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, ${\hbox{{\it Ra}}$, although the usual linear modal stability analysis cannot be performed. Below the critical Rayleigh number for energy stability of a Newtonian fluid, ${\hbox{{\it Ra}}}_E$, the Bingham fluid is also globally asymptotically stable. Above ${\hbox{{\it Ra}}}_E$, we provide stability bounds that are conditional on ${\hbox{{\it Ra}}}\,{-}\,{\hbox{\it Ra}}_E$, as well as on the Bingham number $B$, the Prandtl number $\hbox{\it Pr}$, 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–Bénard cells. When stable, however, the decaying flow pattern is distinctly different to that of a Newtonian perturbation. As $t\rightarrow\infty$, 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.

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We study miscible displacements in long ducts in the dispersive limit of small $\varepsilon \hbox{\it Pe}$, where $\varepsilon \,{\ll}\, 1$ is the inverse aspect ratio and $\hbox{\it Pe}$ the Péclet number. We consider the class of generalized Newtonian fluids, with specified closure laws for the fluid properties of the concentration-dependent mixture. Regardless of viscosity ratio and the constitutive laws of the pure fluids, for sufficiently small $\varepsilon \hbox{\it Pe}$ these displacements are characterized by rapid cross-stream diffusion and slow streamwise dispersion, i.e. the concentration appears to be near-uniform across the duct and spreads slowly as it translates. Using the multiple-scales method we derive the leading-order asymptotic approximation to the average fluid concentration $\bar{c}_0$. We show that $\bar{c}_0$ evolves on the slow timescale $t \sim (\varepsilon \hbox{\it Pe})^{-1}$, and satisfies a nonlinear diffusion equation in a frame of reference moving with the mean speed of the flow. In the case that the two fluids have identical rheologies and the concentration represents a passive tracer, the diffusion equation is linear. For Newtonian fluids we recover the classical results of Taylor (l953), Aris (1956), and for power-law fluids those of Vartuli et al. (1995). In the case that the fluids differ and/or that mixing is non-passive, $\bar{c}_0$ satisfies a nonlinear diffusion equation in the moving frame of reference. Given a specific mixing/closure law for the rheological properties, we are able to compute the dispersive diffusivity $D_T(\bar{c}_0)$ and predict spreading along the channel. We show that $D_T(\bar{c}_0)$ can vary significantly with choice of mixing law and discuss why. This also opens the door to possibilities of controlling streamwise spreading by the rheological design of reactive mixtures, i.e. including chemical additives such that the rheology of the mixture behaves very differently to the rheology of either pure fluid. Computed examples illustrate the potential effects that might be achieved.

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