Triple diffusive convection in a viscoelastic Oldroyd-B fluid layer

Abstract

The stability of a triply diffusive viscoelastic fluid layer in which the fluid density depends on three stratifying agencies possessing different diffusivities is investigated. The viscoelastic fluid is modeled by means of the Oldroyd-B constitutive equation. Analytical expressions are obtained for steady and oscillatory onset by carrying out the linear instability analysis and the crossover boundary between them is demarcated by identifying a codimension-two point in the viscoelastic parameters plane. The occurrence of disconnected closed oscillatory neutral curve lying well below the stationary neutral curve is established for some values of governing parameters indicating the requirement of three critical values of thermal Rayleigh number to specify the linear instability criteria. However, the possibility of quasiperiodic bifurcation from the motionless basic state is not perceived and this is in contradiction to the case of inelastic couple stress and Newtonian fluids. The corresponding weakly nonlinear stability of stationary and oscillatory modes has been carried out using a perturbation method. The cubic Landau equations are derived and the stability of bifurcating solution is discussed. The viscoelastic parameters influence the stability of stationary bifurcation despite their effect is not felt on the stationary onset. The stationary and oscillatory finite amplitude solution is found to bifurcate either subcritical or supercritical depending on the choice of governing parameters. The effect of Prandtl number and viscoelastic parameters on stationary and oscillatory convection modes of heat and mass transfer is analyzed.