Anisotropic bidispersive convection

This paper investigates thermal convection in an anisotropic bidisperse porous medium. A bidisperse porous medium is one which possesses the usual pores, but in addition, there are cracks or fissures in the solid skeleton and these give rise to a second porosity known as micro porosity. The novelty of this paper is that the macro permeability and the micro permeability are each diagonal tensors but the three components in the vertical and in the horizontal directions may be distinct in both the macro and micro phases. Thus, there are six independent permeability coefficients. A linear instability analysis is presented and a fully nonlinear stability analysis is inferred. Several Rayleigh number and wavenumber calculations are presented and it is found that novel cell structures are predicted which are not present in the single porosity case.

Horizontally isotropic double porosity convection

We analyse instability and nonlinear stability in a layer of saturated double porosity medium. In a double porosity or bidisperse porous medium, there are normal pores which give rise to a macroporosity. But, there are also cracks or fissures in the solid skeleton and these give arise to another porosity known as micro porosity. In this paper, the macropermeability is horizontally isotropic, in the sense that the vertical component of permeability is different to the horizontal one which is the same in all horizontal directions. Thus, the permeability is transversely isotropic with the isotropy axis in the vertical direction of gravity. We also allow the micro permeability to be horizontally isotropic, but the permeability ratios of vertical to horizontal are different in the macro- and micro-phases. The effect of the difference of ratios is examined in detail.

Horizontally isotropic bidispersive thermal convection

A bidispersive porous material is one which has usual pores but additionally contains a system of micro pores due to cracks or fissures in the solid skeleton. We present general equations for thermal convection in a bidispersive porous medium when the permeabilities, interaction coefficient and thermal conductivity are anisotropic but symmetric tensors. In this case, we show exchange of stabilities holds and fluid movement will commence via stationary convection, and additionally we show the global nonlinear stability threshold is the same as the linear instability one. Attention is then focused on the case where the interaction coefficient and thermal conductivity are isotropic, and the permeability is isotropic in the horizontal directions, although the permeability in the vertical direction is different. The nonlinear stability threshold is calculated in this case and numerical results are presented and discussed in detail.

Bidispersive vertical convection

A bidispersive porous material is one which has usual pores but additionally contains a system of micro pores. We consider a fluid-saturated bidispersive porous medium in the vertical layer x ∈(−1/2,1/2) with gravity in the − z (downward) direction. The walls of the layer are maintained at different constant temperatures. A suitable Rayleigh number is defined and we derive a global stability threshold below which no instability may arise. We additionally show that the porous layer is stable for all Rayleigh numbers provided the initial temperature gradient is bounded in a precise sense.

Porous convection with local thermal non-equilibrium temperatures and with Cattaneo effects in the solid

There is increasing interest in convection in local thermal non-equilibrium (LTNE) porous media. This is where the solid skeleton and the fluid may have different temperatures. There is also increasing interest in thermal wave motion, especially at the microscale and nanoscale, and particularly in solids. Much of this work has been based on the famous model proposed by Carlo Cattaneo in 1948. In this paper, we develop a model for thermal convection in a fluid-saturated Darcy porous medium allowing the solid and fluid parts to be at different temperatures. However, we base our thermodynamics for the fluid on Fourier's law of heat conduction, whereas we allow the solid skeleton to transfer heat by means of the Cattaneo heat flux theory. This leads to a novel system of partial differential equations involving Darcy's law, a parabolic fluid temperature equation and effectively a hyperbolic solid skeleton temperature equation. This system leads to novel physics, and oscillatory convection is found, whereas for the standard LTNE Darcy model, this does not exist. We are also able to derive a rigorous nonlinear global stability theory, unlike work in thermal convection in other second sound systems in porous media.

Triply resonant penetrative convection

A thermal convection model is considered that consists of a layer of viscous incompressible fluid contained between two horizontal planes. Gravity is acting vertically downward, and the fluid has a density maximum in the active temperature range. A heat source/sink that varies with vertical height is imposed. It is shown that in this situation there are three possible (different) sub-layers that may induce convective overturning instability. The possibility of resonance between the motion in these layers is investigated. A region is discovered where a very sharp increase in Rayleigh number is observed. In addition to a linearized instability analysis, two global (unconditional) nonlinear stability thresholds are derived.

