Yield stress effects on Rayleigh-Bénard convection
Journal title : Journal of Fluid Mechanics
Volume : 566
Pagination : 389-419
Publication date : 01/11/2006
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 characterisation 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, the 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 timescales, i.e. before/after natural convection stops. The two decay timescales are clearly observed in our computational results. We are also able to computationally determine approximate marginal stability parameters, 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 closely appears to resemble 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 the time tends to infinity, a stable Newtonian perturbation decays exponentially and asymptotically resembles the least stable eigenfunction of the linearised problem. By contrast, as the time approaches its stopping value, the Bingham fluid is characterised 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.