In this work, we present a system of equations which describes non-isothermal flow in a bidispersive porous medium under conditions of local thermal non-equilibrium. The porous medium consists of macro pores, and in the solid skeleton are cracks or fissures which give rise to micro pores. The temperatures in the solid skeleton and in the fluids in the macro and micro pores are all allowed to be independent. After presenting the general model, we derive a result of universal stability, which guarantees exponential decay of the solution for all initial data. We further present a concrete example by specializing the model to the problem of thermal convection in a layer heated from below.

We analyse a theory for thermal convection in a Darcy porous material where the skeletal structure is one with macropores, but also cracks or fissures, giving rise to a series of micropores. This is thus thermal convection in a bidisperse, or double porosity, porous body. The theory allows for non-equilibrium thermal conditions in that the temperature of the solid skeleton is allowed to be different from that of the fluid in the macro- or micropores. The model does, however, allow for independent velocities and pressures of the fluid in the macro- and micropores. The threshold for linear instability is shown to be the same as that for global nonlinear stability. This is a key result because it shows that one may employ linearized theory to ensure that the key physics of the thermal convection problem has been captured. It is important to realize that this has not been shown for other theories of bidisperse media where the temperatures in the macro- and micropores may be different. An analytical expression is obtained for the critical Rayleigh number and numerical results are presented employing realistic parameters for the physical values which arise. Article Highlights: A two-temperature regime for a bidisperse Darcy porous medium is proposed to study the thermal convection problem. The optimal result of coincidence between the linear instability and nonlinear stability critical thresholds is proven. Numerical analysis enhances that the scaled heat transfer coefficient between the fluid and solid and the porosity-weighted conductivity ratio stabilize the problem significantly.

In this paper we have proved exponential asymptotic stability for the corotational incompressible diffusive Johnson-Segalman viscolelastic model and a simple decay result for the corotational incompressible hyperbolic Maxwell model. Moreover we have established continuous dependence and uniqueness results for the non-zero equilibrium solution.   In the compressible case, we have proved a Hölder continuous dependence theorem upon the initial data and body force for both models, whence follows a result of continuous dependence on the initial data and, therefore, uniqueness.   For the Johnson-Segalman model we have also dealt with the case of negative elastic viscosities, corresponding to retardation effects. A comparison with other type of viscoelasticity, showing short memory elastic effects, is given.

We develop a theory for thermal convection in a double porosity material of Brinkman–Forchheimer type when there is a single temperature. The saturating fluid is one of Kelvin–Voigt type, and the equation for the temperature is one due to C.I. Christov. It is shown that the global nonlinear stability threshold coincides with the linear stability one. A thoroughly analytical discussion of both linear instability analysis and global nonlinear energy stability is provided. Numerical results show that the relative permeability and Brinkman viscosity between the macro and micro pores are key parameters which play a dominant role in determining the critical Rayleigh number for the onset of convective motions.