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. Numeri- cal 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.

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.