The effects of shear flows on thermal instability of viscoelastic fluids in a fluid-porous system

A mathematical model is set up to describe natural convection in a two-layer system with effects of a plane shear flows and a constant flux heating in the case of a viscoelastic fluid. Through theoretical analysis and numerical simulation, the results will be obtained to explain and predict the physical phenomena of thermal instability. A linear stability analysis will be used to get eigenvalue problems for a plane shear flow. Then, a spectral method will be adopted to compute the neutral curves in thermal convection with the effects of shear flows and viscoelastic fluids. With the help of numerical simulation, it shows that neutral curves have a bi-modal nature for a proper fluid-porous depth ratio. Transverse rolls generated by shear flows are studied as well as longitudinal rolls. The effects of viscoelastic fluids on thermal instability will also be investigated in this paper. For Newtonian fluids, longitudinal rolls always set in before transvers rolls. Transvers rolls under the effects of viscoelastic fluids may be observed early in the flow phenomena.