The Bénard problem for an Oldroyd–Oskolkov fluid of second order

Abstract A model is analysed which generalizes an Oldroyd fluid in line with Oskolkov’s general classification of Oldroyd fluids of order N. We analyse the thermal convection problem where a horizontal layer of fluid is heated from below when that fluid is an Oldroyd fluid of order 2 according to the definition of Oskolkov. The thresholds for linear instability are calculated in detail, and the critical Rayleigh and wave numbers are determined. The key is to employ a very important classification of Anatoly P. Oskolkov to show that the constitutive equation for a generalized Burgers fluid is a natural extension of the constitutive equation for a Oldroyd fluid.