Pergamon 0960-0779(94)00274-6 Chaos, Solitons & Fractals Vol. 6, pp. 341-345, 1995 Copyright © 1995 Elsevier Science Ltd Printed in Great Britain. All fights reserved 0960-0779/95 $9.50 + .00 Convection in Oldroyd-B Fluid: Amplitude Equation J. MARTINEZ-MARDONES, 1 R. TIEMANN 2 and W. ZELLER l 1 Ins%l%u%o de Flslca, Unlversldad Cat611ca de Valparafso, Cas. 4059, Valparaiso, Chile. Facul%ad de Clenelas, Universldad de Playa Aneha (UPLACED), Cas. 34-V, Valparaiso, Chile. Abstract - The near-threshold behavior of thermal convection in viscoelastic poly- meric Oldroyd-B type fluids in an infinite horizontal layel" heated from below is determined for free and conductive boundary conditions, at the upper and lower limiting surfaces. Linear and nonlinear coefficients corresponding to the different bifurcations are evaluated up to third order of the amplitude equation. A rigorous analytical and numerical calculation of the coefficients of the Ginzburg-Landau equation in the Oldroyd-B model convection is presented. INTRODUCTION Rayleigh-B~nard convection in horizontal layers of fluids heated from below has been a sub- ject of much interest recently [1] for binary mixtures and viscoelastic fluids. There seems to be important to study the travelling and standing waves formation in nonlinear states, in pattern formation in a dynamics state, and in the breakdown of simple wave states to more complicated, chaotic states. In this paper, we present the analytical and numerical determination of the coefficients of the generalized Ginzburg-Landau amplitude equation [2] for Oldroyd-B polymeric fluid [31 convec- tion, for free and conducting boundaries, and we discuss the possible stationary and oscillatory pattern formation in fluid behavior. A similar work has been done for binary fluid mixtures [41. VISCOELASTIC DYNAMICS FORMULATION A thin infinite horizontal polymeric fluid layer of depth d, initially at rest, is heated from below, We use the well-known hydrodynamic Boussinesq approximation. The viscoelastic Rayleigh-B6nard phenomenon can be described by a nondimensional system of perturbation equations corresponding to matter, momentum and energy transfer, and the constitutive equation [5,6] atVzq/ = - PR0xO + PAZTxz + PaxzS - J(~,V2~) (l) atO = - ax~ + VzO - J(~,O] (2) FatT - FA8 Azq/ = A2@ - x - F[J(qJ,T ) + 1/2(V21]!)S + 1/2(A2~/J)U] XZ % XZ XZ + FA[j(@,&2@) _ 2(Oxz~ )(Az~)] (3) 341