Thermal plumes in viscoplastic fluids: flow onset and development

Abstract

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 B cr , the fluid is unable to yield. For the flow studied, B cr ≈ 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 → B cr , the plume never starts.

Flow, onset and stability: Qualitative analysis of yield stress fluid flow in enclosures

“…In the situations explored 415 the difference comes from the (slow) evolution of the temperature field which changes the body force. Stability of the flows in §5 has not been considered here, but follows along similar lines as explored in [7,8]. Static steady states A C C E P T E D M A N U S C R I P T are generally stable but conditionally so.…”

“…Fluid is assumed to be yielded in asymptotically thin layers that separate the plugs, allowing Here we have seen delayed onset for B < B c,∞ , and for B ≥ B c,∞ there is no motion at all. A different scenario that also leads to delayed onset has been studied in [8] where the setup involves heating a tank from below. In regimes of 375 sufficiently low B and large R, intermittent pulsation of thermal plumes results.…”

“…Whether time-invariant or not, and whether coupled to the flow or independently prescribed, the body force determines how the momentum balance is affected by purely viscous stresses, yield stress and the body force, and hence allows for different dynamic behaviours; e.g. recent studies have illustrated the possibility of delayed flow 55 onset when the driving force is the buoyancy due to temperature differences in the domain [6,7,8]. In naturally convecting flows, the forcing is coupled with the flow field.…”

“…density changes are neglected everywhere in the governing equations except in evaluating the driving body force, F; see[18] for further discussion of this approximation.For background on use of energy stability/variational methods in this type of flow: Rayleigh-Bénard flows are considered in[19]; control of differentially 260 heated cavity flows via the yield stress are in[7]; onset of thermal plumes is in[8].Steady state flows (static and transient), flow stability (conditional and global) and computations of example transients are illustrated for each configuration. The analytical tools and methodology used here are those outlined in the previous sections and in the above references.…”

Buoyancy-driven destabilization of an immersed granular bed

“…the numerical implementation of adapted efficient algorithms, the coupling with heat transfer, and the regularization problem for vanishing shear rate, were identified and gave birth to a vast literature (see e.g. the recent review by Saramito & Wachs (2017), a comparison of differents modellings (Fraggedakis et al 2016), or the more specific simulation of a thermal plume (Karimfazli et al 2016), a situation which resembles ours). Another recent approach is more specifically dedicated to dense granular flows.…”