Effect of time-periodic vertical oscillations of the Rayleigh–Bénard system on nonlinear convection in viscoelastic liquids

A study of heat transport in Rayleigh–Bénard convection in viscoelastic liquids with/without gravity modulation is made using a most minimal representation of Fourier series and a representation with higher modes. The Oldroyd-B constitutive relation is considered. The resulting non-autonomous Lorenz model (generalized Khayat–Lorenz model of four modes and seven modes) is solved numerically using the adaptive-grid Runge–Kutta–Fehlberg45 method to quantify the heat transport. The effect of gravity modulation is shown to be stabilizing there by leading to a situation of reduced heat transfer. The Deborah number is shown to have an antagonistic influence on convection compared to the stabilizing effect of modulation amplitude and elastic ratio. The results in respect of Maxwell, Rivlin–Ericksen and Newtonian liquids are obtained as particular cases of the present study. A transformation of the momentum equations illustrates the equivalence of present approach and the one due to Khayat that uses normal stresses explicitly.