second important way in which the yield stress affects the flow is that when the flow is asymptotically stable, the velocity perturbation decays to zero in a finite time. We are able to provide estimates for the stopping time for the various types of stability. A consequence of the finite time decay is that the temperature perturbation decays on two distinctly different time scales, i.e. before/after natural convection stops. The two decay time scales are clearly observed in our computational results. We are also able to determine approximate marginal stability parameters via computation, when in the conditional stability regime, although computation is not ideal for this purpose. When just above the marginal stability limits, perturbations grow into a self-sustained cellular motion that appears to resemble closely the Newtonian secondary motion, i.e. Rayleigh-Bénard cells. When stable, however, the decaying flow pattern is distinctly different to that of a Newtonian perturbation. As t → ∞, a stable Newtonian perturbation decays exponentially and asymptotically resembles the least stable eigenfunction of the linearized problem. By contrast, as Ra approaches its stopping value, the Bingham fluid is characterized by growth of a slowly rotating (almost) unyielded core within each convection cell, with fully yielded fluid contained in a progressively narrow layer surrounding the core. Finally, preliminary analyses and remarks are made concerning extension of our results to inclined channels, stability of three-dimensional flows and the inclusion of residual stresses in the analysis. © 2006 Cambridge University Press