Tipping points in Cattaneo–Christov thermohaline convection

A model is proposed for thermohaline convection when the heat flux satisfies a relaxation time law rather than the classical Fourier one. Here, we adopt the recent law due to Christov. That a Cattaneo-like law would be relevant in thermohaline convection in star evolution was proposed in 1995 by Herrera and Falcón. They do not develop a detailed model which we do here. We find that with the Cattaneo–Christov law, there is a transition curve (depending on the salt concentration) such that for Cattaneo numbers greater than this transition, the nature of convection changes dramatically.

Green–Naghdi fluid with non-thermal equilibrium effects

A. E. Green, FRS and P. M. Naghdi developed a new theory of continuum mechanics based on an entropy identity rather than an entropy inequality. In particular, within the framework of this theory, they developed a new set of equations to describe viscous flow. The new theory additionally involves vorticity and spin of vorticity. We here develop the theory of Green and Naghdi to be applicable to thermal convection in a fluid in which is suspended a collection of minute metallic-like particles. Thus, we develop a non-Newtonian theory we believe capable of describing a nanofluid. Numerical results are presented for copper oxide or aluminium oxide particles in water or in ethylene glycol. Such combinations are used in real nanofluid suspensions.

Acoustic acceleration waves in homentropic Green and Naghdi gases

Acceleration and temperature rate waves in lossless Green and Naghdi gases are investigated. The exact equations of motion are also derived and then simplified under the finite-amplitude approximation. Bounds are established for the theory-specific coupling parameter, as well as several other quantities, and results are compared/contrasted with those for classical perfect gases.

Poroacoustic acceleration waves

A model for acoustic waves in a porous medium is investigated. Due to the use of lighter materials in modern buildings and noise concerns in the environment, such models for poroacoustic waves are of much interest to the building industry. The model has been investigated in some detail by P. M. Jordan. Here we present a rational continuum thermodynamic derivation of the Jordan model. We then present results for the amplitude of an acceleration wave making no approximations whatsoever.

Structural stability in porous elasticity

We consider the linearized system of equations for an elastic body with voids as derived by Cowin & Nunziato. We demonstrate that the solution depends continuously on changes in the coefficients, which couple the equations of elastic deformation and of voids. It is also shown that the solution to the coupled system converges, in an appropriate measure, to the solutions of the uncoupled systems as the coupling coefficients tend to zero.

Global nonlinear stability in porous convection with a thermal non-equilibrium model

We show that the global nonlinear stability threshold for convection with a thermal non-equilibrium model is exactly the same as the linear instability boundary. This result is shown to hold for the porous medium equations of Darcy, Forchheimer or Brinkman. This optimal result is important because it shows that linearized instability theory has captured completely the physics of the onset of convection. The equivalence of the linear instability and nonlinear stability boundaries is also demonstrated for thermal convection in a non-equilibrium model with the Darcy law, when the layer rotates with a constant angular velocity about an axis in the same direction as gravity.

Bounds for some non-standard problems in porous flow and viscous Green–Naghdi fluids

A. E. Green, F. R. S. and P. M. Naghdi developed a new theory of continuum mechanics based on an entropy identity rather than an entropy inequality. In particular, within the framework of this theory they developed a new set of equations to describe viscous flow. The new theory additionally involves vorticity and spin of vorticity. We here derive energy bounds for a class of problem in which the ‘initial data’ are given as a combination of data at time t =0 and at a later time t = T . Such problems are in vogue in the mathematical literature and may be used, for example, to give estimates of solution behaviour in an improperly posed problem where one wishes to continue a solution backward in time. In addition, we derive similar energy bounds for a solution to the Brinkman–Forchheimer equations of viscous flow in porous media.

Decay bounds in a model for aggregation of microglia: application to Alzheimer's disease senile plaques

We present a model for chemotaxis, as applied to the aggregation of microglia in Alzheimer's disease. Using biological parameters found in the literature, testable thresholds are derived such that amyloid plaques are predicted not to form if conditions fall below a threshold. The model we use was developed by Luca et al . (Luca et al . 2003 Bull. Math. Biol. 65 , 693–730) and incorporates terms for both attractant and repellent signals. Our analysis can be applied to both two and three-dimensional spatial domains with application to any cell system involving chemotaxis.

Energy bounds for some non-standard problems in thermoelasticity

A.E. Green F.R.S. and P.M. Naghdi developed two theories of thermoelasticity, called type II and type III, which are likely to be more natural candidates for the identification of a thermoelastic body than the usual theory. We here derive energy bounds for a class of problem in which the ‘initial data’ are given as a combination of data at time t =0 and at a later time t = T . Such problems are in vogue in the mathematical literature and may be used, for example, to give estimates of solution behaviour in an improperly posed problem where one wishes to continue a solution backward in time.