I.A. Frigaard

On the stability of Poiseuille flow of a Bingham fluid

The stability to linearized two-dimensional disturbances of plane Poiseuille flow of a Bingham fluid is considered. Bingham fluids exhibit a yield stress in addition to a plastic viscosity and this description is typically applied to drilling muds. A non-zero yield stress results in an additional parameter, a Bingham number, and it is found that the minimum Reynolds number for linear instability increases almost linearly with increasing Bingham number.

Static wall layers in the displacement of two visco-plastic fluids in a plane channel

In a plane-channel displacement flow of two visco-plastic fluids, it is possible for there to be a static residual layer of the displaced fluid left stuck to the walls of the channel. This phenomenon provides an idealized model for the formation of a wet micro-annulus , due to poor mud removal, during the primary cementing of an oil well. Using a lubrication approximation, it is shown that sufficient conditions for the non-existence of a static wall layer can be computed simply in terms of two dimensionless parameters: the Bingham number for the displacing fluid ( B 1 ) and the ratio of the yield stresses of the two fluids (ϕ Y ). When these conditions are not met, it is possible to compute the maximum possible static wall layer thickness h max , which depends on B 1 , ϕ Y and on a third dimensionless parameter ϕ B , a buoyancy to yield stress ratio. On computing displacements using the lubrication approximation, the interface is observed to asymptotically approach the maximum static layer thickness as t → ∞. Results from fully two-dimensional displacement computations are also presented. These indicate that the displacement front propagates at a steady speed along the channel, leaving behind a static layer which is significantly thinner than h max . Surprisingly, the computed static layer thickness is observed to decrease with a parametric increase in the dimensionless yield stress of the displaced fluid. To explain these results we analyse the streamline configuration close to a steadily advancing displacement front. We demonstrate heuristically that the local visco-plastic dissipation functional will be approximately minimized by a critical layer thickness at which the displaced fluid begins to recirculate ahead of the displacement front. Comparison of the critical recirculation limit with the static layer thickness computed from the fully transient model gives a very close agreement, suggesting that a form of energy minimization is responsible in this case for selecting the static layer thickness.

Nonlinear stability of Poiseuille flow of a Bingham fluid: theoretical results and comparison with phenomenological criteria

Abstract We present new results on the nonlinear stability of Bingham fluid Poiseuille flows in pipes and plane channels. These results show that the critical Reynolds number for transition, Re c , increases with Bingham number, B , at least as fast as Re c ∼ B 1/2 as B →∞. Estimates for the rate of increase are also provided. We compare these bounds and existing linear stability bounds with predictions from a series of phenomenological criteria for transition, as B →∞, concluding that only Hanks [AIChE J. 9 (1963) 306; 15 (1) (1963) 25] criteria can possibly be compatible with the theoretical criteria as B →∞. In the more practical range of application, 0≤ B ≤50, we show that there exists a large disparity between the different phenomenological criteria that have been proposed.

Settling of an isolated spherical particle in a yield stress shear thinning fluid

We visualize the flow induced by an isolated non-Brownian spherical particle settling in a shear thinning yield stress fluid using particle image velocimetry. With Re<1, we show a breaking of the fore-aft symmetry and relate this to the rheological properties of the fluid. We find that the shape of the yield surface approximates that of an ovoid spheroid with its major axis approximately five times greater than the radius of the particle. The disagreement of our experimental findings with previous numerical simulations is discussed.

Temperature surges in current-limiting circuit devices

This paper studies the problem of heat transfer in a thermistor, which is used as a switching device in electronic circuits. The temperature field is coupled to the current flow by ohmic heating in the device, and the problem is rendered highly nonlinear by a very rapid variation of electrical conductivity with temperature. Approximate methods based on high activation energy asymptotics are developed to describe the transient heat flow, which occurs when the circuit is switched on. In particular, it is found that a transient “surge” phenomenon (akin to thermal runaway, but self-saturating) occurs, and we conjecture that this phenomenon may be associated with cracking of thermistors, which sometimes occurs during operation.

