Thermosolutal convection in an evolving soluble porous medium

We describe a mathematical model of double-diffusive (thermosolutal) convection in a saturated porous layer, when the solubility of the solute depends on temperature, and the porosity and permeability of the porous medium evolve through dissolution and precipitation. We present the results of linear and weakly nonlinear stability analyses and explore the longer-term development of the system numerically. When the solutal concentration gradient is destabilising, the dynamics are somewhat similar to those previously found for single-species convection [Ritchie & Pritch ard, J. Fluid Mech. 673: 286–317, 2011], including the occurrence of subcritical instabilities driven by a reaction– diffusion mechanism. However, when the solutal concentration gradient is stabilising and the thermal gradient is destabilising, novel dynamics emerge. These include a vertical segregation of circulation cells and porosity perturbations near the onset of convection, and over longer timescales the formation of a low-permeability region in the middle of the layer, pierced by occasional high-permeability channels. Under these conditions, convection may die away to nearly zero for extended periods before resuming vigorously in localised regions at later times

Weak-anchoring effects in a thin pinned ridge of nematic liquid crystal

A theoretical investigation of weak-anchoring effects in a thin two-dimensional pinned static ridge of nematic liquid crystal resting on a flat solid substrate in an atmosphere of passive gas is performed. Specifically, we solve a reduced version of the general system of governing equations recently derived by Cousins et al. [Proc. Roy. Soc. A}, 478(2259):20210849, 2022] valid for a symmetric thin ridge under the one-constant approximation of the Frank--Oseen bulk elastic energy with pinned contact lines to determine the shape of the ridge and the behaviour of the director within it. Numerical investigations covering a wide range of parameter values indicate that the energetically-preferred solutions can be classified in terms of the Jenkins--Barratt--Barbero--Barberi critical thickness into five qualitatively different types of solution. In particular, the theoretical results suggest that anchoring breaking occurs close to the contact lines. The theoretical predictions are supported by the results of physical experiments for a ridge of the nematic 4'-pentyl-4-biphenylcarbonitrile (5CB). In particular, these experiments show that the homeotropic anchoring at the gas--nematic interface is broken close to the contact lines by the stronger rubbed planar anchoring at the nematic--substrate interface. A comparison between the experimental values of and the theoretical predictions for the effective refractive index of the ridge gives a first estimate of the anchoring strength of an interface between air and 5CB to be $(9.80\pm1.12)\times10^{-6}\,{\rm N m}^{-1}$ at a temperature of $(22\pm1.5)^\circ$C

Weak-anchoring effects in a thin pinned ridge of nematic liquid crystal

A theoretical investigation of weak-anchoring effects in a thin two-dimensional pinned static ridge of nematic liquid crystal resting on a flat solid substrate in an atmosphere of passive gas is performed. Specifically, we solve a reduced version of the general system of governing equations recently derived by Cousins et al. [Proc. R. Soc. A 478, 20210849 (2022)] valid for a symmetric thin ridge under the one-constant approximation of the Frank-Oseen bulk elastic energy with pinned contact lines to determine the shape of the ridge and the behavior of the director within it. Numerical investigations covering a wide range of parameter values indicate that the energetically preferred solutions can be classified in terms of the Jenkins-Barratt-Barbero-Barberi critical thickness into five qualitatively different types of solution. In particular, the theoretical results suggest that anchoring breaking occurs close to the contact lines. The theoretical predictions are supported by the results of physical experiments for a ridge of the nematic 4 ′ -pentyl-4-biphenylcarbonitrile (5CB). In particular, these experiments show that the homeotropic anchoring at the gas-nematic interface is broken close to the contact lines by the stronger rubbed planar anchoring at the nematic-substrate interface. A comparison between the experimental values of and the theoretical predictions for the effective refractive index of the ridge gives a first estimate of the anchoring strength of an interface between air and 5CB to be (9.80 ± 1.12) × 10^(− 6) Nm^(−1) at a temperature of (22 ± 1.5)∘C

Maximizing the heat flux in steady unicellular porous media convection

The primary aim of the present study is to determine the maximum heat transport attainable in steady 2D unicellular porous media convection. By focusing on this restricted class of flows we are able to use an efficient iterative numerical scheme to systematically probe the way in which the heat transport depends on the inter-plume spacing. Guided by our numerical results, we also propose a large-Ra asymptotic reduction of the governing equations that yields the asymptotic structure of the solutions giving the maximum heat transport

Structure and stability of steady porous medium convection at large Rayleigh number

A systematic investigation of unstable steady-state solutions of the Darcy–Oberbeck–Boussinesq equations at large values of the Rayleigh number Ra is performed to gain insight into two-dimensional porous medium convection in domains of varying aspect-ratio L. The steady convective states are shown to transport less heat than the statistically steady ‘turbulent’ flow realised at the same parameter values: the Nusselt number Nu∼Ra for turbulent porous medium convection, while Nu∼Ra 0.6 for the maximum heat-transporting steady solutions. A key finding is that the lateral scale of the heat-flux-maximising solutions shrinks roughly as L∼Ra−0.5, reminiscent of the decrease of the mean inter-plume spacing observed in turbulent porous medium convection as the thermal forcing is increased. A spatial Floquet analysis is performed to investigate the linear stability of the fully nonlinear steady convective states, extending a recent study by Hewitt et al. (J. Fluid Mech.737, 2013) by treating a base convective state – and secondary stability modes – that satisfy appropriate boundary conditions along plane parallel walls. As in that study, a bulk instability mode is found for sufficiently small aspect-ratio base states. However, the growth rate of this bulk mode is shown to be significantly reduced by the presence of the walls. Beyond a certain critical Ra-dependent aspect-ratio, the base state is most strongly unstable to a secondary mode that is localised near the heated and cooled walls. Direct numerical simulations, strategically initialised to investigate the fully nonlinear evolution of the most dangerous secondary instability modes, suggest that the (long time) mean inter-plume spacing in statistically-steady porous medium convection results from a balance between the competing effects of these two types of instability

Comment on Sochi's variational method for generalised Newtonian flow

A recently introduced variational method (Sochi, Rheol Acta 53:15–22, 2014) has been used to obtain flow profiles for generalised Newtonian fluids in steady rectilinear flow. This Comment examines the relationship between this variational method and the dynamical equations for such flows