Multiscale modelling of porous convection with a viscoelastic Maxwell fluid

Abstract A model is developed for thermal convection in a porous medium which has multiscale levels of porosity. The saturating fluid is one of Maxwell type. We concentrate on dual porosity where there are the normal macro pores, but also present are cracks or fissures which give rise to micro pores. Thermal convection in the dual porosity case is known as bidispersive convection. We also produce a model for three levels of porosity, namely macro pores, meso pores, and micro pores, and thermal convection in this case is known as tridispersive convection. We use a mixture theory for a solid and a fluid to describe the porous medium and in this way the Darcy terms arise naturally as friction due to flow of fluid past the solid. The Maxwell fluid is incorporated naturally by appealing to the classical constitutive theory of such a fluid and work of Anatoly P. Oskolkov. In this way the Maxwell fluid is introduced via the constitutive equation for the stress tensor and its time derivative as a function of the symmetric part of the velocity gradients in the macropores and in the micropores. The thermal convection problem where a layer of saturated porous material is heated from below is analysed in detail and numerical results are presented displaying the novelties due to the inclusion of a viscoelastic Maxwell fluid. In particular, we also consider the case of a light solid skeleton with nanoscale architecture, this being important due to the current interest in microfluidics.