Mud removal and cement placement during primary cementing of an oil well – Laminar non-Newtonian displacements in an eccentric annular Hele-Shaw cell

A two-dimensional model is derived of the displacement flows that occur during primary cementing of oil and gas wells. The displacement geometry is a long narrow eccentric annulus, between the casing and the rock formation. The model consists of a series of first-order convection equations for the fluid concentrations and a quasi-linear Poisson-type equation for the stream function. Coupling is through the velocity field and the concentration-dependent fluid properties.A range of computed results from this model is presented. One simulation illustrates how a channel of mud can be left behind on the narrow side of the annulus. Another shows that stable steady-state displacements can occur, although conditions under which this occurs are not yet understood. A third simulation captures some of the complexity that occurs in realistic cementing operations.

Buoyancy-dominated displacement flows in near-horizontal channels: the viscous limit

We consider the viscous limit of a plane channel miscible displacement flow of two generalized Newtonian fluids when buoyancy is significant. The channel is inclined close to horizontal. A lubrication/thin-film approximation is used to simplify the governing equations and a semi-analytical solution is found for the flux functions. We show that there are no steady travelling wave solutions to the interface propagation equation. At short times the diffusive effects of the interface slope are dominant and there is a flow reversal, relative to the mean flow. We are able to find a short-time similarity solution governing this initial counter-current flow. At longer times the solution behaviour can be predicted from the associated hyperbolic problem (where diffusive effects are set to zero). Each solution consists of a number N ≥ 1 of steadily propagating fronts of differing speeds, joined together by segments of interface that are stretched between the fronts. Diffusive effects are always present in the propagating fronts. We explore the effects of viscosity ratio, inclinations and other rheological properties on the front height and front velocity. Depending on the competition of viscosity, buoyancy and other rheological effects, it is possible to have single or multiple fronts. More efficient displacements are generally obtained with a more viscous displacing fluid and modest improvements may also be gained with slight positive inclination in the direction of the density difference. Fluids that are considerably shear-thinning may be displaced at high efficiencies by more viscous fluids. Generally, a yield stress in the displacing fluid increases the displacement efficiency and yield stress in the displaced fluid decreases the displacement efficiency, eventually leading to completely static residual wall layers of displaced fluid. The maximal layer thickness of these static layers can be directly computed from a one-dimensional momentum balance and indicates the thickness of static layer found at long times.

Conditions for static bubbles in viscoplastic fluids

We consider the slow motion of a gas bubble in a cylindrical column filled with a viscoplastic fluid, modeled here as a Herschel–Bulkley fluid. Because of the yield stress of the fluid, it is possible that a bubble will remain trapped in the fluid indefinitely. We adapt Prager’s two variational principles to our problem. From these variational principles we develop two general stopping conditions, i.e., for a given bubble we can calculate a critical Bingham number above which the bubble will not move. The first condition is derived by bounding the velocity field and the second condition by bounding the stress field. We illustrate these conditions by considering specific bubble shapes, e.g., axisymmetric bubbles. We also develop a condition for bubble motion.

Yield stress effects on Rayleigh–Bénard convection

enard insta- bility 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 B ,t he Prandtl number Pr, 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-B´ enard 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 t 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.

Super-stable parallel flows of multiple visco-plastic fluids

Abstract We consider the stability of a multi-layer plane Poiseuille flow of two Bingham fluids. It is shown that this two-fluid flow is frequently more stable than the equivalent flow of either fluid alone. This phenomenon of super-stability results only when the yield stress of the fluid next to the channel wall is larger than that of the fluid in the centre of the channel, which need not have a yield stress. Our result is in direct contrast to the stability of analogous flows of purely viscous generalised Newtonian fluids, for which short wavelength interfacial instabilities can be found at relatively low Reynolds numbers. The results imply the existence of parameter regimes where visco-plastic lubrication is possible, permitting transport of an inelastic generalised Newtonian fluid in the centre of a channel, lubricated at the walls by a visco-plastic fluid, travelling in a stable laminar flow at higher flow rates than would be possible for the single fluid alone.