Maurizio Gentile

We develop and analyze a theory for thermal convection in a linearly viscous incompressible fluid employing the heat flux law attributed to Sir Harold Jeffreys, F.R.S. This theory employs two relaxation coefficients known as the Maxwell and Jeffreys coefficients and we derive conditions on these parameters, which allow stationary or oscillatory convection to occur. We present a sharp global nonlinear energy stability theory when the linear Jeffreys constitutive heat model is employed. The nonlinear analysis shows precisely how the Maxwell and Jeffreys parameters affect the conditions for the onset of instability. It is the ratio of the two parameters which is crucial. It is shown that oscillatory convection may only occur when the Maxwell parameter exceeds the Jeffreys one.

The problem of thermal convection in a layer of viscous incompressible fluid is analysed. The heat flux law is taken to be one of Cattaneo type. The time derivative of the heat flux is allowed to be a material derivative, or a general objective derivative. It is shown that only one objective derivative leads to results consistent with what one expects in real life. This objective derivative leads to a Cattaneo–Christov theory, and the results for linear instability theory are in agreement with those for a material derivative. It is further shown that none of the theories allow a standard nonlinear, energy stability analysis. A further heat flux due to P.M. Mariano is added and then an analysis is performed for stationary convection, oscillatory convection, and fully nonlinear theory. For the material derivative case, the analysis proceeds and global nonlinear stability is achieved. For Cattaneo–Christov theory, it appears necessary to add a regularization term in the equation for the heat flux, and even then the analysis only works in two space dimensions, and is conditional upon the size of the initial data. For the three-dimensional situation, it is shown how a nonlinear stability analysis may be achieved with a Navier–Stokes–Voigt fluid rather than a Navier–Stokes one.

A reaction–diffusion model, known as the Sel’kov–Schnakenberg model, is considered. The nonlinear stability of the constant steady state is studied by using a special Liapunov functional and a maximum principle for regular solutions.

We derive linear instability and nonlinear stability thresholds for a problem of thermal convection in a bidispersive porous medium with a single temperature when Darcy theory is employed in the micropores whereas Brinkman theory is utilized in the macropores. It is important to note that we show that the linear instability threshold is the same as the nonlinear stability one. This means that the linear theory is capturing completely the physics of the onset of thermal convection. The coincidence of the linear and nonlinear stability boundaries is established under general thermal boundary conditions.

We obtain the linear instability and nonlinear stability thresholds for a problem of thermal convection in a rotating bidispersive porous medium with a single temperature. We show that the linear instability threshold is the same as the nonlinear stability one. This means that the linear theory is capturing completely the physics of the onset of thermal convection.