Abstract
The onset of convection in a porous layer heated from below is considered, and we determine how the presence of two solid but heat-conducting bounding plates of finite thickness alters the manner in which convection ensues. Heat is generated by the lower plate (with an insulating lower boundary), but the upper one is passive with a fixed upper boundary temperature. It is shown that this composite layer may mimic in turn one of the three different types of classical single-layer onset problems which are well-known in the literature. The type which is selected (or indeed whether it corresponds to a transitional case) depends quite critically on the precise values of the relative thickness of the solid layers and their conductivity ratio. It is also shown that care needs to be taken over declaring that the solid plates are thin: extreme values of the conductivity ratio can yield a stability criterion which appears to be different from that suggested by the imposed boundary conditions.
Highlights
The modelling of convective instability in porous layers has a very strong heritage
The foundational works are by Lapwood (1948) and Horton and Rogers (1945), and these determined the criterion for the onset of convective instability in a horizontal porous layer which is heated from below
We have conducted a linear stability analysis to determine the effect on stability criterion of having two solid but identical bounding sublayers above and below a saturated porous layer where the lower sublayer generates heat internally
Summary
The modelling of convective instability in porous layers has a very strong heritage. Mojtabi and Rees (2011) considered such a three-layer system subject to constant heat flux boundary conditions, and this was motivated by the need to model an experimental system where solid boundaries do not have zero thickness and idealised boundary conditions cannot be applied in practice. It is shown in the Appendix that this may be mapped onto a system where the lower solid sublayer is subject to a constant heat flux but without heat generation within the sublayer While this may seem to be a simple intermediate case between those of Mojtabi and Rees (2011) and Rees and Mojtabi (2011), three different limiting cases for the critical parameters are found in different asymptotic regions of parameter space, and this is a novel finding
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