Transitions and bistability of thermal convection between differentially rotating spherical shells
File(s)
Author(s)
Mannix, Paul
Type
Thesis
Abstract
This thesis investigates the motion of heated fluid between two differentially rotating spherical shells. Although the motion is simple for weak heating and differential rotation, it transitions to more complicated states as both increase. Heating a quiescent fluid from below gives rise to cellular convective motion as the temperature gradient becomes sufficiently steep. Typically this transition increases heat transfer. Differentially rotating spherical shells also generate a state of cellular motion, which in this case transports angular momentum. When both effects are present, it is often assumed the fluid chooses an optimal configuration which maximises the transfer of angular momentum and heat. Depending on how the equilibrium is reached however, this maximisation may not always be achieved, with two different stable equilibria often found to co-exist for the same heating and rotation strengths. We want to understand why the fluid motion in a spherical shell is bistable, and how this scenario might arise.
We find that bistability in the non-rotating problem depends largely on the shell separation, and when differential rotation is introduced, on the relative strength of heating to rotation. In both cases the strength of thermal plume boundary layers appears to play a key role. We find that for differentially rotating problem, the transition between solutions can occur either due to the destabilisation of an equatorial plume by buoyancy forces, or alternatively a polar plume by Ekman pumping. Our results demonstrate that although bistability in this system cannot be simply explained by the flow maximising its torque or heat transfer, the polar and equatorial regions are of particular significance. As bistability is a ubiquitous feature of many fluid systems, particularly those for which the transport of heat and angular momentum is important, it's hoped that this study can provide valuable insights and help towards the understanding of such behaviour.
We find that bistability in the non-rotating problem depends largely on the shell separation, and when differential rotation is introduced, on the relative strength of heating to rotation. In both cases the strength of thermal plume boundary layers appears to play a key role. We find that for differentially rotating problem, the transition between solutions can occur either due to the destabilisation of an equatorial plume by buoyancy forces, or alternatively a polar plume by Ekman pumping. Our results demonstrate that although bistability in this system cannot be simply explained by the flow maximising its torque or heat transfer, the polar and equatorial regions are of particular significance. As bistability is a ubiquitous feature of many fluid systems, particularly those for which the transport of heat and angular momentum is important, it's hoped that this study can provide valuable insights and help towards the understanding of such behaviour.
Version
Open Access
Date Issued
2019-11
Date Awarded
2020-03
Copyright Statement
Creative Commons Attribution NonCommercial Licence
Advisor
Mestel, Andrew Jonathan
Publisher Department
Department of Mathematics
Publisher Institution
Imperial College London
Qualification Level
Doctoral
Qualification Name
Doctor of Philosophy (PhD)