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Abstract. The qualitative effect of nonuniform temperature gradient on the linear stability analysis of the Rayleigh-Benard convection problem in a Boussinesquian, viscoelastic fluid-filled, high-porosity medium is studied numerically using the single-term Galerkin technique. The eigenvalue is obtained for free-free, free-rigid, and rigid-rigid boundary combinations with isothermal temperature conditions. Thermodynamics and also the present stability analysis dictates the strain retardation time to be less than the stress relaxation time for convection to set in as oscillatory motions in a high-porosity medium. Furthermore, the analysis predicts the critical eigenvalue for the viscoelastic problem to be less than that of the corresponding Newtonian fluid problem. In a series of papers [12,13,14], Oldroyd proposed and studied a set of constitutive equations for viscoelastic fluids in an attempt to explain the rheological behaviour of some non-Newtonian fluids. The dynamical behaviour of these practically important fluids under isothermal conditions is better understood than under nonisothermal conditions (see Harris [5]). In the Rayleigh-Benard instability problem with respect to these fluids, the principle of exchange of stability is, in general, not valid and hence oscillatory mode of convection appears.Herbert [6] was the first to analyse the problem of oscillatory convection in Oldroyd viscoelastic fluids. Subsequently, Green [4], Vest and Arpaci [22] studied the problem using the not-very-realistic Jeffrey and Maxwell models. Sokolov and Tanner [21] confirm the unrealistic nature of these models by using an integral form of constitutive equation. Eltayeb

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