Thermal plumes in viscoplastic fluids: flow onset and development

The purely conductive state in configurations such as the Rayleigh–Bénard one is linearly stable for yield stress fluids at all Rayleigh numbers, $Ra$ . However, on changing to localized heater configurations the static background state exists only if the yield stress is sufficiently large. Otherwise, thermal plumes may be induced in a stationary viscoplastic fluid layer, as illustrated in the recent experimental study of Davaille et al. ( J. Non-Newtonian Fluid Mech. , vol. 193, 2013, 144–153). Here, we study an analogous problem both analytically and computationally, from the perspective of an ideal yield stress fluid (Bingham fluid) that is initially stationary in a locally heated rectangular tank. We show that for a non-zero yield stress the onset of flow waits for a start time $t_{s}$ that increases with the dimensionless ratio of yield stress to buoyancy stress, denoted $B$ . We provide a precise mathematical definition of $t_{s}$ and approximately evaluate this for different values of $B$ , using both computational and semianalytical methods. For sufficiently large $B\geqslant B_{cr}$ , the fluid is unable to yield. For the flow studied, $B_{cr}\approx 0.00307$ . The critical value $B_{cr}$ and the start time $t_{s}$ , for $B<B_{cr}$ , are wholly independent of $Ra$ and $Pr$ . For $B<B_{cr}$ , yielding starts at $t=t_{s}$ . The flow develops into either a weakly or a strongly convective flow. In the former case the passage to a steady state is relatively smooth and monotone, resulting eventually in a steady convective plume above the heater, rising and impinging on the upper wall, then recirculating steadily around the tank. With strongly convecting flows, for progressively larger $Ra$ we observe an increasing number of distinct plume heads and a tendency for plumes to develop as short-lived pulses. Over a certain range of $(Ra,B)$ the flow becomes temporarily frozen between two consecutive pulses. Such characteristics are distinctly reminiscent of the experimental work of Davaille et al. ( J. Non-Newtonian Fluid Mech. , vol. 193, 2013, 144–153). The yield stress plays a multifaceted role here as it affects plume temperature, size and velocity through different mechanisms. On the one hand, increasing $B$ tends to increase the maximum temperature of the plume heads. On the other hand, for larger $B\rightarrow B_{cr}$ , the plume never starts.

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