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, ${\hbox{{\it Ra}}$ , 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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